Chaos refers to certaintyDynamic systemUnpredictable and similar due to sensitivity to initial valueRandomnessOfmotion。Also called chaos.The English word Chaos originates from Greek, and its original meaning is the scene before the beginning of the universe. Its basic meaning mainly refers to the state of chaos and disorder.As a scientific term, chaos refers to a form of movement.[1]
The certainty of dynamic system is a mathematical concept, which means that the state of the system at any time is determined by the initial state.Although the motion state at any future time can be calculated according to the initial state data and the motion law of the motion, because the measurement of the initial data cannot be completely accurate, the prediction result is bound to have errors, or even unpredictable.The predictability of motion is a physical concept.Even if a movement is deterministic, it can still be unpredictable, and the two are not contradictory.Newtonian mechanicsThe success, especially its predictionNeptuneTo some extent, the success of the game has led to a misunderstanding, which equates certainty with predictability, thinking that the deterministic movement must be predictable.Studies since the 1970s have shown that a large number ofnonlinear system AlthoughsystemIt is deterministic, but it is very sensitive to the initial value of the motion staterandomThe unpredictable state of motion of chaotic motion.[1]
Chaos meansReal worldThere is a seemingly irregular and complex movement pattern in.The common feature is that the original orderly movement pattern that follows simple physical laws suddenly deviates from the expected regularity under certain conditions and becomes an unordered pattern.Chaos can occur in a wide range of deterministic dynamic systems.Chaos is statistically similar torandom process, considered to beDeterministic systemAn intrinsic randomness in.[2]
DynamicsThe system canenergywhetherconservationDifferentiate intoConservative systemandDissipative system；It can also be divided intoIntegrable systemAnd non integrable systems.In all possible mechanical systems, non integrable systems are ubiquitous, and integrable systems are very rare.Traditional mechanics textbooks only teach integrable systems, notdescribeThe true face of Newtonian mechanics.Typical of non integrable mechanical systemsMoving imageHow to become aMathematical puzzle。At the end of the 19th century, HPoincare Talking about the solar systemstabilityFirst foundThree body problemNon integrable and three body motion orbitcomplexity。Until the early 1960s, three mathematicians A. Kolmogorov and VArnoldAnd J. Morse provedKAM TheoremLater, I answered some questions in a positive way.[1]
Chaos in conservative systems
KAM theorem says that if a system deviates from an integrable system sufficiently small, the overall motion picture is similar to that of an integrable system.But KAM theorem does not answer how the system moves under large deviation.At this time, the system still obeys the deterministic Newtonian mechanical equation, that is, as long as the system starts from a certain initial point accurately, its motiontrackIt is absolutely certain.But ifinitial condition No matter how small a change occurs, some motion orbits of the system will change unexpectedly.This kind of appearance that the motion orbit in the deterministic system is extremely sensitive to the initial valuedisorderAnd chaotic motion.A typical non integrable mechanical system usually has two different regions of regular motion and random motion.With deviationIntegrabilityThe random region gradually expands, and finally replaces the regular region.Therefore, from the perspective of predictability,decisiveThe Newtonian mechanics ofRandomness。[1]
KAM theorem shows that it is nearly integrablehamilton system The nature of the motion of.The research on Hamiltonian system started from this point found that when KAM theorem is not applicable, chaos also occurs in the system.In the 1970s,Dynamic systemInternal randomness theory orchaos theory And relatedStrange attractorThe mathematical theory of the.Some people think that this theory may be the final clarificationfluid mechanicsBut some people think that the current chaos theory is relatively simplemathematical model , for the imageNavier Stokes equationThat waypartial differential equation There is nothing we can do, so it is too early to solve the turbulence mechanism.stayphysicsAnd other scientific fields, there are also various examples of chaotic motion.Chaotic phenomenonThe discovery ofclassical mechanicsandstatistical mechanicsThe communication between deterministic theory and stochastic theory is enlightening in thought.[3]
Chaos of dissipative systems
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Figure 1 A Lorentz attractor shaped like a butterfly wing
