一类分数阶混沌系统在（1，2）区间内的投影同步

摘要

本文研究了一类分数阶混沌系统的投影同步方法<q个<2被证明。通过精确的理论推导，建立了投影同步方法。为了说明该方案的有效性，我们讨论了两个例子：（1）分数阶Lorenz混沌系统q个= 1.1; （2） 分数阶修正蔡氏分数阶混沌系统q个= 1.02. 数值模拟结果表明了该方案的有效性和可行性。

1.简介

许多现实世界的物理系统都可以用分数阶微分方程很好地、更准确地描述[1–4]. 近年来，在许多分数阶非线性系统中发现了混沌现象，如分数阶Lorenz混沌系统[5，6]蔡氏分数阶混沌电路系统[6]分数阶修正Duffing混沌系统[7]分数阶Rössler混沌系统[8，9]，分数阶Chen混沌系统[6–8]分数阶忆阻混沌系统[10]等等。

在过去20年中，由于其在科学和工程领域的潜在应用[11，12]混沌系统的同步越来越受到人们的关注，提出了许多同步方案。其中，Mainieri和Rehacek提出了一种特殊的同步方案，称为投影同步[13]. 主系统和从系统可以同步到PS中的比例因子，这可以用于将二进制数字扩展到M（M）-快速通信的二进制数字通信[13，14].

然而，许多以前的同步方法[6–9，13–16]对于只关注分数阶0的分数阶混沌系统<q个<1，但实际上有许多分数阶系统具有分数阶1<q个现实世界中<2。例如，时间分数热传导方程[17]，分数电报方程[18]，时间分数反应扩散系统[19]分数扩散波方程[20]，时空分数扩散方程[21]超扩散系统[22],等。，但在[17–22]. 同时，基于数值模拟，Ge和Jhuang[23]报道了分数阶旋转机械系统同步的一些结果q个= 1.1. 迄今为止，据我们所知，对于分数阶混沌系统，1<q个<2通过精确理论。那么，如何实现分数阶非线性系统的混沌同步<q个<2通过精确的理论化是一个有趣且具有学术意义和实际意义的开放性问题。

基于上述讨论，本文提出了一类分数阶混沌系统的投影同步方法<q个<2通过精确理论。为了证明该方案的有效性，对分数阶Lorenz混沌系统进行了投影同步q个=1.1和蔡氏分数阶修正混沌系统q个分别讨论=1.02。数值模拟表明了我们方案的有效性和可行性。

2.问题陈述和主要结果

本文中，分数阶的Caputo导数q个用于函数（f）(t吨)，定义为${\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=")">)</trans>}{\displaystyle {<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="0">0</trans>}}^{\mathrm{<trans\; data-src="t">t吨</trans>}}\frac{{\mathrm{<trans\; data-src="f">（f）</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\tau ">\tau </trans>}<trans\; data-src=")">)</trans>}{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\tau ">\tau </trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="m">米</trans>}}}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="\tau ">\tau </trans>}}$哪里米−1<q个<米，D类^q个表示分数阶的卡普托导数q个for函数（f）(t吨),米是大于的最小整数q个，（f）^(米)(t吨)是米-通常意义上的th导数，以及$\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\displaystyle {<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="0">0</trans>}}^{<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{\mathrm{<trans\; data-src="t">t吨</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="t">t吨</trans>}}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}}$表示伽马函数。

现在，考虑以下分数阶混沌系统：

$${\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="A">一个</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$$

（1）

其中1<q个<2是分数阶，并且$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="1">1</trans>}}$是状态向量。$\mathrm{<trans\; data-src="A">一个</trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="n">n个</trans>}}$是一个常数实矩阵。$\mathrm{<trans\; data-src="A">一个</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="1">1</trans>}}$和$\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="1">1</trans>}}$是系统中的线性部分和非线性部分（1）分别是。

在本文中，我们只讨论了一类满足以下条件的分数阶混沌系统：

$$\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">，</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$$

(2)

