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 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.

[1]

Brendan Weickert. Infinite-dimensional complex dynamics: A quantum random walk. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 517-524. doi: 10.3934/dcds.2001.7.517

[2]

Björn Augner, Birgit Jacob. Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. Evolution Equations and Control Theory, 2014, 3 (2) : 207-229. doi: 10.3934/eect.2014.3.207

[3]

Qing Xu. Backward stochastic Schrödinger and infinite-dimensional Hamiltonian equations. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5379-5412. doi: 10.3934/dcds.2015.35.5379

[4]

Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069

[5]

Yangyou Pan, Yuzhen Bai, Xiang Zhang. Dynamics of locally linearizable complex two dimensional cubic Hamiltonian systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (6) : 1761-1774. doi: 10.3934/dcdss.2019116

[6]

John Erik Fornæss. Infinite dimensional complex dynamics: Quasiconjugacies, localization and quantum chaos. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 51-60. doi: 10.3934/dcds.2000.6.51

[7]

Vincent Renault, Michèle Thieullen, Emmanuel Trélat. Optimal control of infinite-dimensional piecewise deterministic Markov processes and application to the control of neuronal dynamics via Optogenetics. Networks and Heterogeneous Media, 2017, 12 (3) : 417-459. doi: 10.3934/nhm.2017019

[8]

Renhai Wang, Bixiang Wang. Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2461-2493. doi: 10.3934/dcdsb.2020019

[9]

Gideon Simpson, Michael I. Weinstein, Philip Rosenau. On a Hamiltonian PDE arising in magma dynamics. Discrete and Continuous Dynamical Systems - B, 2008, 10 (4) : 903-924. doi: 10.3934/dcdsb.2008.10.903

[10]

Eleonora Bardelli, Andrea Carlo Giuseppe Mennucci. Probability measures on infinite-dimensional Stiefel manifolds. Journal of Geometric Mechanics, 2017, 9 (3) : 291-316. doi: 10.3934/jgm.2017012

[11]

Sergey V Lototsky, Henry Schellhorn, Ran Zhao. An infinite-dimensional model of liquidity in financial markets. Probability, Uncertainty and Quantitative Risk, 2021, 6 (2) : 117-138. doi: 10.3934/puqr.2021006

[12]

M. D. König, Stefano Battiston, M. Napoletano, F. Schweitzer. On algebraic graph theory and the dynamics of innovation networks. Networks and Heterogeneous Media, 2008, 3 (2) : 201-219. doi: 10.3934/nhm.2008.3.201

[13]

Tapio Helin. On infinite-dimensional hierarchical probability models in statistical inverse problems. Inverse Problems and Imaging, 2009, 3 (4) : 567-597. doi: 10.3934/ipi.2009.3.567

[14]

Radu Ioan Boţ, Sorin-Mihai Grad. On linear vector optimization duality in infinite-dimensional spaces. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 407-415. doi: 10.3934/naco.2011.1.407

[15]

Kening Lu, Alexandra Neamţu, Björn Schmalfuss. On the Oseledets-splitting for infinite-dimensional random dynamical systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1219-1242. doi: 10.3934/dcdsb.2018149

[16]

Chris Guiver, Mark R. Opmeer. Bounded real and positive real balanced truncation for infinite-dimensional systems. Mathematical Control and Related Fields, 2013, 3 (1) : 83-119. doi: 10.3934/mcrf.2013.3.83

[17]

Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 821-836. doi: 10.3934/dcdsb.2021066

[18]

Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3295-3317. doi: 10.3934/dcds.2020406

[19]

Antonio Ambrosetti, Massimiliano Berti. Applications of critical point theory to homoclinics and complex dynamics. Conference Publications, 1998, 1998 (Special) : 72-78. doi: 10.3934/proc.1998.1998.72

[20]

Philip Schrader. Morse theory for elastica. Journal of Geometric Mechanics, 2016, 8 (2) : 235-256. doi: 10.3934/jgm.2016006

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (78)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]