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Rough basin boundaries in high dimension: Can we classify them experimentally?
Tamás Bódai1, Valerio Lucarini2
1Center for Climate Physics, Institute for Basic Science, Busan 46241, South Korea.
Chaos (Woodbury, N.Y.)
|November 3, 2020
Summary
We found that rough basin boundaries in high-dimensional systems depend on Lyapunov exponents, not just fractal dimensions. Measuring both maximal and cross-boundary Lyapunov exponents is crucial for distinguishing fractal boundary types.
Area of Science:
- Dynamical Systems and Chaos Theory
- Fractal Geometry
- Nonlinear Dynamics
Background:
- Bistable systems often exhibit complex basin boundaries.
- Rough basin boundaries are characterized by a nonattracting chaotic set.
- Previous work established conditions for roughness in 2D systems.
Purpose of the Study:
- To extend the understanding of rough basin boundaries to high-dimensional bistable systems.
- To identify the key dynamical properties governing boundary roughness.
- To develop a method for characterizing fractal basin boundaries.
Main Methods:
- Analysis of Lyapunov exponents, specifically the cross-boundary Lyapunov exponent (λx).
- Development of a generalized formula for the co-dimension of rough basin boundaries.
- Investigation using 2D noninvertible and 3D invertible minimal models.
Main Results:
- The condition for rough basin boundaries in 2D systems is shown to hold for high-dimensional systems.
- A generalized formula for the co-dimension of rough boundaries is presented.
- The partial dimension D0(x) associated with λx is found to be unity for rough boundaries, not measurable by standard fractal dimension techniques.
Conclusions:
- Rough basin boundaries in high-dimensional systems are characterized by a specific Lyapunov exponent condition.
- Distinguishing between rough and filamentary fractal boundaries requires measuring both maximal and cross-boundary Lyapunov exponents.
- Traditional fractal dimension measurements are insufficient for this distinction.
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