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

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

doi:10.3934/DCDSB.2022083

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

Carvalho, Alexandre N. ¹
Rocha, Luciano R. N. ¹
Langa, José A. ³
Obaya, Rafael ²

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 IMUVA, Instituto de Matemáticas, Universidad de Valladolid
3 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 - Sevilla, Spain

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Revista:

Discrete and Continuous Dynamical Systems - B

ISSN: 1531-3492, 1553-524X

Año de publicación: 2023

Volumen: 28

Número: 1

Páginas: 426-448

Tipo: Artículo

DOI: 10.3934/DCDSB.2022083 GOOGLE SCHOLAR Acceso abierto editor

Otras publicaciones en: Discrete and Continuous Dynamical Systems - B

Resumen

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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