# American Institute of Mathematical Sciences

December  2012, 7(4): 617-659. doi: 10.3934/nhm.2012.7.617

## Sturm global attractors for $S^1$-equivariant parabolic equations

 1 Freie Universität Berlin, Institut für Mathematik I, Arnimallee 2-6, D-14195 Berlin, Germany 2 Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Instituto Superior Técnico, Departamento de Matemática, Universidade Técnica de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal, Portugal

Received  January 2012 Published  December 2012

We consider a semilinear parabolic equation of the form $u_t = u_{xx} + f(u,u_x)$ defined on the circle $x ∈ S^1=\mathbb{R}/2\pi\mathbb{Z}$. For a dissipative nonlinearity $f$ this equation generates a dissipative semiflow in the appropriate function space, and the corresponding global attractor $A_f$ is called a Sturm attractor. If $f=f(u,p)$ is even in $p$, then the semiflow possesses an embedded flow satisfying Neumann boundary conditions on the half-interval $(0,\pi)$. This is due to $O(2)$ equivariance of the semiflow and, more specifically, due to reflection at the axis through $x=0,\pi\in S^1$. For general $f=f(u,p)$, where only $SO(2)$ equivariance prevails, we will nevertheless use the Sturm permutation $\sigma$ introduced for the characterization of Neumann flows to obtain a purely combinatorial characterization of the Sturm attractors $A_f$ on the circle. With this Sturm permutation $\sigma$ we then enumerate and describe the heteroclinic connections of all Morse-Smale attractors $A_f$ with $m$ stationary solutions and $q$ periodic orbits, up to $n:=m+2q \le 9$.
Citation: Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617
##### References:
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Fiedler, Connecting orbits in scalar reaction diffusion equations,, Dynamics Reported, 1 (1988), 57. Google Scholar [8] P. Brunovský and B. Fiedler, Connecting orbits in scalar reaction diffusion equations II: The complete solution,, J. Differential Equations, 81 (1989), 106. doi: 10.1016/0022-0396(89)90180-0. Google Scholar [9] E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGraw-Hill, (1955). Google Scholar [10] R. Czaja and C. Rocha, Transversality in scalar reaction-diffusion equations on a circle,, J. Differential Equations, 245 (2008), 692. doi: 10.1016/j.jde.2008.01.018. Google Scholar [11] M. P. do Carmo, "Differential Geometry of Curves and Surfaces,", Prentice-Hall, (1976). Google Scholar [12] B. Fiedler and J. Mallet-Paret, The Poincaré-Bendixson theorem for scalar reaction diffusion equations,, Arch. Rational Mech. Anal., 107 (1989), 325. doi: 10.1007/BF00251553. Google Scholar [13] B. Fiedler, Global attractors of one-dimensional parabolic equations: Sixteen examples,, Tatra Mt. Math. Publ., 4 (1994), 67. Google Scholar [14] B. Fiedler, Do global attractors depend on boundary conditions?,, Doc. Math. J. DMV, 1 (1996), 215. Google Scholar [15] B. Fiedler and C. Rocha, Heteroclinic orbits of semilinear parabolic equations,, J. Differential Equations, 125 (1996), 239. doi: 10.1006/jdeq.1996.0031. Google Scholar [16] B. Fiedler and C. Rocha, Realization of meander permutations by boundary value problems,, J. Differential Equations, 156 (1999), 282. doi: 10.1006/jdeq.1998.3532. Google Scholar [17] B. Fiedler and C. Rocha, Orbit equivalence of global attractors of semilinear parabolic differential equations,, Trans. Amer. Math. Soc., 352 (2000), 257. doi: 10.1090/S0002-9947-99-02209-6. Google Scholar [18] B. Fiedler and C. Rocha, Connectivity and design of planar global attractors of Sturm type. II: Connection graphs,, J. Differential Equations, 245 (2008), 692. doi: 