The intuitive image of chaotic motion followsenergyconstantlydissipationandfreedomReducedDissipative systemIt can be seen more clearly in.1963 United StatesMeteorologistE. Lorenz is studyingThermal convectionThe partial differential equation of thermal convection with infinite degrees of freedom is reduced to threevariableFirst order nonlinear ordinary differential equations of:
dx/dt=－σx+σy
dy/dt=rx－y－xz
dz/dt=bz+xy
Variable in the formulaxIs the intensity of atmospheric convection,yexpressUpflowAndDownflowTemperature difference,zIndicates the vertical temperature profile change.coefficientσbyPrandtl number，rbyRayleigh number，bbymeasurementHorizontal temperature structure and vertical temperature structureAttenuation rateDifferences.Lorenz selectedσ=10，r=28，b=8/3, and then numerically solve the equations.It is found that this extremely simplified system has a very complex form of motion.A slight change in the starting value is enough to make the orbit completely different.The numerical calculation results are calculated byx,y,zThe three-dimensional phase space of the support is drawn.This is athree-dimensional spaceThe continuous smooth curve that seems to rotate from left to right in disorder does not intersect itself, presenting complex structural patterns.No matter where the initial value is selected, the orbits of the system have the same destination, forming the so-called strange attractor.On the singular attractor, if two arbitrarily close points are selected as initial valuespath of particleExponentially, they are rapidly separated, showing extreme sensitivity to initial values.Specifically, the order and times of track jumping from left to right are completely different.The calculation shows that the initial positions of the 10000 points that almost converge together will beattractor Go everywheredistribution, which means that in such a system, the motion is unpredictable due to the slight difference of initial values.[1]
certaintyDissipative systemMovement is ultimately limited to low dimensionsattractor This phenomenon is very common.If the damped pendulum stops due to resistance, its attractor is called a fixed point;Appropriate input energy cancellationdissipation, the pendulum can still keep a certain period of oscillation, and the attractor isLimit cycle。Such attractors do not have initial value sensitivity, so they are called ordinary attractors.[1]
Lorenz attractorIs the first one found in the dissipative systemStrange attractor, and since then in manynonlinear system Various strange attractors such asCelestial motionIn the modelEnon attractor, descriptionnonlinear vibration Uchida attractor of van der Boer equation, descriptionChemical oscillationBrussels attractor, etc.The strange attractor has some unique properties: ① the motion orbit on it is extremely sensitive to the initial value and unpredictable; ②It has a fractal structure, which is similar to the whole.The calculation shows that theFractal dimension2.06.The strange attractor also hasErgodicity of various states, i.ephase spaceThe moving track zigzags back and forth through every point on the attractor.[1]
Two basic characteristics characterizing disorder in chaos are: unpredictability and extremely sensitive dependence on initial values.It was studied by E. Lorenzweather forecastThe problem of atmospheric flow was first revealed.He passed theprogramstaycomputerUpper solution simulationEarth's atmosphereIt is found that as long as there is a negligible difference in the initial value as the starting point of the experiment, the same process will lead to different images in the chaotic state.And because it is impossible to measure the initial value with infinite precision, it is impossible to predict the final results of any chaotic system.Lorenz also found that although chaos seems to be disordered, it still has a certain order, and the number printed according to the thousands of possible solutions output by the computer is only randomly distributed in the range of a certain state.Just as the daily weather can have an infinite number of unpredictable configurations, the annual climate is more or less stable.The source of this internal order is a kind ofattractor Because it has a tendency tosystemOr an equation attracts to a certain final state.The attractors of Lorenz model are a classStrange attractorThe solution of the equation will infinitely approach to this strange attractor, hovering back and forth to form an integrated left and right clusters, just like a pair of butterfly wings in flutter (see Figure 1).[2]
A famous expression of chaos isButterfly effect"If a butterfly in South America flaps its wings, it will cause a hurricane in Florida."[4]
Complex behavior of models
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Simple causes can lead to complex consequences.Many seem to be in disorderRandom fluctuationThe temporal change or spatial pattern of may come from the repeated application of some simple and definiteNonlinearThe result of the basic action.A typical example is the extremely simple one-dimensional iterative wormhole model.[1]
