The quasi-periodic reversible Hopf bifurcation

International Journal of Bifurcation and Chaos

Volumn 17, Issue 8, 2007, Pages 2605-2623

  Broer, Henk W ^a   Ciocci, M Cristina ^b   Hanßmann, Heinz ^c  

^a UNIVERSITY OF GRONINGEN   (Netherlands)

^b IMPERIAL COLLEGE LONDON   (United Kingdom)

^c UTRECHT UNIVERSITY   (Netherlands)

Author keywords

1:1 resonance; KAM theory; Relative equilibria; Versal unfolding

Indexed keywords

CHAOS THEORY; PROBLEM SOLVING; RESONANCE;

CANTOR SETS; KAM THEORY; RELATIVE EQUILIBRIA; VERSAL UNFOLDING;

HOPF BIFURCATION;

EID: 34748899387     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S021812740701866X     Document Type: Article

References (40)

1. Arnold, V. I. [1971] "On matrices depending on parameters," Russ. Math. Surv. 26, 29-43.
2. Braaksma, B. L. J. & Broer, H. W. [1987] "On a quasi-periodic Hopf bifurcation," Ann. Inst. Henri Poincaré 4, 115-168.
3. Bridges, T. [1998] "Poisson structure of the reversible 1:1 resonance," Physica D 112, 40-49.
4. Broer, H. W., Huitema, G. B., Takens, F. & Braaksma, B. L. J. [1990] "Unfoldings and bifurcations of quasi-periodic tori," Mem. Amer. Math. Soc. 83, 1-175.
5. Broer, H. W. & Vegter, G. [1992] "Bifurcational aspects of parametric resonance," Dyn. Rep., New Series 1, 1-51.
6. Broer, H. W. [1993] "Notes on perturbation theory 1991," Erasmus ICP Mathematics and Fundamental Applications (Aristotle University, Thessaloniki).
7. Broer, H. W. & Huitema, G. B. [1995] "Unfoldings of quasi-periodic tori in reversible systems," J. Dyn. Diff. Eq. 7, 191-212.
8. Broer, H. W., Huitema, G. B. & Sevryuk, M. B. [1996] Quasi-Periodic Motions in Families of Dynamical Systems: Order Amidst Chaos, Lecture Notes in Mathematic, Vol. 1645 (Springer, Berlin).
9. Broer, H. W., Cushman, R. H., Fassò, F. & Takens, F. [2007a] "Geometry of KAM-tori for nearly integrable Hamiltonian systems," Erg. Th. Dyn. Syst. 27 725-741.
10. Broer, H. W., Hoo, J. & Naudot, V. [2007b] "Normal linear stability of quasi-periodic tori," J. Diff. Eq. 232, 355-418.
11. Broer, H. W., Hanßmann, H. & Hoo, J. [2007c] "The quasi-periodic Hamiltonian Hopf bifurcation," Nonlinearity 20, 417-460.
12. Broer, H. W., Ciocci, M. C. & Vanderbauwhede, A. [2006] "Normal 1:1 resonance of invariant tori in reversible systems: A study in quasi-periodic bifurcation theory," (in preparation).
13. Ciocci, M. C. [2003] Bifurcations of Periodic Points in Families of Reversible Mappings and of QuasiPeriodic Solutions in Reversible Systems, PhD thesis (Ghent University, Belgium). Available online at: http://www.lib.ugent.be/execl/infod/ethesis.shtml (accessed 28 February 2006).
14. Ciocci, M. C. [2005] "Persistence of quasi-periodic orbits in families of reversible systems with a 1:1 resonance," Proc. Equadiff 03, Hasselt 2003, eds. Dumortier, F. et al. (World Scientific, Singapore), pp. 720-725.
15. Cushman, R. H. & Sanders, J. A. [1986] "Nilpotent normal forms and representation theory of sl(2, ℝ)," Multiparameter Bifurcation Theory (Areata, 1985), Contemp. Math. 56, 31-51.
16. Efstathiou, K., Cushman, R. H. & Sadovskii, D. A. [2004] "Hamiltonian Hopf bifurcation of hydrogen in crossed fields," Physica D 194, 250-274.
