Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit

Journal of Differential Equations

Volumn 218, Issue 2, 2005, Pages 390-443

  Rademacher, Jens D M ^a  

^a UNIVERSITY OF MINNESOTA   (United States)

Author keywords

Heteroclinic cycles; Homoclinic bifurcation; Self organization; Winding number

Indexed keywords

EID: 27744536022     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jde.2005.03.016     Document Type: Article

References (39)

1. A. Back J. Guckenheimer M.R. Myers F.J. Wicklin P.A. Worfolk DsTool: computer assisted exploration of dynamical systems Notices Amer. Math. Soc. 39 1992 303-309
2. M. Bär M. Hildebrand M. Eiswirth M. Falke H. Engel M. Neufeld Chemical turbulence and standing waves in a surface reaction model: The influence of global coupling and wave instabilities Chaos 4 1994 499-507
3. G. Bordiuogov, H. Engel, Unpublished simulations of the three-component light sensitive Oregonator model, see [26] for details.
4. P.L. Buono M. Golubitsky A. Palacios Heteroclinic cycles in rings of coupled cells Physica D 143 2000 74-108
5. V.V. Bykov The bifurcations of separatrix contours and chaos Physica D 62 1993 290-299
6. S.-N. Chow, J.K. Hale, Methods of bifurcation theory, Grundlehren Math. Wiss., vol. 251, Springer, New York, 1982.
7. E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, vol. XII, McGill-Hill Book Company, Inc., New York, Toronto, London, 1955.
8. W.A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics, vol. 629, Springer, Berlin, Heidelberg, New York, 1978.
9. P. Coullet C. Riera C. Tresser Stable static localized structures in one dimension Phys. Rev. Lett. 84 2000 3069-3072
10. B. Deng K. Sakamoto Sil'nikov-Hopf bifurcations J. Differential Equations 119 1995 1-23
11. L.J. Diaz J. Rocha Heterodimensional cycles partial hyperbolicity and limit dynamics, Fund. Math. 174 2002 127-186
12. P. Glendinning C. Sparrow T-points: A codimension two heteroclinic bifurcation J. Statist. Phys. 43 1986 479-488
13. J.K. Hale X.-B. Lin Heteroclinic orbits for retarded functional differential equations J. Differential Equations 65 1986 175-202
14. G. Haller S. Wiggins Geometry and chaos near resonant equilibria of 3-DOF Hamiltonian systems Physica D 90 1996 319-365
15. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer, Berlin, Heidelberg, 1981.
16. P. Hirschberg E. Knobloch Shilnikov-Hopf bifurcation Physica D 62 1993 202-216
17. H. Kokubu, Heteroclinic bifurcations associated with different saddle indices, in: Dynamical Systems and Related Topics, Advanced Series in Dynamic Systems, vol. 9, World Scientific, River Edge, NJ, 1991.
18. H. Kokubu A construction of three-dimensional vector fields which have a codimension two heteroclinic loop at Glendinning-Sparrow T-point Z. Angew. Math. Phys. 44 1993 510-536
19. W.S. Koon M.W. Lo J.E. Marsden S.D. Ross Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics Chaos 10 2000 427-469
20. J. Krishnan I.G. Kevrekidis M. Or-Guil M.G. Zimmermann M. Bär Numerical bifurcation and stability analysis of solitary pulses in an excitable reaction-diffusion medium Comput. Methods Appl. Mech. Eng. 170 1999 253-275
21. R. Labarca Bifurcation of contracting singular cycles Ann. Sci. École Norm. Sup. (4) 28 1995 705-745
22. X.-B. Lin Using Melnikov's method to solve Silnikov's problems Proc. Roy. Soc. Edinburgh A 116 1990 295-325
23. 4 symmetry Physica D 135 2000 263-304
24. S. Nii The accumulation of eigenvalues in a stability problem Physica D 142 2000 70-86
25. J.D.M. Rademacher, Homoclinic bifurcation from heteroclinic cycles with periodic orbits and tracefiring of pulses, Ph.D. Thesis, University of Minnesota, 2004.
26. T. Rieß, Using Lin's method for an almost Shilnikov problem, Diploma Thesis, Technical University of Illmenau, Faculty of Mathematics and the Sciences, 2003.
27. M. Romeo C.K.R.T. Jones The stability of traveling calcium pulses in a pancreatic acinar cell Physica D 177 2003 242-258
28. B. Sandstede Verzweigungstheorie homokliner Verdopplungen 1993 Dissertation University of Stuttgart
29. B. Sandstede A. Scheel Gluing unstable fronts and backs together can produce stable pulses Nonlinearity 13 2000 1465-1482
30. B. Sandstede A. Scheel Absolute and convective instabilities of waves on unbounded and large bounded domains Physica D 145 2000 233-277
31. G.R. Sell, Y. Youngchen, Dynamics of Evolutionary Equations, Applied Mathematical Science, vol. 143, Springer, New York, 2002.
32. A.L. Shilnikov, L.P. Shilnikov, D.V. Turaev, Blue sky catastrophe in singularly perturbed systems, Moscow Math. J. (2003), accepted.
33. L.P. Shil'nikov On the generation of a periodic motion from a trajectory which leaves and re-enters a saddle-saddle state of equilibrium Soviet Math. Dokl. 7 1966 1155-1158
34. J. Sieber Numerical bifurcation analysis for multisection semiconductor lasers SIAM J. Appl. Dyn. Systems 1 2002 248-270
35. D. Simpson, V. Kirk, J. Sneyd. Complex oscillations and waves of calcium in pancreatic acinar cells, Physica D (2004), submitted.
36. J. Sneyd A. LeBeau D. Yule Traveling waves of calcium in pancreatic acinar cells: Model construction and bifurcation analysis Physica D 145 2000 158-179
37. D. Snita P. Hasal J.H. Merkin Electric field induced propagating structures in a model of spatio-temporal signalling Physica D 14 2000 155-169
38. A. Vanderbauwhede B. Fiedler Homoclinic period blow-up in reversible and conservative systems Z. Angew. Math. Phys. 43 1992 292-318
39. Z. Deming X. Zhihong Bifurcations of heteroclinic loops Sci. China Ser. A 41 1998 837-848