在这里${\mathrm{<trans\; data-src="x">x个</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="1">1</trans>}}$是一个真实的变量。H（H）_我(x个′(t吨))∈R（右）^n个×n个和H（H）_n个(x个(t吨)−x个′(t吨),x个′(t吨)) ∈R（右）^n个×1是实矩阵。H（H）_我(x个′(t吨))x个(t吨),−x个′(t吨))和H（H）_n个(x个(t吨)−x个′(t吨),x个′(t吨))是关于的线性部分和非线性部分(x个(t吨)−x个'(t吨))分别是。事实上，非线性部分$\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$在许多分数阶混沌系统中，例如分数阶Lorenz混沌系统[5，6]，蔡氏分数阶修正混沌系统[24]，分数阶Duffing混沌系统[7]分数阶Rossler混沌系统[8，9]，分数阶Chen混沌系统[6–8],等。，都满意方程式（2）.

现在，我们研究如何实现分数阶混沌系统的投影同步（1）.选择分数阶混沌系统（1）作为主系统，并选择以下受控分数阶系统作为从系统：

$${\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="A">一个</trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$$

(3)

其中α≠0是一个命名为比例因子的常数，并且u个(x个(t吨),年(t吨))∈R（右）^n个×1是真正的反馈控制器，稍后确定。

定义。 对于分数阶混沌主系统（1）和从系统(3)，如果存在非零常数α，则称为投影同步$\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{\mathrm{<trans\; data-src="lim">林</trans>}}<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="=">=</trans>\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{\mathrm{<trans\; data-src="lim">林</trans>}}<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$哪里$\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="1">1</trans>}}$ 是同步错误。符号||•||表示矩阵范数。

定理。 从机系统中的实反馈控制器(3)被选为$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$.它被称为主系统之间的投影同步（1）和从系统(3)如果存在实矩阵K，则：

• ${{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$，$\underset{\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="0">0</trans>}}{\mathrm{<trans\; data-src="lim">林</trans>}}\frac{<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src="\Vert ">\Vert </trans>}{<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$对于任何x个(t吨),
• $\mathrm{<trans\; data-src="Re">重新</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">一个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="0">0</trans>}$，$\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src="=">=</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="max">最大值</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="Re">重新</trans>}\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">一个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src=">">></trans>{<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="q">q个</trans>}}$.

哪里$\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>$. ω是矩阵特征值实部的最小绝对值(A+αK).

证明。根据e（电子）(t吨) =αy(t吨)−x个(t吨)分数阶系统之间的误差系统（1）和系统(3)描述如下：

$${\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="A">一个</trans>}\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$$

(4)

自$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$，系统(4)可以重写为：

$${\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="A">一个</trans>}\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$$

(5)

由$\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>$、系统(5)可以更改为：

$${\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">一个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>$$

(6)

自${{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$对于任何x个(t吨)，因此e（电子）(t吨)=0是误差系统中的零解(6)现在，让我们e（电子）₁₀和e（电子）₂₀是系统的初始条件(6)，所以解决方案e（电子）(t吨)用于系统(6)可以显示为：

$$\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="E">电子</trans>}}_{\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="[">[</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">一个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src="]">]</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="10">10</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="t">t吨</trans>}{\mathrm{<trans\; data-src="E">电子</trans>}}_{\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="[">[</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">一个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src="]">]</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="20">20</trans>}}<trans\; data-src="+">+</trans>{\displaystyle \underset{\mathrm{<trans\; data-src="0">0</trans>}}{\overset{\mathrm{<trans\; data-src="t">t吨</trans>}}{<trans\; data-src="\int ">\int </trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="E">电子</trans>}}_{\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src="[">[</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">一个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src="]">]</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}$$

(7)

哪里电子_q个,1，电子_q个,2和电子_{q、 q个}是Mittag-Lefler型的双参数函数，即，${\mathrm{<trans\; data-src="E">电子</trans>}}_{\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="p">第页</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\displaystyle \underset{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}}{\overset{\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\frac{{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="q">q个</trans>}\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=")">)</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans>$、和z（z）是一个变量。

对于Mittag-Lefler函数电子_{q、 第页}(z（z）)，不平等(8)已在参考中获得[25]:

$$<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="E">电子</trans>}}_{\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="p">第页</trans>}}<trans\; data-src="[">[</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">一个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src="]">]</trans><trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="\le ">\le </trans><trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">一个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}}<trans\; data-src="\Vert ">\Vert </trans>$$

