2-D Hamiltonian Duffing oscillator. Elliptic functions from a dynamical systems point of view
نویسنده
چکیده
The bidimensional Duffing oscillator (2DHD) is a parametric Hamiltonian system, integrable by elliptic functions, which includes as particular cases the isotropic oscillator and the classic Hamiltonian Duffing model. The Stark effect, after the KS transformation is used and the parameters are properly chosen, may be studied as a couple of 2DHD. A particular case of 2DHD system has been considered as an example exhibiting focus-focus singularity. Moreover it is a basic model for some perturbed systems under 1-1 resonance. Restricted to bounded trajectories we present a detailed analysis of the solutions in polar variables. Special attention is given to the singular values of the energy-momentum mapping connected with circular orbits. Moreover we carried out the full reduction using the Hamilton-Jacobi equation. Different aspects related to the elliptic integrals involved and their properties are presented. In our opinion the 2DHD model represents an alternative to the pendulum or the free rigid body models for the study of elliptic functions and integrals from a dynamical systems point of view.
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