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.

[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.

[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.

[4]

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

[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.

[6]

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

[7]

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

[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.

[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., ().

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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.

[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.

[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.

[29]

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

[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.

[31]

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

[32]

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

[33]

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

[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.

[35]

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

[36]

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

[37]

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

[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.

[39]

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

[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.

[41]

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

[42]

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

[43]

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

[44]

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

[45]

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

[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.

[47]

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

[48]

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

[49]

M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations,, Cambridge University Press, (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. doi: 10.1088/0951-7715/24/7/010.

[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.

[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.

[4]

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

[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.

[6]

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

[7]

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

[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.

[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., ().

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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.

[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.

[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.

[29]

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

[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.

[31]

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

[32]

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

[33]

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

[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.

[35]

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

[36]

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

[37]

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

[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.

[39]

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

[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.

[41]

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

[42]

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

[43]

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

[44]

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

[45]

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

[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.

[47]

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

[48]

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

[49]

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

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