Summary |
The thesis is on the limit cycle in nonlinear dynamics systems. To begin, I mention a nonlinear system - superlattices. Then, I generally introduce the method to study nonlinear dynamics. I mainly introduce phase plane-a topological analysis to tackle nonlinear system. I particularly discuss two attractors - the fixed point and limit cycle. A self-sustained oscillation in nonlinear system can be understood as the manifestation of one-dimensional attractors - limit cycle in phase plane. For better understanding of nonlinear dynamical systems, I investigate a famous nonlinear system-van der Pol (VDP) system. As the nonlinear term in the VDP equation increases, the shape of the limit cycle changes from a circle to a square-like loop, and the corresponding self-sustained oscillation becomes a pulse-like curve from a sinusoid one. The period of limit cycle also increases. Lastly, the robustness of limit cycles of nonlinear dynamics is investigated by adding a small random velocity field to the VDP equation in its two-dimensional phase plane. Our numerical calculations show that a limit cycle does not change much under a weak random perturbation. Thus it confirms the conjecture that a limit cycle will make only a small deformation under a weak external perturbation. The idea can be used to understand ac response of self-sustained oscillations in nonlinear dynamical systems. |