A Numerical bifurcation function for homoclinic orbits

SIAM Journal on Numerical Analysis

Volumn 35, Issue 5, 1998, Pages 2055-2069

  Ashwin, Peter ^a,b   Mei, Zhen ^c  

^a INSTITUT NON LINÉAIRE DE NICE   (France)

^b UNIVERSITY OF SURREY   (United Kingdom)

^c PHILIPPS UNIVERSITY MARBURG   (Germany)

Author keywords

Homoclinic orbit; Numerical bifurcation function; Periodic solutions

Indexed keywords

EID: 0346964363     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142996298168     Document Type: Article

References (19)

1. P. ASHWIN, K. BÖHMER, AND Z. MEI, Forced symmetry breaking of homoclinic cycles in a PDE with O(2) symmetry, J. Comput. Appl. Math., 70 (1996), pp. 297-310.
2. W.-J. BEYN, Global bifurcations and their numerical computation, in Continuation and Bifurcations: Numerical Techniques and Applications, D. Roose et al., eds., Kluwer Academic Publishers, Norwell, MA, 1990. pp. 169-181.
3. W.-J. BEYN, The numerical computation of connecting orbits in dynamical systems, IMA J. Numer. Anal., 9 (1990), pp. 379-405.
4. A.R. CHAMPNEYS AND YU.A. KUZNETSOV, Numerical detection and continuation of codimension-two homoclinic bifurcations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 4 (1994), pp. 795-822.
5. A.R. CHAMPNEYS, YU.A. KUZNETSOV, AND B. SANDSTEDE, A Numerical Toolbox for Homoclinic Bifurcation Analysis, University of Bristol, Department of Engineering Maths., Bristol. England, Report 1.95. 1995.
6. P. CHOSSAT, Forced symmetry breaking of an O(2)-symmetric homoclinic cycle, Nonlinearity, 6 (1993), pp. 723-731.
7. E. DOEDEL, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congr. Numer., 30 (1981), pp. 265-284.
8. E. DOEDEL AND J.P. KERNEVEZ, AUTO: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations, Tech. report, California Institute of Technology, Pasadena, CA, 1986.
9. E. FREIRE, A. RODRIGUEZ-LUIS, AND E. PONCE, A case study for homoclinic chaos in an autonomous electronic circuit: A trip from Takens-Bogdanov to Hopf-Shilnikov, Physica D, 62 (1993), pp. 230-253.
10. B. HASSARD, Computation of invariant manifolds, in New approaches to Nonlinear Problems, P. Holmes, ed., SIAM, Philadelphia, 1980, pp. 27-42.
11. F.R. DE HOOG AND R. WEISS, An approximation theory for boundary value problems on infinite intervals, Computing, 24 (1980), pp. 227-239.
12. I.G. KEVREKIDIS, B. NICOLAENKO, AND J.C. SCOVEL, Back in the saddle again: A computer assisted study of the Kuramoto-Sivashinsky equation, SIAM J. Appl. Math., 50 (1990). pp. 760-790.
13. Y. A. KUZNETSOV, Elements of Applied Bifurcation Theory, Springer- Verlag, New York, 1995.
14. M. LENTINI AND H.B. KELLER, Boundary value problems on semi- infinite intervals and their numerical solution, SIAM J. Numer. Anal., 17 (1980), pp. 577-604.
15. X.-B. LIN, Using Melnikov's method to solve Silnikov's problems, Proc. Roy. Soc. Edinburgh, 116A (1990), pp. 295-325.
16. G. MOORE, Computation and parametrization of periodic and connecting orbits, IMA J. Numer. Anal., 15 (1995), pp. 245-263.
17. B. SANDSTEDE, Private communication. 1995.
18. R. SEYDEL, From Equilibrium to Chaos-Practical Bifurcation and Stability Analysis, Elsevier, New York, Amsterdam, London, 1988.
19. A. VANDERBAUWHEDE AND B. FIEDLER, Homoclinic period blow-up in reversible and conservative systems, Z. Angew. Math. Phys., 43 (1992), pp. 292-318.