Searching chaotic saddles in high dimensions.

Area of Science:

• Dynamical Systems and Chaos Theory
• Numerical Analysis
• Computational Physics

Background:

• Transiently chaotic systems exhibit trajectories that escape a bounded region in finite time.
• Characterizing non-attracting sets is crucial for understanding the dynamics of these systems.
• Existing methods for approximating these sets face challenges, especially in high dimensions.

Purpose of the Study:

• To propose novel numerical methods for approximating non-attracting sets in transiently chaotic systems.
• To enhance the efficiency and accuracy of finding initial conditions with long escape times.
• To investigate the performance of these methods in high-dimensional systems.

Main Methods:

• Developed an exponential search domain method, scaling with the largest Lyapunov exponent (λ₁).
• Introduced an anisotropic search domain method utilizing singular values of the Jacobian matrix.
• Compared performance against the Stagger-and-Step method using coupled Hénon maps.

Main Results:

• Both proposed methods outperform the Stagger-and-Step method.
• The anisotropic method demonstrates efficiency independent of escape time (τ) in high dimensions.
• Simulations were conducted in systems up to 24 dimensions with multiple positive Lyapunov exponents.

Conclusions:

• The new numerical methods provide significant improvements for approximating non-attracting sets.
• The anisotropic method offers a robust approach for high-dimensional transient chaos.
• These findings suggest potential for characterizing non-attracting sets in complex spatio-temporal systems.