Abstract
Stable fiber bundles are important structures for understanding nonautonomous dynamics. These sets have a hierarchical structure ranging from stable to strong stable fibers. First, we compute corresponding structures for linear systems and prove an error estimate. The spectral concept of choice is the Sacker-Sell spectrum that is based on exponential dichotomies. Secondly, we tackle the nonlinear case and propose an algorithm for the numerical approximation of stable hierarchies in nonautonomous difference equations. This method generalizes the contour algorithm for computing stable fibers from [ 38 , 39 ]. It is based on Hadamard's graph transform and approximates fibers of the hierarchy by zero-contours of specific operators. We calculate fiber bundles and illustrate errors involved for several examples, including a nonautonomous Lorenz model.
Highlights
Invariant manifolds in dynamical systems partition the state space into different areas of attraction
Existence results for invariant manifolds of hyperbolic fixed points in autonomous dynamical systems are given, e.g., in [28, 34, 45, 51] and corresponding results for nonautonomous fiber bundles of bounded trajectories have been developed in [2, 4]
Finding invariant manifolds analytically is hardly possible for many relevant models and numerical approximations are the only feasible alternatives
Summary
Invariant manifolds in dynamical systems partition the state space into different areas of attraction. Existence results for invariant manifolds of hyperbolic fixed points in autonomous dynamical systems are given, e.g., in [28, 34, 45, 51] and corresponding results for nonautonomous fiber bundles of bounded trajectories have been developed in [2, 4]. Proposed techniques are based on numerical continuation, boundary value problems, Taylor expansions, the parameterization method, fixed point iterations and set oriented methods, see [8, 10, 17, 20, 22, 23, 32, 42, 49, 52], where this list is by no means complete Some of these methods allow nonautonomous generalizations, e.g. We determine the stable hierarchy for a nonautonomous variant of the famous Lorenz system [43]
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