(8)

根据方程式（7）和不平等(8)，其中一个具有：

$$\begin{array}{l}<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="\le ">\le </trans><trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="E">电子</trans>}}_{\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="[">[</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">一个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src="]">]</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="10">10</trans>}}<trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="+">+</trans><trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="t">t吨</trans>}{\mathrm{<trans\; data-src="E">电子</trans>}}_{\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="[">[</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">一个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src="]">]</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="20">20</trans>}}<trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="+">+</trans><trans\; data-src="\Vert ">\Vert </trans>{\displaystyle \underset{\mathrm{<trans\; data-src="0">0</trans>}}{\overset{\mathrm{<trans\; data-src="t">t吨</trans>}}{<trans\; data-src="\int ">\int </trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="E">电子</trans>}}_{\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src="[">[</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">一个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src="]">]</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="\Vert ">\Vert </trans>\\ <trans\; data-src="\le ">\le </trans><trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">一个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="10">10</trans>}}<trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="+">+</trans><trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">一个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="20">20</trans>}}<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="+">+</trans>{\displaystyle \underset{\mathrm{<trans\; data-src="0">0</trans>}}{\overset{\mathrm{<trans\; data-src="t">t吨</trans>}}{<trans\; data-src="\int ">\int </trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">一个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="q">q个</trans>}}}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src="\Vert ">\Vert </trans>}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}\end{array}$$

(9)

自Re[λ(一个+αK)]<0，因此(一个+αK（K）)是一个稳定性矩阵。所以，$<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">一个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}\mathrm{<trans\; data-src="t">t吨</trans>}}$、和$<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">一个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}}<trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}\mathrm{<trans\; data-src="t">t吨</trans>}}$，在这里N个₀>0是一个合适的常数。

现在，不平等(9)可以转换为：

$$<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="10">10</trans>}}<trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="20">20</trans>}}<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}{\displaystyle \underset{\mathrm{<trans\; data-src="0">0</trans>}}{\overset{\mathrm{<trans\; data-src="t">t吨</trans>}}{<trans\; data-src="\int ">\int </trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans>}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}$$

(10)

由于${{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$，$\underset{\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="0">0</trans>}}{\mathrm{<trans\; data-src="lim">林</trans>}}\frac{<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src="\Vert ">\Vert </trans>}{<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$对于任何x个(t吨)，所以存在一个常数ε>0，使得$<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="\le ">\le </trans><trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="/">/</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$作为$<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}$.

那么，不平等(10)可以重写为：

$$<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="10">10</trans>}}<trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="20">20</trans>}}<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="+">+</trans>{\displaystyle \underset{\mathrm{<trans\; data-src="0">0</trans>}}{\overset{\mathrm{<trans\; data-src="t">t吨</trans>}}{<trans\; data-src="\int ">\int </trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans>}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}$$

(11)

即：

$$<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{\mathrm{<trans\; data-src="\omega ">\omega </trans>}\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="10">10</trans>}}<trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="20">20</trans>}}<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="+">+</trans>{\displaystyle \underset{\mathrm{<trans\; data-src="0">0</trans>}}{\overset{\mathrm{<trans\; data-src="t">t吨</trans>}}{<trans\; data-src="\int ">\int </trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans>}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="s">秒</trans>}$$

(12)

根据参考文献中的结果[26]，不平等(12)可以更改为：

$$<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{\mathrm{<trans\; data-src="\omega ">\omega </trans>}\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="\le ">\le </trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="10">10</trans>}}<trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="20">20</trans>}}<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="E">电子</trans>}}_{\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src="]">]</trans>$$

(13)

