DynamIC Bifurcation Theory Reading Group, Autumn Term 2002

Local bifurcation of periodic solutions with symmetry



Reading: Jeroen S.W. Lamb, Ian Melbourne and Claudia Wulff. Bifurcation from periodic solutions with spatiotemporal symmetry, including resonances and mode-interactions. Preprint DynamIC 2001-3

Tentative meeting time: Thursdays 15:00 - 17:00. First Meeting Week 1 of Term, October 3rd 2002. Room: TBA

Participation: Participation in the reading group is open to all. It is mainly aimed at PhD students working on or beginning to work on some research project in Dynamics. The reading group can also be used by final year and MSc students as preparation for project work. The reading will be shared between the participants who are expected to present sections of the material.

Background:

Local bifurcation from periodic solutions The general theory of local bifurcation of periodic solutions in general (dissipative, non-symmetric) systems has been developed in the 1960-1970s. The common technique in such studies is the use of a (Poincare) return map, and study the bifurcation properties of such a map.

Bifurcation and Symmetry Dynamical behaviour can be very sensitive to the structure of a dynamical system. One such a structure is symmetry. We distinguish between two types of symmetry in dynamical systems: equivariance and reversibility. We say that a dynamical system is G-equivariant if for every solution x(t) of such a system, gx(t) is also a solution, where g is an element of some group of transformations G. We say that a dynamical system is R-reversible if for every solution x(t) of such a system, Rx(-t) is also a solution, where R is an some (invertible) transformation. Many dynamical systems arising in the context of applications are equivariant and/or reversible, which motivates the setting.

Equivariant Bifurcation Theory The systematic study of equivariant bifurcation problems originated in the 1970 and is still in progress. The book by Golubitsky, Stewart and Schaeffer provides a good introduction to the issues involved, and the route of the equivariant branching lemma for obtaining partial answers. The theory focussed primarily on bifurction from equilibria in equivariant vector fields.

Twisted equivariance In the study of bifurcations from periodic solutions with spatiotemporal symmetry, it has turned out to be useful to recognize the property of twisted equivariance and twisted reversibility for a first-hit/pull-back map that captures the local dynamics near the periodic solution.

Takens normal form A key step in the analysis is the Takens normal form, enabling the study of local bifurcations for a fixed point of the map to the study of a local bifurcation of an equilibrium of an associated reversible equivariant vector field. In combination with the first-hit/pull-back map, this yields a general method for reducing bifurcation from periodic solutions to bifurcation from equilibria.

Background Reading:

Books:

General Bifurcation Theory:

Golubitsky, Martin; Schaeffer, David G. Singularities and groups in bifurcation theory. Vol. I. Applied Mathematical Sciences, 51. Springer-Verlag, New York, 1985 (for Liapunov-Schmidt reduction)

Guckenheimer, John; Holmes, Philip Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Revised and corrected reprint of the 1983 original. Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1990.

Chow, Shui Nee; Hale, Jack K. Methods of bifurcation theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], 251. Springer-Verlag, New York-Berlin, 1982

Kuznetsov, Yuri A. Elements of applied bifurcation theory. Second edition. Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998

Equivariant Bifurcation Theory

Chossat, Pascal; Lauterbach, Reiner Methods in equivariant bifurcations and dynamical systems. Advanced Series in Nonlinear Dynamics, 15. World Scientific Publishing Co., Inc., River Edge, NJ, 2000 (but note that their treatment of periodic solutions is incorrect!)

Golubitsky, Martin; Stewart, Ian; Schaeffer, David G. Singularities and groups in bifurcation theory. Vol. II. Applied Mathematical Sciences, 69. Springer-Verlag, New York, 1988

Golubitsky, Martin; Stewart, Ian The symmetry perspective. From equilibrium to chaos in phase space and physical space. Progress in Mathematics, 200. Birkhduser Verlag, Basel, 2002

Field, Michael Lectures on bifurcations, dynamics and symmetry. Pitman Research Notes in Mathematics Series, 356. Longman, Harlow, 1996

Fiedler, Bernold Global bifurcation of periodic solutions with symmetry. Lecture Notes in Mathematics, 1309. Springer-Verlag, Berlin, 1988.

Relevant Research Papers:

Takens, Floris Forced oscillations and bifurcations. Applications of global analysis, I (Sympos., Utrecht State Univ., Utrecht, 1973), pp. 1--59. Comm. Math. Inst. Rijksuniv. Utrecht, No. 3-1974, Math. Inst. Rijksuniv. Utrecht, Utrecht, 1974. Reprinted in: Global analysis of dynamical systems, 1--61, Inst. Phys., Bristol, 2001.

Vanderbauwhede, A. Period-doublings and orbit-bifurcations in symmetric systems. Dynamical systems and ergodic theory (Warsaw, 1986), 197--208, Banach Center Publ., 23, PWN, Warsaw, 1989

Vanderbauwhede, A. Equivariant period doubling. Advanced topics in the theory of dynamical systems (Trento, 1987), 235--246, Notes Rep. Math. Sci. Engrg., 6, Academic Press, Boston, MA, 1989

Jeroen S.W. Lamb. Local bifurcations in k-symmetric dynamical systems. Nonlinearity 9 (1996), 537--557. MR 97d:58172 ; Preprint(ps) ; Article

Jeroen S.W. Lamb. k-Symmetry and return maps of spacetime symmetric flows. Nonlinearity 11 (1998), 601--629. MR 99c:58140 ; Article

Jeroen S. W. Lamb and Ian Melbourne. Bifurcation from discrete rotating waves. Arch. Ration. Mech. Anal. 149 (1999), 229--270. MR 2001h:37108 ; Preprint(ps) ; Article

Jeroen S.W. Lamb and Ian Melbourne. Bifurcation from periodic solutions with spatiotemporal symmetry. In: Pattern formation in continuous and coupled systems (Minneapolis, MN, 1998), IMA 115 (Martin Golubitsy, Dan Luss and Stephen H. Strogatz ed.), Springer, New York, 1999. pp. 175--191. MR 2000:d37062 ; Preprint(ps) ;

Rucklidge, A. M.; Silber, M. Bifurcations of periodic orbits with spatio-temporal symmetries. Nonlinearity 11 (1998), no. 5, 1435--1455 Article

Some lecture notes from M4A35 (Bifurcation Theory), not lectured in the year 2002-2003, that may be useful:
Thanks to Kevin Webster, who will be coordinating the reading group ( Kevin.Webster@ic.ac.uk).

Jeroen Lamb
Mathematics Department, Imperial College.
Jeroen.Lamb@ic.ac.uk