A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations
Confluentes Mathematici, Tome 3 (2011) no. 3, pp. 471-493.
Publié le :
DOI : 10.1142/S1793744211000369
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Joly, Romain; Raugel, Geneviève. A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations. Confluentes Mathematici, Tome 3 (2011) no. 3, pp. 471-493. doi : 10.1142/S1793744211000369. http://www.numdam.org/articles/10.1142/S1793744211000369/

[1] R. Abraham and J. Robbin, Transversal Mappings and Flows (W. A. Benjamin, 1967).

[2] S. B. Angenent, The Morse–Smale property for a semilinear parabolic equation, J. Differential Equations 62 (1986) 427–442.

[3] S. B. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math. 390 (1988) 79–96.

[4] S. B. Angenent and B. Fiedler, The dynamics of rotating waves in scalar reaction diffusion equations, Trans. Amer. Math. Soc. 307 (1988) 545–568.

[5] D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Imeni V. A. Steklova 90 (1967) 3–210.

[6] A. Arroyo and F. Rodriguez Hertz, Homoclinic bifurcations and uniform hyperbolicity for three-dimensional flows, Ann. Inst. H. Poincaré, Anal. Non Linéaire 20 (2003) 805–841.

[7] I. Bendixson, Sur les courbes définies par des équations différentielles, Acta Math. 24 (1901) 1–88.

[8] C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math. 158 (2004) 33–104.

[9] C. Bonatti and L. D´ıaz, Connexions hétéroclines et généricité d’une infinité de puits et de sources, Ann. Sci. École Norm. Sup. 32 (1999) 135–150.

[10] C. Bonatti, S. Gan and L. Wen, On the existence of non-trivial homoclinic classes, Ergodic Theory Dynam. Systems 27 (2007) 1473–1508.

[11] P. Brunovsk´y, The attractor of the scalar reaction diffusion equation is a smooth graph, J. Dynam. Differential Equations 2 (1990) 293–323.

[12] P. Brunovsk´y and B. Fiedler, Connecting orbits in scalar reaction diffusion equations. II. The complete solution, J. Differential Equations 81 (1989) 106–135.

[13] P. Brunovsk´y, R. Joly and G. Raugel, Generic Kupka–Smale property for the parabolic equations, in preparation.

[14] P. Brunovsk´y and P. Pol´aˇcik, The Morse–Smale structure of a generic reaction- diffusion equation in higher space dimension, J. Differential Equations 135 (1997) 129–181.

[15] P. Brunovsk´y and G. Raugel, Genericity of the Morse–Smale property for damped wave equations, J. Dynam. Differential Equations 15 (2003) 571–658.

[16] X.-Y. Chen, A strong unique continuation theorem for parabolic equations, Math. Ann. 311 (1998) 603–630.

[17] S. Crovisier, Birth of homoclinic intersections: A model for the central dynamics of partially hyperbolic systems, Ann. of Math. 172 (2010) 1641–1677.

[18] R. Czaja and C. Rocha, Transversality in scalar reaction-diffusion equations on a circle, J. Differential Equations 245 (2008) 692–721.

[19] E. N. Dancer and P. Pol´aˇcik, Realization of vector fields and dynamics of spatially homogeneous parabolic equations, Mem. Amer. Math. Soc. 140 (1999) n◦ 668.

[20] B. Fiedler and J. Mallet-Paret, A Poincaré–Bendixson theorem for scalar reaction- diffusion equations, Arch. Rational Mech. Anal. 107 (1989) 325–345.

[21] B. Fiedler and C. Rocha, Heteroclinic orbits of semilinear parabolic equations, J. Differential Equations 125 (1996) 239–281.

[22] B. Fiedler and C. Rocha, Realization of meander permutations by boundary value problems, J. Differential Equations 156 (1999) 282–308.

[23] B. Fiedler and C. Rocha, Orbit equivalence of global attractors of semilinear parabolic differential equations, Trans. Amer. Math. Soc. 352 (2000) 257–284.

[24] 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.

[25] B. Fiedler and A. Scheel, Spatio-temporal dynamics of reaction-diffusion patterns, in Trends in Nonlinear Analysis (Springer-Verlag, 2003), pp. 23–152.

[26] G. Fusco and W. M. Oliva, Jacobi matrices and transversality, Proc. Roy. Soc. Edin- burgh Sect. A 109 (1988) 231–243.

[27] G. Fusco and W. M. Oliva, Transversality between invariant manifolds of periodic orbits for a class of monotone dynamical systems, J. Dynam. Differential Equations 2 (1990) 1–17.

[28] G. Fusco and C. Rocha, A permutation related to the dynamics of a scalar parabolic PDE, J. Differential Equations 91 (1991) 111–137.

[29] G. Fusco and S. M. Verduyn Lunel, Order structures and the heat equation, J. Differential Equations 139 (1997) 104–145.

