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

Erakutsi afiliazioak +

Aldizkaria:

Discrete and Continuous Dynamical Systems - B

ISSN: 1531-3492, 1553-524X

Argitalpen urtea: 2023

Alea: 28

Zenbakia: 1

Orrialdeak: 426-448

Mota: Artikulua

DOI: 10.3934/DCDSB.2022083 GOOGLE SCHOLAR Sarbide irekia editor

Beste argitalpen batzuk: Discrete and Continuous Dynamical Systems - B

Laburpena

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.

Erreferentzia bibliografikoak

E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho, J. A. Langa.Stability of gradient semigroups under perturbation, Nonlinearity, 24 (2011), 2099-2117.
E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho, J. A. Langa.Non-autonomous Morse decomposition and Lyapunov functions for dynamically gradient processes, Trans. Amer. Math. Soc., 365 (2013), 5277-5312.
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam, 1992.
M. C. Bortolan, A. N. Carvalho and J. A. Langa, Attractors Under Autonomous and Non-Autonomous Perturbations, Mathematical Surveys and Monographs, 246. American Mathematical Society, Providence, RI, 2020.
M. C. Bortolan, A. N. Carvalho, J. A. Langa and G. Raugel, Non-autonomous perturbations of Morse-Smale semigroups: Stability of the phase diagram, J. Dyn, Diff. Eq., in press.
M. C. Bortolan, T. Caraballo, A. N. Carvalho, J. A. Langa.Skew-product semiflows and Morse decomposition, J. Differential Equations, 255 (2013), 2436-2462.
T. Caraballo, A. N. Carvalho, J. A. Langa and A. N. Oliveira-Sousa, The effect of a small bounded noise on the hyperbolicity for autonomous semilinear differential equations, J. Math. Anal. Appl., 500 (2021), Paper No. 125134, 27 pp.
T. Caraballo, J. A. Langa, Z. Liu.Gradient infinite-dimensional random dynamical systems, SIAM J. Appl. Dyn. Syst., 11 (2012), 1817-1847.
T. Caraballo, J. A. Langa, R. Obaya, A. M. Sanz.Global and cocycle attractors for non-autonomous reaction-diffusion equations, The case of null upper Lyapunov exponent, J. Differential Equations, 265 (2018), 3914-3951.
A. N. Carvalho, J. A. Langa.An extension of the concept of gradient semigroups which is stable under perturbation, J. Diff. Eq., 246 (2009), 2646-2668.
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013.
A. N. Carvalho, J. A. Langa, J. C. Robinson.Structure and bifurcation of pullback attractors in a non-autonomous Chafee-Infante equation, Proc. Amer. Math. Soc., 140 (2012), 2357-2373.
A. N. Carvalho, J. A. Langa, J. C. Robinson, A. Suárez.Characterization of non-autonomous attractors of a perturbed gradient system, J. Diff. Eq., 236 (2007), 570-603.
N. Chafee, E. F. Infante.A bifurcation problem for a nonlinear partial differential equation of parabolic type, Applicable Anal., 4 (1974/75), 17-37.
V. V. Chepyzhov, M. I. Vishik.Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002.
C. Conley, Isolated Invariant Sets and the Morse Index, American Mathematical Society, Providence, R. I., 1978.
G. Fusco, J. K. Hale.Slow-motion manifolds, dormant instability and singular perturbations, J. Dyn. Diff. Equations, 1 (1989), 75-94.
J. K. Hale, Ordinary Differential Equations, Interscience, New York, 1969.
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.
J. K. Hale, X. B. Lin, G. Raugel.Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. Comp., 50 (1988), 89-123.
J. K. Hale, L. T. Magalhães and W. M. Oliva, An Introduction to Infinite-dimensional Dynamical Systems - Geometric Theory, Applied Mathematical Sciences, 47. Springer-Verlag, New York, 1984.
J. K. Hale, G. Raugel.Lower semi-continuity of attractors of gradient systems and applications, Ann. Mat. Pur. Appl., 154 (1989), 281-326.
J. K. Hale, G. Raugel.Convergence in dynamically gradient systems with applications to PDE, Z. Angew. Math. Phys., 43 (1992), 63-124.
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981.
P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, Providence, RI, 2011.
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.
J. A. Langa, J. C. Robinson.Determining asymptotic behavior from the dynamics on attracting sets, J. Dyn. Diff. Eq., 11 (1999), 319-331.
D. E. Norton.The fundamental theorem of dynamical systems, Comment. Math. Univ. Carolin., 36 (1995), 585-597.
J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001.
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002.
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988.
M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations, Cambridge University Press, Cambridge, 1992.