Bifurcation from periodic solutions with spatiotemporal symmetry, including resonances and mode interactions

Journal of Differential Equations

Volumn 191, Issue 2, 2003, Pages 377-407

  Lamb, J S W ^a   Melbourne, I ^b   Wulff, C ^b,c  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

^b UNIVERSITY OF SURREY   (United Kingdom)

^c UNIVERSITY OF WARWICK   (United Kingdom)

Author keywords

Equivariant bifurcation theory; Mode interactions; Periodic and relative periodic solutions; Spatiotemporal symmetry

Indexed keywords

EID: 0037670939     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0022-0396(03)00019-6     Document Type: Article

References (34)

1. D.Armbruster, J. Guckenheimer, S. Kim, Chaotic dynamics with square symmetry, Phys. Lett. A 140 (1989) 416-420.
2. V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, in: Grundlehren, Vol. 250, Springer, New York, 1988.
3. D.K. Arrowsmith, C.M. Place, An Introduction to Dynamical Systems, Cambridge University Press, Cambridge, 1990.
4. P. Ashwin, I. Melbourne, Noncompact drift for relative equilibria and relative periodic orbits, Nonlinearity 10 (1997) 595-616.
5. D. Barkley, L.S. Tuckerman, M. Golubitsky, Bifurcation theory for three-dimensional flow in the wake of a circular cylinder, Phys. Rev. E 61 (2000) 5247-5252.
6. P.-L. Buono, A model of central pattern generators for quadruped locomotion. Ph.D. Thesis, University of Houston, 1998.
7. P.-L. Buono, J.S.W. Lamb, M. Roberts, Bifurcation and branching of equilibria in reversible equivariant vector fields, in preparation.
8. P. Chossat, M. Golubitsky, Iterates of maps with symmetry, SIAM J. Math. Anal. 19 (1988) 1259-1270.
9. M.-C. Ciocci, A. Vanderbauwhede, On the bifurcation and stability of periodic orbits in reversible and symplectic diffeomorphisms, in: G. Gaeta (Ed.), Symmetry and Perturbation Theory, Rome, 1998, World Scientific, River Edge, NJ, 1999, pp. 159-166.
10. C. Elphick, E. Tirapegui, M.E. Brachet, P. Coullet, G. Iooss, A simple global characterisation for normal forms of singular vector fields, Physics D 29 (1987) 95-127.
11. B. Fiedler, Global Bifurcation of Periodic Solutions with Symmetry, in: Lecture Notes in Mathematics, Vol. 1309, Springer, Berlin, 1988.
12. B. Fiedler, B. Sandstede, A. Scheel, C. Wulff, Bifurcation from relative equilibria to non-compact group actions: Skew products, meanders, and drifts, Doc. Math. J. DMV 1 (1996) 479-505.
13. M.J. Field, Local structure for equivariant dynamics, in: R.M. Roberts, I.N. Stewart (Eds.), Singularity Theory and its Applications, Part II, Lecture Notes in Mathematics, Vol. 1463, Springer, New York, 1991, pp. 142-166.
14. M.J. Field, Symmetry breaking for equivariant maps, in: G. Lehrer et al. (Eds.), Algebraic Groups and Related Subjects, Australian Mathematical Society Lecture Series, Cambridge University Press, Cambridge, 1997, pp. 219-253.
15. M. Golubitsky, V.G. LeBlanc, I. Melbourne, Hopf bifurcation from rotating waves and patterns in physical space, J. Nonlinear Sci. 10 (2000) 69-101.
16. M. Golubitsky, J.N. Stewart, D. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. II, in: Applied Mathematical Science, Vol. 69, Springer, New York, 1988.
17. M. Krupa, Bifurcations of relative equilibria, SIAM J. Math. Anal. 21 (1990) 1453-1486.
18. J.S.W. Lamb, Local bifurcations in k-symmetric dynamical systems, Nonlinearity 9 (1996) 537-557.
19. J.S.W. Lamb, k-Symmetry and return maps of space-time symmetric flows, Nonlinearity 11 (1998) 601-629.
20. J.S.W. Lamb, H. Brands, Symmetries and reversing symmetries in kicked systems, in: P. Chossat (Ed.), Dynamics, Bifurcation and Symmetry, New Trends and New Tools, NATO ASI Series C, Vol. 437, Kluwer, Dordrecht, 1994, pp. 181-196.
21. J.S.W. Lamb, I. Melbourne, Bifurcation from discrete rotating waves, Arch. Rational Mech. Anal. 149 (1999) 229-270.
22. J.S.W. Lamb, I. Melbourne, Bifurcation from periodic solutions with spatiotemporal symmetry, in: M. Golubitsky, D. Luss, S. Strogatz (Eds.), Pattern Formation in Continuous and Coupled Systems, IMA Volumes in Mathematics and its Applications, Vol. 115, Springer, New York, 1999, pp. 175-191.
23. J.S.W. Lamb, I. Melbourne, C. Wulff, Bifurcation from periodic and relative periodic solutions in equivariant dynamical systems. in: B. Fiedler, et al. (Eds.), Equadiff 99, International Conference on Differential Equations, World Scientific, Singapore, 2000.
24. J.S.W. Lamb, C. Wulff, Reversible relative periodic orbits, J. Differential Equations 178 (2002) 60-100.
25. m, m∞gt;3, Comm. Math. Phys. 64 (1978) 35-40.
26. 4 symmetry near the O(2) limit, Phys. Lett. A 284 (2001) 99-111.
27. A.M. Rucklidge, M. Silber, Bifurcations of periodic orbits with spatio-temporal symmetries, Nonlinearity 11 (1998) 1435-1455.
28. D. Ruelle, Bifurcations in the presence of a symmetry group, Arch. Rational Mech. Anal. 51 (1973) 136-152.
29. M.B. Sevryuk, Reversible systems, in: Lecture Notes in Mathematics, Vol. 1211, Springer, Berlin, 1986.
30. J.W. Swift, Hopf bifurcation with the symmetry of the square, Nonlinearity 1 (1988) 333-377.
31. F. Takens, Forced oscillations and bifurcations, Comm. Math. Inst. Univ. Utrecht 3 (1974) 1-59.
32. A. Vanderbauwhede, Subharmonic branching in reversible systems, SIAM J. Math. Anal. 21 (1990) 954-979.
33. C. Wulff, J.S.W. Lamb, I. Melbourne, Bifurcation from relative periodic solutions, Ergod. Theory &Dyn. Systems 21 (2001) 605-635.
34. C. Wulff, R.M. Roberts, Hamiltonian systems near relative periodic orbits, SIAM J. Appl. Dyn. Systems 1 (2002) 1-43.