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Structure of non-autonomous attractors for a class of diffusively coupled ODE

The first author is partially supported by Grants FAPESP 2018/10997-6, 2020/14075-6 and CNPq 306213/2019-2. The second author is supported by CNPq grant 131858/2020-3. The third author has been partially supported by FEDER Ministerio de Economía, Industria y Competitividad grants PGC2018-096540-B-I00 and PGC2018-098308-B-I00, and Proyecto I+D+i Programa Operativo FEDER Andalucia US-1254251 and P20-00592. The fourth author is partially supported by FEDER Ministerio de Economía, Industria y Competitividad grants MTM2015-66330-P and RTI2018-096523-B-I00 and by Universidad de Valladolid under project PIP-TCESC-2020

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  • In this work we will study the structure of the skew-product attractor for a planar diffusively coupled ordinary differential equation, given by $ \dot{x} = k(y-x)+x-\beta(t)x^3 $ and $ \dot{y} = k(x-y)+y-\beta(t)y^3 $, $ t\geq 0 $. We identify the non-autonomous structures that completely describes the dynamics of this model giving a Morse decomposition for the skew-product attractor. The complexity of the isolated invariant sets in the global attractor of the associated skew-product semigroup is associated to the complexity of the attractor of the associated driving semigroup. In particular, if $ \beta $ is asymptotically almost periodic, the isolated invariant sets will be almost periodic hyperbolic global solutions of an associated globally defined problem.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


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  • Figure 1.  Linearization around $ \pm(1, 1) $

    Figure 2.  Linearization around $ \pm (\sqrt{1-2k}, -\sqrt{1-2k}) $

    Figure 3.  Linearization around $ (0, 0) $

    Figure 4.  Phase portrait for $ k\in \left(\frac{1}{3}, \frac{1}{2}\right) $

    Figure 5.  Region $ Q_1 $

    Figure 6.  Region $ \mathrm{T}(Q_1) $

    Figure 7.  Contour lines

    Figure 8.  Representation of the uniform attractor for (1)

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