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 \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.
Citation: Bernold Fiedler, Carlos Rocha. Nonlinear Sturm global attractors: Unstable manifold decompositions as regular CW-complexes. Discrete & Continuous Dynamical Systems - A, 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.  doi: 10.1016/0022-0396(86)90093-8.  Google Scholar

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North Holland, (1992).   Google Scholar

[3]

A. Banyaga and D. Hurtubise, Lectures on Morse Homology,, Springer-Verlag, (2004).  doi: 10.1007/978-1-4020-2696-6.  Google Scholar

[4]

P. Brunovský, The attractor of the scalar reaction diffusion equation is a smooth graph,, J. Dynamics and Differential Equations, 2 (1990), 293.  doi: 10.1007/BF01048948.  Google Scholar

[5]

P. Brunovsky and B. Fiedler, Numbers of zeros on invariant manifolds in reaction-diffusion equations,, Nonlinear Analysis, 10 (1986), 179.  doi: 10.1016/0362-546X(86)90045-3.  Google Scholar

[6]

P. Brunovský and B. Fiedler, Connecting orbits in scalar reaction diffusion equations,, Dynamics Reported, 1 (1988), 57.   Google Scholar

[7]

P. Brunovský and B. Fiedler, Connecting orbits in scalar reaction diffusion equations II: The complete solution,, J. Diff. Eqns., 81 (1989), 106.  doi: 10.1016/0022-0396(89)90180-0.  Google Scholar

[8]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, Colloq. AMS, (2002).   Google Scholar

[9]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations,, Wiley, (1994).   Google Scholar

[10]

B. Fiedler (ed.), Handbook of Dynamical Systems, 2,, Elsevier, (2002).   Google Scholar

[11]

B. Fiedler and C. Rocha, Heteroclinic orbits of semilinear parabolic equations,, J. Diff. Eqns., 125 (1996), 239.  doi: 10.1006/jdeq.1996.0031.  Google Scholar

[12]

B. Fiedler and C. Rocha, Realization of meander permutations by boundary value problems,, J. Diff. Eqns., 156 (1999), 282.  doi: 10.1006/jdeq.1998.3532.  Google Scholar

[13]

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

[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.  doi: 10.1016/j.jde.2007.09.015.  Google Scholar

[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.  doi: 10.1515/CRELLE.2009.076.  Google Scholar

[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.  doi: 10.1007/s10884-009-9149-2.  Google Scholar

[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.  Google Scholar

[18]

B. Fiedler and A. Scheel, Spatio-temporal dynamics of reaction-diffusion patterns, In, Trends in Nonlinear Analysis, (2003), 23.   Google Scholar

[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).   Google Scholar

[20]

H. de Fraysseix, P. O. de Mendez and P. Rosenstiehl, Bipolar orientations revisited,, Discr. Appl. Math., 56 (1995), 157.  doi: 10.1016/0166-218X(94)00085-R.  Google Scholar

[21]

R. Fritsch and R. A. Piccinini, Cellular Structures in Topology,, Cambridge University Press, (1990).  doi: 10.1017/CBO9780511983948.  Google Scholar

[22]

G. Fusco and C. Rocha, A permutation related to the dynamics of a scalar parabolic PDE,, J. Diff. Eqns., 91 (1991), 111.  doi: 10.1016/0022-0396(91)90134-U.  Google Scholar

[23]

V. A. Galaktionov, Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications, , Chapman & Hall, (2004).  doi: 10.1201/9780203998069.  Google Scholar

[24]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Math. Surv., 25 (1988).   Google Scholar

[25]

J. K. Hale, L. T. Magalhães and W. M. Oliva, Dynamics in Infinite Dimensions,, Springer-Verlag, (2002).  doi: 10.1007/b100032.  Google Scholar

[26]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lect. Notes Math. 840, 840 (1981).   Google Scholar

[27]

D. Henry, Some infinite dimensional Morse-Smale systems defined by parabolic differential equations,, J. Diff. Eqns., 59 (1985), 165.  doi: 10.1016/0022-0396(85)90153-6.  Google Scholar

[28]

M. S. Jolly, Explicit construction of an inertial manifold for a reaction diffusion equation,, J. Diff. Eqns. , 78 (1989), 220.  doi: 10.1016/0022-0396(89)90064-8.  Google Scholar

[29]

O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Cambridge University Press, (1991).  doi: 10.1017/CBO9780511569418.  Google Scholar

[30]

H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations,, J. Math. Kyoto Univ., 18 (1978), 221.   Google Scholar

[31]

H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations,, Publ. Res. Inst. Math. Sci., 15 (1979), 401.  doi: 10.2977/prims/1195188180.  Google Scholar

[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.   Google Scholar

[33]

H. Matano and K.-I. Nakamura, The global attractor of semilinear parabolic equations on $S^1$,, Discr. Contin. Dyn. Syst., 3 (1997), 1.   Google Scholar

[34]

W. Oliva, Stability of Morse-Smale Maps,, Technical Report, 1 (1983).   Google Scholar

[35]

J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction,, Springer-Verlag, (1982).   Google Scholar

[36]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[37]

G. Raugel, Global attractors in partial differential equations,, Handbook of dynamical systems, 2 (2002), 885.  doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar

[38]

C. Rocha, Properties of the attractor of a scalar parabolic PDE,, J. Dyn. Differ. Equations, 3 (1991), 575.  doi: 10.1007/BF01049100.  Google Scholar

[39]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Springer-Verlag, (2002).  doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[40]

C. Sturm, Sur une classe d'équations à différences partielles,, J. Math. Pure Appl., 1 (1836), 373.   Google Scholar

[41]

H. Tanabe, Equations of Evolution,, Pitman, (1979).   Google Scholar

[42]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[43]

M. Wolfrum, Geometry of heteroclinic cascades in scalar parabolic differential equations,, J. Dyn. Differ. Equations, 14 (2002), 207.  doi: 10.1023/A:1012967428328.  Google Scholar

[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.   Google Scholar

show all references

References:
[1]

S. Angenent, The Morse-Smale property for a semi-linear parabolic equation,, J. Diff. Eqns., 62 (1986), 427.  doi: 10.1016/0022-0396(86)90093-8.  Google Scholar

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North Holland, (1992).   Google Scholar

[3]

A. Banyaga and D. Hurtubise, Lectures on Morse Homology,, Springer-Verlag, (2004).  doi: 10.1007/978-1-4020-2696-6.  Google Scholar

[4]

P. Brunovský, The attractor of the scalar reaction diffusion equation is a smooth graph,, J. Dynamics and Differential Equations, 2 (1990), 293.  doi: 10.1007/BF01048948.  Google Scholar

[5]

P. Brunovsky and B. Fiedler, Numbers of zeros on invariant manifolds in reaction-diffusion equations,, Nonlinear Analysis, 10 (1986), 179.  doi: 10.1016/0362-546X(86)90045-3.  Google Scholar

[6]

P. Brunovský and B. Fiedler, Connecting orbits in scalar reaction diffusion equations,, Dynamics Reported, 1 (1988), 57.   Google Scholar

[7]

P. Brunovský and B. Fiedler, Connecting orbits in scalar reaction diffusion equations II: The complete solution,, J. Diff. Eqns., 81 (1989), 106.  doi: 10.1016/0022-0396(89)90180-0.  Google Scholar

[8]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, Colloq. AMS, (2002).   Google Scholar

[9]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations,, Wiley, (1994).   Google Scholar

[10]

B. Fiedler (ed.), Handbook of Dynamical Systems, 2,, Elsevier, (2002).   Google Scholar

[11]

B. Fiedler and C. Rocha, Heteroclinic orbits of semilinear parabolic equations,, J. Diff. Eqns., 125 (1996), 239.  doi: 10.1006/jdeq.1996.0031.  Google Scholar

[12]

