Symbolic partition in chaotic maps

Area of Science:

• Dynamical Systems and Chaos Theory
• Nonlinear Dynamics
• Computational Physics

Background:

• Understanding the complex behavior of dynamical systems is crucial.
• Accurate partition boundaries are essential for analyzing chaotic systems.
• Existing methods for locating these boundaries can be computationally intensive or imprecise.

Purpose of the Study:

• To develop and validate a new method for locating partition boundaries in dynamical systems.
• To utilize unstable manifold data for identifying folding points and homoclinic tangencies.
• To demonstrate the applicability of the method to well-known chaotic maps.

Main Methods:

• The method relies solely on data from the unstable manifold.
• Partition boundaries are identified by examining folding points at various levels.
• These folding points are shown to coincide with homoclinic tangencies.
• The technique is applied to the two-dimensional Hénon map and a three-dimensional map.

Main Results:

• The method successfully located partition boundaries in both the Hénon map and the three-dimensional map.
• Comparison with previous results for the Hénon map showed good agreement.
• Lyapunov exponents were computed using metric entropy derived from the new partition.
• These computations confirmed the validity and accuracy of the proposed scheme.

Conclusions:

• The proposed method provides an effective way to determine partition boundaries in dynamical systems using unstable manifold data.
• The coincidence with homoclinic tangencies and successful application to known maps validate the approach.
• This technique offers a reliable tool for chaos analysis and computation of dynamical invariants like Lyapunov exponents.