The paper proposes an algorithm to solve a general Duffing equation, in which a process of transforming the initial equation to a resulting equation is proposed, and then the coupling successive approximation method is applied to solve the resulting equation. By using this algorithm a special physical factor and complex-valued solutions to the general Duffing equation are revealed. | Volume 36 Number 2 2 2014 Vietnam Journal of Mechanics, VAST, Vol. 36, No. 2 (2014), pp. 77 – 93 A COUPLING SUCCESSIVE APPROXIMATION METHOD FOR SOLVING DUFFING EQUATION AND ITS APPLICATION Dao Huy Bich1 , Nguyen Dang Bich2,∗ 1 Hanoi University of Science, VNU, Vietnam 2 Institute for Building Science and Technology (IBST), Hanoi, Vietnam ∗ E-mail: dangbichnguyen@ Received March 15, 2014 Abstract. The paper proposes an algorithm to solve a general Duffing equation, in which a process of transforming the initial equation to a resulting equation is proposed, and then the coupling successive approximation method is applied to solve the resulting equation. By using this algorithm a special physical factor and complex-valued solutions to the general Duffing equation are revealed. The proposed algorithm does not use any assumption of small parameters in the equation solving. The coupling successive procedure provides an analytic approximated solution in both real-valued or complex-valued solution. The procedure also reveals a formula to evaluate the vibration frequency, ϕ, of the non-linear equation. Since the first approximation solution is in a closed-form, the chaos index of the general Duffing equation and the chaotic characteristics of solutions can be predicted. Some examples are used to illustrate the proposed method. In the case of chaotic solution, the Pointcaré conjecture is used for solution verification. Keywords: General Duffing equation, coupling successive approximation method, chaos index, chaotic structures of solutions. 1. INTRODUCTION Many dynamical problems lead to a general Duffing equation [1–6] with not only the third order but also second order nonlinearities. Recently, there have been increasing researches dealing with the Duffing equation with a set of chaotic solutions. The commonly index used to recognize the chaotic solutions of the Duffing equation is Liapunov index [7], which requires a complex calculation. Therefore, there
