# American Institute of Mathematical Sciences

December  2014, 34(12): 5099-5122. doi: 10.3934/dcds.2014.34.5099

## Nonlinear Sturm global attractors: Unstable manifold decompositions as regular CW-complexes

 1 Institut für Mathematik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany 2 Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais, 1049-001 Lisbon, Portugal

Received  February 2014 Revised  May 2014 Published  June 2014

We study global attractors $\mathcal{A}_f$ of scalar partial differential equations $u_t=u_{xx}+f(x,u,u_x)$ on the unit interval with, say, Neumann boundary. Due to nodal properties of differences of solutions, which amount to a nonlinear Sturm property, we call $\mathcal{A}_f$ a Sturm global attractor. We assume all equilibria $v$ to be hyperbolic. Due to a gradient-like structure we can then write $$\mathcal{A}_f = \bigcup\limits_{v}\, W^u(v) (*)$$ as a dynamic decomposition into finitely many disjoint invariant sets: the unstable manifolds $W^u(v)$ of the equilibria $v$. Based on our previous Schoenflies result [17], we prove that the dynamic decomposition $(*)$ is in fact a regular finite CW-complex with cells $W^u(v)$, in the Sturm case. We call this complex the regular dynamic complex or Sturm complex of the Sturm attractor $\mathcal{A}_f$.
We characterize the planar Sturm complexes by bipolar orientations of their 1-skeletons. We also show that any regular finite CW-complex which is the closure of a single 3-cell arises as a Sturm complex. We include a preliminary discussion of the tetrahedron and the octahedron as Sturm complexes.
Citation: Bernold Fiedler, Carlos Rocha. Nonlinear Sturm global attractors: Unstable manifold decompositions as regular CW-complexes. Discrete and Continuous Dynamical Systems, 2014, 34 (12) : 5099-5122. doi: 10.3934/dcds.2014.34.5099
##### References:
 [1] S. Angenent, The Morse-Smale property for a semi-linear parabolic equation, J. Diff. Eqns., 62 (1986), 427-442. doi: 10.1016/0022-0396(86)90093-8. [2] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam, 1992. [3] A. Banyaga and D. Hurtubise, Lectures on Morse Homology, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-1-4020-2696-6. [4] P. Brunovský, The attractor of the scalar reaction diffusion equation is a smooth graph, J. Dynamics and Differential Equations, 2 (1990), 293-323. doi: 10.1007/BF01048948. [5] P. Brunovsky and B. Fiedler, Numbers of zeros on invariant manifolds in reaction-diffusion equations, Nonlinear Analysis, TMA, 10 (1986), 179-193. doi: 10.1016/0362-546X(86)90045-3. [6] P. Brunovský and B. Fiedler, Connecting orbits in scalar reaction diffusion equations, Dynamics Reported, 1 (1988), 57-89. [7] P. Brunovský and B. Fiedler, Connecting orbits in scalar reaction diffusion equations II: The complete solution, J. Diff. Eqns., 81 (1989), 106-135. doi: 10.1016/0022-0396(89)90180-0. [8] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Colloq. AMS, Providence, 2002. [9] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Wiley, Chichester, 1994. [10] B. Fiedler (ed.), Handbook of Dynamical Systems, 2, Elsevier, Amsterdam, 2002. [11] B. Fiedler and C. Rocha, Heteroclinic orbits of semilinear parabolic equations, J. Diff. Eqns., 125 (1996), 239-281. doi: 10.1006/jdeq.1996.0031. [12] B. Fiedler and C. Rocha, Realization of meander permutations by boundary value problems, J. Diff. Eqns., 156 (1999), 282-308. doi: 10.1006/jdeq.1998.3532. [13] B. Fiedler and C. Rocha, Orbit equivalence of global attractors of semilinear parabolic differential equations, Trans. Amer. Math. Soc., 352 (2000), 257-284. doi: 10.1090/S0002-9947-99-02209-6. [14] B. Fiedler and C. Rocha, Connectivity and design of planar global attractors of Sturm type, II: Connection graphs, J. Diff. Eqs., 244 (2008), 1255-1286. doi: 10.1016/j.jde.2007.09.015. [15] 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), 71-96. doi: 10.1515/CRELLE.2009.076. [16] B. Fiedler and C. Rocha, Connectivity and design of planar global attractors of Sturm type, III: Small and Platonic examples, J. Dyn. Differ. Equations, 22 (2010), 121-162. doi: 10.1007/s10884-009-9149-2. [17] B. Fiedler and C. Rocha, Schoenflies speres as boundaries of bounded unstable manifolds in gradient sturm systems, J. Dyn. Differential Eqs., 2013. doi: 10.1007/s10884-013-9311-8. [18] B. Fiedler and A. Scheel, Spatio-temporal dynamics of reaction-diffusion patterns, In Trends in Nonlinear Analysis, M. Kirkilionis et al. (eds.), Springer-Verlag, Berlin, (2003), 23-152. [19] B. Fiedler, C. Grotta-Ragazzo and C. Rocha, An Explicit Lyapunov Function for Reflection Symmetric Parabolic Differential Equations on the Circle, Russ. Math. Surveys., 2014, in press. [20] H. de Fraysseix, P. O. de Mendez and P. Rosenstiehl, Bipolar orientations revisited, Discr. Appl. Math., 56 (1995), 157-179. doi: 10.1016/0166-218X(94)00085-R. [21] R. Fritsch and R. A. Piccinini, Cellular Structures in Topology, Cambridge University Press, 1990. doi: 10.1017/CBO9780511983948. [22] G. Fusco and C. Rocha, A permutation related to the dynamics of a scalar parabolic PDE, J. Diff. Eqns., 91 (1991), 111-137. doi: 10.1016/0022-0396(91)90134-U. [23] V. A. Galaktionov, Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications, Chapman & Hall, Boca Raton, 2004. doi: 10.1201/9780203998069. [24] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surv., 25. AMS Publications, Providence, 1988. [25] J. K. Hale, L. T. Magalhães and W. M. Oliva, Dynamics in Infinite Dimensions, Springer-Verlag, New York, 2002. doi: 10.1007/b100032. [26] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math. 840, Springer-Verlag, New York, 1981. [27] D. Henry, Some infinite dimensional Morse-Smale systems defined by parabolic differential equations, J. Diff. Eqns., 59 (1985), 165-205. doi: 10.1016/0022-0396(85)90153-6. [28] M. S. Jolly, Explicit construction of an inertial manifold for a reaction diffusion equation, J. Diff. Eqns. , 78 (1989), 220-261. doi: 10.1016/0022-0396(89)90064-8. [29] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, 1991. doi: 10.1017/CBO9780511569418. [30] H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ., 18 (1978), 221-227. [31] H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci., 15 (1979), 401-454. doi: 10.2977/prims/1195188180. [32] 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 Math., 29 (1982), 401-441. [33] H. Matano and K.-I. Nakamura, The global attractor of semilinear parabolic equations on $S^1$, Discr. Contin. Dyn. Syst., 3 (1997), 1-24. [34] W. Oliva, Stability of Morse-Smale Maps, Technical Report, Dept. Applied Math. IME-USP, 1, 1983. [35] J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, Springer-Verlag, New York, 1982. [36] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [37] G. Raugel, Global attractors in partial differential equations, Handbook of dynamical systems, 2 (2002), 885-982. doi: 10.1016/S1874-575X(02)80038-8. [38] C. Rocha, Properties of the attractor of a scalar parabolic PDE, J. Dyn. Differ. Equations, 3 (1991), 575-591. doi: 10.1007/BF01049100. [39] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9. [40] C. Sturm, Sur une classe d'équations à différences partielles, J. Math. Pure Appl., 1 (1836), 373-444. [41] H. Tanabe, Equations of Evolution, Pitman, Boston, 1979. [42] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8. [43] M. Wolfrum, Geometry of heteroclinic cascades in scalar parabolic differential equations, J. Dyn. Differ. Equations, 14 (2002), 207-241. doi: 10.1023/A:1012967428328. [44] T. I. Zelenyak, Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable, Diff. Eqns., 4 (1968), 34-45.

