Series Solutions Of Some Classical Dyanmical Sysmtems By Means Of Homotopy Analysis Method

Analytical methods are really important for complex nonlinear problems. People never stop working for developing them. Especially, searching for analytical solution of strongly and complex nonlinear systems is the key to reveal the dynamical hehaviors of them. However, classical perturbation methods achieve great successes for weak nonlinear problems which solved many classical nonlinear problems. But due to their basic hypothesis, depending on small parameters, they cannot apply to strongly nonlinear problems. Very recently, homotopy analysis method is presented as a tool for handling nonlinear problems, which regarded as a outstanding method among non-perturbation methods. The method does not depends on small parameters, thus it can be used to solve both strongly and weak nonlinear problems. Moreover, it provides a certain auxiliary parameter to adjust and control the convergence of the solutions. In this dissertation, we apply the homotopy analysis method to some classical dynamical systems and obtained the following results:Firstly, we apply homotopy analysis method to solve the Kolmogorov equation in turbulence theory. They are two kinds of nonlinear differential equations with fractional nonlinear terms, and the solutions of them are never been found. In Chapter 2, we expand the Kolmogorov equation as a Taylor series, thus the equation is reduced to a nonlinear differential equation with integer term. We choose the polynomial as basic functions to express the solutions of Kolmogorov equations. The obtained solutions have simple series forms. By comparing with numerical solutions, we found this proposed solutions correct the analytical one given by Obukhoff and Yaglom.Secondly, we did some pioneer works on how to obtain the analytical expression of chaotic attractors. In chapter 3, we apply the multistage homotopy analysis method to Lüsystem, united system and Liu system. We successfully obtained the analytical solution of the chaotic attractors in Lüsystem and Liu system, respectively. We found the solutions can be used to re-construct the butterfly shape attractors. By comparing with numerical solution, we found the analytical solutions are exact. Moreover, we used the same method to re-construct the limit cycles of Liu system., and the analytical solutions agree with numerical solution very well.Furthermore, in chapter 4 we apply analytical method to find the analytical solution of the chaotic attractors in some fractional dynamical system. We obtain the analytical solution of the chaotic attractor o fractional Lorenz system and the limit cycle of fractional Brusselator system. Besides, we solve fractional logistic equation by means of standard homotopy analysis method. The solution agrees well with literatures on this equation and exact even for low fractional derivatives.Finally, we apply homotopy analysis method to some strongly nonlinear vibration of engineering structures. Based Galerkin's truncation, the governing equation is reduced to an ordinary differential equation. After applying the homotopy analysis method, the approximate solution of the transverse vibration of the beams is obtained. Numerical results show that the approximate solution is in good agreement with numerical results even the nonlinear coefficient blows up to 100 times. Following the same analytical route, we applied the homotopy analysis method to a pendulum with periodically varying length. Our homotopy analysis solution has a self-adjust parameter to control the convergence interval. Thus, this result is uniform valid for large time interval.