Generic Morse–Smale property for the parabolic equation on the circle
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 6, p. 1397-1440
In this paper, we show that, for scalar reaction–diffusion equations u t =u xx +f(x,u,u x ) on the circle S 1 , the Morse–Smale property is generic with respect to the non-linearity f. In Czaja and Rocha (2008) [13], Czaja and Rocha have proved that any connecting orbit, which connects two hyperbolic periodic orbits, is transverse and that there does not exist any homoclinic orbit, connecting a hyperbolic periodic orbit to itself. In Joly and Raugel (2010) [31], we have shown that, generically with respect to the non-linearity f, all the equilibria and periodic orbits are hyperbolic. Here we complete these results by showing that any connecting orbit between two hyperbolic equilibria with distinct Morse indices or between a hyperbolic equilibrium and a hyperbolic periodic orbit is automatically transverse. We also show that, generically with respect to f, there does not exist any connection between equilibria with the same Morse index. The above properties, together with the existence of a compact global attractor and the Poincaré–Bendixson property, allow us to deduce that, generically with respect to f, the non-wandering set consists in a finite number of hyperbolic equilibria and periodic orbits. The main tools in the proofs include the lap number property, exponential dichotomies and the Sard–Smale theorem. The proofs also require a careful analysis of the asymptotic behavior of solutions of the linearized equations along the connecting orbits.
Dans cet article, nous démontrons qu'il existe un ensemble générique de non-linéarités f pour lesquelles les équations de réaction-diffusion u t =u xx +f(x,u,u x ), sur le cercle S 1 , ont la propriété de Morse–Smale. Dans Czaja et Rocha (2008) [13], Czaja et Rocha avaient montré que toute connexion entre deux orbites périodiques hyperboliques est transverse et qu'il n'existe pas d'orbite homocline à une orbite périodique hyperbolique. Dans Joly et Raugel (2010) [31], nous avons démontré qu'il existe un ensemble générique de non-linéarités f pour lesquelles tous les points d'équilibre et toutes les orbites périodiques sont hyperboliques. Dans ce travail, nous prouvons que toute connexion entre deux points d'équilibre hyperboliques d'indices de Morse distincts ou entre un point d'équilibre et une orbite périodique hyperboliques est transverse. Nous montrons également qu'il existe un ensemble générique de non-linéarités f pour lesquelles il n'existe pas de connexions entre points d'équilibre ayant même indice de Morse. Grâce à la propriété de Poincaré–Bendixson, nous déduisons des propriétés ci-dessus et de l'existence d'un attracteur global compact que, génériquement en la non-linéarité f, l'ensemble non-errant se réduit à un nombre fini de points d'équilibre et d'orbites périodiques hyperboliques. Dans nos démonstrations, les propriétés du nombre de zéros, les dichotomies exponentielles, le comportement asymptotique des solutions des équations linéarisées et évidemment le théorème de Sard–Smale jouent un rôle crucial.
DOI : https://doi.org/10.1016/j.anihpc.2010.09.001
Classification:  35B10,  35B30,  35K57,  37D05,  37D15,  37L45,  35B40
Keywords: Transversality, Hyperbolicity, Periodic orbits, Morse–Smale, Poincaré–Bendixson, Exponential dichotomy, Lap-number, Genericity, Sard–Smale
@article{AIHPC_2010__27_6_1397_0,
     author = {Joly, Romain and Raugel, Genevi\`eve},
     title = {Generic Morse--Smale property for the parabolic equation on the circle},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {27},
     number = {6},
     year = {2010},
     pages = {1397-1440},
     doi = {10.1016/j.anihpc.2010.09.001},
     zbl = {1213.35046},
     mrnumber = {2738326},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2010__27_6_1397_0}
}
Joly, Romain; Raugel, Geneviève. Generic Morse–Smale property for the parabolic equation on the circle. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 6, pp. 1397-1440. doi : 10.1016/j.anihpc.2010.09.001. http://www.numdam.org/item/AIHPC_2010__27_6_1397_0/