10.1016/j.jde.2007.09.015. Google Scholar [19] B. Fiedler and C. Rocha, Connectivity and design of planar global attractors of Sturm type. I: Bipolar orientations and Hamiltonian paths,, J. Reine Angew. Math., 635 (2009), 71. doi: 10.1515/CRELLE.2009.076. Google Scholar [20] B. Fiedler and C. Rocha, Connectivity and design of planar global attractors of Sturm type. III: Small and platonic examples,, J. Dynam. Differential Equations, 22 (2010), 509. doi: 10.1007/s10884-009-9149-2. Google Scholar [21] B. Fiedler, C. Rocha, D. Salazar and J. Solà-Morales, Dynamics of peacewise-autonomous bistable parabolic equations,, in, 31 (2002), 151. Google Scholar [22] B. Fiedler and A. Scheel, Dynamics of reaction-diffusion patterns,, in, (2002), 23. Google Scholar [23] 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. doi: 10.1016/j.jde.2003.10.027. Google Scholar [24] B. Fiedler, C. Rocha and M. Wolfrum, A permutation characterization of Sturm global attractors of Hamiltonian type,, J. Differential Equations, 252 (2012), 588. doi: 10.1016/j.jde.2011.08.013. Google Scholar [25] G. Fusco and C. Rocha, A permutation related to the dynamics of a scalar parabolic PDE,, J. Differential Equations, 91 (1991), 75. doi: 10.1016/0022-0396(91)90134-U. Google Scholar [26] J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", Math. Surv., 25 (1988). Google Scholar [27] J. K. Hale, L. T. Magalhães and W. M. Oliva, "Dynamics in Infinite Dimensions,", Second edition, 47 (2002). Google Scholar [28] J. K. Hale and G. Raugel, Convergence in gradient-like systems with applications to PDE,, Z. Angew. Math. Phys., 43 (1992), 63. doi: 10.1007/BF00944741. Google Scholar [29] J. Härterich and M. Wolfrum, Convergence in gradient-like systems with applications to PDE,, Discrete and Contin. Dyn. Syst., 12 (2005), 531. Google Scholar [30] P. Hartman, "Ordinary Differential Equations,", Birkhäuser, (1982). Google Scholar [31] D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lect. Notes in Math, 840 (1981). Google Scholar [32] D. Henry, Some infinite dimensional Morse-Smale systems defined by parabolic differential equations,, J. Differential Equations, 59 (1985), 165. doi: 10.1016/0022-0396(85)90153-6. Google Scholar [33] 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. doi: 10.1090/S0002-9947-2010-04890-1. Google Scholar [34] 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. doi: 10.1016/j.anihpc.2010.09.001. Google Scholar [35] T. Krisztin and H.-O. Walther, Unique periodic orbits for delay positive feedback and the global attractor,, J. Dynam. Differential Equations, 13 (2001), 1. doi: 10.1023/A:1009091930589. Google Scholar [36] S. K. Lando, "Lectures on Generating Functions,", Stud. Math. Lib., 23 (2003). Google Scholar [37] S. K. Lando and A. K. Zvonkin, Meanders,, Selecta Math. Soviet., 11 (1992), 117. Google Scholar [38] H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations,, J. Math. Kyoto Univ., 18 (1878), 221. Google Scholar [39] 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. Google Scholar [40] H. Matano, Asymptotic behavior of solutions of semilinear heat equations on $S^1$,, in, (1988), 139. doi: 10.1007/978-1-4613-9608-6_8. Google Scholar [41] H. Matano and K.