一类受迫Liénard系统的最终零解

Eventually Vanished Solutions of a Forced Liénard System

寻找一类带有时间依赖强迫项的Linard系统的最终零解,这是一种当t→±∞时趋于0的特殊有界解.由于不是微扰的Hamilton系统,所以不能使用Melnikov方法来判断最终零解的存在性.研究了一个逼近原系统的周期受迫系统序列的周期解序列,并且证明这个周期解序列有一个收敛子列,其极限就是原受迫Linard系统的最终零解.其中使用Schauder不动点定理解决了由非Hamilton造成的困难.

Eventually vanished solutions, a special class of bounded solutions which tend to 0 as t→±∞, of a Lienard system with a time-dependent force were found. Not assuming it to be a small perturbation of a Hamiltonian system, the well-known Melnikov method could not be employed to determine the existence of eventually vanished solutions. A sequence of periodically forced systems was applied to approximate the considered system and their periodic solutions were found, where the difficulties caused by the non-Hamiltonian form were overcome by applying the Schauder＇s fixed point theorem. The fact that the sequence of those periodic solutions has an accumulation gave the existence of an eventually vanished solution of the forced Lienard system.