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)

Lecture 3  4D Rössler system  Other HD Lorenz models  Another HD Duffing system  Double pendula  Other interesting HD systems  Routes to chaos  Summary Objective: Review other systems that show high-dimensional chaos (HDC) and determine basic routes to HDC (HDC) and determine basic routes to HDC

4D Rössler System First HD system with two positive Lyapunov First HD system with two positive Lyapunov exponents was introduced by Rössler (1979) exponents was introduced by Rössler (1979)

Strange Attractor I Plane projection of the strange attractor First-return map to a Poincaré section Strange attractor is characterized by two positive, one negative and one zero Lyapunov exponents.

Strange Attractor II AIHARE, Electrical Eng. Co., Japan

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 Lorenz’s equations and derived a 4D Lorenz model. They designed a circuit that approximates the 4D system.

Theory vs Experiment Li, Tang and Chen (2005)

HD Lorenz Models II 9D Lorenz model (Reiterer et al. 1998) Model describes a 3D Rayleigh-Benard convection Model does not conserve energy in dissipationless limit (Roy & Musielak 2006) Hyperchaos at R = 43.3 Period-doubling cascade

Another HD Duffing System Savi & Pacheco (2002)

Phase Portraits Savi & Pacheco (2002)

Double Pendula I Ross Bannister: Initial speeds, left: Initial speeds, right: main arm = degrees/sec main arm = degrees/sec secondary arm = 0.0 degrees/sec

Double Pendula II Bannister (2005)

Coupled Logistic Maps General route to HDC - Harrison & Lai (1999, 2000) Pazo et al (2001)

Coupled Rössler Systems Harrison & Lai (2000)

Modified Chua’s Circuit Thamilmaran et al (2004) Original Chua’s circuit Modified Chua’s circuit

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

Theoretical Results Thamilmaran et al. (2004)

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

Types and Properties of HD Systems 1. Strange attractors with dimensions d > 3 but only one positive Lyapunov exponent - no hyperchaos. 1. Strange attractors with dimensions d cor > 3 but only one positive Lyapunov exponent - no hyperchaos. 2. Strange attractors with dimensions d > 3 and two or more positive Lyapunov exponents - systems with hyperchaos. 2. Strange attractors with dimensions d cor > 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)

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. to HD chaos. Harrison & Lai (1999) and Pazo et al (2001)

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) Musielak et al (2005)  Sequence of Neimark-Sacker bifurcations Alberts & Sprott (2004)

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 d > 3.  High dimensional chaos (HDC) is observed in HD nonlinear systems whose strange attractors have dimensions d cor > 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 known for LD systems or via new routes; four new routes have been identified, others still remain to be routes have been identified, others still remain to be discovered. discovered.

Acknowledgments Special thanks to Professor Ahmad M. Harb 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. 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.

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