B. Fiedler and C. Rocha, Realization of meander permutations by boundary value problems,, J. Diff. Eqns., 156 (1999), 282.  doi: 10.1006/jdeq.1998.3532.  Google Scholar

[13]

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

[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.  doi: 10.1016/j.jde.2007.09.015.  Google Scholar

[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.  doi: 10.1515/CRELLE.2009.076.  Google Scholar

[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.  doi: 10.1007/s10884-009-9149-2.  Google Scholar

[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.  Google Scholar

[18]

B. Fiedler and A. Scheel, Spatio-temporal dynamics of reaction-diffusion patterns, In, Trends in Nonlinear Analysis, (2003), 23.   Google Scholar

[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).   Google Scholar

[20]

H. de Fraysseix, P. O. de Mendez and P. Rosenstiehl, Bipolar orientations revisited,, Discr. Appl. Math., 56 (1995), 157.  doi: 10.1016/0166-218X(94)00085-R.  Google Scholar

[21]

R. Fritsch and R. A. Piccinini, Cellular Structures in Topology,, Cambridge University Press, (1990).  doi: 10.1017/CBO9780511983948.  Google Scholar

[22]

G. Fusco and C. Rocha, A permutation related to the dynamics of a scalar parabolic PDE,, J. Diff. Eqns., 91 (1991), 111.  doi: 10.1016/0022-0396(91)90134-U.  Google Scholar

[23]

V. A. Galaktionov, Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications, , Chapman & Hall, (2004).  doi: 10.1201/9780203998069.  Google Scholar

[24]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Math. Surv., 25 (1988).   Google Scholar

[25]

J. K. Hale, L. T. Magalhães and W. M. Oliva, Dynamics in Infinite Dimensions,, Springer-Verlag, (2002).  doi: 10.1007/b100032.  Google Scholar

[26]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lect. Notes Math. 840, 840 (1981).   Google Scholar

[27]

D. Henry, Some infinite dimensional Morse-Smale systems defined by parabolic differential equations,, J. Diff. Eqns., 59 (1985), 165.  doi: 10.1016/0022-0396(85)90153-6.  Google Scholar

[28]

M. S. Jolly, Explicit construction of an inertial manifold for a reaction diffusion equation,, J. Diff. Eqns. , 78 (1989), 220.  doi: 10.1016/0022-0396(89)90064-8.  Google Scholar

[29]

O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Cambridge University Press, (1991).  doi: 10.1017/CBO9780511569418.  Google Scholar

[30]

H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations,, J. Math. Kyoto Univ., 18 (1978), 221.   Google Scholar

[31]

H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations,, Publ. Res. Inst. Math. Sci., 15 (1979), 401.  doi: 10.2977/prims/1195188180.  Google Scholar

[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.   Google Scholar

[33]

H. Matano and K.-I. Nakamura, The global attractor of semilinear parabolic equations on $S^1$,, Discr. Contin. Dyn. Syst., 3 (1997), 1.   Google Scholar

[34]

W. Oliva, Stability of Morse-Smale Maps,, Technical Report, 1 (1983).   Google Scholar

[35]

J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction,, Springer-Verlag, (1982).   Google Scholar

[36]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[37]

G. Raugel, Global attractors in partial differential equations,, Handbook of dynamical systems, 2 (2002), 885.  doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar

[38]

C. Rocha, Properties of the attractor of a scalar parabolic PDE,, J. Dyn. Differ. Equations, 3 (1991), 575.  doi: 10.1007/BF01049100.  Google Scholar

[39]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Springer-Verlag, (2002).  doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[40]

C. Sturm, Sur une classe d'équations à différences partielles,, J. Math. Pure Appl., 1 (1836), 373.   Google Scholar

[41]

H. Tanabe, Equations of Evolution,, Pitman, (1979).   Google Scholar

[42]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[43]

M. Wolfrum, Geometry of heteroclinic cascades in scalar parabolic differential equations,, J. Dyn. Differ. Equations, 14 (2002), 207.  doi: 10.1023/A:1012967428328.  Google Scholar

[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.   Google Scholar

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