[1] S. Agmon, Unicité et convexité dans les problèmes différentiels, Presses de l'Université de Montreal (1966) | MR 252808 | Zbl 0147.07702

[2] A. Andronov, L.S. Pontrjagin, Systèmes grossiers, Dokl. Akad. Nauk 14 (1937), 247-250 | JFM 63.0728.01

[3] S. Angenent, The Morse–Smale property for a semi-linear parabolic equation, J. Differential Equations 62 (1986), 427-442 | MR 837763 | Zbl 0581.58026

[4] S. Angenent, The zero set of a solution of a parabolic equation, Journal für die Reine und Angewandte Mathematik 390 (1988), 79-96 | MR 953678 | Zbl 0644.35050

[5] S. Angenent, B. Fiedler, The dynamics of rotating waves in scalar reaction diffusion equations, Trans. Amer. Math. Soc. 307 (1988), 545-568 | MR 940217 | Zbl 0696.35086

[6] C. Bardos, L. Tartar, Sur l'unicité rétrograde des équations paraboliques et quelques questions voisines, Arch. Rational Mech. Analysis 50 (1973), 10-25 | MR 338517 | Zbl 0258.35039

[7] P. Brunovský, R. Joly, G. Raugel, Genericity of the Kupka–Smale property for scalar parabolic equations, manuscript.

[8] P. Brunovský, P. Poláčik, The Morse–Smale structure of a generic reaction–diffusion equation in higher space dimension, J. Differential Equations 135 (1997), 129-181 | MR 1434918 | Zbl 0868.35062

[9] P. Brunovský, G. Raugel, Genericity of the Morse–Smale property for damped wave equations, Journal of Dynamics and Differential Equations 15 no. 2 (2003), 571-658 | MR 2046732 | Zbl 1053.35099

[10] M. Chen, X.-Y. Chen, J.K. Hale, Structural stability for time periodic one dimensional parabolic equations, J. Differential Equations 96 (1992), 355-418 | MR 1156666 | Zbl 0779.35061

[11] S.-N. Chow, J.K. Hale, J. Mallet-Paret, An example of bifurcation to homoclinic orbits, J. Differential Equations 37 (1980), 351-373 | MR 589997 | Zbl 0439.34035

[12] W.A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Math. vol. 629, Springer-Verlag (1978) | MR 481196 | Zbl 0376.34001

[13] R. Czaja, C. Rocha, Transversality in scalar reaction–diffusion equations on a circle, J. Differential Equations 245 (2008), 692-721 | MR 2422524 | Zbl 1157.35004

[14] B. Fiedler, J. Mallet-Paret, A Poincaré–Bendixson theorem for scalar reaction–diffusion equations, Arch. Rational Mech. Analysis 107 (1989), 325-345 | MR 1004714 | Zbl 0704.35070

[15] B. Fiedler, C. Rocha, M. Wolfrum, Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle, J. Differential Equations 201 (2004), 99-138 | MR 2057540 | Zbl 1064.35076

[16] G. Fusco, W.M. Oliva, Transversality between invariant manifolds of periodic orbits for a class of monotone dynamical systems, Journal of Dynamics and Differential Equations 2 (1990), 1-17 | Zbl 0702.34038

[17] M. Golubitsky, V. Guillemin, Stable Mappings and their Singularities, Graduate Texts in Mathematics vol. 14, Springer-Verlag, New York, Heidelberg (1973) | MR 341518 | Zbl 0294.58004

[18] J.K. Hale, R. Joly, G. Raugel, book in preparation.

[19] J.K. Hale, X.-B. Lin, Heteroclinic orbits for retarded functional differential equations, J. Differential Equations 65 (1986), 175-202 | MR 861515 | Zbl 0611.34074

[20] J.K. Hale, L.T. Magalhães, W.M. Oliva, Dynamics in Infinite Dimensions, Applied Mathematical Sciences vol. 47, Springer-Verlag (2002) | MR 1914080 | Zbl 1002.37002

[21] J.K. Hale, G. Raugel, Behaviour near a non-degenerate periodic orbit, manuscript, 2010.

