Poincar\'e Maps for Multiscale Physics Discovery and Nonlinear Floquet Theory

TL;DR

This paper introduces a data-driven method using sparse regression to construct Poincaré maps, enabling analysis of multiscale nonlinear systems and stability of periodic orbits, with applications to various dynamical systems.

Contribution

It presents a novel approach combining sparse regression with Poincaré maps for discovering system dynamics and nonlinear Floquet theory from data.

Findings

Effective in modeling invariant manifolds and coarse-grained dynamics.

Applicable to systems with periodic, quasi-periodic, and chaotic behavior.

Provides a formalism for nonlinear Floquet stability analysis.

Abstract

Poincar\'e maps are an integral aspect to our understanding and analysis of nonlinear dynamical systems. Despite this fact, the construction of these maps remains elusive and is primarily left to simple motivating examples. In this manuscript we propose a method of data-driven discovery of Poincar\'e maps based upon sparse regression techniques, specifically the sparse identification of nonlinear dynamics (SINDy) algorithm. This work can be used to determine the dynamics on and near invariant manifolds of a given dynamical system, as well as provide long-time forecasting of the coarse-grained dynamics of multiscale systems. Moreover, the method provides a mathematical formalism for determining nonlinear Floquet theory for the stability of nonlinear periodic orbits. The methods are applied to a range of examples including both ordinary and partial differential equations that exhibit periodic, quasi-periodic, and chaotic behavior.