17. Efstathiou, K. [2005] Metamorphoses of Hamiltonian Systems with Symmetries, Lecture Notes in Mathematics, Vol. 1864 (Springer, Berlin).
18. Gibson, C. G. [1979] Singular Points of Smooth Mappings (Pitman, London).
19. Hanßmann, H. [1998] "The quasi-periodic centre-saddle bifurcation," J. Diff. Eq. 142, 305-370.
20. Hanßmann, H. & Meer, J. C. van der [2002] "On the Hamiltonian Hopf bifurcations in the 3D Hénon-Heiles family," J. Dyn. Diff. Eq. 14, 675-695.
21. Hirsch, M. W. & Smale, S. [1974] Differential Equations, Dynamical Systems, and Linear Algebra (Academic Press, NY).
22. Hoo, J. [2005] Quasi-Periodic Bifurcations in a Strong Resonance, PhD thesis (Rijksuniversiteit Groningen, The Netherlands).
23. Huitema, G. B. [1988] Unfoldings of Quasi-Periodic Tori, PhD thesis (Rijksuniversiteit Groningen, The Netherlands) .
24. Iooss, G., Mielke, A. & Demay, Y. [1989] "Theory of steady Ginzburg-Landau equation in hydrodynamic stability problems," European J. Mech. B/Fluids 8, 229-268.
25. Iooss, G. & Los, J. [1990] "Bifurcation of spatially quasi-periodic solutions in hydrodynamic stability problems," Nonlinearity 3, 851-871.
26. Jorba, À. & Villanueva, J. [1997] "On the normal behaviour of partially elliptic lower-dimensional tori of Hamiltonian systems," Nonlinearity 10, 783-822.
27. Knobloch, J. & Vanderbauwhede, A. [1995] "Hopf bifurcation at a k-fold resonances in reversible systems," preprint N. M16/95 (Technische Universität Ilmenau).
28. Knobloch, J. & Vanderbauwhede, A. [1996] "Hopf bifurcation at k-fold resonances in conservative systems," Nonlinear Dynamical Systems and Chaos, Groningen 1995, eds. Broer, H. W. et al., Progress in Nonlinear Differential Equations and Their Applications, Vol. 19 (Birkhäuser, Basel), pp. 155-170.
29. Lamb, J. S. W. [1994] Reversing Symmetries in Dynamical Systems, PhD thesis (Universiteit van Amsterdam, The Netherlands).
30. 4," preprint (Imperial College London).
31. Mielke, A. [1986] "A reduction principle for nonautonomous systems in infinite dimensional spaces," J. Diff. Eq. 65, 68-88.
32. Moser, J. [1973] Stable and Random Motions in Dynamical Systems (Princeton University Press, Princeton).
33. Rink, B. [2002] "Direction-reversing traveling waves in the even Fermi-Pasta-Ulam lattice," J. Nonlin. Sci. 12, 479-504.
34. Sevryuk, M. B. [1986] Reversible Systems, Lecture Notes in Mathematics, Vol. 1211 (Springer, Berlin).
35. Sevryuk, M. B. [1993] "Invariant tori of reversible systems of intermediate dimensions," Russ. Akad. Sci. Dokl. Math. 47, 129-133.
36. Sevryuk, M. B. [1997] "Invariant tori of intermediate dimensions in Hamiltonian systems," Reg. Chaot. Dyn. 2, 30-40.
37. van der Meer, J. C. [1985] The Hamiltonian Hopf Bifurcation, Lecture Notes in Mathematics, Vol. 1160 (Springer, Berlin).
38. van der Meer, J. C., Sanders, J. A. & Vanderbauwhede, A. [1994] "Hamiltonian structure of the reversible nonsemisimple 1:1 resonance," Dynamics, Bifurcation and Symmetry, Cargése 1993, ed. Chossat, P. (Kluwer, Dordrecht), pp. 221-240.
39. Vanderbauwhede, A. [1990] "Hopf bifurcation for equivariant conservative and time-reversible systems," Proc. Roy. Soc. Edinburgh A 116, 103-128.
40. Wei, B. [2001] "Perturbations of lower dimensional tori in the resonant zone for reversible systems," JMAA 253, 558-577.