Application Of Homotopy Analysis Method To Some Nonlinear Periodic Oscillation Problems

In everyday life, oscillations occur everywhere, such as the swinging pendulum, mechanical oscillations, oscillations in optical and sound waves and stock market. According to the characteristics, oscillations can be divided into linear oscillations and nonlinear oscillations. Strictly speaking, most oscillation systems are nonlinear. Because there is a distinction between linear and nonlinear oscillations, in many cases, linear theory can not explain parameter oscillation, super-harmonic and sub-harmonic resonance, jump phenomena and many other complex phenomena, in that nonlinear oscillation problems can only be well understood using nonlinear oscillation theories and methods.In this thesis, the analytic method named "Homotopy Analysis Method" (HAM) proposed by Prof. Shijun Liao to solve nonlinear differential equations is employed to study nonlinear periodic oscillations. In summary, the HAM has the following advantages:1) it is independent of any small/large physical parameters; 2) it provides us great freedom to choose equation type and solution expression of linear high-order approximation equations; 3) the convergence of approximation series is guaranteed by a introduced non-zero convergence-control parameter. Hence, the HAM can be applied to different types of equations with high nonlinearity.With the HAM, the smooth and non-smooth periodic oscillations are studied. The main work of the thesis is as follows:For the smooth periodic oscillation system such as Van der Pol-Duffing forced os-cillator:firstly, the stable/unstable periodic series solutions of the Van der Pol-Duffing forced oscillator are obtained by the HAM. Note the numerical methods such as the fourth-order Runge Kutta are invalid for the unstable periodic solutions here. The sta-bility of the periodic solutions obtained is analyzed in the context of Floquet theory; secondly, the spectral analysis and the bifurcation theory also support the above discus-sion; thirdly, two branches of the frequency-response curve of the Van der Pol-Duffing forced oscillator are unstable in the case of a set of parameters, so the jump phenome-na cannot be observed from the Van der Pol-Duffing forced oscillator; lastly, the idea of multiple scales method is combined into the HAM. Based on that, the periodic solu-tion, period-doubling solution and quasi-periodic solution of the Van der Pol-Duffing forced oscillator are obtained and convergent for a considerable time. Meanwhile, the reliable time Tc is proposed to measure the effective time domain of the approximation to the quasi-periodic solution. Comparing the numerical results with the analytical ap-proximation solutions, the feasibility and effectiveness of this approach are verified by five examples.For non-smooth periodic oscillation system:Based on the HAM, an analytic ap-proach is proposed for the highly nonlinear periodic oscillator with absolute value terms. The non-smoothness of absolute value terms is handled by means of Fourier expansion, and the convergence is accelerated using the iteration method. Therefore, the complexity of the determination about plus or minus sign of the absolute value term can be avoided. The periodic solutions with large amplitudes, multiple stable/un-stable limit cycles and sub-harmonic resonance of this kind of non-smooth periodic oscillator are obtained and studied. It is found that the average method is invalid when natural frequency ω0= 0 and the multiple unstable limit cycle cannot be obtained by the numerical methods such as the fourth-order Runge Kutta. However, the analytic approach proposed in this thesis can overcome all the above limitations.