使用参考中的结果[4]:$<trans\; data-src="|">|</trans>{\mathrm{<trans\; data-src="E">电子</trans>}}_{\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="p">第页</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="|">|</trans><trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans><trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="|">|</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="p">第页</trans>}<trans\; data-src=")">)</trans><trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="q">q个</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{\mathrm{<trans\; data-src="Re">重新</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src=")">)</trans>}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans><trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="|">|</trans><trans\; data-src=")">)</trans>}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}$，其中N个_我>0(我=1,2), |z（z）|≥0，arg(z（z）)≤ρ和0.5q个<ρ<最小值（π，πq个)因此，不等式(13)可以更改为：

$$<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{\mathrm{<trans\; data-src="\omega ">\omega </trans>}\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="\le ">\le </trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="10">10</trans>}}<trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="20">20</trans>}}<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="E">电子</trans>}}_{\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src="]">]</trans><trans\; data-src="\le ">\le </trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="10">10</trans>}}<trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="20">20</trans>}}<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{\mathrm{<trans\; data-src="t">t吨</trans>}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="q">q个</trans>}}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="/">/</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>$$

(14)

即：

$$\begin{array}{l}<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="\le ">\le </trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="10">10</trans>}}<trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="20">20</trans>}}<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{\mathrm{<trans\; data-src="t">t吨</trans>}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="q">q个</trans>}}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="/">/</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}\mathrm{<trans\; data-src="t">t吨</trans>}}\\ <trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="10">10</trans>}}<trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="20">20</trans>}}<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="[">[</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src="]">]</trans>}<trans\; data-src="+">+</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="10">10</trans>}}<trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="20">20</trans>}}<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="/">/</trans><trans\; data-src="\{">\{</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src="]">]</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{\mathrm{<trans\; data-src="\omega ">\omega </trans>}\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="\}">\}</trans>\end{array}$$

(15)

自$\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src="=">=</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="max">最大值</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="Re">重新</trans>}\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">一个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src=">">></trans>{<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="q">q个</trans>}}$，其中一个ω>0，并且${<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="0">0</trans>}$因此，$\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{\mathrm{<trans\; data-src="lim">林</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="10">10</trans>}}<trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="20">20</trans>}}<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="[">[</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src="]">]</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">，</trans>\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{\mathrm{<trans\; data-src="lim">林</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="10">10</trans>}}<trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="20">20</trans>}}<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="N">N个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="/">/</trans><trans\; data-src="\{">\{</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="t">t吨</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src="]">]</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{\mathrm{<trans\; data-src="\omega ">\omega </trans>}\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="\}">\}</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$因此：

$$\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{\mathrm{<trans\; data-src="lim">林</trans>}}<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$$

(16)

公式（16）表示误差系统中的零解(6)是渐近稳定的，所以$\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{\mathrm{<trans\; data-src="lim">林</trans>}}<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="=">=</trans>\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\infty ">\infty </trans>}}{\mathrm{<trans\; data-src="lim">林</trans>}}<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$因此，分数阶混沌系统之间的投影同步（1）和系统(3)将获得。证明已完成。□

我们注意到一些学者[27–29]讨论了具有非延迟和延迟耦合的复杂动力学网络、混沌同步在保密通信中的应用以及具有多状态延迟的Takagi-Sugeno模糊系统。将我们的结果应用于这些问题是我们正在进行的工作。

3.示例

为了证明第二节提出的投影同步方法的有效性，我们将该同步方案应用于分数阶Lorenz混沌系统q个=1.1和分数阶修正Chua混沌系统q个分别为1.02。

3.1. 分数阶Lorenz混沌系统与1的投影同步<q个<2

分数阶Lorenz系统[30]由以下人员提供：

$$<trans\; data-src="(">(</trans>\begin{array}{c}{\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}\mathrm{<trans\; data-src="x">x个</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\\ {\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\\ {\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}\end{array}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\begin{array}{c}\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>\\ \mathrm{<trans\; data-src="\gamma ">\gamma </trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}\\ {\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}\end{array}<trans\; data-src=")">)</trans>$$

(17)

其中σ=10，γ=28，β=8/3。分数阶Lorenz系统(17)显示一个混沌吸引子q个= 1.1. 混沌吸引子如所示图1.

