Distinguishing between Regular and Chaotic orbits of Flows by the Weighted Birkhoff Average

TL;DR

This paper demonstrates that the weighted Birkhoff average can effectively distinguish between regular and chaotic flows, achieving high accuracy for quasiperiodic orbits and providing an efficient criterion for chaos detection.

Contribution

It extends the application of the weighted Birkhoff average to flows, showing super-convergence and practical effectiveness in identifying orbit types.

Findings

WBA achieves machine precision for quasiperiodic orbits within ~1000 periods.

Chaotic trajectories show slower convergence, enabling discrimination.

Validated on three diverse dynamical systems.

Abstract

This paper investigates the utility of the weighted Birkhoff average (WBA) for distinguishing between regular and chaotic orbits of flows, extending previous results that applied the WBA to maps. It is shown that the WBA can be super-convergent for flows when the dynamics and phase space function are smooth, and the dynamics is conjugate to a rigid rotation with Diophantine rotation vector. The dependence of the accuracy of the average on orbit length and width of the weight function width are investigated. In practice, the average achieves machine precision of the rotation frequency of quasiperiodic orbits for an integration time of O(10^3) periods. The contrasting, relatively slow convergence for chaotic trajectories allows an efficient discrimination criterion. Three example systems are studied: a two-wave Hamiltonian system, a quasiperiodically forced, dissipative system that has a strange attractor with no positive Lyapunov exponents, and a model for magnetic field line flow.