[22] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. vol. 840, Springer-Verlag (1981) | MR 610244 | Zbl 0456.35001

[23] D. Henry, Some infinite dimensional Morse–Smale systems defined by parabolic differential equations, J. Differential Equations 59 (1985), 165-205 | MR 804887 | Zbl 0572.58012

[24] D. Henry, Exponential dichotomies, the shadowing lemma and homoclinic orbits in Banach spaces, Resenhas IME-USP 1 (1994), 381-401 | MR 1357942 | Zbl 0906.58034

[25] D. Henry, Perturbation of the Boundary for Boundary Value Problems of Partial Differential Operators, London Mathematical Society Lecture Note Series vol. 318, Cambridge University Press, Cambridge, UK (2005) | MR 2160744 | Zbl 1170.35300

[26] M.W. Hirsch, Stability and convergence on strongly monotone dynamical systems, J. Reine Angew. Math. 383 (1988), 1-53 | MR 921986 | Zbl 0624.58017

[27] M.W. Hirsch, C.C. Pugh, M. Shub, Invariant manifolds, Bull. Amer. Math. Soc. 76 (1970), 1015-1019 | MR 292101 | Zbl 0226.58009

[28] M.W. Hirsch, C.C. Pugh, M. Shub, Invariant Manifolds, Lecture Notes in Math. vol. 583, Springer-Verlag, Berlin, New York (1977) | MR 501173 | Zbl 0355.58009

[29] R. Joly, Generic transversality property for a class of wave equations with variable damping, Journal de Mathématiques Pures et Appliquées 84 (2005), 1015-1066 | MR 2155898 | Zbl 1082.35109

[30] R. Joly, Adaptation of the generic PDE's results to the notion of prevalence, Journal of Dynamics and Differential Equations 19 (2007), 967-983 | MR 2357534 | Zbl 1130.35013

[31] R. Joly, G. Raugel, Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle, Trans. Amer. Math. Soc. 362 (2010), 5189-5211 | MR 2657677 | Zbl 1205.35151

[32] R. Joly, G. Raugel, A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations, preprint, submitted for publication. | MR 2847240 | Zbl 1241.35001

[33] I. Kupka, Contribution à la théorie des champs génériques, Contributions to Differential Equations 2 (1963), 457-484 I. Kupka, Contribution à la théorie des champs génériques, Contributions to Differential Equations 3 (1964), 411-420 | MR 165536 | Zbl 0149.41002

[34] S. Lang, Introduction to Differentiable Manifolds, John Wiley and Sons, USA (1962) | MR 155257 | Zbl 0103.15101

[35] P. Lax, A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations, Communications on Pure and Applied Mathematics IX (1956), 747-766 | MR 86991 | Zbl 0072.33004

[36] X.B. Lin, Exponential dichotomies and homoclinic orbits in functional differential equations, J. Differential Equations 63 (1986), 227-254 | MR 848268 | Zbl 0589.34055

[37] H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations, Journal of Mathematics of Kyoto University 18 (1978), 221-227 | MR 501842 | Zbl 0387.35008

[38] H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semi-linear parabolic equation, J. Fac. Sci. Univ. Tokyo Sec. IA 29 (1982), 401-441 | MR 672070 | Zbl 0496.35011

[39] K. Nickel, Gestaltaussagen über Lösungen parabolischer Differentialgleichungen, Journal für die Reine und Angewandte Mathematik 211 (1962), 78-94 | MR 146534 | Zbl 0127.31801

[40] W.M. Oliva, Morse–Smale semiflows. Openness and A-stability, Differential Equations and Dynamical Systems, Lisbon, 2000, Fields Inst. Commun. vol. 31, Amer. Math. Soc., Providence, RI (2002), 285-307 | MR 1904521 | Zbl 1220.37068

[41] W. Ott, J.A. Yorke, Prevalence, Bulletin of the American Mathematical Society 42 (2005), 263-290 | MR 2149086 | Zbl 1111.28014

