Boundary transformation representation of attractor shape deformation

Area of Science:

• Nonlinear Dynamics
• Chaos Theory
• Geometric Analysis

Background:

• Characterizing strange attractors is crucial for detecting parameter changes in chaotic systems.
• Existing methods may not fully capture the geometric deformations of attractors.

Purpose of the Study:

• To introduce a novel method for comparing the geometry of two attractors.
• To quantify attractor deformation using shape context analysis of their boundaries.

Main Methods:

• Poincaré sections of attractors are sampled along their outer limits.
• A boundary transformation is computed to warp one set of points into another.
• Shape context analysis is applied to the boundary points.

Main Results:

• The boundary transformation effectively describes attractor deformation.
• This transformation is approximately proportional to system parameter changes.
• The method demonstrates effectiveness on both simulated and experimental data with noise.

Conclusions:

• The proposed shape context analysis of attractor boundaries offers a robust method for detecting parameter variations in chaotic systems.
• This technique provides a new geometric descriptor for understanding attractor deformation and system dynamics.