-I. Nakamura, The global attractor of semilinear parabolic equations on $S^1$,, Discrete Contin. Dyn. Syst., 3 (1997), 1. Google Scholar [42] K. Mischaikow, Conley index theory,, in, 1609 (1995), 119. doi: 10.1007/BFb0095240. Google Scholar [43] Y. Miyamoto, On connecting orbits of semilinear parabolic equations on $S^1$,, Documenta Math., 9 (2004), 435. Google Scholar [44] N. Nadirashvili, Connecting orbits for nonlinear parabolic equations,, Asian J. Math., 2 (1998), 135. Google Scholar [45] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [46] C. Ragazzo, Scalar autonomous second order ordinary differential equations,, Preprint, (2010). doi: 10.1007/s12346-011-0063-8. Google Scholar [47] G. Raugel, Global attractors in partial differential equations,, in, 2 (2002), 885. doi: 10.1016/S1874-575X(02)80038-8. Google Scholar [48] C. Rocha, Properties of the attractor of a scalar parabolic PDE,, J. Dynam. Differential Equations, 3 (1991), 575. doi: 10.1007/BF01049100. Google Scholar [49] C. Rocha, Bifurcations in discretized reaction-diffusion equations,, Resenhas IME-USP, 1 (1994), 403. Google Scholar [50] C. Rocha, Realization of period maps of planar Hamiltonian systems,, J. Dynam. Differential Equations, 19 (2007), 571. doi: 10.1007/s10884-007-9081-2. Google Scholar [51] B. Sandstede and B. Fiedler, Dynamics of periodically forced parabolic equations on the circle,, Ergodic Theory Dynam. Systems, 12 (1992), 559. doi: 10.1017/S0143385700006933. Google Scholar [52] R. Schaaf, "Global Solution Branches of Two Point Boundary Value Problems,", Lect. Notes in Math, 1458 (1990). Google Scholar [53] J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", Springer-Verlag, (1983). Google Scholar [54] C. Sturm, Sur une classe d'équations à différences partielles,, J. Math. Pure Appl., 1 (1836), 373. Google Scholar [55] M. Urabe, Relations between periods and amplitudes of periodic solutions of $\ddot x + g(x) = 0$,, Funkcial. Ekvac., 6 (1964), 63. Google Scholar [56] M. Wolfrum, Geometry of heteroclinic cascades in scalar parabolic differential equations,, J. Dynam. Differential Equations, 14 (2002), 207. doi: 10.1023/A:1012967428328. Google Scholar [57] M. Wolfrum, A sequence of order relations, encoding heteroclinic connections in scalar parabolic PDE,, J. Differential Equations, 183 (2002), 56. doi: 10.1006/jdeq.2001.4114. Google Scholar [58] J. A. Yorke, Periods of periodic solutions and the Lipschitz constant,, Proc. Amer. Math. Soc., 22 (1969), 509. Google Scholar [59] T. I. Zelenyak, Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable,, Differential Equations, 4 (1968), 34. Google Scholar

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##### References:
 [1] R. Abraham and J. Robbin, "Transversal Mappings and Flows,", Benjamin, (1967). Google Scholar [2] S. Angenent, The Morse-Smale property for a semi-linear parabolic equation,, J. Differential Equations, 62 (1986), 427. doi: 10.1016/0022-0396(86)90093-8. Google Scholar [3] S. Angenent, The zero set of a solution of a parabolic equation,, J. Reine Angew. Math., 390 (1988), 79. doi: 10.1515/crll.1988.390.79. Google Scholar [4] S. Angenent and B. Fiedler, The dynamics of rotating waves in scalar reaction diffusion equations,, Trans. Amer. Math. Soc., 307 (1988), 545. doi: 10.2307/2001188. Google Scholar [5] V. I. Arnold, A branched covering $CP^2 \rightarrow S^4$, hyperbolicity and projective topology,, Siberian Math. J., 29 (1988), 717. doi: 10.1007/BF00970265. Google Scholar [6] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North Holland, (1992). Google Scholar [7] P. Brunovský and B. Fiedler, Connecting orbits in scalar reaction diffusion equations,, Dynamics Reported, 1 (1988), 57. Google Scholar [8] P. Brunovský and B. Fiedler, Connecting orbits in scalar reaction diffusion equations II: The complete solution,, J. Differential Equations, 81 (1989), 106. doi: 10.1016/0022-0396(89)90180-0. Google Scholar [9] E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGraw-Hill, (1955). Google Scholar [10] R. Czaja and C. Rocha, Transversality in scalar reaction-diffusion equations on a circle,, J. Differential Equations, 245 (2008), 692. doi: 10.1016/j.jde.2008.01.018. Google Scholar [11] M. P. do Carmo, "Differential Geometry of Curves and Surfaces,", Prentice-Hall, (1976). Google Scholar [12] B. Fiedler and J. Mallet-Paret, The Poincaré-Bendixson theorem for scalar reaction diffusion equations,, Arch. Rational Mech. Anal., 107 (1989), 325. doi: 10.1007/BF00251553. Google Scholar [13] B. Fiedler, Global attractors of one-dimensional parabolic equations: Sixteen examples,, Tatra Mt. Math. Publ., 4 (1994), 67. Google Scholar [14] B. Fiedler, Do global attractors depend on boundary conditions?,, Doc. Math. J. DMV, 1 (1996), 215. Google Scholar [15] B. Fiedler and C. Rocha, Heteroclinic orbits of semilinear parabolic equations,, J. Differential Equations, 125 (1996), 239. doi: 10.1006/jdeq.1996.0031. Google Scholar [16] B. Fiedler and C. Rocha, Realization of meander permutations by boundary value problems,, J. Differential Equations, 156 (1999), 282. doi: 10.1006/jdeq.1998.3532. Google Scholar [17] B. Fiedler and C. Rocha, Orbit equivalence of global attractors of semilinear parabolic differential equations,, Trans. Amer. Math. Soc., 352 (2000), 257. doi: 10.1090/S0002-9947-99-02209-6. Google Scholar [18] B. Fiedler and C. Rocha, Connectivity and design of planar global attractors of Sturm type. II: Connection graphs,, J. Differential Equations, 245 (2008), 692. doi: 10.1016/j.jde.2007.09.015. Google Scholar [19] B. Fiedler and C. Rocha, Connectivity and design of planar global attractors of Sturm type. I: Bipolar orientations and Hamiltonian paths,, J. Reine Angew. Math., 635 (2009), 71. doi: 10.1515/CRELLE.2009.076. Google Scholar [20] B. Fiedler and C. Rocha, Connectivity and design of planar global attractors of Sturm type. III: Small and platonic examples,, J. Dynam. Differential Equations, 22 (2010), 509. doi: 10.1007/s10884-009-9149-2. Google Scholar [21] B. Fiedler, C. Rocha, D. Salazar and J. Solà-Morales, Dynamics of peacewise-autonomous bistable parabolic equations,, in, 31 (2002), 151. Google Scholar [22] B. Fiedler and A. Scheel, Dynamics of reaction-diffusion patterns,, in, (2002), 23. Google Scholar [23] 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. doi: 10.1016/j.jde.2003.10.027. Google Scholar [24] B. Fiedler, C. Rocha and M. Wolfrum, A permutation characterization of Sturm global attractors of Hamiltonian type,, J. Differential Equations, 252 (2012), 588. doi: 10.1016/j.jde.2011.08.013. Google Scholar [25] G. Fusco and C. Rocha, A permutation related to the dynamics of a scalar parabolic PDE,, J. Differential Equations, 91 (1991), 75. doi: 10.1016/0022-0396(91)90134-U. Google Scholar [26] J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", Math. Surv., 25 (1988). Google Scholar [27] J. K. Hale, L. T. Magalhães and W. M. Oliva, "Dynamics in Infinite Dimensions,", Second edition, 47 (2002). Google Scholar [28] J. K. Hale and G. Raugel, Convergence in gradient-like systems with applications to PDE,, Z. Angew. Math. Phys., 43 (1992), 63. doi: 10.1007/BF00944741. Google Scholar [29] J. Härterich and M. Wolfrum, Convergence in gradient-like systems with applications to PDE,, Discrete and Contin. Dyn. Syst., 12 (2005), 531. Google Scholar [30] P. Hartman, "Ordinary Differential Equations,", Birkhäuser, (1982). Google Scholar [31] D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lect. Notes in Math, 840 (1981). Google Scholar [32] D. Henry, Some infinite dimensional Morse-Smale systems defined by parabolic differential equations,, J. Differential Equations, 59 (1985), 165. doi: 10.1016/0022-0396(85)90153-6. Google Scholar [33] 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. doi: 10.1090/S0002-9947-2010-04890-1. Google Scholar [34] 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. doi: 10.1016/j.anihpc.2010.09.001. Google Scholar [35] T. Krisztin and H.