Friday 17 August, 2001,  11:00 am 

Link Seminar Room

Professor J. D. Meiss

University of Colorado, USA

Twistless Bifurcations in Hamiltonian Systems

ABSTRACT:

Near a nonresonant, linearly stable periodic orbit, the dynamics of Hamiltonian system can be represented by angle-action coordinates. The actions are constant and the angles rotate with a frequency depending upon the action. Thus, in these coordinates, the
dynamics is represented entirely by the Lagrangian ``frequency map'' which gives the rotation number as a function of the action. The twist matrix, given by the Jacobian of the rotation number, describes the anharmonicity in the system. When the twist is singular the frequency map is in general not locally one-to-one. Here we investigate the occurrence of fold and cusp singularities in the frequency map. We show that these necessarily occur near third order resonances. Consequences of this multiple island chains and exotic reconnection bifurcations. We illustrate the results by numerical computations of frequency maps for a quadratic, symplectic map