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##### References:
 [1] S. Angenent, The Morse-Smale property for a semi-linear parabolic equation, J. Diff. Eqns., 62 (1986), 427-442. doi: 10.1016/0022-0396(86)90093-8. [2] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam, 1992. [3] A. Banyaga and D. Hurtubise, Lectures on Morse Homology, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-1-4020-2696-6. [4] P. Brunovský, The attractor of the scalar reaction diffusion equation is a smooth graph, J. Dynamics and Differential Equations, 2 (1990), 293-323. doi: 10.1007/BF01048948. [5] P. Brunovsky and B. Fiedler, Numbers of zeros on invariant manifolds in reaction-diffusion equations, Nonlinear Analysis, TMA, 10 (1986), 179-193. doi: 10.1016/0362-546X(86)90045-3. [6] P. Brunovský and B. Fiedler, Connecting orbits in scalar reaction diffusion equations, Dynamics Reported, 1 (1988), 57-89. [7] P. Brunovský and B. Fiedler, Connecting orbits in scalar reaction diffusion equations II: The complete solution, J. Diff. Eqns., 81 (1989), 106-135. doi: 10.1016/0022-0396(89)90180-0. [8] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Colloq. AMS, Providence, 2002. [9] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Wiley, Chichester, 1994. [10] B. Fiedler (ed.), Handbook of Dynamical Systems, 2, Elsevier, Amsterdam, 2002. [11] B. Fiedler and C. Rocha, Heteroclinic orbits of semilinear parabolic equations, J. Diff. Eqns., 125 (1996), 239-281. doi: 10.1006/jdeq.1996.0031. [12] B. Fiedler and C. Rocha, Realization of meander permutations by boundary value problems, J. Diff. Eqns., 156 (1999), 282-308. doi: 10.1006/jdeq.1998.3532. [13] B. Fiedler and C. Rocha, Orbit equivalence of global attractors of semilinear parabolic differential equations, Trans. Amer. Math. Soc., 352 (2000), 257-284. doi: 10.1090/S0002-9947-99-02209-6. [14] B. Fiedler and C. Rocha, Connectivity and design of planar global attractors of Sturm type, II: Connection graphs, J. Diff. Eqs., 244 (2008), 1255-1286. doi: 10.1016/j.jde.2007.09.015. [15] 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), 71-96. doi: 10.1515/CRELLE.2009.076. [16] B. Fiedler and C. Rocha, Connectivity and design of planar global attractors of Sturm type, III: Small and Platonic examples, J. Dyn. Differ. Equations, 22 (2010), 121-162. doi: 10.1007/s10884-009-9149-2. [17] B. Fiedler and C. Rocha, Schoenflies speres as boundaries of bounded unstable manifolds in gradient sturm systems, J. Dyn. Differential Eqs., 2013. doi: 10.1007/s10884-013-9311-8. [18] B. Fiedler and A. Scheel, Spatio-temporal dynamics of reaction-diffusion patterns, In Trends in Nonlinear Analysis, M. Kirkilionis et al. (eds.), Springer-Verlag, Berlin, (2003), 23-152. [19] B. Fiedler, C. Grotta-Ragazzo and C. Rocha, An Explicit Lyapunov Function for Reflection Symmetric Parabolic Differential Equations on the Circle, Russ. Math. Surveys., 2014, in press. [20] H. de Fraysseix, P. O. de Mendez and P. Rosenstiehl, Bipolar orientations revisited, Discr. Appl. Math., 56 (1995), 157-179. doi: 10.1016/0166-218X(94)00085-R. [21] R. Fritsch and R. A. Piccinini, Cellular Structures in Topology, Cambridge University Press, 1990. doi: 10.1017/CBO9780511983948. [22] G. Fusco and C. Rocha, A permutation related to the dynamics of a scalar parabolic PDE, J. Diff. Eqns., 91 (1991), 111-137. doi: 10.1016/0022-0396(91)90134-U. [23] V. A. Galaktionov, Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications, Chapman & Hall, Boca Raton, 2004. doi: 10.1201/9780203998069. [24] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surv., 25. AMS Publications, Providence, 1988. [25] J. K. Hale, L. T. Magalhães and W. M. Oliva, Dynamics in Infinite Dimensions, Springer-Verlag, New York, 2002. doi: 10.1007/b100032. [26] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math. 840, Springer-Verlag, New York, 1981. [27] D. Henry, Some infinite dimensional Morse-Smale systems defined by parabolic differential equations, J. Diff. Eqns., 59 (1985), 165-205. doi: 10.1016/0022-0396(85)90153-6. [28] M. S. Jolly, Explicit construction of an inertial manifold for a reaction diffusion equation, J. Diff. Eqns. , 78 (1989), 220-261. doi: 10.1016/0022-0396(89)90064-8. [29] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, 1991. doi: 10.1017/CBO9780511569418. [30] H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ., 18 (1978), 221-227. [31] H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci., 15 (1979), 401-454. doi: 10.2977/prims/1195188180. [32] 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 Math., 29 (1982), 401-441. [33] H. Matano and K.-I. Nakamura, The global attractor of semilinear parabolic equations on $S^1$, Discr. Contin. Dyn. Syst., 3 (1997), 1-24. [34] W. Oliva, Stability of Morse-Smale Maps, Technical Report, Dept. Applied Math. IME-USP, 1, 1983. [35] J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, Springer-Verlag, New York, 1982. [36] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [37] G. Raugel, Global attractors in partial differential equations, Handbook of dynamical systems, 2 (2002), 885-982. doi: 10.1016/S1874-575X(02)80038-8. [38] C. Rocha, Properties of the attractor of a scalar parabolic PDE, J. Dyn. Differ. Equations, 3 (1991), 575-591. doi: 10.1007/BF01049100. [39] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9. [40] C. Sturm, Sur une classe d'équations à différences partielles, J. Math. Pure Appl., 1 (1836), 373-444. [41] H. Tanabe, Equations of Evolution, Pitman, Boston, 1979. [42] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8. [43] M. Wolfrum, Geometry of heteroclinic cascades in scalar parabolic differential equations, J. Dyn. Differ. Equations, 14 (2002), 207-241. doi: 10.1023/A:1012967428328. [44] T. I. Zelenyak, Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable, Diff. Eqns., 4 (1968), 34-45.
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