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August  2017, 37(7): 3601-3623. doi: 10.3934/dcds.2017155

On a definition of Morse-Smale evolution processes

Radosław Czaja 1, , Waldyr M. Oliva 2,3, and  Carlos Rocha 3,,

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

* Corresponding author: Carlos Rocha

Received  June 2016 Revised  February 2017 Published  April 2017

Fund Project: This work was partially supported by FCT/Portugal through the project UID/MAT/04459/2013 and FAPESP thematic project 2015/21049-3.

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.

Keywords: Pullback attractors, nonautonomous dynamical systems, Morse-Smale systems.
Mathematics Subject Classification: Primary: 37B55, 35B41; Secondary: 35B10, 37B20, 37B35.
Citation: Radosław Czaja, Waldyr M. Oliva, Carlos Rocha. On a definition of Morse-Smale evolution processes. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3601-3623. doi: 10.3934/dcds.2017155
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.

Figure 1.  Phase portrait of the autonomous equation (10) for $\delta>0$ .
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Figure 2.  Recurrent behavior for $\xi=(t, z(t))$ . Here $t_+-\bar t>\overline T$ and $\bar{t}-t_->\overline T$ .
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