arXiv subjects

E. Sander

Publications and source records attributed to E. Sander.

6 recordsLinked to original sources

arXiv2024

Computing Lyapunov Exponents using Weighted Birkhoff Averages

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. 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 convergence at similar, slow rates. Though the original MEGNO method does not compute Lyapunov exponents, we show how to reformulate it as a weighted average that does.

nlin.CD↗

arXiv2024

Proportions of Incommensurate, Resonant, and Chaotic Orbits for Torus Maps

This paper focuses on distinguishing classes of dynamical behavior for one- and two-dimensional torus maps, in particular between orbits that are incommensurate, resonant, periodic, or chaotic. We first consider Arnold's circle map, for which there is a universal power law for the fraction of nonresonant orbits as a function of the amplitude of the nonlinearity. Our methods give a more precise calculation of the coefficients for this power law. For two-dimensional torus maps, we show that there is no such universal law for any of the classes of orbits. However, we find different categories of maps with qualitatively similar behavior. Our results are obtained using three fast and high precision numerical methods: weighted Birkhoff averages, Farey trees, and resonance orders.

math.DS↗

arXiv2023

Resonance and Weak Chaos in Quasiperiodically-Forced Circle Maps

In this paper, we focus on a numerical technique, the weighted Birkhoff average (WBA) to distinguish between four categories of dynamics for quasiperiodically-forced circle maps. Regular dynamics can be classified by rotation vectors, and these can be rapidly computed to machine precision using the WBA. Regular orbits can be resonant or incommensurate and we distinguish between these by computing their "resonance order." When the dynamics is chaotic the WBA converges slowly. Such orbits can be strongly chaotic, when they have a positive Lyapunov exponent or weakly chaotic, when the maximal Lyapunov exponent is zero. The latter correspond to the strange nonchaotic attractors (SNA) that have been observed in quasiperiodically-forced circle maps beginning with the models introduced by Ding, Grebogi, and Ott. The WBA provides a new technique to find SNAs, and allows us to accurately compute the proportions of each of the four orbit types as a function of map parameters.

nlin.CD↗

arXiv2021

Birkhoff Averages and the Breakdown of Invariant Tori in Volume-Preserving Maps

In this paper, we develop numerical methods based on the weighted Birkhoff average for studying two-dimensional invariant tori for volume-preserving maps. The methods do not rely on symmetries, such as time-reversal symmetry, nor on approximating tori by periodic orbits. The rate of convergence of the average gives a sharp distinction between chaotic and regular dynamics and allows accurate computation of rotation vectors for regular orbits. Resonant and rotational tori are distinguished by computing the resonance order of the rotation vector to a given precision. Critical parameter values, where tori are destroyed, are computed by a sharp decrease in convergence rate of the Birkhoff average. We apply these methods for a three-dimensional generalization of Chirikov's standard map: an angle-action map with two angle variables. Computations on grids in frequency and perturbation amplitude allow estimates of the critical set. We also use continuation to follow tori with fixed rotation vectors. We test three conjectures for cubic fields that have been proposed to give locally robust invariant tori.

math.DS↗

arXiv2019

Birkhoff Averages and Rotational Invariant Circles for Area-Preserving Maps

Rotational invariant circles of area-preserving maps are an important and well-studied example of KAM tori. John Greene conjectured that the locally most robust rotational circles have rotation numbers that are noble, i.e., have continued fractions with a tail of ones, and that, of these circles, the most robust has golden mean rotation number. The accurate numerical confirmation of these conjectures relies on the map having a time reversal symmetry, and these methods cannot be applied to more general maps. In this paper, we develop a method based on a weighted Birkhoff average for identifying chaotic orbits, island chains, and rotational invariant circles that do not rely on these symmetries. We use Chirikov's standard map as our test case, and also demonstrate that our methods apply to three other, well-studied cases.

nlin.CD↗

arXiv2007

A classification of explosions in dimension one

A discontinuous change in the size of an attractor is the most easily observed type of global bifurcation. More generally, an explosion is a discontinuous change in the set of recurrent points. An explosion often results from heteroclinic and homoclinic tangency bifurcations. Newhouse and Palis conjectured in 1976 that planar explosions are generically the result of either tangency or saddle node bifurcations. In this paper, we prove this conjecture for one-dimensional maps. Furthermore, we give a full classification for all possible tangency bifurcations and whether they lead to explosions.

math.DS↗