# American Institute of Mathematical Sciences

December  2012, 7(4): 617-659. doi: 10.3934/nhm.2012.7.617

## Sturm global attractors for $S^1$-equivariant parabolic equations

 1 Freie Universität Berlin, Institut für Mathematik I, Arnimallee 2-6, D-14195 Berlin, Germany 2 Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Instituto Superior Técnico, Departamento de Matemática, Universidade Técnica de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal, Portugal

Received  January 2012 Published  December 2012

We consider a semilinear parabolic equation of the form $u_t = u_{xx} + f(u,u_x)$ defined on the circle $x ∈ S^1=\mathbb{R}/2\pi\mathbb{Z}$. For a dissipative nonlinearity $f$ this equation generates a dissipative semiflow in the appropriate function space, and the corresponding global attractor $A_f$ is called a Sturm attractor. If $f=f(u,p)$ is even in $p$, then the semiflow possesses an embedded flow satisfying Neumann boundary conditions on the half-interval $(0,\pi)$. This is due to $O(2)$ equivariance of the semiflow and, more specifically, due to reflection at the axis through $x=0,\pi\in S^1$. For general $f=f(u,p)$, where only $SO(2)$ equivariance prevails, we will nevertheless use the Sturm permutation $\sigma$ introduced for the characterization of Neumann flows to obtain a purely combinatorial characterization of the Sturm attractors $A_f$ on the circle. With this Sturm permutation $\sigma$ we then enumerate and describe the heteroclinic connections of all Morse-Smale attractors $A_f$ with $m$ stationary solutions and $q$ periodic orbits, up to $n:=m+2q \le 9$.
Citation: Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617
##### References:
 [1] R. Abraham and J. Robbin, "Transversal Mappings and Flows," Benjamin, New York, 1967.  Google Scholar [2] S. Angenent, The Morse-Smale property for a semi-linear parabolic equation, J. Differential Equations, 62 (1986), 427-442. doi: 10.1016/0022-0396(86)90093-8.  Google Scholar [3] S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96. doi: 10.1515/crll.1988.390.79.  Google Scholar [4] S. Angenent and B. Fiedler, The dynamics of rotating waves in scalar reaction diffusion equations, Trans. Amer. Math. Soc., 307 (1988), 545-568. doi: 10.2307/2001188.  Google Scholar [5] V. I. Arnold, A branched covering $CP^2 \rightarrow S^4$, hyperbolicity and projective topology, Siberian Math. J., 29 (1988)(1989), 717-726. doi: 10.1007/BF00970265.  Google Scholar [6] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North Holland, Amsterdam, 1992.  Google Scholar [7] P. Brunovský and B. Fiedler, Connecting orbits in scalar reaction diffusion equations, Dynamics Reported, 1 (1988), 57-89.  Google Scholar [8] P. Brunovský and B. Fiedler, Connecting orbits in scalar reaction diffusion equations II: The complete solution, J. Differential Equations, 81 (1989), 106-135. doi: 10.1016/0022-0396(89)90180-0.  Google Scholar [9] E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill, New York, 1955.  Google Scholar [10] R. Czaja and C. Rocha, Transversality in scalar reaction-diffusion equations on a circle, J. Differential Equations, 245 (2008), 692-721. doi: 10.1016/j.jde.2008.01.018.  Google Scholar [11] M. P. do Carmo, "Differential Geometry of Curves and Surfaces," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976.  Google Scholar [12] B. Fiedler and J. Mallet-Paret, The Poincaré-Bendixson theorem for scalar reaction diffusion equations, Arch. Rational Mech. Anal., 107 (1989), 325-345. doi: 10.1007/BF00251553.  