American Institute of Mathematical Sciences

Advanced
September  2003, 9(5): 1293-1322. doi: 10.3934/dcds.2003.9.1293

Bifurcation from a homoclinic orbit in partial functional differential equations

Shigui Ruan 1, , Junjie Wei 2, and  Jianhong Wu 3,

1. 

Department of Mathematics, The University of Miami, P.O. Box 249085, Coral Gables, Florida 33124, United States
2. 

Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China
3. 

Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada

Received  May 2002 Revised  November 2002 Published  June 2003

We consider a family of partial functional differential equations which has a homoclinic orbit asymptotic to an isolated equilibrium point at a critical value of the parameter. Under some technical assumptions, we show that a unique stable periodic orbit bifurcates from the homoclinic orbit. Our approach follows the ideas of Šil'nikov for ordinary differential equations and of Chow and Deng for semilinear parabolic equations and retarded functional differential equations.
Keywords: stable and unstable manifolds, bifurcation, Homoclinic orbit, variation of constant formula, periodic solutions., partial functional differential equations.
Mathematics Subject Classification: 35R10, 34K15, 58F1.
Citation: Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293
[1]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167
[2]

Vera Ignatenko. Homoclinic and stable periodic solutions for differential delay equations from physiology. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3637-3661. doi: 10.3934/dcds.2018157
[3]

Nguyen Thieu Huy, Pham Van Bang. Invariant stable manifolds for partial neutral functional differential equations in admissible spaces on a half-line. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2993-3011. doi: 10.3934/dcdsb.2015.20.2993
[4]

E. N. Dancer, Norimichi Hirano. Existence of stable and unstable periodic solutions for semilinear parabolic problems. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 207-216. doi: 10.3934/dcds.1997.3.207
[5]

Michihiro Hirayama, Naoya Sumi. Hyperbolic measures with transverse intersections of stable and unstable manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1451-1476. doi: 10.3934/dcds.2013.33.1451
[6]

Eric Benoît. Bifurcation delay - the case of the sequence: Stable focus - unstable focus - unstable node. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 911-929. doi: 10.3934/dcdss.2009.2.911
[7]

Pablo Aguirre, Bernd Krauskopf, Hinke M. Osinga. Global invariant manifolds near a Shilnikov homoclinic bifurcation. Journal of Computational Dynamics, 2014, 1 (1) : 1-38. doi: 10.3934/jcd.2014.1.1
[8]

C. M. Groothedde, J. D. Mireles James. Parameterization method for unstable manifolds of delay differential equations. Journal of Computational Dynamics, 2017, 4 (1&2) : 21-70. doi: 10.3934/jcd.2017002
[9]

Raoul-Martin Memmesheimer, Marc Timme. Stable and unstable periodic orbits in complex networks of spiking neurons with delays. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1555-1588. doi: 10.3934/dcds.2010.28.1555
[10]

Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure & Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291
[11]

Changrong Zhu, Bin Long. The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3793-3808. doi: 10.3934/dcdsb.2016121
[12]

Yong Li, Zhenxin Liu, Wenhe Wang. Almost periodic solutions and stable solutions for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5927-5944. doi: 10.3934/dcdsb.2019113
[13]

David L. Finn. Noncompact manifolds with constant negative scalar curvature and singular solutions to semihnear elliptic equations. Conference Publications, 1998, 1998 (Special) : 262-275. doi: 10.3934/proc.1998.1998.262
[14]

S. Secchi, C. A. Stuart. Global bifurcation of homoclinic solutions of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1493-1518. doi: 10.3934/dcds.2003.9.1493
[15]

Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031
[16]

Luis Barreira, Claudia Valls. Stable manifolds with optimal regularity for difference equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1537-1555. doi: 10.3934/dcds.2012.32.1537
[17]

Pierluigi Benevieri, Alessandro Calamai, Massimo Furi, Maria Patrizia Pera. On general properties of retarded functional differential equations on manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 27-46. doi: 10.3934/dcds.2013.33.27
[18]

Marat Akhmet. Quasilinear retarded differential equations with functional dependence on piecewise constant argument. Communications on Pure & Applied Analysis, 2014, 13 (2) : 929-947. doi: 10.3934/cpaa.2014.13.929
[19]

Samir Adly, Daniel Goeleven, Dumitru Motreanu. Periodic and homoclinic solutions for a class of unilateral problems. Discrete & Continuous Dynamical Systems - A, 1997, 3 (4) : 579-590. doi: 10.3934/dcds.1997.3.579
[20]

Xiao Wen. Structurally stable homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1693-1707. doi: 10.3934/dcds.2016.36.1693

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (28)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences