Abstract: Birkhoff’s ergodic theorem implies that when an orbit is ergodic on an invariant set, spatial averages of a phase-space function can be computed as time averages. However the convergence of a time average can be very slow. In 2016, Das et al introduced a C^∞ weighting technique that they later showed can give super-polynomial convergence of averages for orbits that lie on (Diophantine) invariant tori. Evelyn Sander and I showed that this Weighted Birkhoff Average (WBA) can give a sharp distinction between chaotic and regular dynamics and that it allows accurate computation of rotation vectors for regular orbits.
Nathan Duignan and I applied* this to several flows: the two-wave Hamiltonian system, the Paul-Hudson-Helander model magnetic field line flow and a quasiperiodically forced, dissipative system with a “strange nonchaotic attractor”. In practice the WBA is shown to achieve machine precision for quasiperiodic orbits after an integration time of O(10^3) periods. The contrasting, relatively slow convergence for chaotic trajectories allows an efficient discrimination criterion. We propose that the WBA could be more efficient than visualizing Poincare sections or computing Lyapunov exponents.
*Duignan, N. and J. D. Meiss (2023). “Distinguishing between Regular and Chaotic orbits of Flows by the Weighted Birkhoff Average.” Physica D 449(July): 133749.
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