dynamic system(dynamic system) ismathematicsA concept on.There is a fixed rule in the dynamic system, which describes the evolution of a point in the geometric space with time.For example, mathematical models describing the pendulum shaking, the flow of water in the pipeline, or the number of fish in the lake every spring are all dynamic systems.[1]
In the dynamic system, there is a concept called state, which is a set of real numbers that can be determined.The small change of state corresponds to the small change of this group of real numbers.This group of real numbers is also the geometric space coordinates of a manifold.The evolution rule of dynamic system is controlled by a set of functions, which describes how the future state depends on the current state.This rule is deterministic, that is, for a given time interval, the state can only evolve into a future state.[1]
In the power systemstateThe concept of state is a group that can be determinedreal number。The small change of state corresponds to the small change of this group of real numbers.This group of real numbers is also a kind ofmanifoldGeospatial coordinates of the.The evolution rules of dynamic system are a set offunctionOfFixed Rules, which describes how the future state depends on the current state.This rule is deterministic, that is, for a given time interval, only one future state can evolve from the current state.
If the state of the system is only examined at a series of discontinuous time points, then the dynamic system isDiscrete dynamic system；If the time is continuous, you get aContinuous power system。If the system depends on time in a continuous and differentiable way, we call it aSmooth dynamical system。
theoretical development
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There are often some systems that evolve with time in nature, such asPlanetary systemIf there are mathematical models for such systems, their common basic mathematical model is that there is a set X composed of all possible states and a dynamic law φ t: X → X.Thus, a state x ∈ X changes with time t and becomes state φ t (x).If X isEuclidean spaceOr generally atopological space , the time t occupies the full area (-,), and the dynamic law φ t also meets other simple and natural conditions (seeTopological dynamical system）, a power system is obtained.At this time, there is a trajectory through each point x ∈ X, that is, the set {φ t (x) | t ∈ (-,)}.If X is oneeuclidean space , or, more broadly, a smooth manifold, and the dynamic system φ t: X → X is differentiable for t at every x ∈ X:ordinary differential equation Group or ordinary micro system S.On the contrary, if X is a compact smooth manifold and there is a C1 constant differential system S on it, then according to the basic theory of ordinary differential equations, S always generates a dynamic system.Here, S is C 1, that is, S is continuously differentiable with respect to x.As mentioned above, dynamic system theory and ordinary differentialequationThe contents discussed in the qualitative theory seem to be indistinguishable, but there are different aspects. Dynamic systems focus on the qualitative research of abstract systems rather than specific equations. Its research approach focuses on the interrelationship between a family of trajectories, in other words, it is holistic.Some of the integrity istopologySome are statistical;The latter is mainly ergodic.Dynamic system theory is a development of classical ordinary differential equation theory.
Research History
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The study of dynamic systems began at the end of the 19th century,(J. -) H. PoincareHere we goQualitative Theory of Ordinary Differential EquationsThe research of, the topics discussed (such as stability, the existence and regression of periodic orbits, etc.) and the focus of the research methods used are the origination of the later mathematical branch of dynamic systems.G. D. BerkhovIn several years since 1912, with the three body problem as the background, the research on dynamic systems has been expanded, including his ergodicity theorem.In the celestial mechanics orhamilton system Many years later, the Kolmogorov Arnold Moser torsion theorem with the stability of the solar system as the background appeared in the field of.For several years from 1931Fra. Fra. MarkovStarting from summarizing Berkhov's theory and formally proposing the abstract concept of dynamic system, Soviet scholars further promoted the development of dynamic system theory.
In the past twenty years, the research of dynamic systems has undergone qualitative changes.This stems from the study of structural stability.Many of the major achievements in this regard areXIs compact and smoothmanifoldM.The C1 constant microsystem S on M is said to be structurally stable if a sufficiently small C1 disturbance does not change the phase diagram structure of S.That is, if any C1 constant microsystem Z on M is sufficiently close to S, then there is a topological transformation from M to itself that maps the trajectory of S to the trajectory of Z (here, the so-called sufficiently close is in the sense of C).The reason why the concept of structural stability is widely accepted is that the mathematical model used in practical application is often simplified compared with the real phenomenon. Therefore, in order to make the model used effective, it is hoped that there will be some degree of unchanged structure despite small disturbances.Obviously, the dynamic system theory starting from the stability in this sense involves not only the integrity of the phase diagram of each single ordinary and differential system, but also the integrity of the set made of many ordinary and differential systems on the same manifold, in other words, this is a large range.The concept of structural stability of ordinary microsystems is first defined by A. A. Andronov andЛ. С. PontryaginIn 1937, it was put forward on some kind of plane ordinary differential equations, but after more than 20 years, M. Pexoto gave2D structureStabilization systemAfter the density theorem, people began to pay attention to it, because the structurally stable system on the two-dimensional closed surface not only has a simpler phase diagram structure, but also any C1 ordinary microsystem can be arbitrarily approached by the structurally stable system.In manifolddimensionWhen it is greater than 2, whether there is the same conclusion? This question has inspired people toDifferential dynamical systemLater, it became clear that the phase diagram of structurally stable systems in high-dimensional cases is generally very complex, and the density theorem is no longer tenable.
Mathematicians led by S. Smeier have made important contributions to the study of differential dynamic systems, and their influence has been enduring.For example, compact invariant subset with hyperbolic structure is still the root of many specific topics.Since the density theorem is no longer valid in high-dimensional cases, this involves the bifurcation problem with unusual complexity, but it may be more consistent with some "chaos" phenomena in nature.Lorenz people care aboutStrange attractorThe Fegenbaum phenomenon is very enlightening, and the research in this field has penetrated into many scientific fields such as physics, chemistry, biology, etc.
