American Institute of Mathematical Sciences

Advanced
May  2008, 9(3&4, May): 431-461. doi: 10.3934/dcdsb.2008.9.431

Heteroclinic connections in singularly perturbed systems

Flaviano Battelli 1, and  Ken Palmer 2,

1. 

Dipartimento di Scienze Matematiche, Facoltà di Ingegneria, Università, Via Brecce Bianche, 1, 60100 Ancona, Italy
2. 

Department of Mathematics, National Taiwan University, Taipei 106, Taiwan

Received  October 2006 Revised  July 2007 Published  February 2008

We consider a singularly perturbed system with two normally hyperbolic centre manifolds. We derive one bifurcation function, the zeros of which correspond to heteroclinic connections near such a connection for the unperturbed system, and a second bifurcation function the zeros of which correspond to the vectors in the intersection of the tangent spaces to the centre-unstable and centre-stable manifolds along the heteroclinic connections.
Keywords: Singular perturbation, Melnikov function., homoclinic bifurcation, invariant manifolds.
Mathematics Subject Classification: Primary: 34E15, 34C23; Secondary: 37G2.
Citation: Flaviano Battelli, Ken Palmer. Heteroclinic connections in singularly perturbed systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 431-461. doi: 10.3934/dcdsb.2008.9.431
[1]

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
[2]

Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part II. Networks & Heterogeneous Media, 2015, 10 (4) : 897-948. doi: 10.3934/nhm.2015.10.897
[3]

Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part I. Networks & Heterogeneous Media, 2013, 8 (4) : 1009-1034. doi: 10.3934/nhm.2013.8.1009
[4]

Flaviano Battelli. Saddle-node bifurcation of homoclinic orbits in singular systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 203-218. doi: 10.3934/dcds.2001.7.203
[5]

Yulin Zhao, Siming Zhu. Higher order Melnikov function for a quartic hamiltonian with cuspidal loop. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 995-1018. doi: 10.3934/dcds.2002.8.995
[6]

Rovella Alvaro, Vilamajó Francesc, Romero Neptalí. Invariant manifolds for delay endomorphisms. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 35-50. doi: 10.3934/dcds.2001.7.35
[7]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805
[8]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233
[9]

Leonardo Mora. Homoclinic bifurcations, fat attractors and invariant curves. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1133-1148. doi: 10.3934/dcds.2003.9.1133
[10]

Rui Dilão, András Volford. Excitability in a model with a saddle-node homoclinic bifurcation. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 419-434. doi: 10.3934/dcdsb.2004.4.419
[11]

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
[12]

Ilona Gucwa, Peter Szmolyan. Geometric singular perturbation analysis of an autocatalator model. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 783-806. doi: 10.3934/dcdss.2009.2.783
[13]

S. Bautista, C. Morales, M. J. Pacifico. On the intersection of homoclinic classes on singular-hyperbolic sets. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 761-775. doi: 10.3934/dcds.2007.19.761
[14]

José F. Alves, Davide Azevedo. Statistical properties of diffeomorphisms with weak invariant manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 1-41. doi: 10.3934/dcds.2016.36.1
[15]

Henk Broer, Aaron Hagen, Gert Vegter. Numerical approximation of normally hyperbolic invariant manifolds. Conference Publications, 2003, 2003 (Special) : 133-140. doi: 10.3934/proc.2003.2003.133
[16]

George Osipenko. Indestructibility of invariant locally non-unique manifolds. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 203-219. doi: 10.3934/dcds.1996.2.203
[17]

Christopher K. R. T. Jones, Siu-Kei Tin. Generalized exchange lemmas and orbits heteroclinic to invariant manifolds. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 967-1023. doi: 10.3934/dcdss.2009.2.967
[18]

Arturo Echeverría-Enríquez, Alberto Ibort, Miguel C. Muñoz-Lecanda, Narciso Román-Roy. Invariant forms and automorphisms of locally homogeneous multisymplectic manifolds. Journal of Geometric Mechanics, 2012, 4 (4) : 397-419. doi: 10.3934/jgm.2012.4.397
[19]

Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 579-596. doi: 10.3934/dcds.2006.15.579
[20]

Roberto Castelli. Efficient representation of invariant manifolds of periodic orbits in the CRTBP. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 563-586. doi: 10.3934/dcdsb.2018197

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences