Higher-Order Resonances in Dynamical Systems

Abstract

This thesis is a collection of studies on higher-order resonances in an important class of dynamical systems called coupled oscillators systems. After giving an overview of the mathematical background, we start in Chapter 1 by presenting a study on resonances in two degrees of freedom, autonomous, Hamiltonian systems. Due to the presence of a symmetry condition on one of the degrees of freedom, we show that some of the resonances vanish as lower order resonances. After giving a sharp estimate of the resonance domain, we investigate this order change of resonance in a rather general potential problem with discrete symmetry and consider as an example the H´enon-Heiles family of Hamiltonians. We also study a classical example of a mechanical system with symmetry, the elastic pendulum, which leads to a natural hierarchy of resonances with the 4 : 1-resonance as the most prominent after the 2 : 1-resonance and which explains why the 3 : 1- resonance is neglected. In chapter 2 of this thesis we study the performance of a symplectic numerical integrator based on the splitting method. This method is applied to a subtle problem i.e. higher order resonance of the elastic pendulum. In order to numerically study the phase space of the elastic pendulum at higher order resonance, a numerical integrator which preserves qualitative features after long integration times is needed. We show by means of an example that our symplectic method offers a relatively cheap and accurate numerical integrator. In chapter 3 we study two degree of freedom Hamiltonian systems and applications to nonlinear wave equations. Near the origin, we assume that near the linearized system has purely imaginary eigenvalues: ±i[omega]1 and ±i[omega]2 with 0 <[omega]2/[omega]1«1 or w2/w1»1, which is interpreted as a perturbation of a problem with double zero eigenvalues. Using the averaging method, we compute the normal form and show that the dynamics differs from the usual one for Hamiltonian systems at higher order resonances. Under certain conditions, the normal form is degenerate which forces us to normalize to higher degree. The asymptotic character of the normal form and the corresponding invariant tori is validated using KAM theorem. This analysis is then applied to widely separated mode-interaction in a family of nonlinear wave equations containing various degeneracies. In chapter 4 we present an analysis of a system of coupled-oscillators. We make two assumptions for our system. The first assumption is that the frequencies of the characteristic oscillations are widely separated, and the second is that the nonlinear part of the vector field preserves the distance to the origin. Using the first assumption, we prove that the reduced normal form of our system, exhibits an invariant manifold which, exists for all values of the parameters and cannot be perturbed away by including higher order terms in the normal form. Using the second assumption, we view the normal form as an energy-preserving three-dimensional system which is linearly perturbed. Restricting our selves to a small perturbation, the flow of the energy-preserving system is used to study the flow in general. We present a complete study of the flow of energy-preserving system and the bifurcations in it. Using these results, we provide the condition for having a Hopf bifurcation of one of the two equilibria. We also numerically follow the periodic solution created via the Hopf bifurcation and find a sequence of period-doubling and fold bifurcations, and also a torus (or Naimark-Sacker) bifurcation. This chapter is a sequel to the study of the previous chapter, where a system of coupled oscillators with widely separated frequencies and energy-preserving quadratic nonlinearity is studied. However, in this paper we are more concerned with the energy-preserving nature of the nonlinearity. We also study a singularly perturbed conservative system in R [exp. n], which is a generalization of our system, and derive a condition for the existence of nontrivial equilibrium of such a system. Returning to the original system we start with for a different set of parameter values compared with those in [?]. Numerically, we find interesting bifurcations and dynamics such as torus (Naimark-Sacker) bifurcation, chaos and heteroclinic-like behaviour