AN ANALYSIS OF THE SINGULARLY PERTURBED CONSERVATIVE SYSTEM - ADI KUSUMO (NIM: 30104004), FAJAR
In this dissertation a singularly perturbed conservative system is studied. The system is a normal form of a coupled-oscillator system with widely-separated frequencies. The widely-separated frequencies case is the extreme type in higher order resonance class. The nonlinear terms of the coupled-oscillator system is quadratic and preserves the energy. Analysis of the system is focused to continue and to complete the previous results, especially for the degenerate case. If the assumption of the frequencies is changed into the strong resonance case, the normal form of the system becomes a special case of the one in the widely-separated frequencies case. It shows that a coupled-oscillator system with widely-separated frequencies can have a similar dynamics with the system in a strong resonance class. A chaotic dynamics is found in the degenerate system. The mechanism of chaos is the sequence of period-doubling bifurcations which is accumulated at a value of parameter. This bifurcation produce many periodic solutions with various period at the parameter value. Therefore, the shape of the strange attractor is generated by the periodic solutions. The system has some symmetries. One of them is the similarity between the dynamics in the upper-half plane of the phase space and the dynamics in the lower-half plane. By adding a forcing term in the system which breaks the symmetry, the existence of the equilibria will be influenced. The sequence of period-doubling bifurcations is also found when the forcing term is varied. In this case, there are two attractors coexist for a specific value of the forcing. The boundary of the initial value of each attractor will be studied.