Chaotic Dynamics

This paper investigates how state-space partitions affect data-driven approximations of transfer operators for chaotic systems. A modified bisecting K-means algorithm builds a hierarchy of regions while keeping the number of samples in each region more balanced. Tests on the Lorenz and Kuramoto-Sivashinsky systems track the convergence of stationary statistics, Koopman eigenfunctions, and temporal correlations as the partition is refined. The results reveal different convergence rates across the two systems and show that the irreversible part of the operator can better approximate some temporal correlations. Readers get a practical partitioning strategy and evidence for deciding whether additional states improve a statistical model enough to justify their cost.

This paper asks whether the local stretching and contraction of chaotic dynamics can be inferred from a statistical transition model. It estimates the Jacobian of a discrete-time system from a matrix approximating the Perron-Frobenius transfer operator. The approach links transitions between state-space regions to local instability and dissipation without requiring explicit governing equations. Numerical experiments on one- and two-dimensional chaotic maps illustrate the reconstruction and its dependence on the available representation. Readers get a method for extracting local dynamical information from transfer operators and a clear starting point for assessing its use in sensitivity analysis.