Google Scholar [13] B. Fiedler, Global attractors of one-dimensional parabolic equations: Sixteen examples, Tatra Mt. Math. Publ., 4 (1994), 67-92.  Google Scholar [14] B. Fiedler, Do global attractors depend on boundary conditions?, Doc. Math. J. DMV, 1 (1996), 215-228.  Google Scholar [15] B. Fiedler and C. Rocha, Heteroclinic orbits of semilinear parabolic equations, J. Differential Equations, 125 (1996), 239-281. doi: 10.1006/jdeq.1996.0031.  Google Scholar [16] B. Fiedler and C. Rocha, Realization of meander permutations by boundary value problems, J. Differential Equations, 156 (1999), 282-308. doi: 10.1006/jdeq.1998.3532.  Google Scholar [17] 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.  Google Scholar [18] B. Fiedler and C. Rocha, Connectivity and design of planar global attractors of Sturm type. II: Connection graphs, J. Differential Equations, 245 (2008), 692-721. doi: 10.1016/j.jde.2007.09.015.  Google Scholar [19] 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.  Google Scholar [20] B. Fiedler and C. Rocha, Connectivity and design of planar global attractors of Sturm type. III: Small and platonic examples, J. Dynam. Differential Equations, 22 (2010), 509-532. doi: 10.1007/s10884-009-9149-2.  Google Scholar [21] B. Fiedler, C. Rocha, D. Salazar and J. Solà-Morales, Dynamics of peacewise-autonomous bistable parabolic equations, in "Differential Equations and Dynamical Systems" (Lisbon, 2000), 151-163, (eds. A. Galves, J. K. Hale, C. Rocha), Fields Inst. Commun., 31, Amer. Math. Soc., Providence, RI, (2002).  Google Scholar [22] B. Fiedler and A. Scheel, Dynamics of reaction-diffusion patterns, in "Trends in Nonlinear Analysis, Festschrift Dedicated to Willi Jäger for His 60th Birthday, 23-152" (eds. M. Kirkilionis, R. Rannacher and F. Tomi), Springer-Verlag, Heidelberg, (2002).  Google Scholar [23] B. Fiedler, C. Rocha and M. Wolfrum, Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle, J. Differential Equations, 201 (2004), 99-138. doi: 10.1016/j.jde.2003.10.027.  Google Scholar [24] B. Fiedler, C. Rocha and M. Wolfrum, A permutation characterization of Sturm global attractors of Hamiltonian type, J. Differential Equations, 252 (2012), 588-623. doi: 10.1016/j.jde.2011.08.013.  Google Scholar [25] G. Fusco and C. Rocha, A permutation related to the dynamics of a scalar parabolic PDE, J. Differential Equations, 91 (1991), 75-94. doi: 10.1016/0022-0396(91)90134-U.  Google Scholar [26] J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Math. Surv., 25. AMS Publications, Providence, 1988.  Google Scholar [27] J. K. Hale, L. T. Magalhães and W. M. Oliva, "Dynamics in Infinite Dimensions," Second edition, Applied Mathematical Sciences, 47. Springer-Verlag, New York, 2002.  Google Scholar [28] J. K. Hale and G. Raugel, Convergence in gradient-like systems with applications to PDE, Z. Angew. Math. Phys., 43 (1992), 63-124. doi: 10.1007/BF00944741.  Google Scholar [29] J. Härterich and M. Wolfrum, Convergence in gradient-like systems with applications to PDE, Discrete and Contin. Dyn. Syst., 12 (2005), 531-554.  Google Scholar [30] P. Hartman, "Ordinary Differential Equations," Birkhäuser, Boston, 1982. (first edition Wiley, New York, 1964)  Google Scholar [31] D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lect. Notes in Math, 840. Springer-Verlag, New York, 1981.  Google Scholar [32] D. Henry, Some infinite dimensional Morse-Smale systems defined by parabolic differential equations, J. Differential Equations, 59 (1985), 165-205. doi: 10.1016/0022-0396(85)90153-6.  