The Parametrically Forced Pendulum: A Case Study in 1 1/2 Degree of Freedom

Abstract

This paper is concerned with the global coherent (i.e., non-chaotic) dynamics of the parametrically forced pendulum. The system is studied in a 1 1 2 degree of freedom Hamiltonian setting with two parameters, where a spatio-temporal symmetry is taken into account. Our explorations are restricted to large regions of coherent dynamics in phase space and parameter plane. At any given parameter point we restrict to a bounded subset of phase space, using KAM theory to exclude an infinitely large region with rather trivial dynamics. In the absence of forcing the system is integrable. Analytical and numerical methods are used to study the dynamics in a parameter region away from integrability, where the analytic results of a perturbation analysis of the nearly integrable case are used as a starting point. We organize the dynamics by dividing the parameter plane in fundamental domains, guided by the linearized system at the upper and lower equilibria. Away from integrability some features of the nearly integrable coherent dynamics persist, while new bifurcations arise. On the other hand, the chaotic region increases.

Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering

“…It is clear from Eqs. (13) and (14) that the kick operator and the fixed Floquet Hamiltonian are not completely independent. Usually, one uses the freedom in the definition of the kick operator to obtainĤ F in its simplest form.…”

“…The behavior of such systems is very rich -they can display interesting integrability-to-chaos transitions, as well as counter-intuitive effects, such as dynamical localization [4][5][6][7][8][9][10][11][12][13] and dynamical stabilization [3,14,15]. The latter manifests itself in reduced ionisation rates in atomic systems irradiated by electromagnetic fields in the regime of high frequencies and high intensities [16][17][18][19][20][21], or as diminished spreading of wave packets in systems subject to periodic driving [22,23].…”

Quasiperiodic Dynamics in Bose-Einstein Condensates in Periodic Lattices and Superlattices

“…Our numerical simulations, which show the same scaling that our theoretical results indicate, suggest the presence of (parameter-dependent) integrable dynamics of positive but small measure inside the "chaotic sea." [48] Remark 11 : A similar combination of islands of invariant tori within a chaotic sea occurs in the example of a parametrically forced planar pendulum [7]. The division of phase space into a mostly quasiperiodic and a mostly chaotic region is the typical behavior that one expects to observe in a large class of forced one dof Hamiltonian systems [15].…”

Dynamical stability of slip-stacking particles

“…We find that this Hamiltonian is isomorphic to that of a pendulum under a sinusoidal driving force (in the absence of gravity). The driven pendulum is a type of nonlinear Mathieu equation that is a subject of ongoing research in computational mathematics [19,20]. A canonical form for the driven pendulum may be parametrized as [19] …”