It is assumed that all adults die after reproduction, and then hatch out the next generation without overlapping generations.If the population of the next generation is simply proportional to the population of the previous generation, as long as the average number of eggs laid is greater than 1, the population will be full after several generations of reproduction.This is it.malthusInsect Mouth Theory: Insect Mouth PressGeometric seriesGrowth.However, with the increase of the population, the population of insects scrambles for limited food and mates, and the spread of infectious diseases caused by contact among insects, the population will decrease.The number of eggs laid is proportional to the number of population, and the struggle and contact between insects is proportional to the square of population.availablexn+1=λxn(1-xn）OfiterationProcess description of insect population changes, wherexnRepresents the nth generation of insect population, λ is agrowth rateOfparameter, the value range is 0 ≤λ≤4。Corresponding to oneλValue, any initial valuexzeroAccording to the above iteration relationship, calculate thexone,xtwo,... Don't look at the first limitedxValue.It shows the simplicityIterative modelComplex behavior.At 0 ≤λWhen ≤ 1, the population number is finally 0, indicating that the insect species are extinct within this range.At 1 ≤λWhen ≤ 3, the number of insects will change withλSingle value rise,x(λ)=1－1/λ, the iteration value is fixed point.From 3<λ, there are two different types of insect population changes: firstx(λ）Jump between 2 points, and then make periodic jump between 4, 8, 16,..., 2n points, showing thatPeriod doubling bifurcationRule, thisλThe region is not sensitive to the initial value;Whenλ≥λ∞There are certainλIn the region, if the initial value is slightly changed, thex(λ）Specific experiencenumerical valueIt is totally different. This is the chaotic region sensitive to the initial valueaccuracyIn this area, small periodic variation areas insensitive to initial values can be seen.This periodic region embedded in the chaotic region is called periodic window, and its bifurcation diagram existsSelf similarityStructure.It is easy to see that even ifxn+1=λxn(1－xn）Such a simple iteration, due to the nonlinear effect, will also show the change process from bifurcation to chaos and the complex picture of periodic motion and chaotic motion intertwined, chaotic and methodical.[1]
The road to chaos
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For various nonlinearitiesmathematical model Theoretical research onnonlinear system Ofexperimental study , revealing a variety of typical ways for the system to move from regular motion to chaotic motion with the change of control parameters, among which the representative ones are:
① Period doubling bifurcation road.The system appears 2, 4, 8,... times the period successively, and finally enters the chaotic state.Near the limit point, this series of bifurcationsparameter space andphase spaceAre invariant under scale transformation, namelySelf similarity。useRenormalization groupA set of these bifurcation processes can be obtained by calculationUniversal constantThey are consistent with the experimental facts.[1]
② Quasi periodic roads.With the change of control parameters, the system appears one after anotherFixed point、Limit cycleQuasi periodic two-dimensional torus, and then enter into a chaotic state.This is a chaos generation mechanism proposed by D. Rueller and F. Tackens in 1975.The occurrence mechanism is availableCircular mappingIt shows that some scaling laws and universal constants are also found here.[1]
③ Bursts of chaotic roads.This kind of road is characterized by periodic motion and chaotic motion appearing alternately.As the control parameters approach the transition point, the random motion fragments that burst from time to time in regular motion become more and more frequent, and finally enter a completely chaotic state.The analysis shows that the mechanism of chaos can be explained by the process of tangent bifurcation of discrete map.[1]
Development direction of chaos research
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A Lorentz attractor in the shape of butterfly wings
chaotic motion 、Strange attractorThe concept of "road to chaos" has broadened the thinking of theoretical and experimental workers.Since the 1980sPlasmaDischarge systemNonlinear circuit、acousticsHarmony optical coupling systemLaserAnd optical bistable devices, chemical oscillating reactions, animalsCardiac myocyteOfforced vibration 、wild animalpopulationChaos is found everywhere in such fields as the number of fluctuations of, human brain wave signals, and even social and economic activities, which shows that chaotic movement is a lot ofnonlinear system Typical behavior of.As the main research field of nonlinear science, chaos research focuses on the following aspects: ① spatiotemporal chaos; ②Quantum chaos；③Further classification of chaotic motion; ④Fine characterization of chaotic attractors; ⑤Synchronization and control of chaos.[1]
Although there are some strict researches on chaosMathematical methodHowever, a large number of researches mainly rely on computer numerical experiments.The study of chaos is related to many disciplines.stayAnalytical mechanicsKAM theorem can be used to judge a class of approximately integrablehamilton system (a nonlinear dynamic system).open systemThe study of chaotic motion ofDissipative structure theoryThere are close connections.Research on Chaos andSynergeticsThey are also closely related. They both study the transformation of systems from order to disorder and from disorder to order.staysystem science In China, more and more attention has been paid to the study of chaos.The application prospect of chaos research needs to be further revealed.Chaotic phenomenonThe discovery of also makes people get new enlightenment on the relationship between epistemology and randomness.[2]
Significance of chaos research
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The practical significance of chaos research is multifaceted. ①The discovery of chaotic motion makes people see that there arenatureA form of movement that has been ignored for a long time, so as to understand many phenomena that were difficult to understand in the past.For example, it was found after 1977 that the superconducting tunnel junction placed in the microwave cavity appeared abnormal noise with the increase of gainexperimentinNoiseEquivalent temperature of up to 5 × 10fourK above, which cannot be explained by any known mechanism at that time.Later, it was understood that this was the system entering the chaos area, and the noise came fromdynamicsItself.high energyParticle acceleratorBeam loss inMagnetic confinement fusion Leakage of plasma in the devicenuclear power plantThe possible harmful backflow in the circulating water system is related to chaos. ②The discovery of chaotic motion provides a new perspective to consider the problem.For example, the problem of long-term weather forecast, the discovery of Lorenz attractorAtmospheric dynamicsThe sensitivity of the solution of the equations to the initial value has shaken the original idea that long-term weather forecasting can be solved by improving the calculation accuracy.The ergodic property of the chaotic attractor can guarantee the stability of the mean for a long time and the independence of the initial conditions.Because long-term weather forecast is concerned about the futurerainfall、temperatureThe average value of, chaos increases insteadlong-range weather forecastReliability.In addition,geomagnetic fieldThe multiple random turns that have affected global weather changes in the past million yearsel nino phenomenon, can be accessed fromDeterministic systemFrom the perspective of chaotic motion. ③Research on Chaotic Motion Applied PhysicsmathematicsAnd other precise scientific methods to study complexLife phenomenonIt has an important enlightening effect.Such as variousBiological rhythm, neither completely periodic nor purely random.It is not only affected by natural periodic processes such as season, day and night, but also maintains its intrinsic characteristics.Using mathematical models such as coupled nonlinear oscillators to simulate and cooperate with physiological experiments can reveal variousArrhythmiaThe possible relationship between atrioventricular conduction obstruction, ventricular fibrillation and chaotic motion.AnthropomorphicElectroencephalogram, found that the onset of epilepsyElectroencephalogramObviouslyPeriodicityThe brain waves of normal people are closer to random signals.These are found by dimension measurementsignalNot really random, but fromdimensionDynamic behavior on attractors that are not very high. ④Chaos research has changed human'sView of nature。There have always been determinism andprobability theoryTwo sets of opposite description systems.Newtonian mechanicsSince its establishment, the scientific tradition has praised the determinism system, andprobability theoryThe description is regarded as a necessary supplement.Chaotic motion pairDeterministic systemThere is inherent randomness in itself, which will undoubtedly make people from this artificial oppositiondescribeGet rid of the system, deepen the understanding of necessity and contingency, and more comprehensively understand the unity of nature.[1]
The discovery of chaos andchaos theory The establishment of is the same asrelativityandQuantum theoryIt is also a major breakthrough in Newton's deterministic classical theory.Many scientists believe that,Physics in the 20th CenturyThree brilliant scientific miracles are the creation of relativity, quantum theory and chaos theory.
