DYNAMICS AND BIFURCATIONS OF A NONHOLONOMIC HEISENBERG SYSTEM

Abstract

We analyze both theoretically and numerically the dynamical behavior of a modification of a nonholonomic Hamiltonian system known as the Heisenberg system. The equations of motion are derived by computing the Lagrange multiplier in terms of the Poisson commutator. The presence of the constraint induces a nontrivial dynamical behavior that has been investigated by plotting the Poincaré section for the reduced system and taking advantage of a conserved quantity and the reversibilities. The dynamics is organized around two Lyapunov families of periodic orbits whose bifurcation behavior has been analyzed with a continuation technique both on the conserved quantity and on the parameters of the problem. The nongeneric branching behavior of the normal modes is theoretically explained by studying the variational equations that reduces to a Hill equation and its well known coexistence property. Invariant tori around the elliptic periodic orbits have been numerically detected but not further analyzed.