Complete reduction of oscillators in resonance p:q.

This paper extends to any type of resonance (p:q) the Lissajous transformation that handles the resonance (1:1) in a Hamiltonian composed of two harmonic oscillators. The manifolds of constant energy for such a system are two-dimensional surfaces of revolution that are spheres for the resonance 1:1, spheres pinched once for the resonances (1:q) when 1 View Article and Find Full Text PDF

Bifurcations in biparametric quadratic potentials. II.

Quadratic Hamiltonians with the phase space on the S (2) sphere represent numerous dynamical systems. There are only two classes of quadratic Hamiltonians depending on two parameters. We analyze the occurrence of bifurcations and we obtain the bifurcation lines in the parameter plane for one of these classes, thus complementing the work done in a previous paper where the other class was analyzed.

Bifurcations in biparametric quadratic potentials.

Numerous dynamical systems are represented by quadratic Hamiltonians with the phase space on the S (2) sphere. For a class of these Hamiltonians depending on two parameters, we analyze the occurrence of bifurcations and we obtain the bifurcation lines in the parameter plane. As the parameters evolve, the appearance-disappearance of homoclinic orbits in the phase portrait is governed by three types of bifurcations, the pitchfork, the teardrop and the oyster bifurcations.