为了实现分数阶Lorenz混沌系统的投影同步(17)、系统(17)被选为主系统，因此从系统可以如下构造：

$$<trans\; data-src="(">(</trans>\begin{array}{c}{\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\\ {\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\\ {\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}\end{array}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\begin{array}{c}\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>\\ \mathrm{<trans\; data-src="\gamma ">\gamma </trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\\ <trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}\end{array}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}$$

(18)

哪里$\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="=">=</trans>{<trans\; data-src="(">(</trans>\begin{array}{ccc}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}& {\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}& {\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}\end{array}<trans\; data-src=")">)</trans>}^{\mathbf{<trans\; data-src="T">T型</trans>}}$、和${\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>$.T型表示矩阵的转置：

显然，$\mathrm{<trans\; data-src="A">一个</trans>}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\begin{array}{ccc}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}& \mathrm{<trans\; data-src="\sigma ">\sigma </trans>}& \mathrm{<trans\; data-src="0">0</trans>}\\ \mathrm{<trans\; data-src="\gamma ">\gamma </trans>}& <trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}& \mathrm{<trans\; data-src="0">0</trans>}\\ \mathrm{<trans\; data-src="0">0</trans>}& \mathrm{<trans\; data-src="0">0</trans>}& <trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\beta ">\beta </trans>}\end{array}<trans\; data-src=")">)</trans>$，$\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\begin{array}{c}\mathrm{<trans\; data-src="0">0</trans>}\\ <trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}\\ {\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\end{array}<trans\; data-src=")">)</trans>$.

使用小时(αy)−小时(x个) =H（H）_我(x个)e（电子）+H（H）_n个(e（电子），x个)，矩阵H（H）_我(x个)和矩阵H（H）_n个(e（电子），x个)可以推导如下：

$${\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\begin{array}{ccc}\mathrm{<trans\; data-src="0">0</trans>}& \mathrm{<trans\; data-src="0">0</trans>}& \mathrm{<trans\; data-src="0">0</trans>}\\ <trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}& \mathrm{<trans\; data-src="0">0</trans>}& <trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\\ {\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}& {\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}& \mathrm{<trans\; data-src="0">0</trans>}\end{array}<trans\; data-src=")">)</trans><trans\; data-src=",">，</trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\begin{array}{c}\mathrm{<trans\; data-src="0">0</trans>}\\ {\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}\\ <trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\end{array}<trans\; data-src=")">)</trans>$$

根据$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}$、控制器u个(x个，年)在从属系统中(18)被选择为：

$$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\begin{array}{ccc}\mathrm{<trans\; data-src="0">0</trans>}& \mathrm{<trans\; data-src="0">0</trans>}& \mathrm{<trans\; data-src="0">0</trans>}\\ <trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}& \mathrm{<trans\; data-src="0">0</trans>}& <trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\\ {\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}& {\mathrm{<trans\; data-src="x">x个</trans>}}_{}& \mathrm{<trans\; data-src="0">0</trans>}\end{array}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}$$

现在，很容易验证以下内容：

$${{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>{<trans\; data-src="(">(</trans>\begin{array}{c}\mathrm{<trans\; data-src="0">0</trans>}\\ {\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}\\ <trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\end{array}<trans\; data-src=")">)</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$$

和：

$$\frac{<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans>}{<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="\Vert ">\Vert </trans>}<trans\; data-src="=">=</trans>\frac{\sqrt{{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}}}{\sqrt{{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}{}^{\mathrm{<trans\; data-src="2">2</trans>}}}}<trans\; data-src="\le ">\le </trans>\frac{\sqrt{{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}}}{\sqrt{{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}{}^{\mathrm{<trans\; data-src="2">2</trans>}}}}<trans\; data-src="=">=</trans><trans\; data-src="|">|</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="|">|</trans>$$

因此：

$${{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">，</trans>\underset{\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="0">0</trans>}}{\mathrm{<trans\; data-src="lim">林</trans>}}\frac{<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans>}{<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="\Vert ">\Vert </trans>}<trans\; data-src="\le ">\le </trans>\underset{\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="0">0</trans>}}{\mathrm{<trans\; data-src="lim">林</trans>}}<trans\; data-src="|">|</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="|">|</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$$

