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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  September 2021 Revised  March 2022 Early access May 2022

Fund Project: 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

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: Alexandre N. Carvalho, Luciano R. N. Rocha, José A. Langa, Rafael Obaya. Structure of non-autonomous attractors for a class of diffusively coupled ODE. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022083
References:
[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.

show all references

References:
[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.

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)
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