Recognition of the bifurcation type of resonance in mildly degenerate Hopf-Nemark-Sacker families

Nonlinearity

Volumn 21, Issue 10, 2008, Pages 2463-2482

  Broer, H W ^a   Holtman, S J ^a   Vegter, G ^a  

^a UNIVERSITY OF GRONINGEN   (Netherlands)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 56349097337     PISSN: 09517715     EISSN: 13616544     Source Type: Journal    
DOI: 10.1088/0951-7715/21/10/014     Document Type: Article

References (24)

1. Arnol'd V I 1982 Geometrical Methods in the Theory of Ordinary Differential Equations (Berlin: Springer)
2. Broer H W, Golubitsky M and Vegter G 2003 The geometry of resonance tongues: a singularity theory approach Nonlinearity 16 1511-38
3. Broer H W, Golubitsky M and Vegter G 2007 Geometry of resonance tongues Singularity Theory. Proc. 2005 Marseille Singularity School and Conf. (Marseille) pp 327-56
4. Broer H W, Holtman S J, Vegter G and Vitolo R 2008 The 4D resonance set of a mildly degenerate Hopf-Nemark-Sacker family in preparation
5. Broer H W, Hoveijn I, Lunter G and Vegter G 2003 Bifurcations in Hamiltonian Systems (Springer Lecture Notes in Mathematics) (Berlin: Springer)
6. Broer H W, Simó C and Tatjer J-C 1998 Towards global models near homoclinic tangencies of dissipative diffeomorphisms Nonlinearity 11 667-770
7. Broer H W, Simó C and Vitolo R 2002 Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing Nonlinearity 15 1205-67
8. Broer H W, Simó C and Vitolo R 2008 The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms, analysis of a resonance bubble Physica D 237 1773-99
9. Broer H W, Simó C and Vitolo R 2008 The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms, the Arnol'd resonance web Bull. Belgian Math. Soc. Simon Stevin to be published
10. Broer H W and Vegter G 1992 Bifurcational aspects of parametric resonance Dynamics Reported (New Series vol 1) pp 1-51
11. Broer H W and Vegter G 2008 Generic Hopf-Nemark-Sacker bifurcations in feed-forward systems Nonlinearity 21 1547-78
12. Ciocci M C, Litvak-Hinenzon A and Broer H W 2005 Survey on dissipative KAM theory including quasi-periodic bifurcation theory based on lectures by Henk Broer Geometric Mechanics and Symmetry: The Peyresq Lectures (LMS Lecture Notes Series vol 306) ed J Montaldi and T Ratiu (Cambridge: Cambridge University Press) pp 303-55
13. Devaney R L 1989 An Introduction to Chaotic Dynamical Systems (Redwood City, CA: Addison-Wesley)
14. Golubitsky M and Schaeffer D G 1985 Singularities and Groups in Bifurcation Theory (Applied Mathematical Sciences vol I) (Berlin: Springer)
15. Krauskopf B 1994 Bifurcation sequences at 1:4 resonance: an inventory Nonlinearity 7 1073-91
16. Kuznetsov Y A 1995 Elements of Applied Bifurcation Theory (Applied Mathematical Sciences vol 112) (Berlin: Springer)
17. McGehee R P and Peckham B B 1995 Determining the global topology of resonance surfaces for periodically forced oscillator families Normal Forms and Homoclinic Chaos (Fields Institute Communications vol 4) ed W F Langford and W Nagata (Providence, RI: American Mathematical Soceity) pp 233-54
18. McGehee R P and Peckham B B 1996 Arnold flames and resonance surface folds Int. J. Bifurc. Chaos 6 315-36
19. Newhouse S E, Palis J and Takens F 1983 Bifurcation and stability of families of diffeomorphisms Publ. Math. Inst. Hautes Etudes Sci. 57 1-71
20. Peckham B B, Frouzakis C E and Kevrekidis I G 1995 Bananas and banana splits: a parametric degeneracy in the hopf bifurcation for maps SIAM J. Math. Anal. 26 190-217
21. Peckham B B and Kevrekidis I G 1991 Period doubling with higher-order degeneracies SIAM J. Math. Anal. 22 1552-74
22. Peckham B B and Kevrekidis I G 2002 Lighting Arnold flames: resonance in doubly forced periodic oscillators Nonlinearity 15 405-28
23. Takens F 1974 Forced oscilations and bifurcations Applications of Global Analysis I (Communications of the Mathematical Institute Rijksuniversiteit Groningen vol 3) pp 1-59
24. Vanderbauwhede A 1992 Branching of periodic solutions in time-reversible systems Geometry and Analysis in Non-linear Dynamics (Pitman Research Notes in Mathematics vol 222) (London: Pitman) pp 97-113