微分方程(週期解和極限圈)

Question

In the problem the autonomous system is expressed in polar coordinates

Determine the periodic solutions, the limit cycles, and determine the

stability characteristics.

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dr/dt = sinπr,

dθ/dt = 1

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thanks for help~~!

已更新項目:

大大您好，有一個地方有疑問:

-1/2π*∫{1/((1-cos(πr))+1/(1+cos(πr))}d(cos(πr))

=-1/2π*{ln((1+cos(πr))(1-cos(πr)))}

我覺得 ln 裡面的應該是(1+cos(πr)) / (1-cos(πr))吧?

不知道對不對，謝謝~!

Answer 1

最佳解答

In the problem the autonomous system is expressed in polar coordinates

Determine the periodic solutions, the limit cycles, and determine the

stability characteristics. Thanks for help~~!dr/dt=sin(πr)dθ/dt=1Ans:Let w=angular speeddθ/dt=1 => ∫dθ=∫dt+a (a.b=積分常數)t+a=θ=w*t => t=a/(w-1)∫dt=∫dr/sin(πr)t+b=∫sin(πr)d(πr)/πsin^2(πr)=-1/π*∫d(cos(πr))/(1-cos^2(πr))=-1/2π*∫{1/((1-cos(πr))+1/(1+cos(πr))}d(cos(πr))=-1/2π*{ln((1+cos(πr))(1-cos(πr)))}=-1/2π*ln(1-cos^2(πr))=-1/2π*ln(sin^2(πr))=-1/π*ln(sin(πr))ln(sin(πr))=-π(t+b)periodic solutions: sin(πr)=e^(-π(t+b))=e^(-π(a/(w-1)+b))=e^(-c).........c=-π(a/(w-1)+b)limit cycles:πr=asin(e^(-c)) => r=asin(e^(-c))/πstability characteristics:lim(c->infinity)e^(-c)=0sin(πr)=0 => r=0,1,2,3,....

2013-05-21 14:05:11 補充：

Update for limit cycles:

πr=nπ+asin(e^(-c)) (n=0.1.2.3.....)

=> r=n+asin(e^(-c))/π

2013-05-21 16:49:05 補充：

-(t+b)π=∫d(cos(πr))/(1-cos^2(πr))

=∫du/(1-u^2).....u=cos(πr)

=0.5∫(1/(1+u)+1/(1-u))du

=0.5ln((1+u)(1-u))

Check: ((1-u)+(1+u))/2(1-u^2)=1/(1-u^2)