[30] V. A. Galaktianov and P. J. Harwin, Sturm’s theorems on zero sets in nonlinear parabolic equations, in Sturm Liouville Theory: Past and Present, eds. W. O. Amrein, A. M. Hinz and D. B. Pearson (Birkhäuser-Verlag, 2005).

[31] M. Golubitsky and V. Guillemin, Stable Mapping and Their Singularities, Graduate Texts in Mathematics, Vol. 14 (Springer-Verlag, 1973).

[32] R. W. Ghrist and R. C. Vandervorst, Scalar parabolic PDEs and braids, Trans. Amer. Math. Soc. 361 (2009) 2755–2788.

[33] J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Publ. Math. Inst. Hautes Études Sci. 50 (1979) 59–72.

[34] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, Vol. 25 (Amer. Math. Soc., 1988).

[35] J. K. Hale, Dynamics of a scalar parabolic equation, Canad. Appl. Math. Quart. 5 (1997) 209–305.

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

[37] J. K. Hale, L. Magalh˜aes and W. Oliva, An Introduction to Infinite Dimensional Dynamical Systems, Applied Mathematical Sciences, Vol. 47 (Springer-Verlag, 1984); Second edition, Dynamics in Infinite Dimensions (Springer-Verlag, 2002).

[38] R. Hardt and L. Simon, Nodal sets for solutions of elliptic equations, J. Differential Geom. 30 (1989) 505–522.

[39] S. Hayashi, Connecting invariant manifolds and the solution of the C1 stability and Ω-stability conjectures for flows, Ann. Math. 145 (1997) 81–137.

[40] D. B. Henry, Some infinite-dimensional Morse–Smale systems defined by parabolic partial differential equations, J. Differential Equations 59 (1985) 165–205.

[41] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840 (Springer-Verlag, 1981).

[42] M. W. Hirsch, Differential equations and convergence almost everywhere in strongly monotone semiflows, Contemp. Math. 17 (1983) 267–285.

[43] M. W. Hirsch, Systems of differential equations which are competitive or cooperative. I. Limit sets, SIAM J. Math. Anal. 13 (1982) 167–179.

[44] M. W. Hirsch, Systems of differential equations that are competitive or cooperative. II. Convergence almost everywhere, SIAM J. Math. Anal. 16 (1985) 423–439.

[45] M. W. Hirsch, Stability and convergence on strongly monotone dynamical systems, J. Reine Angew. Math. 383 (1988) 1–53.

[46] M. Jolly, Explicit construction of an inertial manifold for a reaction diffusion equation, J. Differential Equations 78 (1989) 220–261.

[47] R. Joly, Generic transversality property for a class of wave equations with variable damping, J. Math. Pures Appl. 84 (2005) 1015–1066.

[48] 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.

[49] R. Joly and G. Raugel, Generic Morse–Smale property for the parabolic equation on the circle, Ann. Inst. H. Poincaré, Anal. Non Linéaire 27 (2010) 1397–1440.

[50] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Sys- tems, with a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, Vol. 54 (Cambridge Univ. Press, 1995).

[51] I. Kupka, Contribution à la théorie des champs génériques, Differential Equations 2 (1963) 457–484 [Addendum and corrections, ibid. 3 (1964) 411–420].

[52] J. Mallet-Paret and H. L. Smith, The Poincaré–Bendixson theorem for monotone cyclic feedback systems, J. Dynam. Differential Equations 2 (1990) 367–421.

[53] H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equa- tions, J. Math. Kyoto Univ. 18 (1978) 221–227.

[54] H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semi- linear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982) 401–441.

[55] H. Matano and K.-I. Nakamura, The global attractor of semilinear parabolic equations on T1 , Disc. Cont. Dynam. Syst. 3 (1997) 1–24.

[56] S. E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology 13 (1974) 9–18.

[57] S. E. Newhouse, Lectures on dynamical systems, in Dynamical Systems: C.I.M.E. Lectures, Bressanone, Italy, June 1978, eds. J. Guckenheimer, J. Moser and S. E. Newhouse, Progress in Mathematic, Vol. 8 (Birkhäuser-Verlag, 1980).

[58] K. Nickel, Gestaltaussagen über Lösungen parabolischer Differentialgleichungen, J. Reine Angew. Math. 211 (1962) 78–94.

[59] Z. Nitecki, Differentiable Dynamics (MIT Press, 1971).

[60] W. M. Oliva, Morse–Smale semiflows. Openess and A-stability, in Differential Equa- tions and Dynamical Systems (Lisbon, 2000), Fields Inst. Commun., Vol. 31 (Amer. Math. Soc., 2002), pp. 285–307.

[61] J. Palis, On Morse–Smale dynamical systems, Topology 8 (1968) 385–404.

[62] J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, Translated from the Portuguese by A. K. Manning (Springer-Verlag, 1982).

[63] J. Palis and S. Smale, Structural stability theorems, in Global Analysis, Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968 (Amer. Math. Soc., 1970), pp. 223–231.

[64] M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology 1 (1962) 101–120.

