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On a definition of Morse-Smale evolution processes
1. | Institute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland |
2. | Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010,05508-090 São Paulo, Brazil |
3. | Center for Mathematical Analysis, Geometry, and Dynamical Systems, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal |
In this paper we consider a definition of Morse-Smale evolution process that extends the notion of Morse-Smale dynamical system to the nonautonomous framework. In particular we consider nonautonomous perturbations of autonomous systems. In this case our definition of Morse-Smale evolution process holds for perturbations of Morse-Smale autonomous systems with or without periodic orbits. We establish that small nonautonomous perturbations of autonomous Morse-Smale evolution processes derived from certain nonautonomous differential equations are Morse-Smale evolution processes. We apply our results to examples of scalar parabolic semilinear differential equations generating evolution processes and possessing periodic orbits.
References:
[1] |
S. Angenent and B. Fiedler,
The dynamics of rotating waves in scalar reaction diffusion equations, Trans. Amer. Math. Soc., 307 (1988), 545-568.
doi: 10.1090/S0002-9947-1988-0940217-X. |
[2] |
L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations, Lecture Notes in Math. 1926, Springer, 2008.
doi: 10.1007/978-3-540-74775-8. |
[3] |
N. N. Bogoliubov and Y. A. Mitropolski, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York, 1961. |
[4] |
M. Bortolan, A. N. Carvalho and J. A. Langa,
Structure of attractors for skew product semiflows, J. Differential Equations, 257 (2014), 490-522.
doi: 10.1016/j.jde.2014.04.008. |
[5] |
A. N. Carvalho and J. A. Langa,
Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and unstable manifolds, J. Differential Equations, 233 (2007), 622-653.
doi: 10.1016/j.jde.2006.08.009. |
[6] |
A. N. Carvalho and J. A. Langa,
An extention of the concept of gradient semigroups which is stable under perturbation, J. Differential Equations, 246 (2009), 2646-2668.
doi: 10.1016/j.jde.2009.01.007. |
[7] |
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. |
[8] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Nonautonomous Dynamical Systems, Applied Math. Sciences, 182, Springer, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[9] |
A. N. Carvalho, J. A. Langa and J. C. Robinson,
Non-autonomous dynamical systems, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 703-747.
doi: 10.3934/dcdsb.2015.20.703. |
[10] |
A. N. Carvalho, J. A. Langa, J. C. Robinson and Su′arez A.,
Characterization of nonautonomous attractors of a perturbed gradient system, J. Differential Equations, 236 (2007), 570-603.
doi: 10.1016/j.jde.2007.01.017. |
[11] |
V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Colloquium Publications 49, American Mathematical Society, Providence, R. I. , 2002. |
[12] |
C. Chicone and Y. Latushkin,
Center manifolds for infinite dimensional nonautonomous differential equations, J. Differential Equations, 141 (1997), 356-399.
doi: 10.1006/jdeq.1997.3343. |
[13] |
J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, London Mathematical Society, Lecture Note Series 278, Cambridge University Press, Cambridge, 2000.
doi: 10.1017/CBO9780511526404. |
[14] |
N. S.-Chow and K. Lu,
Invariant manifolds for flows in Banach spaces, J. Differential Equations, 74 (1988), 285-317.
doi: 10.1016/0022-0396(88)90007-1. |
[15] |
C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics 38, American Mathematical Society, Providence, R. I. , 1978. |
[16] |
W. A. Coppel and K. J. Palmer,
Averaging and integral manifolds, Bull. Austral. Math. Soc., 2 (1970), 197-222.
doi: 10.1017/S0004972700041812. |
[17] |
R. Czaja and C. Rocha,
Transversality in scalar reaction-diffusion equations on a circle, J. Differential Equations, 245 (2008), 692-721.
doi: 10.1016/j.jde.2008.01.018. |
[18] |
N. Fenichel,
Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971), 193-226.
doi: 10.1512/iumj.1972.21.21017. |
[19] |
B. Fiedler, C. Rocha and M. Wolfrum,
Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle, J. Differential Equations, 201 (2004), 99-138.
doi: 10.1016/j.jde.2003.10.027. |
[20] |
J. K. Hale,
Integral manifolds of perturbed differential systems, Ann. Math., 73 (1961), 496-531.
