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

The first author is supported by by the NCN grant 2018/29/B/ST1/00109. The work has been conducted during his visit to FAU sponsored by the Fulbright Foundation. The third author was partially supported by National Science Foundation grant DMS 1813501

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  • This work studies existence and regularity questions for attracting invariant tori in three dimensional dissipative systems of ordinary differential equations. Our main result is a constructive method of computer assisted proof which applies to explicit problems in non-perturbative regimes. We obtain verifiable lower bounds on the regularity of the attractor in terms of the ratio of the expansion rate on the torus with the contraction rate near the torus. We consider separately two important cases of rotational and resonant tori. In the rotational case we obtain $ C^k $ lower bounds on the regularity of the embedding. In the resonant case we verify the existence of tori which are only $ C^0 $ and neither star-shaped nor Lipschitz.

    Mathematics Subject Classification: 34C45, 70K43, 37G35, 65P20, 65G20.

    Citation:

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  • 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

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