\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

Abstract Full Text(HTML) Figure(8) Related Papers Cited by
  • 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.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • 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)

  • [1] E. R. Aragão-CostaT. CaraballoA. 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-CostaT. CaraballoA. 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. BortolanT. CaraballoA. 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. CaraballoJ. 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. CaraballoJ. A. LangaR. 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. CarvalhoJ. 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. CarvalhoJ. A. LangaJ. 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. HaleX. 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.
  • 加载中

Figures(8)

SHARE

Article Metrics

HTML views(325) PDF downloads(307) Cited by(0)

Access History

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return