Bifurcation From Periodic Solutions with Spatiotemporal Symmetry
In this paper, we discuss some recent developments in the understanding of generic bifurcation from periodic solutions with spatiotemporal symmetries. We focus mainly on the theory for bifurcation from isolated periodic solutions in dynamical systems with a compact symmetry group. Moreover, we discuss how our theory justifies certain heuristic assumptions underlying previous approaches towards period preserving and period doubling bifurcation from periodic solutions.
KeywordsPeriodic Solution Irreducible Representation Hopf Bifurcation Period Doubling Period Doubling Bifurcation
Unable to display preview. Download preview PDF.
- P. L. BUONO, A model of central pattern generators for quadruped locomotion. Ph. D. Thesis, University of Huston, (1998).Google Scholar
- M. VAN DYKE, An album of fluid motion, Parabolic Press, Stanford CA, (1982).Google Scholar
- B. FIEDLER, Global Bifurcations of Periodic Solutions with Symmetry, Lecture Notes in Math, 1309, Springer, Berlin, (1988).Google Scholar
- M. J. FIELD, Symmetry breaking for equivariant maps, In: ‘Algebraic Groups and Related Subjects’, (G. Lehrer et al, eds) Australian Math. Soc. Lecture Series, Cambridge Univ. Press, (1996), 219–253.Google Scholar
- M. GOLUBITSKY, I. STEWART AND D. SCHAEFFER, Singularities and Groups in Bifurcation Theory: Vol. II, Applied Mathematical Sciences, 69, Springer, New York, (1988).Google Scholar
- J. GUCKENHEIMER AND P. HOLMES, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42, Springer, New York, (1983).Google Scholar
- J. S. W. LAMB AND I. MELBOURNE, Bifurcation from discrete rotating waves, To appear in Arch. Rat. Mech. Anal.Google Scholar
- C. WULFF, J. S. W. LAMB AND I. MELBOURNE, Bifurcation from relative periodic solutions, submitted.Google Scholar
- E. V. NIKOLAEV, Periodic motions in systems with a finite symmetry group, Preprint, Pushchino, (1994).Google Scholar
- A. VANDERBAUWHEDE, Equivariant period doubling, In Advanced Topics in the Theory of Dynamical Systems, (G. Fusco et aI, eds.), Notes Rep. Math. Sci. Engrg., 6, (1989), 235–246.Google Scholar