Computer assisted proofs of two-dimensional attracting invariant tori for ODEs

Figure 1.  On the left we have a cone attached at the point $ \gamma _U^{-1}(q) $ in the case when $ k = 2 $ and $ n = 3 $. Note that the cone is not the blue (cone shaped) set. The cone $ \mathbf{Q}(\gamma_U^{-1}(q)) $ is the complement of the blue set in $ \mathbb{R}^3 $; i.e. the white region outside of the blue set. On the right we have an example of a Lipschitz manifold

Figure 2.  The set $ U $ is a collection of boxes, and we prove the existence of a star-shaped invariant closed curve around $ q^{\ast} $ which satisfies the cone conditions

Figure 3.  A well aligned cone

Figure 4.  Construction of $ \mathcal{G}(h) $

Figure 5.  Since $ \mathcal{G}^{n}(h) $ and $ \mathcal{G}^{n-1}(h) $ satisfy cone conditions, we can find an angle $ \alpha $, such that the isosceles triangle, with base joining $ q_{n} $ and $ q_{n-1} $, as in above plot, will fit between $ \mathcal{G}^{n}(h) $ and $ \mathcal{G}^{n-1}(h) $. By compactness of $ U $ and the fact that we have a finite number of $ C^{1} $ local maps $ \gamma_{i} $, the $ \alpha $ can be chosen independently of $ n, \theta, q_{n} $ and of $ q_{n-1} $. This means that the area between $ \mathcal{G}^{n}(h) $ and $ \mathcal{G}^{n-1}(h) $ is bounded from below by $ C\Vert q_{n}-q_{n-1}\Vert^{2} $, where $ C>0 $ is some constant independent from $ n $ and $ \theta $

Figure 6.  Generalization to a vector bundle setting. The vector bundle $ E $ is in grey, its base is the curve $ p^* $, which is in black, with the fibers $ E_{\theta} $ represented as the grey lines. The set $ U $ consists of the union of the small rectangles. Note that in this picture $ p^* $ is not the invariant curve, rather it is the base of the vector bundle

Figure 7.  Intersections of the invariant Lipschitz tori for the Van der Pol system with the $ t = 0 $ section for each parameter from (22). The smaller the $ \mu $ the more circular/smooth the curve

Figure 8.  At $ \alpha = 0.75 $ we have an attracting limit cycle of $ P^2 $ on $ \Sigma $ (figure on the left) which is the intersection of the two dimensional $ C^k $ torus of the ODE with $ \Sigma \cap \{y>0\} $. On the right we plot half of the torus. In black we have both components of the torus intersection with $ \Sigma $; one for $ y<0 $ and the other for $ y>0 $

Figure 10.  Colors have the same meaning as in Figure 9. At $ \alpha = 0.8225 $ we have transverse intersections of $ W^{u}(c_{h}) $ and $ W^{s}(c_{h}) $ which leads to chaotic dynamics

Figure 9.  The plot of the periodic orbits $ c_{h} $ and $ c_{s} $ on $ \Sigma \cap \{y>0\} $. (The orbits are of period $ 6 $, but we plot only half of the points with $ y>0 $.) The hyperbolic orbit $ c_{u} $ is in blue, and the attracting orbit $ c_{s} $ is in green. The manifold $ W^{s}(c_{h}) $ is in red and $ W^{u}(c_{h}) $ is blue. (Left) at $ \alpha = 0.815 $, we see that a branch of $ W^{u}(c_{h}) $ goes inside and wraps around the attracting invariant circle. (Right) at $ \alpha = 0.835 $, we observe that a branch of $ W^{u}(c_{h}) $ goes to the other side of $ W^{s}(c_{h}) $ and gets caught in the basin of attraction of $ c_{s} $

Figure 11.  A resonance torus for $ \alpha = 0.85 $. In red we plot the periodic orbit from which the tori have initially originated through the Hopf type bifurcation