doi: 10.2307/1970314. |
[21] |
J. K. Hale, Oscillations in Nonlinear Systems, Dover Publications, Inc., New York, 1992. Originally published by McGraw-Hill, New York, 1963. |
[22] |
J. K. Hale, Ordinary Differential Equations, 2nd edition, Robert E. Krieger Publishing Co. , Inc. , Huntington, N. Y. , 1980. |
[23] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, American Mathematical Society, Providence, R. I. , 1988. |
[24] |
J. K. Hale, L. T. Magalh˜aes and W. M. Oliva, Dynamics in Infinite Dimensions, Second Edition, Applied Math. Sciences 47, Springer 2002.
doi: 10.1007/b100032. |
[25] |
D. Henry, Geometric Theory of Semilinear Equations, Lecture Notes in Math. 840, SpringerVerlag, 1981. |
[26] |
D. Henry,
Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations, J. Differential Equations, 59 (1985), 165-205.
doi: 10.1016/0022-0396(85)90153-6. |
[27] |
M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math. 583, Springer-Verlag, 1977. |
[28] |
R. Joly and G. Raugel,
Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle, Trans. Amer. Math. Soc., 362 (2010), 5189-5211.
doi: 10.1090/S0002-9947-2010-04890-1. |
[29] |
R. Joly and G. Raugel,
Morse-Smale property for the parabolic equation on the circle, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 27 (2010), 1397-1440.
doi: 10.1016/j.anihpc.2010.09.001. |
[30] |
P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs 176, American Mathematical Society, Providence, R. I. , 2011.
doi: 10.1090/surv/176. |
[31] |
S. G. Kryzhevich and V. A. Pliss,
Structural stability of nonautonomous systems, Differential Equations, 39 (2003), 1395-1403.
doi: 10.1023/B:DIEQ.0000017913.79915.b1. |
[32] |
W. M. Oliva,
Morse-Smale semiflows, openness and A-stability, Fields Inst. Comm., 31 (2002), 285-307.
|
[33] |
J. Palis,
On Morse-Smale dynamical systems, Topology, 8 (1969), 305-404.
doi: 10.1016/0040-9383(69)90024-X. |
[34] |
J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, SpringerVerlag, 1982. |
[35] |
V. Pliss and G. R. Sell,
Perturbations of attractors of differential equations, J. Differential Equations, 92 (1991), 100-124.
doi: 10.1016/0022-0396(91)90066-I. |
[36] |
R. J. Sacker,
A perturbation theorem for invariant manifolds and Hölder continuity, J. Math. Mech., 18 (1969), 705-762.
|
[37] |
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002.
doi: 10.1007/978-1-4757-5037-9. |
[38] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences 68, Springer, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[39] |
M. Urabe, Nonlinear Autonomous Oscillations, Mathematics in Science and Engineering 34, Academic Press, New York, 1967. |
[40] |
A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, Dynam. Report. , (N. S. ) 1, Springer (1992), 125-163. |
show all references
References:
[1] |
S. Angenent and B. Fiedler,
The dynamics of rotating waves in scalar reaction diffusion equations, Trans. Amer. Math. Soc., 307 (1988), 545-568.
doi: 10.1090/S0002-9947-1988-0940217-X. |
[2] |
L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations, Lecture Notes in Math. 1926, Springer, 2008.
doi: 10.1007/978-3-540-74775-8. |
[3] |
N. N. Bogoliubov and Y. A. Mitropolski, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York, 1961. |
[4] |
M. Bortolan, A. N. Carvalho and J. A. Langa,
Structure of attractors for skew product semiflows, J. Differential Equations, 257 (2014), 490-522.
doi: 10.1016/j.jde.2014.04.008. |
[5] |
A. N. Carvalho and J. A. Langa,
Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and unstable manifolds, J. Differential Equations, 233 (2007), 622-653.
doi: 10.1016/j.jde.2006.08.009. |
[6] |
A. N. Carvalho and J. A. Langa,
An extention of the concept of gradient semigroups which is stable under perturbation, J. Differential Equations, 246 (2009), 2646-2668.
doi: 10.1016/j.jde.2009.01.007. |
[7] |
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. |
[8] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Nonautonomous Dynamical Systems, Applied Math. Sciences, 182, Springer, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[9] |
A. N. Carvalho, J. A. Langa and J. C. Robinson,
Non-autonomous dynamical systems, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 703-747.
doi: 10.3934/dcdsb.2015.20.703. |
[10] |
A. N. Carvalho, J. A. Langa, J. C. Robinson and Su′arez A.,
Characterization of nonautonomous attractors of a perturbed gradient system, J. Differential Equations, 236 (2007), 570-603.
