American Institute of Mathematical Sciences

Advanced
May  2003, 9(3): 585-602. doi: 10.3934/dcds.2003.9.585

Exponentially small splitting of homoclinic orbits of parabolic differential equations under periodic forcing

Karsten Matthies 1,

1. 

Freie Universität Berlin, Institut für Mathematik I, Arnimallee 2-6, 14195 Berlin, Germany

Received  November 2001 Revised  December 2002 Published  February 2003

Homoclinic orbits of semilinear parabolic partial differential equations can split under time-periodic forcing as for ordinary differential equations. The stable and unstable manifold may intersect transverse at persisting homoclinic points. The size of the splitting is estimated to be exponentially small of order exp$(-c/\epsilon)$ in the period $\epsilon$ of the forcing with $\epsilon \rightarrow 0$.
Keywords: exponential smallness., semilinear parabolic equations, splitting, Homoclinic orbit.
Mathematics Subject Classification: 37L99, 37C29, 37G2.
Citation: Karsten Matthies. Exponentially small splitting of homoclinic orbits of parabolic differential equations under periodic forcing. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 585-602. doi: 10.3934/dcds.2003.9.585
[1]

Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293
[2]

Zhengce Zhang, Yanyan Li. Gradient blowup solutions of a semilinear parabolic equation with exponential source. Communications on Pure and Applied Analysis, 2013, 12 (1) : 269-280. doi: 10.3934/cpaa.2013.12.269
[3]

Sérgio S. Rodrigues. Semiglobal exponential stabilization of nonautonomous semilinear parabolic-like systems. Evolution Equations and Control Theory, 2020, 9 (3) : 635-672. doi: 10.3934/eect.2020027
[4]

Zhengce Zhang, Yan Li. Global existence and gradient blowup of solutions for a semilinear parabolic equation with exponential source. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 3019-3029. doi: 10.3934/dcdsb.2014.19.3019
[5]

Flank D. M. Bezerra, Jacson Simsen, Mariza Stefanello Simsen. Convergence of quasilinear parabolic equations to semilinear equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3823-3834. doi: 10.3934/dcdsb.2020258
[6]

Changrong Zhu, Bin Long. The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3793-3808. doi: 10.3934/dcdsb.2016121
[7]

Jian-Wen Sun, Seonghak Kim. Exponential decay for quasilinear parabolic equations in any dimension. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021280
[8]

A. V. Rezounenko. Inertial manifolds with delay for retarded semilinear parabolic equations. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 829-840. doi: 10.3934/dcds.2000.6.829
[9]

Vladimir V. Chepyzhov, Anna Kostianko, Sergey Zelik. Inertial manifolds for the hyperbolic relaxation of semilinear parabolic equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1115-1142. doi: 10.3934/dcdsb.2019009
[10]

Norimichi Hirano, Wen Se Kim. Multiplicity and stability result for semilinear parabolic equations. Discrete and Continuous Dynamical Systems, 1996, 2 (2) : 271-280. doi: 10.3934/dcds.1996.2.271
[11]

Jong-Shenq Guo, Satoshi Sasayama, Chi-Jen Wang. Blowup rate estimate for a system of semilinear parabolic equations. Communications on Pure and Applied Analysis, 2009, 8 (2) : 711-718. doi: 10.3934/cpaa.2009.8.711
[12]

Hiroshi Matano, Ken-Ichi Nakamura. The global attractor of semilinear parabolic equations on $S^1$. Discrete and Continuous Dynamical Systems, 1997, 3 (1) : 1-24. doi: 10.3934/dcds.1997.3.1
[13]

Júlia Matos. Unfocused blow up solutions of semilinear parabolic equations. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 905-928. doi: 10.3934/dcds.1999.5.905
[14]

Luisa Moschini, Guillermo Reyes, Alberto Tesei. Nonuniqueness of solutions to semilinear parabolic equations with singular coefficients. Communications on Pure and Applied Analysis, 2006, 5 (1) : 155-179. doi: 10.3934/cpaa.2006.5.155
[15]

Francis Ribaud. Semilinear parabolic equations with distributions as initial data. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 305-316. doi: 10.3934/dcds.1997.3.305
[16]

Soohyun Bae, Yūki Naito. Separation structure of radial solutions for semilinear elliptic equations with exponential nonlinearity. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4537-4554. doi: 10.3934/dcds.2018198
[17]

A. Rodríguez-Bernal. Perturbation of the exponential type of linear nonautonomous parabolic equations and applications to nonlinear equations. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 1003-1032. doi: 10.3934/dcds.2009.25.1003
[18]

Oksana Koltsova, Lev Lerman. Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 883-913. doi: 10.3934/dcds.2009.25.883
[19]

Benoît Grébert, Tiphaine Jézéquel, Laurent Thomann. Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3485-3510. doi: 10.3934/dcds.2014.34.3485
[20]

W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy