May  2015, 20(3): 703-747. doi: 10.3934/dcdsb.2015.20.703

Non-autonomous dynamical systems

1. 

Instituto de Ciências Matemáticas e de Computaçao, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP

2. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla

3. 

Mathematical Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL

Received  June 2014 Revised  September 2014 Published  January 2015

This review paper treats the dynamics of non-autonomous dynamical systems. To study the forwards asymptotic behaviour of a non-autonomous differential equation we need to analyse the asymptotic configurations of the non-autonomous terms present in the equations. This fact leads to the definition of concepts such as skew-products and cocycles and their associated global, uniform, and cocycle attractors. All of them are closely related to the study of the pullback asymptotic limits of the dynamical system, from which naturally emerges the concept of a pullback attractor. In the first part of this paper we want to clarify these different dynamical scenarios and the relations between their corresponding attractors.
    If the global attractor of an autonomous dynamical system is given as the union of a finite number of unstable manifolds of equilibria, a detailed understanding of the continuity of the local dynamics under perturbation leads to important results on the lower-semicontinuity and topological structural stability for the pullback attractors of evolution processes that arise from small non-autonomous perturbations, with respect to the limit regime. Finally, continuity with respect to global dynamics under non-autonomous perturbation is also studied, for which appropriate concepts for Morse decomposition of attractors and non-autonomous Morse--Smale systems are introduced. All of these results will also be considered for uniform attractors. As a consequence, this paper also makes connections between different approaches to the qualitative theory of non-autonomous differential equations, which are often treated independently.
Citation: Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703
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. doi: 10.1088/0951-7715/24/7/010. Google Scholar

[2]

E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Continuity of Lyapunov functions and of energy level for a generalized gradient system,, Topological Methods Nonl. Anal., 39 (2012), 57. Google Scholar

[3]

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. doi: 10.1090/S0002-9947-2013-05810-2. Google Scholar

[4]

L. Arnold, Random Dynamical Systems,, Springer Monographs in Mathematics, (1998). doi: 10.1007/978-3-662-12878-7. Google Scholar

[5]

J. Arrieta and A. N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations,, Trans. Amer. Math. Soc., 352 (2000), 285. doi: 10.1090/S0002-9947-99-02528-3. Google Scholar

[6]

A. V. Babin and M. Vishik, Regular attractors of semigroups and evolution equations,, J. Math. Pures Appl., 62 (1983), 441. Google Scholar

[7]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, Studies in Mathematics and its Applications, (1992). Google Scholar

[8]

M. C. Bortolan, T. Caraballo, A. N. Carvalho and J. A. Langa, Skew-product flows and Morse decomposition,, J. Diff. Equations, 255 (2013), 2436. doi: 10.1016/j.jde.2013.06.023. Google Scholar

[9]

M. C. Bortolan, A. N. Carvalho, J. A. Langa and G. Raugel, Non-autonomous perturbations of Morse-Smale semigroups: Stability of the phase diagram,, preprint., (). Google Scholar

[10]

M. C. Bortolan, A. N. Carvalho, J. A. Langa, Structural stability of skew-product semiflows,, J. Diff. Equations, 257 (2014), 490. doi: 10.1016/j.jde.2014.04.008. Google Scholar

[11]

T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems,, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 491. Google Scholar

[12]

T. Caraballo, J. A. Langa and J. C. Robinson, Upper semicontinuty of attractors for small random perturbations of dynamical systems,, Comm. Partial Diff. Eq., 23 (1998), 1557. doi: 10.1080/03605309808821394. Google Scholar

[13]

T. Caraballo, J. C. Jara, J. A. Langa and Z. Liu, Morse decomposition of attractors for non-autonomous dynamical systems,, Advanced Nonlinear Studies, 13 (2013), 309. Google Scholar

[14]

A. N. Carvalho and J. A. Langa, The existence and continuity of stable and unstable manifolds for semilinear problems under non-autonomous perturbation in Banach spaces,, J. Diff. Eq., 233 (2007), 622. doi: 10.1016/j.jde.2006.08.009. Google Scholar

[15]

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. doi: 10.1016/j.jde.2009.01.007. Google Scholar

[16]

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. doi: 10.1016/j.jde.2007.01.017. Google Scholar

[17]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Lower semi-continuity of attractors for non-autonomous dynamical systems,, Erg. Th. Dyn. Sys., 29 (2009), 1765. Google Scholar

[18]

A. N. Carvalho, J. A. Langa and J. C. Robinson, On the continuity of pullback attractors for evolution processes,, Nonlin. Anal., 71 (2009), 1812. doi: 10.1016/j.na.2009.01.016. Google Scholar

[19]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems,, Applied Mathematical Sciences, (2013). doi: 10.1007/978-1-4614-4581-4. Google Scholar

[20]

A. N. Carvalho and S. Piskarev, A general approximation scheme for attractors of abstract parabolic problems,, Numer. Funct. Anal. Optim., 27 (2006), 785. doi: 10.1080/01630560600882723. Google Scholar

[21]

V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension,, J. Math. Pures Appl., 73 (1994), 279. Google Scholar

[22]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society Colloquium Publications, (2002). Google Scholar

[23]

C. Conley, Isolated Invariant Sets and the Morse Index,, CBMS Regional Conference Series in Mathematics Vol. 38, (1978). Google Scholar

[24]

H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Prob. Th. Rel. Fields, 100 (1994), 365. doi: 10.1007/BF01193705. Google Scholar

[25]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Math. Surveys and Monographs, (1988). Google Scholar

[26]

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. doi: 10.1090/S0025-5718-1988-0917820-X. Google Scholar

[27]

J. K. Hale and G. Raugel, Lower semi-continuity of attractors of gradient systems and applications,, Ann. Mat. Pura Appl., 154 (1989), 281. doi: 10.1007/BF01790353. Google Scholar

[28]

J. K. Hale and G. Raugel, A damped hyperbolic equation on thin domains,, Trans. Amer. Math. Soc., 329 (1992), 185. doi: 10.1090/S0002-9947-1992-1040261-1. Google Scholar

[29]

J. K. Hale and G. Raugel, Convergence in dynamically gradient systems with applications to PDE,, Z. Angew. Math. Phys., 43 (1992), 63. Google Scholar

[30]

J. K. Hale, L. T. Magalhães and W. M. Oliva, An Introduction to Infinite-Dimensional Dynamical Systems - Geometric Theory,, Applied Mathematical Sciences, (1984). doi: 10.1007/0-387-22896-9_9. Google Scholar

[31]

A. Haraux, Systèmes Dynamiques Dissipatifs et Applications,, Masson, (1991). Google Scholar

[32]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981). Google Scholar

[33]

D. Henry, Semigroups,, Handwritten Notes, (1981). Google Scholar

[34]

D. Henry, Perturbation of the Boundary in Boundary-Valued Problems of Partial Differential Equations,, London Mathematical Society Lecture Note Series, (2005). doi: 10.1017/CBO9780511546730. Google Scholar

[35]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems,, Mathematical Surveys and Monographs, (2011). doi: 10.1090/surv/176. Google Scholar

[36]

J. Palis Jr., Introdução aos Sistemas Dinâmicos,, IMPA, (1977). Google Scholar

[37]

O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Cambridge University Press, (1991). doi: 10.1017/CBO9780511569418. Google Scholar

[38]

J. A. Langa, J. C. Robinson, A. Suárez and A. Vidal-López, The stability of attractors for non-autonomous perturbations of gradient-like systems,, J. Diff. Eq., 234 (2007), 607. doi: 10.1016/j.jde.2006.11.016. Google Scholar

[39]

K. Lu, Structural stability for scalar parabolic equations,, J. Diff. Eq., 114 (1994), 253. doi: 10.1006/jdeq.1994.1150. Google Scholar

[40]

K. Mischaikow, H. Smith and H. R. Thieme, Asymptotically autonomous semiflows: Chain recurrent and Lyapunov functions,, Trans. Amer. Math. Soc., 347 (1995), 1669. doi: 10.1090/S0002-9947-1995-1290727-7. Google Scholar

[41]

J. C. Robinson, Infinite-Dimensional Dynamical Systems,, Cambridge University Press, (2001). doi: 10.1007/978-94-010-0732-0. Google Scholar

[42]

K. P. Rybakowski, The Homotopy Index and Partial Differential Equations,, Universitext, (1987). doi: 10.1007/978-3-642-72833-4. Google Scholar