Google Scholar [33] R. Joly and G. Raugel, Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle, Trans. Amer. Math. Soc., 362 (2010), 5189-5211. doi: 10.1090/S0002-9947-2010-04890-1.  Google Scholar [34] R. Joly and G. Raugel, Generic Morse-Smale property for the parabolic equation on the circle, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1397-1440. doi: 10.1016/j.anihpc.2010.09.001.  Google Scholar [35] T. Krisztin and H.-O. Walther, Unique periodic orbits for delay positive feedback and the global attractor, J. Dynam. Differential Equations, 13 (2001), 1-57. doi: 10.1023/A:1009091930589.  Google Scholar [36] S. K. Lando, "Lectures on Generating Functions," Stud. Math. Lib., 23, American Mathematical Society, 2003.  Google Scholar [37] S. K. Lando and A. K. Zvonkin, Meanders, Selecta Math. Soviet., 11 (1992), 117-144.  Google Scholar [38] H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ., 18 (1878), 221-227.  Google Scholar [39] H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semi-linear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math, 29 (1982), 401-441.  Google Scholar [40] H. Matano, Asymptotic behavior of solutions of semilinear heat equations on $S^1$, in "Nonlinear Diffusion Equations and their Equilibrium States II, 139-162" (eds. W.-M. Ni, L. A. Peletier, J. Serrin). Springer-Verlag, New York, (1988). doi: 10.1007/978-1-4613-9608-6_8.  Google Scholar [41] H. Matano and K.-I. Nakamura, The global attractor of semilinear parabolic equations on $S^1$, Discrete Contin. Dyn. Syst., 3 (1997), 1-24.  Google Scholar [42] K. Mischaikow, Conley index theory, in "Dynamical Systems (Montecatini Terme, 1994), 119-207" (eds. L. Arnold, K. Mischaikow and G. Raugel), Lecture Notes in Math., 1609, Springer, Berlin, (1995). doi: 10.1007/BFb0095240.  Google Scholar [43] Y. Miyamoto, On connecting orbits of semilinear parabolic equations on $S^1$, Documenta Math., 9 (2004), 435-469.  Google Scholar [44] N. Nadirashvili, Connecting orbits for nonlinear parabolic equations, Asian J. Math., 2 (1998), 135-140.  Google Scholar [45] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [46] C. Ragazzo, Scalar autonomous second order ordinary differential equations, Preprint, (2010). doi: 10.1007/s12346-011-0063-8.  Google Scholar [47] G. Raugel, Global attractors in partial differential equations, in "Handbook of Dynamical Systems 2, 885-982" (ed. B. Fiedler), North-Holland, Amsterdam, (2002). doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar [48] C. Rocha, Properties of the attractor of a scalar parabolic PDE, J. Dynam. Differential Equations, 3 (1991), 575-591. doi: 10.1007/BF01049100.  Google Scholar [49] C. Rocha, Bifurcations in discretized reaction-diffusion equations, Resenhas IME-USP, 1 (1994), 403-419.  Google Scholar [50] C. Rocha, Realization of period maps of planar Hamiltonian systems, J. Dynam. Differential Equations, 19 (2007), 571-591. doi: 10.1007/s10884-007-9081-2.  Google Scholar [51] B. Sandstede and B. Fiedler, Dynamics of periodically forced parabolic equations on the circle, Ergodic Theory Dynam. Systems, 12 (1992), 559-571. doi: 10.1017/S0143385700006933.  Google Scholar [52] R. Schaaf, "Global Solution Branches of Two Point Boundary Value Problems," Lect. Notes in Math, 1458, Springer-Verlag, New York, 1990.  Google Scholar [53] J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Springer-Verlag, New York, 1983.  Google Scholar [54] C. Sturm, Sur une classe d'équations à différences partielles, J. Math. Pure Appl., 1 (1836), 373-444. Google Scholar [55] M. Urabe, Relations between periods and amplitudes of periodic solutions of $\ddot x + g(x) = 0$, Funkcial. Ekvac., 6 (1964), 63-88.  