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November  2017, 22(9): 3341-3367. doi: 10.3934/dcdsb.2017140

## Computing stable hierarchies of fiber bundles

 Department of Mathematics, Bielefeld University, POB 100131,33501 Bielefeld, Germany

Received  September 2016 Revised  December 2016 Published  April 2017

Fund Project: Supported by CRC 701 'Spectral Structures and Topological Methods in Mathematics'

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.

Citation: Thorsten Hüls. Computing stable hierarchies of fiber bundles. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3341-3367. doi: 10.3934/dcdsb.2017140
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Illustration of spectral and resolvent intervals
Optimal choices of $\gamma_i$ and $\gamma_{i+1}$
Sacker-Sell spectrum and resolvent set of (26)
Errors of approximate spectral bundles of (26) for $n_-=-400$, $n_+ =10,\dots,100$
Errors of approximate spectral bundles of (26) for $n_+=400$, $n_- =-10,\dots,-100$
Illustration of strong and weak stable fibers in nonlinear systems
Approximation of the Lorenz manifold (left) and of its intersection with the $(x_2,x_3)$ plane (right)
Intersection of the stable manifold of (36) with threedimensional cubes
Intersection of the stable manifold of (36) with twodimensional coordinate planes
Approximation of $\mathcal{H}^2$ for (39) w.r.t. $\xi = 0$ (red ball). Zero-contour $\mathcal{N}_2$ (left) and interpolated graph representation $\tilde g_2$ w.r.t. the tangent space $\text{span}\{v_1,v_2\}$ (right). The approximate strong stable manifold $\mathcal{N}_3$ is shown in white
Approximation error of $\mathcal{N}_2$ for $m=10$ w.r.t. the parameterization (40) (left). Distance of $\mathcal{H}^3$ to its numerical approximation for $m=1,\dots,20$ (right)
Approximation of $\mathcal{H}^2$ and $\mathcal{H}^3$ for (41) (left) and distance of $\mathcal{H}^3$ to its approximation w.r.t. $m=1,\dots,15$ (right)
Approximation of $\mathcal{L}^2$ and $\mathcal{H}^3$ for (42) (left) and distance of $\mathcal{H}^3$ to its approximation w.r.t. $m=1,\dots,15$ (right)
Parametrizations of $\mathcal{H}^2$, computed using the cutoff function (44) with $\mu = \tfrac 14$
Zero-contours (38) w.r.t. the parametrizations from Figure 14
Approximation of the stable manifold $\mathcal{H}^2$ of (43) w.r.t. the fixed point $0$ (red ball). The red lines are parts of the strong stable manifold $\mathcal{H}^3$
Illustration of parts of the strong stable manifold of the fixed point 0
Approximation of the stable and strong stable fiber (in red) of the fixed point $0$ (red ball) for the nonautonomous Lorenz system (45) at time $t = 0$
Approximation of the stable and strong stable fiber of the fixed point $0$ (red ball) for the nonautonomous Lorenz system (45) at times $t\in \{\tfrac 12, 1,\tfrac 32, 2\}$. In the right diagrams, the strong stable manifold is hidden by parts of the stable manifold
Approximation of the stable fiber (left: solid, right: transparent) and of the strong stable fiber (in red) of the fixed point $0$ for the nonautonomous Lorenz system (45) at time $t = 0$
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