[43]

G. R. Sell, Nonautonomous differential equations and dynamical systems,, Trans. Amer. Math. Soc., 127 (1967), 241. Google Scholar

[44]

G. R. Sell, Topological Dynamics and Ordinary Differential Equations,, Van Nostrand Reinhold, (1971). Google Scholar

[45]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Applied Mathematical Sciences, (2002). doi: 10.1007/978-1-4757-5037-9. Google Scholar

[46]

B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations,, in International Seminar on Applied Mathematics - Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V. Reitmann, (1992), 185. Google Scholar

[47]

A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis,, Cambridge University Press, (1996). Google Scholar

[48]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Applied Mathematical Sciences, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar

[49]

M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations,, Cambridge University Press, (1992). Google Scholar

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. doi: 10.1088/0951-7715/24/7/010. Google Scholar

[2]

E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Continuity of Lyapunov functions and of energy level for a generalized gradient system,, Topological Methods Nonl. Anal., 39 (2012), 57. Google Scholar

[3]

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. doi: 10.1090/S0002-9947-2013-05810-2. Google Scholar

[4]

L. Arnold, Random Dynamical Systems,, Springer Monographs in Mathematics, (1998). doi: 10.1007/978-3-662-12878-7. Google Scholar

[5]

J. Arrieta and A. N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations,, Trans. Amer. Math. Soc., 352 (2000), 285. doi: 10.1090/S0002-9947-99-02528-3. Google Scholar

[6]

A. V. Babin and M. Vishik, Regular attractors of semigroups and evolution equations,, J. Math. Pures Appl., 62 (1983), 441. Google Scholar

[7]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, Studies in Mathematics and its Applications, (1992). Google Scholar

[8]

M. C. Bortolan, T. Caraballo, A. N. Carvalho and J. A. Langa, Skew-product flows and Morse decomposition,, J. Diff. Equations, 255 (2013), 2436. doi: 10.1016/j.jde.2013.06.023. Google Scholar

[9]

M. C. Bortolan, A. N. Carvalho, J. A. Langa and G. Raugel, Non-autonomous perturbations of Morse-Smale semigroups: Stability of the phase diagram,, preprint., (). Google Scholar

[10]

M. C. Bortolan, A. N. Carvalho, J. A. Langa, Structural stability of skew-product semiflows,, J. Diff. Equations, 257 (2014), 490. doi: 10.1016/j.jde.2014.04.008. Google Scholar

[11]

T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems,, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 491. Google Scholar

[12]

T. Caraballo, J. A. Langa and J. C. Robinson, Upper semicontinuty of attractors for small random perturbations of dynamical systems,, Comm. Partial Diff. Eq., 23 (1998), 1557. doi: 10.1080/03605309808821394. Google Scholar

[13]

T. Caraballo, J. C. Jara, J. A. Langa and Z. Liu, Morse decomposition of attractors for non-autonomous dynamical systems,, Advanced Nonlinear Studies, 13 (2013), 309. Google Scholar

[14]

A. N. Carvalho and J. A. Langa, The existence and continuity of stable and unstable manifolds for semilinear problems under non-autonomous perturbation in Banach spaces,, J. Diff. Eq., 233 (2007), 622. doi: 10.1016/j.jde.2006.08.009. Google Scholar

[15]

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. doi: 10.1016/j.jde.2009.01.007. Google Scholar

[16]

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. doi: 10.1016/j.jde.2007.01.017. Google Scholar

[17]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Lower semi-continuity of attractors for non-autonomous dynamical systems,, Erg. Th. Dyn. Sys., 29 (2009), 1765. Google Scholar

[18]

A. N. Carvalho, J. A. Langa and J. C. Robinson, On the continuity of pullback attractors for evolution processes,, Nonlin. Anal., 71 (2009), 1812. doi: 10.1016/j.na.2009.01.016. Google Scholar

[19]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems,, Applied Mathematical Sciences, (2013). doi: 10.1007/978-1-4614-4581-4. Google Scholar

[20]

A. N. Carvalho and S. Piskarev, A general approximation scheme for attractors of abstract parabolic problems,, Numer. Funct. Anal. Optim., 27 (2006), 785. doi: 10.1080/01630560600882723. Google Scholar

[21]

