Lin's method for heteroclinic chains involving periodic orbits

Nonlinearity

Volumn 23, Issue 1, 2010, Pages 23-54

  Knobloch, Jürgen ^a   Riebß, Thorsten ^b  

^a ILMENAU UNIVERSITY OF TECHNOLOGY   (Germany)

^b Department of Obstetrics Gynecology   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 76149123202     PISSN: 09517715     EISSN: 13616544     Source Type: Journal    
DOI: 10.1088/0951-7715/23/1/002     Document Type: Article

References (26)

1. Ashwin P, Rucklidge A M and Sturman R 2004 Two-state intermittency near a symmetric interaction of saddle-node and Hopf bifurcations: a case study from dynamo theory Physica D 194 30-48
2. Bamón R, Labarca R, Mãé R and Pacífico M J 1993 The explosion of singular cycles Publ. Math. Inst. Hautes tud. Sci. 78 207-32
3. Beck M, Knobloch J, Lloyd D J B, Sandstede B and Wagenknecht T 2009 Snakes, ladders, and isolas of localised patterns SIAM J. Math. Anal. 41 936-72
4. Beyn W-J 1994 On well-posed problems for connecting orbits in dynamical systems Cont. Math. Chaotic Numer. 172 131-68
5. Champneys A R, Kirk V, Knobloch E, Oldeman B E and Sneyd J 2007 When Shil'nikov Meets Hopf in Excitable Systems SIAM J. Appl. Dyn. Syst. 6 663-93
6. Champneys A R, Kirk V, Knobloch E, Oldeman B E and Rademacher J D M 2009 Unfolding a tangent equilibrium-to-periodic heteroclinic cycle SIAM J. Appl. Dyn. Syst. 8 1261-304
7. Chow S-N and Lin X-B 1990 Bifurcation of a homoclinic orbit with a saddle-node equilibrium J. Diff. Integral Eqns 3 435-66
8. Coppel W A 1978 Dichotomies in Stability Theory (Berlin: Springer)
9. 1-Linearization, and Homoclinic Bifurcation J. Diff. Eqns 79 189-231
10. Hale J and Lin X-B 1986 Heteroclinic orbits for retarded functional differential equations J. Diff. Eqns 65 175-202
11. Homburg A J and Sandstede B 2009 Homoclinic and heteroclinic bifurcations in vector fields Handbook of Dynamical Systems III (Elsevier) to appear
12. Knobloch J 2004 Lin's method for discrete and continuous dynamical systems and applications Habilitationsschrift TU Ilmenau
13. Krauskopf B and Oldeman B E 2006 Bifurcations of global reinjection orbits near a saddle-node Hopf bifurcation Nonlinearity 19 2149-67
14. Krauskopf B and Rie T 2008 A Lin's method approach to finding and continuing heteroclinic connections involving periodic orbits Nonlinearity 21 1655-90
15. Krauskopf B, Tollenaar N and Lenstra D 1998 Tori and their bifurcations in an optically injected semiconductor laser Opt. Commun. 156 158-69
16. Labarca R 1995 Bifurcation of contracting singular cycles Ann. Sci. c. Norm. Supér. 28 705-45
17. Lin X-B 1990 Using Melnikov's method to solve Shilnikov's problems Proc. R. Soc. Edinb. A 116 295-325
18. Lin X-B 2008 Lin's method Scholarpedia 3 6972
19. Pacífico M J and Rovella A 1993 Unfolding contracting singular cycles Ann. Sci. c. Norm. Supér. 26 691-700
20. Rademacher J D 2005 Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit J. Diff. Eqns 218 390-443
21. Rademacher J D 2009 Lyapunov-Schmidt reduction for unfolding heteroclinic networks of equilibria and periodic orbits with tangencies Preprint arXiv:0907.3660
22. Rie T 2008 A Lin's method approach to heteroclinic connections involving periodic orbits-analysis and numerics Dissertationsschrift TU Ilmenau
23. Sandstede B 1993 Verzweigungstheorie homokliner Verdopplungen Dissertationsschrift Universität Stuttgart
24. Vitolo R 2003 Bifurcations of attractors in 3D diffeomorphisms: A study in experimental mathematics PhD Thesis University of Groningen
25. Wieczorek S M, Krauskopf B, Simpson T B and Lenstra D 2005 The dynamical complexity of optically injected semiconductor lasers Phys. Rep. 416 1-128 (Pubitemid 41235265)
26. Zimmermann M, Natiello M and Solari H 2001 Global bifurcations in a laser with injected signal: beyond Adler's approximation Chaos 11 500-13