n-维分段光滑微分系统的周期轨分支

Bifurcation of Periodic Orbits of an n-Dimensional Piecewise Smooth Differential System

研究了n-维分段光滑扰动微分系统{x1=x2+εg1^+(x),x2=-x1+εg3^+(x),x3=εg3^+(x),…xn=εgn^+(x),x1≥0,{x1=x2+εg1^-(x),x2=-x1+εg2^-(x),x3=εg3^-(x),x1<0,…xn=εgn^-(x)x1<0,其中x=(x1,x2,…,xn)^T,0<ε≤1,且gi^±(x),i=1,2,…,n是关于x的m次实系数多项式.应用一阶Melnikov向量函数,得到了从其未扰动系统分支出周期轨个数的上界.

In this paper,we study the following n-dimensional piecewise smooth differential system{x1=x2+εg1^+(x),x2=-x1+εg3^+(x),x3=εg3^+(x),…xn=εgn^+(x),x1≥0,{x1=x2+εg1^-(x),x2=-x1+εg2^-(x),x3=εg3^-(x),x1<0,…xn=εgn^-(x) x1<0,where x=(x1,x2,…,xn)^T,0<ε≤1,and gi^±(x),i=1,2,…,n are real polynomials of x with degree m.By using the first order Melnikov vector function,we obtain the upper bound of the number of periodic orbits bifurcating from the unperturbed system.