关于Duffing方程周期解的研究

Study on Periodic Solutions to Some Duffing Equation

证明了 Duffing方程 x¨ + g( x) =p( t)至少有一个调和解和无穷多的次调和解 .其中 g( x)是导数大于零的奇函数 ,且当 x趋于正无穷大时存在正的极限值 ;p( t)是连续的 2π周期函数 ,满足∫2π0 p( t) dt=0 .在证明中使用了 Poincare′-Birkhoff定理和一个不动点定理 .

Prove the existence of harmonic solution and infinitely many subharmonic solutions for Duffing's equation x¨+g(x)=p(t) where g(x) is an odd function satisfying g′(x)>0 and limx→+∞g(x)=a>0;continuous 2π-periodic function p(t) satisfies:∫ 2π 0p(t)dt=0.Using a generalized form of the Poincare′-Birkhoff theorem and a fixed point theorem.