The Weighted Birkhoff Average as an Efficient Test for Chaos
When
Where
Speakers: James Meiss, Department of Applied Mathematics, UC Boulder
Title: The Weighted Birkhoff Average as an Efficient Test for Chaos
Abstract: Traditional methods to detect chaotic trajectories include visualization and the computation of Lyapunov exponents. Both of these require long time computations of the trajectory and—for Lyapunov—its linearization. Lyapunov exponents converge slowly, if at all, and a number of Lyapunov indicators, including the “Fast” (FLI) and “Mean Exponential Growth” (MEGNO) have been developed to attempt to get around this. An alternative technique, “Frequency Analysis” was developed to indicate trajectories that lie on tori with dynamics conjugate to a rotation, thus giving an indictor for lack of chaos. A contrasting idea is the “0-1 Test” which maps the trajectory onto a random walk to show chaos.
I will discuss work with Evelyn Sander and Nathan Duignan that uses Weighted Birkhoff Averages as a test for chaos. 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 $C^\infty$ weighting technique that they later showed gives super-polynomial convergence when the dynamics is conjugate to a rigid rotation with frequency that is sufficiently incommensurate (Diophantine). We show that the Weighted Birkhoff Average (WBA) gives a sharp distinction between chaotic and regular dynamics and allows accurate computation of rotation vectors for regular orbits. This allows one to find critical parameter values at which invariant tori are destroyed in Hamiltonian systems and symplectic maps. We apply these methods to circle, torus, area-preserving and three-dimensional angle-action maps. Comparisons with other techniques show that the WBA is more efficient. It also has the advantage of accurately computing physically relevant quantities like rotation vectors. We also show that the WBA can detect “strange non-chaotic attractors”, invariant sets that are geometrically strange but have zero Lyapunov exponents.