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.
Citation: |
[1] |
E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Stability of gradient semigroups under perturbation, Nonlinearity, 24 (2011), 2099-2117.
doi: 10.1088/0951-7715/24/7/010.![]() ![]() ![]() |
[2] |
E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Non-autonomous Morse decomposition and Lyapunov functions for dynamically gradient processes, Trans. Amer. Math. Soc., 365 (2013), 5277-5312.
doi: 10.1090/S0002-9947-2013-05810-2.![]() ![]() ![]() |
[3] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam, 1992.
![]() ![]() |
[4] |
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.
![]() ![]() |
[5] |
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.
![]() |
[6] |
M. C. Bortolan, T. Caraballo, A. N. Carvalho and J. A. Langa, Skew-product semiflows and Morse decomposition, J. Differential Equations, 255 (2013), 2436-2462.
doi: 10.1016/j.jde.2013.06.023.![]() ![]() ![]() |
[7] |
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.
doi: 10.1016/j.jmaa.2021.125134.![]() ![]() ![]() |
[8] |
T. Caraballo, J. A. Langa and Z. Liu, Gradient infinite-dimensional random dynamical systems, SIAM J. Appl. Dyn. Syst., 11 (2012), 1817-1847.
doi: 10.1137/120862752.![]() ![]() ![]() |
[9] |
T. Caraballo, J. A. Langa, R. Obaya and 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.
doi: 10.1016/j.jde.2018.05.023.![]() ![]() ![]() |
[10] |
A. N. Carvalho and J. A. Langa, An extension of the concept of gradient semigroups which is stable under perturbation, J. Diff. Eq., 246 (2009), 2646-2668.
doi: 10.1016/j.jde.2009.01.007.![]() ![]() ![]() |
[11] |
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.
doi: 10.1007/978-1-4614-4581-4.![]() ![]() ![]() |
[12] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Structure and bifurcation of pullback attractors in a non-autonomous Chafee-Infante equation, Proc. Amer. Math. Soc., 140 (2012), 2357-2373.
doi: 10.1090/S0002-9939-2011-11071-2.![]() ![]() ![]() |
[13] |
A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Suárez, Characterization of non-autonomous attractors of a perturbed gradient system, J. Diff. Eq., 236 (2007), 570-603.
doi: 10.1016/j.jde.2007.01.017.![]() ![]() ![]() |
[14] |
N. Chafee and E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, Applicable Anal., 4 (1974/75), 17-37.
doi: 10.1080/00036817408839081.![]() ![]() ![]() |
[15] |
V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.
![]() ![]() |
[16] |
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.
![]() ![]() |
[17] |
C. Conley, Isolated Invariant Sets and the Morse Index, American Mathematical Society, Providence, R. I., 1978.
![]() ![]() |
[18] |
G. Fusco and J. K. Hale, Slow-motion manifolds, dormant instability and singular perturbations, J. Dyn. Diff. Equations, 1 (1989), 75-94.
doi: 10.1007/BF01048791.![]() ![]() ![]() |
[19] |
J. K. Hale, Ordinary Differential Equations, Interscience, New York, 1969.
![]() ![]() |
[20] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.
doi: 10.1090/surv/025.![]() ![]() ![]() |
[21] |
J. K. Hale, X. B. Lin and G. Raugel, Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. Comp., 50 (1988), 89-123.
doi: 10.1090/S0025-5718-1988-0917820-X.![]() ![]() ![]() |
[22] |
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.
![]() ![]() |
[23] |
J. K. Hale and G. Raugel, Lower semi-continuity of attractors of gradient systems and applications, Ann. Mat. Pur. Appl., 154 (1989), 281-326.
doi: 10.1007/BF01790353.![]() ![]() ![]() |
[24] |
J. K. Hale and G. Raugel, Convergence in dynamically gradient systems with applications to PDE, Z. Angew. Math. Phys., 43 (1992), 63-124.
doi: 10.1007/BF00944741.![]() ![]() ![]() |
[25] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981.
![]() ![]() |
[26] |
P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, Providence, RI, 2011.
doi: 10.1090/surv/176.![]() ![]() ![]() |
[27] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.
doi: 10.1017/CBO9780511569418.![]() ![]() ![]() |
[28] |
J. A. Langa and J. C. Robinson, Determining asymptotic behavior from the dynamics on attracting sets, J. Dyn. Diff. Eq., 11 (1999), 319-331.
doi: 10.1023/A:1021933514285.![]() ![]() ![]() |
[29] |
D. E. Norton, The fundamental theorem of dynamical systems, Comment. Math. Univ. Carolin., 36 (1995), 585-597.
![]() ![]() |
[30] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001.
![]() ![]() |
[31] |
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002.
doi: 10.1007/978-1-4757-5037-9.![]() ![]() ![]() |
[32] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8.![]() ![]() ![]() |
[33] |
M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations, Cambridge University Press, Cambridge, 1992.
![]() ![]() |
Linearization around
Linearization around
Linearization around
Phase portrait for
Region
Region
Contour lines
Representation of the uniform attractor for (1)