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Mathematical theory of solids: From quantum mechanics to continuum models
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 |
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.
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] | |
[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] | |
[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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