Dynamical Systems and Control
Vol.3 No.03(2014), Article ID:13842,8 pages
DOI:10.12677/DSC.2014.33005

Homotopy Analysis Method for Heterclinic Orbit of Michelson System

Wankai Liu, Youhua Qian*

College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua

Email: *qyh2004@zjnu.cn

Received: May 25th, 2014; revised: Jun. 4th, 2014; accepted: Jun. 12th, 2014

In this paper, we use the homotopy analysis method (HAM) to obtain the analytic approximation of heterclinic orbit in Michelson system. Comparisons are made between the results of the proposed method and exact solutions. The results show that the HAM is an effective and practical technique of analytic approximation for the heterclinic orbit. The proof of convergence theorems for the present method is elucidated as well.

Keywords:Homotopy Analysis Method, Heterclinic Orbit, Convergence Theorems

Email: *qyh2004@zjnu.cn

(1.1)

(1.2)

(2.1)

, (2.2)

, (2.3)

, (2.4)

, (2.5)

, (2.6)

，由零阶形变方程(2.6)易知，因此得到：当由0增大到1时，由初值猜测变化到精确解

, (2.7)

. (2.8)

. (2.9)

. (2.10)

, (2.11)

, (2.12)

. (2.13)

(2.14)

(3.1)

(3.2)

(3.3)

Michelson系统(3.1)满足初始条件(3.3)的精确解为

(3.4a)

(3.4b)

(3.4c)

(3.5)

(3.6a)

(3.6b)

(3.6c)

(3.7)

(3.8)

(3.9)

(3.10)

(3.11)

(3.12)

(3.13a)

(3.13b)

(3.13c)

(3.14a)

(3.14b)

(3.14c)

(3.15a)

(3.15b)

(3.15c)

(3.16)

(3.17a)

(3.17b)

(3.17c)

Figure 1.The -curve of, and obtained from the tenth-order approximation of Equation (3.1)

30-order approximate solution … 42-order approximate solution exact solution

Figure 2. Comparison of the phase portrait curves of the 30 and 42-order approximate solution with the exact solution

Approximatesolution exactsolution

Figure 3.Comparison of the phase portrait curves of the 42-order approximate solution with the exact solution

, (4.1)

. (4.2)

(4.3)

. (4.4)

. (4.5)

. (4.6)

. (4.7)

. (4.8)

(4.9)

. (4.10)

. (4.11)

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*通讯作者。