-O. Walther, Unique periodic orbits for delay positive feedback and the global attractor,, J. Dynam. Differential Equations, 13 (2001), 1. doi: 10.1023/A:1009091930589. Google Scholar [36] S. K. Lando, "Lectures on Generating Functions,", Stud. Math. Lib., 23 (2003). Google Scholar [37] S. K. Lando and A. K. Zvonkin, Meanders,, Selecta Math. Soviet., 11 (1992), 117. Google Scholar [38] H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations,, J. Math. Kyoto Univ., 18 (1878), 221. Google Scholar [39] 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. Google Scholar [40] H. Matano, Asymptotic behavior of solutions of semilinear heat equations on $S^1$,, in, (1988), 139. doi: 10.1007/978-1-4613-9608-6_8. Google Scholar [41] H. Matano and K.-I. Nakamura, The global attractor of semilinear parabolic equations on $S^1$,, Discrete Contin. Dyn. Syst., 3 (1997), 1. Google Scholar [42] K. Mischaikow, Conley index theory,, in, 1609 (1995), 119. doi: 10.1007/BFb0095240. Google Scholar [43] Y. Miyamoto, On connecting orbits of semilinear parabolic equations on $S^1$,, Documenta Math., 9 (2004), 435. Google Scholar [44] N. Nadirashvili, Connecting orbits for nonlinear parabolic equations,, Asian J. Math., 2 (1998), 135. Google Scholar [45] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [46] C. Ragazzo, Scalar autonomous second order ordinary differential equations,, Preprint, (2010). doi: 10.1007/s12346-011-0063-8. Google Scholar [47] G. Raugel, Global attractors in partial differential equations,, in, 2 (2002), 885. doi: 10.1016/S1874-575X(02)80038-8. Google Scholar [48] C. Rocha, Properties of the attractor of a scalar parabolic PDE,, J. Dynam. Differential Equations, 3 (1991), 575. doi: 10.1007/BF01049100. Google Scholar [49] C. Rocha, Bifurcations in discretized reaction-diffusion equations,, Resenhas IME-USP, 1 (1994), 403. Google Scholar [50] C. Rocha, Realization of period maps of planar Hamiltonian systems,, J. Dynam. Differential Equations, 19 (2007), 571. doi: 10.1007/s10884-007-9081-2. Google Scholar [51] B. Sandstede and B. Fiedler, Dynamics of periodically forced parabolic equations on the circle,, Ergodic Theory Dynam. Systems, 12 (1992), 559. doi: 10.1017/S0143385700006933. Google Scholar [52] R. Schaaf, "Global Solution Branches of Two Point Boundary Value Problems,", Lect. Notes in Math, 1458 (1990). Google Scholar [53] J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", Springer-Verlag, (1983). Google Scholar [54] C. Sturm, Sur une classe d'équations à différences partielles,, J. Math. Pure Appl., 1 (1836), 373. Google Scholar [55] M. Urabe, Relations between periods and amplitudes of periodic solutions of $\ddot x + g(x) = 0$,, Funkcial. Ekvac., 6 (1964), 63. Google Scholar [56] M. Wolfrum, Geometry of heteroclinic cascades in scalar parabolic differential equations,, J. Dynam. Differential Equations, 14 (2002), 207. doi: 10.1023/A:1012967428328. Google Scholar [57] M. Wolfrum, A sequence of order relations, encoding heteroclinic connections in scalar parabolic PDE,, J. Differential Equations, 183 (2002), 56. doi: 10.1006/jdeq.2001.4114. Google Scholar [58] J. A. Yorke, Periods of periodic solutions and the Lipschitz constant,, Proc. Amer. Math. Soc., 22 (1969), 509. Google Scholar [59] T. I. Zelenyak, Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable,, Differential Equations, 4 (1968), 34. Google Scholar
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