上述结果表明，满足定理中的条件（i）。

基于第二节的定理，一个合适的非零常数α和常数实矩阵K（K）可以选择Re[λ(一个+αK)]<0和−最大[Reλ(一个+αK)]>[Γ(q个)]^1/q个因此分数阶Lorenz混沌系统之间的投影同步(17)和受控分数阶Lorenz混沌系统(18)可以实现。

例如，让$\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\begin{array}{ccc}\mathrm{<trans\; data-src="0">0</trans>}& <trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="10">10</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}& \mathrm{<trans\; data-src="0">0</trans>}\\ <trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="28">28</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}& \mathrm{<trans\; data-src="0">0</trans>}& \mathrm{<trans\; data-src="0">0</trans>}\\ \mathrm{<trans\; data-src="0">0</trans>}& \mathrm{<trans\; data-src="0">0</trans>}& \mathrm{<trans\; data-src="0">0</trans>}\end{array}<trans\; data-src=")">)</trans>$，α=3，所以λ₁(一个+αK)]=−10,λ₂(一个+αK)]=−1,λ_三(一个+αK)]=−8/3和−最大值[Reλ(一个+αK)]=1>[Γ（1.1）]^1/1.1分别为0.9557。仿真结果如所示图2，其中初始条件为(x个₁₀，x个₂₀，x个₃₀)=（10、20、30），以及(年₁₀，年₂₀，年₃₀)分别=（10、20、30）。在图2a-c，实线表示系统的吸引子(17)，虚线表示系统的吸引子(18).系统之间的投影同步错误(17)和系统(18)如所示图2d.

3.2. 分数阶修正蔡氏混沌系统与1的投影同步<q个<2

1971年，蔡氏发现了蔡氏混沌电路[30]. 2010年，Muthuswamy和Chua[24]报道了最简单的改进Chua混沌电路，该电路由线性无源电感器、线性无源电容器和非线性有源忆阻器组成。最简单的修正蔡氏混沌电路系统[24]可以显示为：

$$<trans\; data-src="(">(</trans>\begin{array}{c}\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}\\ \mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}\\ \mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}\end{array}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\begin{array}{c}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\\ <trans\; data-src="-">-</trans><trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1.7">1.7</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="]">]</trans><trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3.3">3.3</trans>}\\ <trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="0.2">0.2</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}\end{array}<trans\; data-src=")">)</trans>$$

其分数阶系统称为分数阶修正蔡氏混沌系统，可描述如下：

$$<trans\; data-src="(">(</trans>\begin{array}{c}{\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\\ {\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\\ {\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}\end{array}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\begin{array}{c}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\\ <trans\; data-src="-">-</trans><trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1.7">1.7</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="]">]</trans><trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3.3">3.3</trans>}\\ <trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="0.2">0.2</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}\end{array}<trans\; data-src=")">)</trans>$$

(19)

分数阶修正蔡氏系统(19)显示一个混沌吸引子q个= 1.02. 混沌吸引子如所示图3.

为了实现分数阶混沌系统的投影同步(19)、系统(19)被选为主系统，因此从系统可以如下构造：

$$<trans\; data-src="(">(</trans>\begin{array}{c}{\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\\ {\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\\ {\mathrm{<trans\; data-src="D">D类</trans>}}^{\mathrm{<trans\; data-src="q">q个</trans>}}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}\end{array}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\begin{array}{c}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\\ <trans\; data-src="-">-</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1.7">1.7</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3.3">3.3</trans>}\\ <trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="0.2">0.2</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}\end{array}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}$$

(20)

哪里$\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="=">=</trans>{<trans\; data-src="(">(</trans>\begin{array}{ccc}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}& {\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}& {\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}\end{array}<trans\; data-src=")">)</trans>}^{\mathbf{<trans\; data-src="T">T型</trans>}}$、和${\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>$.

