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Linearization around $ \pm(1, 1) $
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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 |
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.
Keywords:
Structure of attractors for non-autonomous dynamical systems,
gradient skew product attractor,
hyperbolic global solutions,
robustness of skew-product attractors under perturbation.
Mathematics Subject Classification:
Primary: 58F15, 58F17; Secondary: 53C35.
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
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References:
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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. |
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A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam, 1992. |
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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. |
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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. |
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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. |
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V. V. Chepyzhov and M. I. Vishik,
Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.
|
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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. |
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C. Conley, Isolated Invariant Sets and the Morse Index, American Mathematical Society, Providence, R. I., 1978. |
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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. |
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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. |
show all references
References:
| [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. |
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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