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
Lecture 3
Objective Review other systems that show
high-dimensional chaos (HDC) and
determine basic routes to HDC

• 4D Rössler system
• Other HD Lorenz models
• Another HD Duffing system
• Double pendula
• Other interesting HD systems
• Routes to chaos
• Summary

3
4D Rössler System

• First HD system with two positive Lyapunov
• exponents was introduced by Rössler (1979)

4
Strange Attractor I
First-return map to a Poincaré section
Plane projection of the strange attractor
Strange attractor is characterized by two
positive, one negative and one zero Lyapunov
exponents.
5
Strange Attractor II
AIHARE, Electrical Eng. Co., Japan
6
HD Lorenz Models I
Li, Tang and Chen (2005) generalized the 3D
Lorenz model by adding a new variable that
couples to the second equation of Lorenzs
equations and derived a 4D Lorenz model.
They designed a circuit that approximates the 4D
system.
7
Theory vs Experiment
Li, Tang and Chen (2005)
8
HD Lorenz Models II
9D Lorenz model (Reiterer et al. 1998) Model
describes a 3D Rayleigh-Benard convection
Hyperchaos at R 43.3
Period-doubling cascade
Model does not conserve energy in dissipationless
limit (Roy Musielak 2006)
9
Another HD Duffing System

Savi Pacheco (2002)
10
Phase Portraits

Savi Pacheco (2002)
11
Double Pendula I

Initial speeds, left Initial speeds, right
main arm 400.0 degrees/sec main arm 400.1
degrees/sec secondary arm 0.0 degrees/sec
secondary arm 0.0 degrees/sec
Ross Bannister www.rdg.ac.uk/ross
12
Double Pendula II

Bannister (2005)
13
Coupled Logistic Maps
General route to HDC - Harrison Lai (1999,
2000)
Pazo et al (2001)
14
Coupled Rössler Systems
Harrison Lai (2000)

15
Modified Chuas Circuit
Original Chuas circuit
Modified Chuas circuit
Thamilmaran et al (2004)

16
Experimental Results
Phase portraits
Poincaré sections
Power spectra
Thamilmaran et al (2004)

17
Theoretical Results
Thamilmaran et al. (2004)

18
Other Systems with HDC

• Coupled Ikeda maps
• Chaotically driven Zaslavsky map
• Delayed Henon maps
  
• Coupled three or more Lorenz systems
• Coupled two or more lasers
• Phonic integrated circuits
• Miniature eye movements
• Excitable physiological systems
• Spreading of rumor

19
Types and Properties of HD Systems

• 1. Strange attractors with dimensions dcor gt 3
  but only one positive Lyapunov exponent - no
  hyperchaos.
• 2. Strange attractors with dimensions dcor gt 3
  and two or more positive Lyapunov exponents -
  systems with hyperchaos.

HD and LD systems behave differently and chaos is
persistent (no windows of periodicity) in HD
dynamical systems (Albers et al 2005)
20
Routes to HDC I

• Same as routes for LD systems
• (a) Period-doubling
• (b) Quasi-periodicity
• (c) Intermittency
• (d) Chaotic transients
• (e) Crisis
• First LD chaos by one of the above routes and
  then
• to HD chaos.
• Harrison Lai (1999) and Pazo et al (2001)

21
Routes to HDC II

• Quasi-periodicity torus doubling torus
  merging chaos
• Venkatesan Lakshmanan (1998)
• Quasi-periodicity torus 3-period window
  chaos
• Musielak et al (2005)
• Sequence of Neimark-Sacker bifurcations
• Alberts Sprott (2004)
  

22
SUMMARY

• High-dimensional (HD) dynamical systems that
  exhibit chaos can be constructed by adding
  degrees of freedom to low-dimensional dynamical
  systems.
• High dimensional chaos (HDC) is observed in HD
  nonlinear systems whose strange attractors have
  dimensions dcor gt 3.
• Two types of systems with HDC have been
  identified, those with and without hyperchaos.
• HD systems may transition to chaos via one of the
  routes
• known for LD systems or via new routes
  four new
• routes have been identified, others
  still remain to be
• discovered.

23
Acknowledgments

• Special thanks to Professor Ahmad M. Harb
• and the organizers of the IASTED
  International Workshop on Modern Nonlinear Theory
  for the invitation to present this tutorial.

Support for this work was provided by NASA /
MSFC, US Army and The Alexander von Humboldt
Foundation in Germany.
24
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