显然，$\mathrm{<trans\; data-src="A">一个</trans>}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\begin{array}{ccc}\mathrm{<trans\; data-src="0">0</trans>}& \mathrm{<trans\; data-src="1">1</trans>}& \mathrm{<trans\; data-src="0">0</trans>}\\ <trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="10">10</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="33">33</trans>}& \mathrm{<trans\; data-src="17">17</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="33">33</trans>}& \mathrm{<trans\; data-src="0">0</trans>}\\ \mathrm{<trans\; data-src="0">0</trans>}& <trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}& <trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="0.2">0.2</trans>}\end{array}<trans\; data-src=")">)</trans>$，$\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\begin{array}{c}\mathrm{<trans\; data-src="0">0</trans>}\\ <trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="17">17</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="33">33</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}\\ {\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}\end{array}<trans\; data-src=")">)</trans>$.

使用小时(αy)−小时(x个)=H（H）_我(x个)e（电子）+H（H）_n个(e（电子），x个)，矩阵H（H）_我(x个)和矩阵H（H）_n个(e（电子），x个)可以推导如下：

$${\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\begin{array}{ccc}\mathrm{<trans\; data-src="0">0</trans>}& \mathrm{<trans\; data-src="0">0</trans>}& \mathrm{<trans\; data-src="0">0</trans>}\\ \mathrm{<trans\; data-src="0">0</trans>}& <trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="17">17</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="33">33</trans>}& <trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="34">34</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="33">33</trans>}\\ \mathrm{<trans\; data-src="0">0</trans>}& <trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}& <trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\end{array}<trans\; data-src=")">)</trans><trans\; data-src=",">，</trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\begin{array}{c}\mathrm{<trans\; data-src="0">0</trans>}\\ \mathrm{<trans\; data-src="17">17</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="2">2</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="+">+</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="]">]</trans><trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="33">33</trans>}\\ {\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}\end{array}<trans\; data-src=")">)</trans>$$

根据$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}$，控制器u个(x个，年)在从属系统中(20)选择为：

$$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\begin{array}{ccc}\mathrm{<trans\; data-src="0">0</trans>}& \mathrm{<trans\; data-src="0">0</trans>}& \mathrm{<trans\; data-src="0">0</trans>}\\ \mathrm{<trans\; data-src="0">0</trans>}& <trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="17">17</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="33">33</trans>}& <trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="34">34</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="33">33</trans>}\\ \mathrm{<trans\; data-src="0">0</trans>}& <trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}& <trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\end{array}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}$$

现在，很容易验证以下内容：

$${{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>{<trans\; data-src="(">(</trans>\begin{array}{c}\mathrm{<trans\; data-src="0">0</trans>}\\ \mathrm{<trans\; data-src="17">17</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="2">2</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="+">+</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="]">]</trans><trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="33">33</trans>}\\ {\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}\end{array}<trans\; data-src=")">)</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$$

和：

$$\begin{array}{l}\frac{<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans>}{<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="\Vert ">\Vert </trans>}<trans\; data-src="=">=</trans>\frac{\sqrt{{<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="1.7">1.7</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="2">2</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="+">+</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="]">]</trans><trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3.3">3.3</trans>}<trans\; data-src="\}">\}</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}}}{\sqrt{{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}{}^{\mathrm{<trans\; data-src="2">2</trans>}}}}\\ <trans\; data-src="\le ">\le </trans>\frac{\sqrt{{<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="2">2</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="+">+</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="]">]</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}}}{\sqrt{{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}{}^{\mathrm{<trans\; data-src="2">2</trans>}}}}<trans\; data-src="\le ">\le </trans>\frac{\sqrt{{<trans\; data-src="(">(</trans><trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="2">2</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="|">|</trans><trans\; data-src="+">+</trans><trans\; data-src="|">|</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="|">|</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}}}{\sqrt{{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}{}^{\mathrm{<trans\; data-src="2">2</trans>}}}}\end{array}$$

根据修正蔡氏混沌系统的有界性，存在一个实正常数M（M）这样的话$\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="max">最大值</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="|">|</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="|">|</trans><trans\; data-src=",">，</trans><trans\; data-src="|">|</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="|">|</trans><trans\; data-src="=">=</trans><trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="|">|</trans><trans\; data-src=")">)</trans>$符号max是最大值。

