Umbilical torus bifurcations in Hamiltonian systems

Transcritical bifurcations in nonintegrable Hamiltonian systems.

We report on transcritical bifurcations of periodic orbits in nonintegrable two-dimensional Hamiltonian systems. We discuss their existence criteria and some of their properties using a recent mathematical description of transcritical bifurcations in families of symplectic maps. We then present numerical examples of transcritical bifurcations in a class of generalized Hénon-Heiles Hamiltonians ...

Hopf bifurcations for Near-Hamiltonian Systems

In this paper, we consider bifurcation of limit cycles in near-Hamiltonian systems. A new method is developed to study the analytical property of the Melnikov function near the origin for such systems. Based on the new method, a computationally efficient algorithm is established to systematically compute the coefficients of Melnikov function. Moreover, we consider the case that the Hamiltonian ...

Extended Abstract: Bifurcations in Hamiltonian Systems with Symmetry

This is an extended a.bstract for the lecture of J .E. Marsden to be presented at the ENOC-93 conference, Hamburg, August, 1993. The goal of the lecture is to outline some recent work of the lecturer, M. Golubitsky,1. Stewart, E. Knobloch, A. Mahalov and T. Ratiu on bifurcations in Hamiltonian systems with symmetry. We will survey some earlier work with D. Lewis, T. Ratiu and others to set the ...

Bifurcations of Normally Parabolic Tori in Hamiltonian Systems

Non-Hamiltonian symplectic torus actions

An action of a torus T on a compact connected symplectic manifold M is Hamiltonian if it admits a momentum map, which is a map on M taking values in the dual of the Lie algebra of T. In 1982, Atiyah and GuilleminSternberg proved that the image of the momentum map is a convex polytope, now called the momentum polytope. In 1988, Delzant showed that if the dimension of T is half of the dimension o...