Google Scholar [56] M. Wolfrum, Geometry of heteroclinic cascades in scalar parabolic differential equations, J. Dynam. Differential Equations, 14 (2002), 207-241. doi: 10.1023/A:1012967428328.  Google Scholar [57] M. Wolfrum, A sequence of order relations, encoding heteroclinic connections in scalar parabolic PDE, J. Differential Equations, 183 (2002), 56-78. doi: 10.1006/jdeq.2001.4114.  Google Scholar [58] J. A. Yorke, Periods of periodic solutions and the Lipschitz constant, Proc. Amer. Math. Soc., 22 (1969), 509-512.  Google Scholar [59] T. I. Zelenyak, Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable, Differential Equations, 4 (1968), 34-45.  Google Scholar

show all references

##### References:
 [1] R. Abraham and J. Robbin, "Transversal Mappings and Flows," Benjamin, New York, 1967.  Google Scholar [2] S. Angenent, The Morse-Smale property for a semi-linear parabolic equation, J. Differential Equations, 62 (1986), 427-442. doi: 10.1016/0022-0396(86)90093-8.  Google Scholar [3] S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96. doi: 10.1515/crll.1988.390.79.  Google Scholar [4] S. Angenent and B. Fiedler, The dynamics of rotating waves in scalar reaction diffusion equations, Trans. Amer. Math. Soc., 307 (1988), 545-568. doi: 10.2307/2001188.  Google Scholar [5] V. I. Arnold, A branched covering $CP^2 \rightarrow S^4$, hyperbolicity and projective topology, Siberian Math. J., 29 (1988)(1989), 717-726. doi: 10.1007/BF00970265.  Google Scholar [6] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North Holland, Amsterdam, 1992.  Google Scholar [7] P. Brunovský and B. Fiedler, Connecting orbits in scalar reaction diffusion equations, Dynamics Reported, 1 (1988), 57-89.  Google Scholar [8] P. Brunovský and B. Fiedler, Connecting orbits in scalar reaction diffusion equations II: The complete solution, J. Differential Equations, 81 (1989), 106-135. doi: 10.1016/0022-0396(89)90180-0.  Google Scholar [9] E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill, New York, 1955.  Google Scholar [10] R. Czaja and C. Rocha, Transversality in scalar reaction-diffusion equations on a circle, J. Differential Equations, 245 (2008), 692-721. doi: 10.1016/j.jde.2008.01.018.  Google Scholar [11] M. P. do Carmo, "Differential Geometry of Curves and Surfaces," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976.  Google Scholar [12] B. Fiedler and J. Mallet-Paret, The Poincaré-Bendixson theorem for scalar reaction diffusion equations, Arch. Rational Mech. Anal., 107 (1989), 325-345. doi: 10.1007/BF00251553.  Google Scholar [13] B. Fiedler, Global attractors of one-dimensional parabolic equations: Sixteen examples, Tatra Mt. Math. Publ., 4 (1994), 67-92.  Google Scholar [14] B. Fiedler, Do global attractors depend on boundary conditions?, Doc. Math. J. DMV, 1 (1996), 215-228.  Google Scholar [15] B. Fiedler and C. Rocha, Heteroclinic orbits of semilinear parabolic equations, J. Differential Equations, 125 (1996), 239-281. doi: 10.1006/jdeq.1996.0031.  Google Scholar [16] B. Fiedler and C. Rocha, Realization of meander permutations by boundary value problems, J. Differential Equations, 156 (1999), 282-308. doi: 10.1006/jdeq.1998.3532.  Google Scholar [17] 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.  Google Scholar [18] B. Fiedler and C. Rocha, Connectivity and design of planar global attractors of Sturm type. II: Connection graphs, J. Differential Equations, 245 (2008), 692-721. doi: 10.1016/j.jde.2007.09.015.  