Computing stable hierarchies of fiber bundles

Figure 1.  Illustration of spectral and resolvent intervals

Figure 2.  Optimal choices of $\gamma_i$ and $\gamma_{i+1}$

Figure 3.  Sacker-Sell spectrum and resolvent set of (26)

Figure 4.  Errors of approximate spectral bundles of (26) for $n_-=-400$, $n_+ =10,\dots,100$

Figure 5.  Errors of approximate spectral bundles of (26) for $n_+=400$, $n_- =-10,\dots,-100$

Figure 6.  Illustration of strong and weak stable fibers in nonlinear systems

Figure 7.  Approximation of the Lorenz manifold (left) and of its intersection with the $(x_2,x_3)$ plane (right)

Figure 8.  Intersection of the stable manifold of (36) with threedimensional cubes

Figure 9.  Intersection of the stable manifold of (36) with twodimensional coordinate planes

Figure 10.  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

Figure 11.  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)

Figure 12.  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)

Figure 13.  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)

Figure 14.  Parametrizations of $\mathcal{H}^2$, computed using the cutoff function (44) with $\mu = \tfrac 14$

Figure 15.  Zero-contours (38) w.r.t. the parametrizations from Figure 14

Figure 16.  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 $\mathcal{H}^3$

Figure 17.  Illustration of parts of the strong stable manifold of the fixed point 0

Figure 18.  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 $t = 0$

Figure 19.  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

Figure 20.  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 $t = 0$