[65] M. M. Peixoto, On an approximation theorem of Kupka and Smale, J. Differential Equations 3 (1967) 214–227.

[66] M. Percie du Sert, in preparation.

[67] H. Poincaré, Sur les courbes définies par une équation différentielle, C. R. Acad. Sci. Paris 90 (1880) 673–675 [see also Œuvres, Gauthier-Villars, Paris, Vol. 1 (1928)].

[68] P. Pol´aˇcik, Convergence in smooth strongly monotone flows defined by semilinear parabolic equations, J. Differential Equations 79 (1989) 89–110.

[69] P. Pol´aˇcik, Imbedding of any vector field in a scalar semilinear parabolic equation, Proc. Amer. Math. Soc. 115 (1992) 1001–1008.

[70] P. Pol´aˇcik, High-dimensional ω-limit sets and chaos in scalar parabolic equations, J. Differential Equations 119 (1995) 24–53.

[71] P. Pol´aˇcik, Reaction-diffusion equations and realization of gradient vector fields, in International Conference on Differential Equations (Lisboa, 1995), (World Scientific, 1998), pp. 197–206.

[72] P. Pol´aˇcik, Persistent saddle connections in a class of reaction-diffusion equations, J. Differential Equations 56 (1999) 182–210.

[73] P. Pol´aˇcik, Parabolic equations: Asymptotic behavior and dynamics on invariant man- ifolds, in Handbook of Dynamical Systems, Vol. 2 (North-Holland, 2002), pp. 835–883.

[74] M. Prizzi and K. Rybakowski, Complicated dynamics of parabolic equations with simple gradient dependence, Trans. Amer. Math. Soc. 350 (1998) 3119–3130.

[75] C. C. Pugh, The closing lemma, Amer. J. Math. 89 (1967) 956–1009.

[76] C. C. Pugh, An improved closing lemma and a general density theorem, Amer. J. Math. 89 (1967) 1010–1021.

[77] C. C. Pugh and C. Robinson, The C1 closing lemma, including Hamiltonians, Ergodic Theory Dynam. Systems 3 (1983) 261–313.

[78] E. R. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Ann. Math. 151 (2000) 961–1023.

[79] G. Raugel, Global attractors in partial differential equations, in Handbook of Dynam- ical Systems, Vol. 2 (North-Holland, 2002), pp. 885–982.

[80] C. Robinson, Introduction to the closing lemma, in The Structure of Attractors in Dynamical Systems (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), Lec- ture Notes in Mathematics, Vol. 668 (Springer, 1978), pp. 225–230.

[81] C. Robinson, Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, 2nd edn., Studies in Advanced Mathematics (CRC Press, 1999).

[82] J. C. Robinson, Infinite-Dimensional Dynamical Systems. An Introduction to Dissipa- tive Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics (Cambridge Univ. Press, 2001).

[83] C. Rocha, Properties of the attractor of a scalar parabolic PDE, J. Dynam. Equations 3 (1991) 575–591.

[84] C. Rocha, Realization of period maps of planar Hamiltonian systems, J. Dynam. Differential Equations 19 (2007) 571–591.

[85] D. Ruelle, Elements of Differentiable Dynamics and Bifurcation Theory (Academic Press, 1989).

[86] B. Sandstede and B. Fiedler, Dynamics of periodically forced parabolic equations on the circle, Ergod. Th. Dynam. Syst. 2 (1992) 559–571.

[87] S. Smale, On gradient dynamical systems, Ann. Math. 74 (1961) 199–206.

[88] S. Smale, Stable manifolds for differential equations and diffeomorphisms, Ann. Scuola Nor. Super. Pisa 17 (1963) 97–116.

[89] S. Smale, Diffeomorphisms with many periodic points, in Differential and Combina- torial Topology, A Symposium in Honor of Marston Morse (Princeton Univ. Press, 1965), pp. 63–80.

[90] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967) 747–817.

[91] S. Smale, On the differential equations of species in competition, J. Math. Biol. 3 (1976) 5–7.

[92] J. Smillie, Competitive and cooperative tridiagonal systems of differential equations, SIAM J. Math. Anal. 15 (1984) 530–534.

[93] H. L. Smith, A discrete Lyapunov function for a class of linear differential equations, Pac. J. Math. 144 (1990) 345–360.

[94] H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Com- petitive and Cooperative Systems, Mathematical Surveys and Monographs, Vol. 41 (Amer. Math. Soc., 1995).

[95] C. Sturm, Sur une classe d’équations à différences partielles, J. Math. Pures Appl. 1 (1836) 373–444.

[96] M. Wolfrum, A sequence of order relations: Encoding heteroclinic connections inscalar parabolic PDE, J. Differential Equations 183 (2002) 56–78.

[97] T. I. Zelenyak, Stabilization of solutions of boundary value problems for a second- order parabolic equation with one space variable, Differencialnye Uravnenija 4 (1968) 34–45 [Translated in Differential Equations 4, 17–22].

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