The entwined wiggling of homoclinic curves emerging from saddle-node/Hopf instabilities

Physica D: Nonlinear Phenomena

Volumn 195, Issue 1-2, 2004, Pages 77-105

  Champneys, A R ^a   Kirk, V ^b  

^a UNIVERSITY OF BRISTOL   (United Kingdom)

^b UNIVERSITY OF AUCKLAND   (New Zealand)

Author keywords

Beyond all orders; Codimension two point; Saddle node Hopf; Shil'nikov homoclinic orbit

Indexed keywords

BIFURCATION (MATHEMATICS); CONVERGENCE OF NUMERICAL METHODS; DIFFERENTIAL EQUATIONS; EIGENVALUES AND EIGENFUNCTIONS; INTEGRATION; PARAMETER ESTIMATION; PERTURBATION TECHNIQUES;

CODIMENSION-TWO POINT; HOMOCLINIC BIFURCATIONS; SADDLE-NODE/HOPF;

WAVE PROPAGATION;

EID: 3042854569     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2004.03.004     Document Type: Article

References (21)

1. Balmforth N.J., Ierley G.R., Spiegel E.A., Chaotic pulse trains. SIAM J. Appl. Math. 54:1994;1291-1334.
2. Beyn W.-J., Numerical analysis of homoclinic orbits emanating from a Takens-Bogdanov point. IMA J. Numer. Anal. 14:1994;381-410.
3. W.J. Beyn, A.R. Champneys, E.J. Doedel, W. Goverts, Yu.A. Kuznetsov, B. Sandstede, Numerical continuation and computation of normal forms, in: B. Fiedler (Ed.), Handbook of Dynamical Systems, vol. III, Towards Applications, Elsevier, 1992, pp. 149-220.
4. Broer H.W., Vegter G., Subordinate Shil'nikov bifurcations near some singularities of vector fields having low codimension. Ergod. Theory Dyn. Syst. 4:1984;509-525.
5. E. Doedel, A. Champneys, T. Fairgrieve, Yu. Kuznetsov, B. Sandstede, X. Wang, AUTO 97: Continuation and bifurcation software for ordinary differential equations. http://indy.cs.concordia.ca/auto.
6. P. Gaspard, Tangences homoclines dans les systèmes dynamiques dissipatifs, PhD Thesis, Université Libre de Bruxelles, 1987.
7. Gaspard P., Local birth of homoclinic chaos. Physica D. 62:1993;94-122.
8. I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, New York, 1980.
9. J. Guckenheimer, On a codimension two bifurcation, in: D. Rand, L.-S. Young (Eds.), Lecture Notes in Mathematics, vol. 898, Springer-Verlag, New York, 1981, pp. 99-142.
10. J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, 2nd ed., Springer-Verlag, New York, 1986.
11. Kirk V., Breaking of symmetry in the saddle-node Hopf bifurcation. Phys. Lett. A. 154:1991;243-248.
12. Kirk V., Merging of resonance tongues. Physica D. 66:1993;267-281.
13. Yu.A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd ed., Springer-Verlag, New York, 1998.
14. Lorenz E.N., Irregularity - a fundamental property of the atmosphere. Tellus A. 36:1984;98-110.
15. Romeo M.M., Jones C.K.R.T., The stability of traveling calcium pulses in a pancreatic acinar cell. Physica D. 177:2003;242-258.
16. Shil'nikov A., Nicolis G., Nicolis C., Bifurcation and predictability analysis of low-order atmospheric circulation model. Int. J. Bifurc. Chaos. 5:1995;1701-1711.
17. L.P. Shilnikov, A.L. Shilnikov, D.V. Turaev, L.O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, Part II, World Scientific, Singapore, 2001.
18. Sieber J., Numerical bifurcation analysis for multi-section semiconductor lasers. SIAM J. Appl. Dyn. Syst. 1:2002;248-270.
19. J. Sim, Saddle-node/Hopf Bifurcations in a Physiological Model, M.Sc. Thesis, University of Auckland, 2002.
20. Sneyd J., LeBeau A., Yule D., Travelling waves of calcium in pancreatic acinar cells: model construction and bifurcation analysis. Physica D. 145:2000;158-179.
21. Van Veen L., Opsteegh T., Verhulst F., Active and passive ocean regimes in a low-order climate model. Tellus Ser. A: Dyn. Meteorol. Oceanogr. 53:2001;616-628.