Chaotic-Dynamics---TAU混沌动力学的头-课件

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1、Chaos vs.RandomnessDo not confuse chaotic with random:Random:irreproducible and unpredictableChaotic:deterministic -same initial conditions lead to same final state but the final state is very different for small changes to initial conditions difficult or impossible to make long-term predictionsCloc

2、kwork(Newton)vs.Chaotic(Poincar)UniverseSuppose the Universe is made of particles of matter interacting according to Newton laws this is just a dynamical system governed by a(very large though)set of differential equationsGiven the starting positions and velocities of all particles,there is a unique

3、 outcome P.Laplaces Clockwork Universe(XVIII Century)!Can Chaos be Exploited?Brief Chaotic History:PoincarBrief Chaotic History:LorenzChaos in the Brave New World of ComputersPoincar created an original method to understand such systems,and discovered a very complicated dynamics,but:It is so complic

4、ated that I cannot even draw the figure.An Example A Pendulumstarting at 1,1.001,and 1.000001 rad:Changing the Driving Forcef=1,1.07,1.15,1.35,1.45Chaos in PhysicsChaos is seen in many physical systems:Fluid dynamics(weather patterns),some chemical reactions,Lasers,Particle accelerators,Conditions n

5、ecessary for chaos:system has 3 independent dynamical variablesthe equations of motion are non-linearWhy Nonlinearity and 3D Phase Space?Dynamical SystemsA dynamical system is defined as a deterministic mathematical prescription for evolving the state of a system forward in timeExample:A system of N

6、 first-order,autonomous ODEDamped Driven Pendulum:Part IThis system demonstrates features of chaotic motion:Convert equation to a dimensionless form:)cos(sin 22fwqqq+tAmgdtdcdtdmlD)cos(sin0tfdtdqdtdDwqqw+Damped Driven Pendulum:Part II3 dynamic variables:,tthe non-linear term:sin this system is chaot

7、ic only for certain values of q,f0,and DIn these examples:D=2/3,q=1/2,and f0 near 1Damped Driven Pendulum:Part IIIto watch the onset of chaos(as f0 is increased)we look at the motion of the system in phase space,once transients die awayPay close attention to the period doubling that precedes the ons

8、et of chaos.07.10f15.10ff0=1.35f0=1.48f0=1.45f0=1.49f0=1.47f0=1.50Forget About Solving Equations!New Language for Chaos:Attractors(Dissipative Chaos)KAM torus(Hamiltonian Chaos)Poincare sectionsLyapunov exponents and Kolmogorov entropyFourier spectrum and autocorrelation functionsPoincar SectionPoin

9、car Section:ExamplesPoincar Section:PendulumThe Poincar section is a slice of the 3D phase space at a fixed value of:Dt mod 2This is analogous to viewing the phase space development with a strobe light in phase with the driving force.Periodic motion results in a single point,period doubling results

10、in two points.Poincar MovieTo visualize the 3D surface that the chaotic pendulum follows,a movie can be made in which each frame consists of a Poincar section at a different phase.Poincare Map:Continuous time evolution is replace by a discrete map f0=1.07f0=1.48f0=1.50f0=1.15q=0.25AttractorsThe surf

11、aces in phase space along which the pendulum follows(after transient motion decays)are called attractorsExamples:for a damped undriven pendulum,attractor is just a point at=0.(0D in 2D phase space)for an undamped pendulum,attractor is a curve(1D attractor)Strange AttractorsChaotic attractors of diss

12、ipative systems(strange attractors)are fractals Our Pendulum:2 dim 3The fine structure is quite complex and similar to the gross structure:self-similarity.non-integer dimensionWhat is Dimension?Capacity dimension of a line and square:1 L2 L/24 L/48 L/82n L/2nN 1 L4 L/216 L/422n L/2nN )/1log(/)(log 0

13、lim)/1()(eeeeeNdLNcddTrivial Example:Point,Line,Surface,Non-Trivial Example:Cantor SetThe Cantor set is produced as follows:N 1 12 1/34 1/98 1/2713log/2log3log/2log lim 0For the pendulum:i=-q (damp coeff.)no contraction or expansion along t direction so that exponent is zerocan be shown that the dim

14、ension of the attractor is:d=2-1/2Dissipative vs Hamiltonian ChaosAttractor:An attractor is a set of states(points in the phase space),invariant under the dynamics,towards which neighboring states in a given basin of attraction asymptotically approach in the course of dynamic evolution.An attractor

15、is defined as the smallest unit which cannot be itself decomposed into two or more attractors with distinct basins of attraction.This restriction is necessary since a dynamical system may have multiple attractors,each with its own basin of attraction.Conservative systems do not have attractors,since

16、 the motion is periodic.For dissipative dynamical systems,however,volumes shrink exponentially so attractors have 0 volume in n-dimensional phase space.Strange Attractors:Bounded regions of phase space(corresponding to positive Lyapunov characteristic exponents)having zero measure in the embedding p

17、hase space and a fractal dimension.Trajectories within a strange attractor appear to skip around randomlyDissipative vs Conservative Chaos:Lyapunov Exponent PropertiesFor Hamiltonian systems,the Lyapunov exponents exist in additive inverse pairs,while one of them is always 0.In dissipative systems i

18、n an arbitrary n-dimensional phase space,there must always be one Lyapunov exponent equal to 0,since a perturbation along the path results in no divergence.Logistic Map:Part IThe logistic map describes a simpler system that exhibits similar chaotic behaviorCan be used to model population growth:For

19、some values of,x tends to a fixed point,for other values,x oscillates between two points(period doubling)and for other values,x becomes chaotic.)1(11-nnnxxxmLogistic Map:Part IITo demonstrate)1(11-nnnxxxmxn-1xnBifurcation Diagrams:Part IBifurcation:a change in the number of solutions to a differenti

20、al equation when a parameter is variedTo observe bifurcatons,plot long term values of,at a fixed value of Dt mod 2 as a function of the force term f0Bifurcation Diagrams:Part IIIf periodic single valuePeriodic with two solutions(left or right moving)2 valuesPeriod doubling double the numberThe onset

21、 of chaos is often seen as a result of successive period doublings.Bifurcation of the Logistic MapBifurcation of PendulumFeigenbaum NumberThe ratio of spacings between consecutive values of at the bifurcations approaches a universal constant,the Feigenbaum number.This is universal to all differentia

22、l equations(within certain limits)and applies to the pendulum.By using the first few bifurcation points,one can predict the onset of chaos.-+-kkkkk.669201.4lim11dmmmm Chaos in PHYS 306/638Aperiodic motion confined to strange attractors in the phase spacePoints in Poincare section densely fill some r

23、egionAutocorrelation function drops to zero,while power spectrum goes into a continuumDeterministic Chaos in PHYS 306/638Deterministic Chaos in PHYS 306/638Deterministic Chaos in PHYS 306/638Deterministic Chaos in PHYS 306/638Deterministic Chaos in PHYS 306/638Chaos in PHYS 306/638Deterministic Chao

24、s in PHYS 306/638Deterministic Chaos in PHYS 306/638Do Computers in Chaos Studies Make any Sense?Shadowing Theorem:Although a numerically computed chaotic trajectory diverges exponentially from the true trajectory with the same initial coordinates,there exists an errorless trajectory with a slightly different initial condition that stays near(shadows)the numerically computed one.Therefore,the fractal structure of chaotic trajectories seen in computer maps is real.谢谢你的阅读v知识就是财富v丰富你的人生