因此：

$$\frac{<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans>}{<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="\Vert ">\Vert </trans>}<trans\; data-src="\le ">\le </trans>\frac{\sqrt{{<trans\; data-src="(">(</trans><trans\; data-src="|">|</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="|">|</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="M">M（M）</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}}}{\sqrt{{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}{}^{\mathrm{<trans\; data-src="2">2</trans>}}}}<trans\; data-src="\le ">\le </trans>\sqrt{\frac{{<trans\; data-src="(">(</trans><trans\; data-src="|">|</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="|">|</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="M">M（M）</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}}{{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}{}^{\mathrm{<trans\; data-src="2">2</trans>}}}}<trans\; data-src="=">=</trans>\sqrt{{<trans\; data-src="(">(</trans><trans\; data-src="|">|</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="|">|</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="M">M（M）</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{}^{\mathrm{<trans\; data-src="2">2</trans>}}}$$

因此：

$${{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="|">|</trans>}_{\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">，</trans>\underset{\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="0">0</trans>}}{\mathrm{<trans\; data-src="lim">林</trans>}}\frac{<trans\; data-src="\Vert ">\Vert </trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=",">，</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\Vert ">\Vert </trans>}{<trans\; data-src="\Vert ">\Vert </trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="\Vert ">\Vert </trans>}<trans\; data-src="\le ">\le </trans>\underset{\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="0">0</trans>}}{\mathrm{<trans\; data-src="lim">林</trans>}}\sqrt{{<trans\; data-src="(">(</trans><trans\; data-src="|">|</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="|">|</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="M">M（M）</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{}^{\mathrm{<trans\; data-src="2">2</trans>}}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$$

上述结果表明，满足了定理中的条件（i）。

基于第2节中的上述定理，一个合适的非零常数α=-0.5和常数实矩阵K（K）可以选择Re[λ(一个+αK)]<0和$<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="max">最大值</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="Re">重新</trans>}\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">一个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src=">">></trans>{<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="q">q个</trans>}}\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}$hold，所以分数阶修正Chua混沌系统之间的投影同步(19)和受控分数阶修正蔡氏混沌系统(20)可以实现。

例如，选择$\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\begin{array}{ccc}\mathrm{<trans\; data-src="4">4</trans>}& \mathrm{<trans\; data-src="0">0</trans>}& \mathrm{<trans\; data-src="0">0</trans>}\\ \mathrm{<trans\; data-src="46">46</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="33">33</trans>}& \mathrm{<trans\; data-src="166">166</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="33">33</trans>}& \mathrm{<trans\; data-src="0">0</trans>}\\ \mathrm{<trans\; data-src="0">0</trans>}& <trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="2">2</trans>}& \mathrm{<trans\; data-src="4">4</trans>}\end{array}<trans\; data-src=")">)</trans>$、和α=$<trans\; data-src="-">-</trans>$分别为0.5。因此，λ_±(一个+αK)=−2±我, λ_三(一个+αK)=−2.2，以及$<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="max">最大值</trans>}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="Re">重新</trans>}\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">一个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}\mathrm{<trans\; data-src="K">K（K）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=">">></trans>{<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1.02">1.02</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="1.02">1.02</trans>}}$分别为0.9891。数值结果如所示图4，其中初始条件为(x个₁₀，x个₂₀，x个₃₀)=（0.1、0、0.1），以及(年₁₀，年₂₀，年₃₀)$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="10">10</trans>}}<trans\; data-src=",">，</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="20">20</trans>}}<trans\; data-src=",">，</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="30">30</trans>}}<trans\; data-src=")">)</trans>$分别为（-1，2，1）。

在图4a–c，实线表示系统的吸引子(19)，虚线表示系统的吸引子(20)分别是。系统之间的投影同步错误(19)和系统(20)如所示图4d.

4.结论

本文针对一类分数阶混沌系统，提出了一种投影同步方法<q个<2.我们的方法可以应用于一类非线性分数阶混沌系统，其中混沌系统中的非线性项满足方程式（2）为了证明所提出的投影同步方案的有效性，我们将该同步方案应用于分数阶Lorenz混沌系统q个=1.1和蔡氏分数阶修正混沌系统q个分别=1.02。数值模拟表明了该方案的有效性和可行性。