doi: 10.1016/j.jde.2007.01.017. |
[11] |
V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Colloquium Publications 49, American Mathematical Society, Providence, R. I. , 2002. |
[12] |
C. Chicone and Y. Latushkin,
Center manifolds for infinite dimensional nonautonomous differential equations, J. Differential Equations, 141 (1997), 356-399.
doi: 10.1006/jdeq.1997.3343. |
[13] |
J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, London Mathematical Society, Lecture Note Series 278, Cambridge University Press, Cambridge, 2000.
doi: 10.1017/CBO9780511526404. |
[14] |
N. S.-Chow and K. Lu,
Invariant manifolds for flows in Banach spaces, J. Differential Equations, 74 (1988), 285-317.
doi: 10.1016/0022-0396(88)90007-1. |
[15] |
C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics 38, American Mathematical Society, Providence, R. I. , 1978. |
[16] |
W. A. Coppel and K. J. Palmer,
Averaging and integral manifolds, Bull. Austral. Math. Soc., 2 (1970), 197-222.
doi: 10.1017/S0004972700041812. |
[17] |
R. Czaja and C. Rocha,
Transversality in scalar reaction-diffusion equations on a circle, J. Differential Equations, 245 (2008), 692-721.
doi: 10.1016/j.jde.2008.01.018. |
[18] |
N. Fenichel,
Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971), 193-226.
doi: 10.1512/iumj.1972.21.21017. |
[19] |
B. Fiedler, C. Rocha and M. Wolfrum,
Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle, J. Differential Equations, 201 (2004), 99-138.
doi: 10.1016/j.jde.2003.10.027. |
[20] |
J. K. Hale,
Integral manifolds of perturbed differential systems, Ann. Math., 73 (1961), 496-531.
doi: 10.2307/1970314. |
[21] |
J. K. Hale, Oscillations in Nonlinear Systems, Dover Publications, Inc., New York, 1992. Originally published by McGraw-Hill, New York, 1963. |
[22] |
J. K. Hale, Ordinary Differential Equations, 2nd edition, Robert E. Krieger Publishing Co. , Inc. , Huntington, N. Y. , 1980. |
[23] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, American Mathematical Society, Providence, R. I. , 1988. |
[24] |
J. K. Hale, L. T. Magalh˜aes and W. M. Oliva, Dynamics in Infinite Dimensions, Second Edition, Applied Math. Sciences 47, Springer 2002.
doi: 10.1007/b100032. |
[25] |
D. Henry, Geometric Theory of Semilinear Equations, Lecture Notes in Math. 840, SpringerVerlag, 1981. |
[26] |
D. Henry,
Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations, J. Differential Equations, 59 (1985), 165-205.
doi: 10.1016/0022-0396(85)90153-6. |
[27] |
M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math. 583, Springer-Verlag, 1977. |
[28] |
R. Joly and G. Raugel,
Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle, Trans. Amer. Math. Soc., 362 (2010), 5189-5211.
doi: 10.1090/S0002-9947-2010-04890-1. |
[29] |
R. Joly and G. Raugel,
Morse-Smale property for the parabolic equation on the circle, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 27 (2010), 1397-1440.
doi: 10.1016/j.anihpc.2010.09.001. |
[30] |
P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs 176, American Mathematical Society, Providence, R. I. , 2011.
doi: 10.1090/surv/176. |
[31] |
S. G. Kryzhevich and V. A. Pliss,
Structural stability of nonautonomous systems, Differential Equations, 39 (2003), 1395-1403.
doi: 10.1023/B:DIEQ.0000017913.79915.b1. |
[32] |
W. M. Oliva,
Morse-Smale semiflows, openness and A-stability, Fields Inst. Comm., 31 (2002), 285-307.
|
[33] |
J. Palis,
On Morse-Smale dynamical systems, Topology, 8 (1969), 305-404.
doi: 10.1016/0040-9383(69)90024-X. |
[34] |
J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, SpringerVerlag, 1982. |
[35] |
V. Pliss and G. R. Sell,
Perturbations of attractors of differential equations, J. Differential Equations, 92 (1991), 100-124.
doi: 10.1016/0022-0396(91)90066-I. |
[36] |
R. J. Sacker,
A perturbation theorem for invariant manifolds and Hölder continuity, J. Math. Mech., 18 (1969), 705-762.
|
[37] |
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002.
doi: 10.1007/978-1-4757-5037-9. |
[38] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences 68, Springer, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[39] |
M. Urabe, Nonlinear Autonomous Oscillations, Mathematics in Science and Engineering 34, Academic Press, New York, 1967. |
[40] |
A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, Dynam. Report. , (N. S. ) 1, Springer (1992), 125-163. |


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