[42] J. Palis, On Morse–Smale dynamical systems, Topology 8 (1969), 385-405 | MR 246316 | Zbl 0189.23902

[43] J. Palis, W. De Melo, Geometric Theory of Dynamical Systems, Springer-Verlag, Berlin (1982) | MR 669541 | Zbl 0491.58001

[44] J. Palis, S. Smale, Structural stability theorems, Global Analysis, Proc. Symp. Pure Math. vol. 14, AMS, Providence, RI (1970), 223-231 | MR 267603 | Zbl 0214.50702

[45] K.J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations 55 (1984), 225-256 | MR 764125 | Zbl 0508.58035

[46] K.J. Palmer, Exponential dichotomies, the shadowing lemma and transversal homoclinic points, U. Kirchgraber, H.O. Walther (ed.), Dynamics Reported, vol. 1, John Wiley and Sons and B.G. Teubner (1988), 265-306 | Zbl 0676.58025

[47] K.J. Palmer, Shadowing in Dynamical Systems. Theory and Applications, Kluwer, Dordrecht, Boston, London (2000) | MR 1885537 | Zbl 0997.37001

[48] M.M. Peixoto, Structural stability on two-dimensional manifolds, Topology 1 (1962), 101-120 | MR 142859 | Zbl 0107.07103

[49] M.M. Peixoto, On an approximation theorem of Kupka and Smale, J. Differential Equations 3 (1967), 214-227 | MR 209602 | Zbl 0153.40901

[50] P. Poláčik, Parabolic equations: Asymptotic behavior and dynamics on invariant manifolds, Handbook of Dynamical Systems, vol. 2, North-Holland, Amsterdam (2002), 835-883 | MR 1901067 | Zbl 1002.35001

[51] D. Ruelle, Elements of Differentiable Dynamics and Bifurcation Theory, Academic Press, London (1989) | MR 982930 | Zbl 0684.58001

[52] F. Quinn, Transversal approximation on Banach manifolds, Global Analysis, Berkeley, 1968, Proceedings of Symposia in Pure Mathematics vol. 15, Amer. Math. Soc., Providence (1970), 213-222 | MR 264713 | Zbl 0206.25705

[53] J.W. Robbin, Algebraic Kupka–Smale theory, Dynamical Systems and Turbulence, Warwick 1980 (Coventry, 1979/1980), Lecture Notes in Math. vol. 898, Springer, Berlin, New York (1981), 286-301 | MR 654896 | Zbl 0487.58018

[54] J.-C. Saut, R. Temam, Generic properties of nonlinear boundary value problems, Communications in PDE 4 (1979), 293-319 | MR 522714 | Zbl 0462.35016

[55] R.J. Sacker, Existence of dichotomies and invariant splitting for linear differential systems IV, J. Differential Equations 27 (1978), 106-137 | MR 477315 | Zbl 0359.34044

[56] D. Salamon, Morse theory, the Conley index and Floer homology, Bulletin of London Mathematical Society 22 (1990), 113-140 | MR 1045282 | Zbl 0709.58011

[57] B. Sandstede, B. Fiedler, Dynamics of periodically forced parabolic equations on the circle, Ergodic Theory and Dynamical Systems 12 (1992), 559-571 | MR 1182662 | Zbl 0754.35066

[58] S. Smale, Morse inequalities for a dynamical system, Bulletin of the AMS 66 (1960), 43-49 | MR 117745 | Zbl 0100.29701

[59] S. Smale, Stable manifolds for differential equations and diffeomorphisms, Annali della Scuola Normale Superiore di Pisa 17 (1963), 97-116 | Numdam | MR 165537 | Zbl 0113.29702

[60] S. Smale, Diffeomorphisms with many periodic points, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, NJ (1965), 63-80 | MR 182020 | Zbl 0142.41103

[61] C. Sturm, Sur une classe d'équations à différences partielles, Journal de Mathématiques Pures et Appliquées 1 (1826), 373-444

[62] T.J. Zelenyak, Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable, Diff. Equations 4 (1968), 17-22 | MR 223758 | Zbl 0232.35053