常微分方程ppt (19).ppt

§5.6 二维自治微分方程组的周期解和 极限环,本节讨论二维自治微分方程组的周期解。,5.6.1 周期解与极限环,如果系统,出发的解,从,则称解 (5.6.2) 是系统 (5.6.1) 的周期解，,周期解在相平面的轨线是一条封闭曲线。,线性系统,的轨线当原点,(0,0) 是中心时，是一族包围原点的封闭曲线，,此时方程组得解都是周期解。,非线性系统,有周期解,除过从原点 (0,0)出发的解外,其它解轨线当时间趋,于无穷时都趋于周期解,Maple 程序(中心),with(DEtools); DEplot([diff(x(t),t)=-y(t)-x(t)*(x(t)^2+y(t)^2-1), diff(y(t),t)=x(t)+y(t)*(x(t)^2+y(t)^2-1)], [x(t),y(t)],t=-1010, [[x(0)=0,y(0)=1],[x(0)=0,y(0)=2], [x(0)=0,y(0)=3],[x(0)=0,y(0)=4], [x(0)=0,y(0)=5],[x(0)=0,y(0)=6], [x(0)=0,y(0)=7]], x=-88,y=-88, stepsize=0.01, linecolor=blue);,Maple 程序(稳定极限环),with(DEtools): DEplot([diff(x(t),t)=-y(t)-x(t)*(x(t)^2+y(t)^2-1), diff(y(t),t)=x(t)-y(t)*(x(t)^2+y(t)^2-1)], [x(t),y(t)],t=-1010, [[x(0)=0.5,y(0)=0],[x(0)=-0.5,y(0)=0], [x(0)=0,y(0)=0.5],[x(0)=0,y(0)=-0.5], [x(0)=0,y(0)=1],[x(0)=4,y(0)=0], [x(0)=-4,y(0)=0],[x(0)=0,y(0)=-4], [x(0)=0,y(0)=4],[x(0)=4,y(0)=4], [x(0)=-4,y(0)=-4],[x(0)=-4,y(0)=4], [x(0)=4,y(0)=-4]], x=-44,y=-44, stepsize=0.01, linecolor=blue);,Maple 程序(不稳定极限环),with(DEtools): DEplot([diff(x(t),t)=-y(t)+x(t)*(x(t)^2+y(t)^2-1), diff(y(t),t)=x(t)+y(t)*(x(t)^2+y(t)^2-1)], [x(t),y(t)],t=-1010, [[x(0)=0.5,y(0)=0],[x(0)=-0.5,y(0)=0], [x(0)=0,y(0)=0.5],[x(0)=0,y(0)=-0.5], [x(0)=0,y(0)=1],[x(0)=4,y(0)=0], [x(0)=-4,y(0)=0],[x(0)=0,y(0)=-4], [x(0)=0,y(0)=4],[x(0)=4,y(0)=4], [x(0)=-4,y(0)=-4],[x(0)=-4,y(0)=4], [x(0)=4,y(0)=-4]], x=-44,y=-44, stepsize=0.01, linecolor=blue);,Maple 程序(半稳定极限环),with(DEtools): DEplot([diff(x(t),t)=-y(t)-x(t)*(x(t)^2+y(t)^2-1)^2, diff(y(t),t)=x(t)-y(t)*(x(t)^2+y(t)^2-1)^2], [x(t),y(t)],t=-1010, [[x(0)=0.5,y(0)=0],[x(0)=-0.5,y(0)=0], [x(0)=0,y(0)=0.5],[x(0)=0,y(0)=-0.5], [x(0)=0,y(0)=1],[x(0)=2,y(0)=0], [x(0)=-2,y(0)=0],[x(0)=0,y(0)=-2], [x(0)=0,y(0)=2],[x(0)=2,y(0)=2], [x(0)=-2,y(0)=-2],[x(0)=-2,y(0)=2], [x(0)=2,y(0)=-2]], x=-22,y=-22, stepsize=0.01, linecolor=blue);,由此可见(5.6.1)中，当,时，情况是比较复杂的。,例 5.6.1 讨论非线性方程组,(5.6.4),在相平面上的轨线分布情况。,是非线性函数,解 引入极坐标,系统 (5.6.4),化为等价系统,系统 (5.6.5) 有三个特解,解 (5.6.6) 即为原点，是一个奇点，解 (5.6.7) 和,(5.6.8) 在相平面上分别是以 (0,0) 为心，半径为,1 和 2 的圆，它们都是系