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

Alexandre N. Carvalho; Luciano R. N. Rocha; José A. Langa; Rafael Obaya

doi:10.3934/dcdsb.2022083

Article Contents

2023, Volume 28, Issue 1: 426-448. Doi: 10.3934/dcdsb.2022083

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

1.
Instituto de Ciências Matemáticas e de Computaçao, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 - São Carlos SP, Brazil
2.
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 - Sevilla, Spain
3.
Departamento de Matemática Aplicada, E. Ingenierías Industriales, Universidad de Valladolid, 47011 Valladolid, and member of IMUVA, Instituto de Matemáticas, Universidad de Valladolid, Spain

Received: August 31, 2021

Revised: February 28, 2022

Early access: May 2022

Published: January 2023

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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Abstract

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.

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

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

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

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

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

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Figure 5. Region $ Q_1 $

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

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Figure 7. Contour lines

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

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