Learning dissipation and instability fields from chaotic dynamics

TL;DR

This paper introduces a method to estimate local instability and dissipation fields in chaotic systems by analyzing the Jacobian derived from the Perron-Frobenius operator, aiding predictions and control.

Contribution

It presents a novel approach to determine local sensitivity and dissipation rates from transition matrices, bypassing the need for explicit governing equations.

Findings

Effective estimation of Jacobian from transition matrices in chaotic maps

Promising results demonstrated on 1D and 2D chaotic systems

Potential for improved prediction and control in chaotic dynamics

Abstract

To make predictions or design control, information on local sensitivity of initial conditions and state-space contraction is both central, and often instrumental. However, it is not always simple to reliably determine instability fields or local dissipation rates, due to computational challenges or ignorance of the governing equations. Here, we construct an alternative route towards that goal, by estimating the Jacobian of a discrete-time dynamical system locally from the entries of the transition matrix that approximates the Perron-Frobenius operator for a given state-space partition. Numerical tests on one- and two-dimensional chaotic maps show promising results.