Google Scholar [19] 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.  Google Scholar [20] B. Fiedler and C. Rocha, Connectivity and design of planar global attractors of Sturm type. III: Small and platonic examples, J. Dynam. Differential Equations, 22 (2010), 509-532. doi: 10.1007/s10884-009-9149-2.  Google Scholar [21] B. Fiedler, C. Rocha, D. Salazar and J. Solà-Morales, Dynamics of peacewise-autonomous bistable parabolic equations, in "Differential Equations and Dynamical Systems" (Lisbon, 2000), 151-163, (eds. A. Galves, J. K. Hale, C. Rocha), Fields Inst. Commun., 31, Amer. Math. Soc., Providence, RI, (2002).  Google Scholar [22] B. Fiedler and A. Scheel, Dynamics of reaction-diffusion patterns, in "Trends in Nonlinear Analysis, Festschrift Dedicated to Willi Jäger for His 60th Birthday, 23-152" (eds. M. Kirkilionis, R. Rannacher and F. Tomi), Springer-Verlag, Heidelberg, (2002).  Google Scholar [23] B. Fiedler, C. Rocha and M. Wolfrum, Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle, J. Differential Equations, 201 (2004), 99-138. doi: 10.1016/j.jde.2003.10.027.  Google Scholar [24] B. Fiedler, C. Rocha and M. Wolfrum, A permutation characterization of Sturm global attractors of Hamiltonian type, J. Differential Equations, 252 (2012), 588-623. doi: 10.1016/j.jde.2011.08.013.  Google Scholar [25] G. Fusco and C. Rocha, A permutation related to the dynamics of a scalar parabolic PDE, J. Differential Equations, 91 (1991), 75-94. doi: 10.1016/0022-0396(91)90134-U.  Google Scholar [26] J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Math. Surv., 25. AMS Publications, Providence, 1988.  Google Scholar [27] J. K. Hale, L. T. Magalhães and W. M. Oliva, "Dynamics in Infinite Dimensions," Second edition, Applied Mathematical Sciences, 47. Springer-Verlag, New York, 2002.  Google Scholar [28] J. K. Hale and G. Raugel, Convergence in gradient-like systems with applications to PDE, Z. Angew. Math. Phys., 43 (1992), 63-124. doi: 10.1007/BF00944741.  Google Scholar [29] J. Härterich and M. Wolfrum, Convergence in gradient-like systems with applications to PDE, Discrete and Contin. Dyn. Syst., 12 (2005), 531-554.  Google Scholar [30] P. Hartman, "Ordinary Differential Equations," Birkhäuser, Boston, 1982. (first edition Wiley, New York, 1964)  Google Scholar [31] D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lect. Notes in Math, 840. Springer-Verlag, New York, 1981.  Google Scholar [32] D. Henry, Some infinite dimensional Morse-Smale systems defined by parabolic differential equations, J. Differential Equations, 59 (1985), 165-205. doi: 10.1016/0022-0396(85)90153-6.  Google Scholar [33] R. Joly and G. Raugel, Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle, Trans. Amer. Math. Soc., 362 (2010), 5189-5211. doi: 10.1090/S0002-9947-2010-04890-1.  Google Scholar [34] R. Joly and G. Raugel, Generic Morse-Smale property for the parabolic equation on the circle, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1397-1440. doi: 10.1016/j.anihpc.2010.09.001.  Google Scholar [35] T. Krisztin and H.-O. Walther, Unique periodic orbits for delay positive feedback and the global attractor, J. Dynam. Differential Equations, 13 (2001), 1-57. doi: 10.1023/A:1009091930589.  Google Scholar [36] S. K. Lando, "Lectures on Generating Functions," Stud. Math. Lib., 23, American Mathematical Society, 2003.  Google Scholar [37] S. K. Lando and A. K. Zvonkin, Meanders, Selecta Math. Soviet., 11 (1992), 117-144.  