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

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

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

Bernold Fiedler ^1, and Carlos Rocha ^2,

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

Abstract
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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 \begin{equation} \mathcal{A}_f = \bigcup\limits_{v}\, W^u(v) (*) \end{equation} 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.

Keywords: Hamiltonian path, bipolar graph, Parabolic PDE, infinite-dimensional dissipative dynamics, meander, Morse theory, cell complex..

Mathematics Subject Classification: 35B41, 05C9.

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.

show all references

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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