求解Duffing方程周期解的大范围收敛方法

A GLOBAL METHOD FOR SOLVING PERIODIC SOLUTION OF DUFFING EQUATION

在条件１ａ（ｔ）ｇｘ（ｔ，ｘ）ｂ（ｔ）４下，构造性地证明了Ｄｕｆｉｎｇ方程２π－周期解的存在唯一性，证明方法同时提供了一种计算周期解的方法．本文利用具全局收敛性的数值延拓算法给出了计算实例．

The existence and uniqueness of 2π periodic solution of Duffing equation have been investigated theoretically by many authors, but few researchers have considered the computational method for approximate solution. In this paper, the following initial value problemx(t)+Cx(t)+g(t,x(t))=e(t) x(0)=α,x(t)=β(*) is firstly considered under the condition1a(t)g x(t,x)b(t)4(**)Denote the solution of (*) by x(t,v), where v=(α,β) T.Define f(v)=(x(t,v),x(t,v)) T, and F(v)=f(v)-v. With this preparation, finding the periodic solution of Duffing equation is transformed into solving F(v)=0. For any given v 0∈R 2, denote H(v,λ,v 0)=F(v)-(1-λ)F(v 0). The following main theorem is constructively proved. Theorem If the continuous function g(t,x) satisfies (**), the solution v=v(λ) of the following initial value problem d v d λ=- -1 F(v 0) v(0)=v 0exists for 0≤λ≤1 and satisfies H(υ,λ,υ 0)≡0. Hence, Duffing equation has a unique 2π periodic solution. Lastly, with the use of the numerical continuation method, some examples are computed. This approach provides a global method for finding solutions of Duffing equation.