Computing Lyapunov exponents using weighted Birkhoff averages

Abstract The Lyapunov exponents of a dynamical system measure the average rate of exponential stretching along an orbit. Positive exponents are often taken as a defining characteristic of chaotic dynamics, with the size of the exponent indicating the strength of the chaos, or in the case of a negative exponent, a measure of the how far an orbit is from being chaotic. However, the standard orthogonalization-based method for computing Lyapunov exponents converges slowly—if at all. Many alternatively techniques have been developed to distinguish between regular and chaotic orbits, though most do not compute the exponents. We compute the Lyapunov spectrum in three ways: the standard method, the weighted Birkhoff average (WBA), and the ‘mean exponential growth rate for nearby orbits’ (MEGNO). The latter two improve convergence for nonchaotic orbits, but the WBA is fastest. However, for chaotic orbits the three methods converge at similar, slow rates. Since there is little computational cost for the WBA, we argue it should be used in any case. Though the original MEGNO method does not compute Lyapunov exponents, we show how to reformulate it as a weighted average that does.