Symmetry-breaking Analysis On The Parameterically Excited Pendulum And The Adaptive-feedback Controller On Helmholtz-Duffing Oscillator

In this thesis, we at first investigate the effect of the symmetry-breaking on the parameterically excited pendulum including a bias term by using the theory of bifurcations and chaos. Then, we introduce a simple adaptive-feedback control method to the Helmholtz-Duffing oscillator, which is an archetype of a class of asymmetric oscillators having distinct homoclinic bifurcations corresponding to different parameters of the oscillator. Finally, we end our thesis with summary and suggestions.The layout of this paper is organized as follows.In Chapter 1, a brief review concerning the theory of nonlinear dynamical systems is introduced, such as Poincare map, Melnikov method, theory of bifurcation and chaos. Meanwhile, a useful numerical simulation software "Dynamics" is introduced in this thesis.In Chapter 2, the effect of the symmetry-breaking on the parameterically excited pendulum including a bias term is investigated. At first, our qualitative analyses and numerical simulations show that the area of the safe region of the uncxcited pendulum (without damping and without forcing) will decrease with the increasing of the bias term. Due to the variation, the critical homoclinic bifurcation of the excited pendulum will increase, and the region where the homoclinic transversal intersection occurs between the stable and unstable manifolds in the Poincarémap will be enlarged. Second, as the bias term increases, our analysis demonstrates that the number and the type of attractors of the Poincarémap, the phase portraits, the basins of attraction, and the bifurcation diagrams will produce a considerable variation. In particular, the stability of the parameterically excited pendulum will lose once the bias term exceeds a critical value. In this case there is no longer any steady state existing. These results suggest that much attention should be paid on controlling the increasing of bias term, especially when the parameterically excited pendulum as a main device is applied to some practical systems.In Chapter 3, a simple adaptive-feedback control method is applied to the Helmholtz-Duffing oscillator, which includes a pair of asymmetric homoclinic orbits when the damped term and the perturbed excitation are removed. At first, the fixed points and homoclinic orbits of the unperturbed Helmholtz-Duffing oscillator for different values of the symmetric parameter are investigated. Then, using the Melnikov's method, the distinct homoclinic bifurcations of the perturbed Helmholtz-Duffing oscillator arc obtained. Finally, we add the adaptive-feedback control term into the perturbed system and get a safe region where there is no longer the transverse intersections between the stable and unstable manifolds in the Poincare map. The subsequent basin of attractions and the maximum Lyapunov exponents are in the agreement with the above theoretical predictions. In Chapter 4, we end this thesis with summary and suggestions.