High Dimensional Chaos

1. High Dimensional Chaos Tutorial Session IASTED International Workshop on Modern Nonlinear Theory (Bifurcation and Chaos) ~Montreal 2007~ Zdzislaw Musielak, Ph.D. and Dora Musielak, Ph.D. University of Texas at Arlington (UTA) Arlington, Texas (USA)

2. High Dimensional Chaos (HDC) • Lecture 1: Basic Concepts and Definitions • Lecture 2: Extension of Lorenz and Duffing systems to high dimensions • Lecture 3: Other high-dimensional (HD) systems with chaos and hyperchaos

3. Lecture 1 • Chaos and Predictability • Basic Techniques • Main Routes to Chaos • Low-Dimensional (LD) Systems • Lorenz and Duffing Systems • High-Dimensional Chaos (HDC) • Summary Objective: Review basic concepts and define “high dimensional chaos”

4. Chaos in Deterministic Systems Deterministic dynamical systems (Newton 1687) Present and future states of these systems are fully determined by their initial conditions Chaos in deterministic systems (Poincaré 1889) Sensitivity of some systems to small changes in their initial conditions is so strong that no reasonable prediction in time is possible

5. Fundamental Paradox System’s chaotic behavior is generated by the fixed deterministic rules that do not themselves involve any element of chance! Since the system is deterministic, its future is in principle completely determined by the past. However, in practice, small uncertainties in the system’s initial conditions are quickly amplified and the system’s behavior becomes unpredictable.

6. Predictability in Time Distance between two initially close trajectories is where λ is the Lyapunov exponent. If λmax is the largest Lyapunov exponent, then the so-called Lyapunov time TL is given by where Ds is the characeristic size of the system.

7. Basic Techniques • Phase portraits • Poincaré sections • Power spectra • Lyapunov exponents

8. Phase Portraits Time series and attractors

9. Low and High Dimensional Systems Low-dimensional systems: (i) dissipative and driven systems described in three-dimensional (3D) phase space (ii) 1D and 2D iterative maps High-dimensional systems: (i) dissipative and driven systems described in phase space with dimension higher than 3D (ii) iterative maps with dimensions higher than 2D

10. Attractors and Lyapunov Exponents Fixed point: all three Lyapunov exponents are negative Limit cycle: two Lyapunov exponents are negative and one is zero Torus: two Lyapunov exponents are zeroand one is negative Strange attractor: one Lyapunov exponent is positive, one is zeroand one is negative

11. Poincaré Sections I Thompson & Stewart (1986)

12. Poincaré Sections II Stroboscopic method Thompson & Stewart (1986)

13. Power Spectra Roy and Musielak (2007)

14. Lyapunov Exponents Rossler (1983)

15. Routes to Chaos • Via local bifurcations: Period-doubling Quasi-periodicity Intermittency • Via global bifurcations: Chaotic transients Crisis

16. Period-Doubling Cascade I Feigenbaum (1979)

17. Period-Doubling Cascade II Logistic map

18. Quasi-Periodicity I Landau (1944) Ruelle & Takens (1971) Argyris, Faust and Haase (1993)

19. Quasi-Periodicity II In some systems a third distinct frequency occurs NRT theorem: no more than 3 frequencies can be observed Newhouse, Ruelle and Takens (NRT 1978)

20. Quasi-Periodicity III Rayleigh-Benard convection Swinney and Gollub (1978)

21. Quasi-Periodicity IV Quasi-periodic route to chaos shown by Poincare sections Transition to chaos

22. Intermittency Time-history response of the vertical velocity component in three different Rayleigh-Benard convection experiments Berge et al. (1980) Pomeau & Manneville (1980)

23. Chaotic transients and Crisis Chaotic transients – system’s trajectories interact with various unstable fixed points and limit cycles; homoclinic and heteroclinic orbits may suddenly appear and strongly influence the trajectories Crisis – a strange attractor may suddenly disappear as a result of its interaction with an unstable fixed point or an unstable limit cycle

24. Low-Dimensional Systems • Lorenz model • Duffing oscillator • Van der Pol oscillator • Dissipative and driven pendulum • Belousov-Zhabotinsky reactions • Rossler model • Logistic map • Hanon map

25. Lorenz Model Convective rolls HEAT  Continuity equation for an incompressible fluid  Navier-Stokes equation with constant viscosity  Heat transfer equation with constant thermal conductivity Lorenz (1963)

26. Mathematical Description Double Fourier expansion in the vertical (z) and horizontal (x) direction of the stream function and the temperature deviation Modes of expansion: and Saltzman (1962) Lorenz truncation:

27. Lorenz Strange Attractor Generated by the “stretch-tear-squeeze” mechanism (Gilmore 1998) Capacity dimension of the strange attractor dcap = 2.06 (Lorenz 1984) Rigorous proof of the existence of the attractor (Tucker 1999)

28. Lorenz Equations where and 3D model in phase space

29. Route to Chaos Chaotic transients Kennamer (1995)

30. Lorenz-like Models • Generalized Lorenz models describing turbulent flows • Lorenz-Maxwell-Haken model describing lasers • Reversal of Earth’s magnetic field • Several models used in communication • Models used to describe chaotic cryptosystems • Models used to control chaos

31. Duffing Oscillator Duffing (1918) Ueda (1979,1980)

32. Duffing Equations

33. Lyapunov Exponents

34. Route to Chaos Benner (1997)

35. Duffing Strange Attractor Generated by the “stretch and roll” mechanism (Gilmore 1998) Capacity dimension: 2.17 for ω = 0.04 2.38 for ω = 0.20 2.53 for ω = 0.32 Burke (1998)

36. Out of Chaos Crisis route Benner (1997)

37. Duffing-like Systems • Electric circuits with nonlinear inductance • Nonlinear mechanical oscillators • Mechanical systems that contain gears, backlash and deadband regions • Large deflections of elastic continua • Buckling of beam-colums • Rigid and flexible missiles

38. From Low to High-Dimensional Systems • Extending the 3D Lorenz model to higher dimensions • Extending the 3D Duffing system to higher dimensions • Other high-dimensional (HD) dynamical systems • High-dimensional chaos (HDC) in these systems

39. High-Dimensional Chaos (HDC) Previously suggested definitions involve: (a) two or more positive Lyapunov exponents (also called hyperchaos) (b) high-dimensional strange attractors (c) multiple strange attractors Definition of HDC used in this tutorial: HDC refers to chaos observed in dynamical systems with phase space dimensions D > 3 and with strange attractors whose correlation dimension dcor > 3

40. SUMMARY • Basic techniques: phase portraits, Poincare sections, power spectra and Lyapunov exponents • Routes to chaos: period-doubling, quasi-periodicity, intermittency, chaotic transients and crisis • Chaotic behavior in 3D Lorenz system and chaotic transients as the system’s route to fully developed chaos • Chaotic behavior in 3D Duffing system and period-doubling as the system’s route to fully developed chaos • Methods for constructing high-dimensional (HD) systems and definition of high dimensional chaos (HDC)