Predicting the bounds of large chaotic systems using low-dimensional manifolds

Area of Science:

• Chaos Theory
• Dynamical Systems
• Computational Physics

Background:

• Predicting long-term extrema in high-dimensional chaotic systems is computationally challenging.
• Existing methods are limited to low-variable models.
• The computational cost of discretizing manifolds scales exponentially with dimension.

Purpose of the Study:

• To present a novel method for predicting extrema of chaotic systems in high-dimensional phase space.
• To reduce the computational burden associated with analyzing chaotic systems.
• To enable the analysis of complex chaotic systems on standard hardware.

Main Methods:

• Treating extrema as belonging to low-dimensional (low-D) discretized manifolds embedded in high-dimensional (high-D) phase space.
• Exploiting the typically low strange attractor dimension relative to the phase space dimension.
• Calculating bounding manifolds for high-D modifications of the Duffing system.

Main Results:

• The developed method can tackle systems with significant computational challenges on a laptop.
• Bounding manifolds were successfully calculated for high-dimensional Duffing system modifications.
• Harmonic behavior can be observed in the bounding manifold even for chaotic underlying systems.

Conclusions:

• The presented method offers an efficient approach to predict chaotic system extrema.
• Solving for one cycle of the bounding manifold allows indefinite prediction of the chaotic system's extrema.
• This technique significantly advances the analysis of complex dynamical systems.