Global angle-action variables for Duffing system.pdf

a r X i v : m a t h / 0 1 0 5 1 4 7 v 1 [ m a t h .D S ] 1 7 M a y 2 0 0 1 Global action-angle variables for Duffing system I. Kunin1 and A. Runov2 February 1, 2008 Abstract The classical representation of Hamiltonian systems in terms of action-angle variables are defined for simply connected domains such as an interior of a homoclinic orbit. On this basis methods of (lo- cal) perturbations leading, in particular, to chaotic systems have been studied in literature. We are describing a new method for constructing global action- angle variables and successive perturbations based on a topological covering of the phase space. The method is demonstrated for repre- sentative example of the Duffing system. The choice of variables for solutions of different problems is typically related to matters of convenience. The question of “the best” variables looks from the first glance as not well posed. But starting from the works of Poincare the special preference has been given to the variables action (I) and angle (θ) for Hamiltonian systems and their perturbations [1, 2]. At the same time the method of action-angle variables is typically restricted to simply connected domains, mainly to Rn. We are generalizing this method to global action-angle variables defined globally for topologically nontrivial phase spaces. The approach is based on topological transformations (covering) of the phase space plus additional 1 Department of Mechanical Engineering, University of Houston, Houston, TX 77204, USA, e-mail kunin@uh.edu 2 Department of Theoretical Physics, St. Petersburg University, Uljanovskaja 1, St. Pe- tersburg, Petrodvorez, 198504, Russia 1 changes of geometry. In this publication the method is demonstrated for the popular conservative and dissipative Duffing system. The (local) action-angle variables for the system and their perturbations for chaos are considered, e. g. in [3, 4]. Let us consider the well known Duffing equation (fig. 1):{ x? = y y? = x? x3 ? μy (1) Let