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Persistent chaos in high dimensions.

D J Albers¹, J C Sprott, J P Crutchfield

¹Max Plank Institute for Mathematics in the Sciences, Leipzig 04103, Germany. albers@cse.ucdavis.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|February 7, 2007

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As system dimension increases, positive Lyapunov exponents grow, reducing periodic behavior. Deterministic chaos persists in high dimensions due to a geometric mechanism preventing catastrophic topological changes.

Area of Science:

Dynamical Systems Theory
Chaos Theory
Statistical Physics

Background:

Universal approximators are key to understanding complex systems.
Dissipative dynamical systems exhibit rich behaviors.
Lyapunov exponents quantify system divergence and chaos.

Purpose of the Study:

To investigate the relationship between system dimension and chaotic behavior.
To explain the persistence of deterministic chaos in high-dimensional systems.
To identify the mechanisms behind topological changes in parameter space.

Main Methods:

Extensive statistical survey of universal approximators.
Analysis of dissipative dynamical systems across varying dimensions.
Geometric analysis of parameter space and topological changes.

Related Experiment Videos

Main Results:

Monotonic increase in positive Lyapunov exponents with system dimension.
Decrease in parameter windows exhibiting periodic behavior.
Identification of inevitable non-catastrophic topological changes in a subset of parameter space.

Conclusions:

A geometric mechanism explains expected, non-catastrophic topological changes.
Deterministic chaos persists in high dimensions due to this mechanism.
Understanding these dynamics is crucial for high-dimensional system analysis.