V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension,, J. Math. Pures Appl., 73 (1994), 279. Google Scholar

[22]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society Colloquium Publications, (2002). Google Scholar

[23]

C. Conley, Isolated Invariant Sets and the Morse Index,, CBMS Regional Conference Series in Mathematics Vol. 38, (1978). Google Scholar

[24]

H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Prob. Th. Rel. Fields, 100 (1994), 365. doi: 10.1007/BF01193705. Google Scholar

[25]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Math. Surveys and Monographs, (1988). Google Scholar

[26]

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. doi: 10.1090/S0025-5718-1988-0917820-X. Google Scholar

[27]

J. K. Hale and G. Raugel, Lower semi-continuity of attractors of gradient systems and applications,, Ann. Mat. Pura Appl., 154 (1989), 281. doi: 10.1007/BF01790353. Google Scholar

[28]

J. K. Hale and G. Raugel, A damped hyperbolic equation on thin domains,, Trans. Amer. Math. Soc., 329 (1992), 185. doi: 10.1090/S0002-9947-1992-1040261-1. Google Scholar

[29]

J. K. Hale and G. Raugel, Convergence in dynamically gradient systems with applications to PDE,, Z. Angew. Math. Phys., 43 (1992), 63. Google Scholar

[30]

J. K. Hale, L. T. Magalhães and W. M. Oliva, An Introduction to Infinite-Dimensional Dynamical Systems - Geometric Theory,, Applied Mathematical Sciences, (1984). doi: 10.1007/0-387-22896-9_9. Google Scholar

[31]

A. Haraux, Systèmes Dynamiques Dissipatifs et Applications,, Masson, (1991). Google Scholar

[32]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981). Google Scholar

[33]

D. Henry, Semigroups,, Handwritten Notes, (1981). Google Scholar

[34]

D. Henry, Perturbation of the Boundary in Boundary-Valued Problems of Partial Differential Equations,, London Mathematical Society Lecture Note Series, (2005). doi: 10.1017/CBO9780511546730. Google Scholar

[35]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems,, Mathematical Surveys and Monographs, (2011). doi: 10.1090/surv/176. Google Scholar

[36]

J. Palis Jr., Introdução aos Sistemas Dinâmicos,, IMPA, (1977). Google Scholar

[37]

O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Cambridge University Press, (1991). doi: 10.1017/CBO9780511569418. Google Scholar

[38]

J. A. Langa, J. C. Robinson, A. Suárez and A. Vidal-López, The stability of attractors for non-autonomous perturbations of gradient-like systems,, J. Diff. Eq., 234 (2007), 607. doi: 10.1016/j.jde.2006.11.016. Google Scholar

[39]

K. Lu, Structural stability for scalar parabolic equations,, J. Diff. Eq., 114 (1994), 253. doi: 10.1006/jdeq.1994.1150. Google Scholar

[40]

K. Mischaikow, H. Smith and H. R. Thieme, Asymptotically autonomous semiflows: Chain recurrent and Lyapunov functions,, Trans. Amer. Math. Soc., 347 (1995), 1669. doi: 10.1090/S0002-9947-1995-1290727-7. Google Scholar

[41]

J. C. Robinson, Infinite-Dimensional Dynamical Systems,, Cambridge University Press, (2001). doi: 10.1007/978-94-010-0732-0. Google Scholar

[42]

K. P. Rybakowski, The Homotopy Index and Partial Differential Equations,, Universitext, (1987). doi: 10.1007/978-3-642-72833-4. Google Scholar

[43]

G. R. Sell, Nonautonomous differential equations and dynamical systems,, Trans. Amer. Math. Soc., 127 (1967), 241. Google Scholar

[44]

G. R. Sell, Topological Dynamics and Ordinary Differential Equations,, Van Nostrand Reinhold, (1971). Google Scholar

[45]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Applied Mathematical Sciences, (2002). doi: 10.1007/978-1-4757-5037-9. Google Scholar

[46]

B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations,, in International Seminar on Applied Mathematics - Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V. Reitmann, (1992), 185. Google Scholar

[47]

A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis,, Cambridge University Press, (1996). Google Scholar

[48]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Applied Mathematical Sciences, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar

[49]

M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations,, Cambridge University Press, (1992). Google Scholar

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