Computations for Computations for Dynamical Systems Dynamical Systems

Computations for Computations for Dynamical Systems Dynamical Systems

張 書 銘

交通大學應用數學系 交通大學應用數學系

[email protected]

2010 年 7 月 27 日 2010 年 7 月 27 日

Outline

‡

Dynamical System y y

‡

Computational Dynamical System

‡

Chaos

‡

Chaos

‡

Examine Chaos

Dynamical System

動態系統是要研究運動方程的解。

Computational Dynamical System

（1）離散動態系統:

（1）離散動態系統:

連續動態系統

（2）連續動態系統 :(ode45,... in MatLab)

Computational Dynamical System

（1）離散動態系統的解:

（1）離散動態系統的解:

發散(infinity) 、固定點、週期解、

擬週期(quasi periodic) ? 擬週期(quasi-periodic)、?

（2）連續動態系統的解:

（2）連續動態系統的解:

發散(infinity) 、平衡點、週期解、

(li it l )

(limit cycle) 、

Computational Dynamical System

•0-D: equilibrium points (radial spiral saddle) (radial, spiral, saddle)

•1-D: limit cycles (closed loops)y ( p )

•2-D: 2-toruses (quasiperiodic surfaces)

•

N

N

•Non integer D: strange attractors (fractal)

•Non-integer D: strange attractors (fractal)

Chaos

混沌理論認為在混沌系統中，初始

混沌理論認為在混沌系統中 初始

條件十分敏感，其微小的變化，在經過

不斷放大 對未來狀態會造成極其巨大

不斷放大，對未來狀態會造成極其巨大 的差別。

的差別

Devanvey's chaos

•敏感性 (sensitivity)： ( y)

對初始條件非常敏感，差之毫釐失之千里。

•傳遞性 (transitivity)：

可到處遍歷。

可到處遍歷。

•週期解稠密性 週期解稠密性 (density)： (density)：

Examine Chaos

• bifurcation diagram

( i d d bli bif l i ti

(period doubling bif.: logistic map, intermittence: tent map)

• Feigenbaum constant

δ= 4.66920160910299067185320382…

FFT

FFT

Examine Chaos

• Poincaré map Poincaré map

(conti. D.S.)

Poincaré map

Poincaré map

•發散 (infinity)、平衡點

•週期解

•極限環 (limit cycle)

•擬週期 (quasi-periodic)

Poincaré map

Poincaré map

Poincaré map

Poincaré map

Examine Chaos

• Lyapunov exponent (Lyapunov y p p ( y p characteristic exponent)

• Poincaré recurrence

• homoclinic orbit

(snapback repellor)

(snapback repellor)

Lyapunov exponent

Lyapunov exponent

• [ Definition] global Lyapunov exponent

• [Computation] local Lyapunov exponent

(average the phase-space volume expansion (average the phase-space volume expansion

along trajectory)

Local Lyapunov exponent

Local Lyapunov exponent

Poincaré recurrence

positive topological entropy

positive topological entropy

Homoclinic orbit

MLM: modified logistic map

Logistic map

Modified Logistic map

Properties of MLM

Chaotic map

• Chaotic map

• No windows

• Uniform distribution E i l t

• Equivalent

• Pseudorandom Pseudorandom

MLM: chaotic map

MLM: no windows

MLM: no windows

MLM: Poincaré recurrence

MLM: uniform distribution (FFT)

r = 5.9

MLM: equivalent (bits error rate analysis)

MLM: pseudorandom

MLM: pseudorandom (SP 800-22)

Random vs. Chaos

( ^x

^{, x}

^{,..., x}

^,... )

( )

Random numbers:

Ch ti i l ( ^y

^{, y}

^{,..., y}

^,... )

Identity：

Chaotic signals:

y

1. Continuous Spectrum 2 C l ti F ti ： 2. Correlation Function：

∑

=

Random vs. Chaos

Distinction：

3 2D charged particles

3 2D charged particles

3 2D charged particles

3 2D charged particles

3 2D charged particles

3 2D charged particles

3 2D charged particles

3 2D charged particles

3 2D charged particles

3 2D charged particles

3 2D charged particles

3 2D charged particles

3 vortices system

3 vortices system

3 vortices system

References

„V. Afraimovich, J. Schmeling, E. Ugalde, J. Urias, Spectra of dimensions for Poincare recurrences, Discrete Contin.

S 6 (4) (2000) 901 914 Dyn. Syst. 6 (4) (2000) 901-914.

„S. M. Chang, M. C. Li and W. W. Lin, Asymptotic synchronization of modified logistic hyper-chaotic synchronization of modified logistic hyper chaotic systems and its applications. Nonlinear Analysis: Real World Applications, Vol. 10, Issue 2 (2009), pp. 869–880.

„S. M. Chang, T. C. Lin and W. W. Lin, Chaotic and

Quasiperiodic Motions of Three Planar Charged Particles.

Int J Bifurcation Chaos Vol 11 No 7 (2001) pp 1937 Int. J. Bifurcation Chaos, Vol. 11, No. 7 (2001), pp. 1937–

References

„S. M. Chang, T. C. Lin and W. W. Lin, Dynamics of Vortices in Two-Dimensional Bose-Einstein

Condensates Int J Bifurcation Chaos Vol 12 No 4 Condensates. Int. J. Bifurcation Chaos, Vol. 12, No. 4 (2002), pp. 739–764.

„S. L. Chen, S. M. Chang, T. T. Hwang and W. W. Lin,

„S. L. Chen, S. M. Chang, T. T. Hwang and W. W. Lin,

Digital secure-communication using robust hyper-chaotic systems. Int. J. Bifurcation Chaos, Vol. 18, No. 11 (2008),

1 14 pp. 1–14.

„T. S. Parker & L. O. Chua, Practical Numerical Algorithm for Chaotic Systems Ch 3 Springer-Verlag 1989

for Chaotic Systems, Ch.3, Springer Verlag, 1989.

References

„List of chaotic maps.

http://en.wikipedia.org/wiki/List_of_chaotic_maps

„動態系統, 動力系統, 混沌理論.

http://zh.wikipedia.org/zh-tw/

„Lyapunov Exponents, Chaos and Time-Series Analysis.

http://sprott.physics.wisc.edu/phys505/lect05.htm

Thank you for your attention!

Thank you for your attention!