Google Scholar [38] H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ., 18 (1878), 221-227.  Google Scholar [39] H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semi-linear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math, 29 (1982), 401-441.  Google Scholar [40] H. Matano, Asymptotic behavior of solutions of semilinear heat equations on $S^1$, in "Nonlinear Diffusion Equations and their Equilibrium States II, 139-162" (eds. W.-M. Ni, L. A. Peletier, J. Serrin). Springer-Verlag, New York, (1988). doi: 10.1007/978-1-4613-9608-6_8.  Google Scholar [41] H. Matano and K.-I. Nakamura, The global attractor of semilinear parabolic equations on $S^1$, Discrete Contin. Dyn. Syst., 3 (1997), 1-24.  Google Scholar [42] K. Mischaikow, Conley index theory, in "Dynamical Systems (Montecatini Terme, 1994), 119-207" (eds. L. Arnold, K. Mischaikow and G. Raugel), Lecture Notes in Math., 1609, Springer, Berlin, (1995). doi: 10.1007/BFb0095240.  Google Scholar [43] Y. Miyamoto, On connecting orbits of semilinear parabolic equations on $S^1$, Documenta Math., 9 (2004), 435-469.  Google Scholar [44] N. Nadirashvili, Connecting orbits for nonlinear parabolic equations, Asian J. Math., 2 (1998), 135-140.  Google Scholar [45] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [46] C. Ragazzo, Scalar autonomous second order ordinary differential equations, Preprint, (2010). doi: 10.1007/s12346-011-0063-8.  Google Scholar [47] G. Raugel, Global attractors in partial differential equations, in "Handbook of Dynamical Systems 2, 885-982" (ed. B. Fiedler), North-Holland, Amsterdam, (2002). doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar [48] C. Rocha, Properties of the attractor of a scalar parabolic PDE, J. Dynam. Differential Equations, 3 (1991), 575-591. doi: 10.1007/BF01049100.  Google Scholar [49] C. Rocha, Bifurcations in discretized reaction-diffusion equations, Resenhas IME-USP, 1 (1994), 403-419.  Google Scholar [50] C. Rocha, Realization of period maps of planar Hamiltonian systems, J. Dynam. Differential Equations, 19 (2007), 571-591. doi: 10.1007/s10884-007-9081-2.  Google Scholar [51] B. Sandstede and B. Fiedler, Dynamics of periodically forced parabolic equations on the circle, Ergodic Theory Dynam. Systems, 12 (1992), 559-571. doi: 10.1017/S0143385700006933.  Google Scholar [52] R. Schaaf, "Global Solution Branches of Two Point Boundary Value Problems," Lect. Notes in Math, 1458, Springer-Verlag, New York, 1990.  Google Scholar [53] J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Springer-Verlag, New York, 1983.  Google Scholar [54] C. Sturm, Sur une classe d'équations à différences partielles, J. Math. Pure Appl., 1 (1836), 373-444. Google Scholar [55] M. Urabe, Relations between periods and amplitudes of periodic solutions of $\ddot x + g(x) = 0$, Funkcial. Ekvac., 6 (1964), 63-88.  Google Scholar [56] M. Wolfrum, Geometry of heteroclinic cascades in scalar parabolic differential equations, J. Dynam. Differential Equations, 14 (2002), 207-241. doi: 10.1023/A:1012967428328.  Google Scholar [57] M. Wolfrum, A sequence of order relations, encoding heteroclinic connections in scalar parabolic PDE, J. Differential Equations, 183 (2002), 56-78. doi: 10.1006/jdeq.2001.4114.  Google Scholar [58] J. A. Yorke, Periods of periodic solutions and the Lipschitz constant, Proc. Amer. Math. Soc., 22 (1969), 509-512.  Google Scholar [59] T. I. Zelenyak, Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable, Differential Equations, 4 (1968), 34-45.  Google Scholar
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