Either by hand or using a computer software package, sketch the phase plane diagrams for the given system. Discuss the stability of each limit cycle and critical point. Here $x=r \cos \theta$ and $y=r \sin \theta$ .
$\frac{d x}{d t}=y+x \sin (1 / r)$ ,
$\frac{d y}{d t}=-x+y \sin (1 / r)$

Solution

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Multiplying the first equation by $x$ , the second one by $y$ and adding, we get

\begin{align*} x\frac{dx}{dt}+y\frac{dy}{dt}&=x\big(y+x\sin(1/r)\big)+y\big(-x+y\sin(1/r)\big)\\ &=xy+x^2\sin(1/r)-xy+y^2\sin(1/r)\\ &\qquad\text{Simplify}\\ &=x^2\sin(1/r)+y^2\sin(1/r)\\ &\qquad\text{Take out common factor}\\ &=(x^2+y^2)\sin(1/r)\\ &\qquad\text{Use the fact that $x^2+y^2=r^2$}\\ &=r^2\sin(1/r) \end{align*}

Using the identity

x\frac{dx}{dt}+y\frac{dy}{dt}=r\frac{dr}{dt},

we have

r\frac{dr}{dt}=r^2\sin(1/r).

Simplify

\frac{dr}{dt}=r\sin(1/r).

We have that $dr/dt=0$ when $\sin(1/r)=0\implies 1/r=n\pi\implies r=1/(n\pi),~n=1,2,\ldots$ . Thus, the origin ( $r=0$ ) is not an isolated critical point. Note that $dr/dt>0$ ( $\sin(1/r)>0$ ) for $2n\pi<1/r<(2n+1)\pi$ or $1\big/[(2n+1)\pi]<r<1\big/(2n\pi)$ , and $dr/dt<0$ for $1\big/(2n\pi)<r<1\big/[(2n-1)\pi]$ .

Consequently, the trajectories spiral into the limit cycles $r=1/(2n\pi)$ (hence, stable) and away from the limit cycles $r=1\big/[(2n+1)\pi]$ (hence, unstable). {\bf The origin is a stable critical point}. To realize this, notice that for every $\epsilon>0$ there exists a positive integer $n_\epsilon$ such that $1/(2n_\epsilon\pi)<\epsilon$ , so we can simply choose $\delta_\epsilon=1/(2n_\epsilon\pi)$ . Then, the fact that $r=1/(2n_\epsilon\pi)$ is a stable limit cycle guarantees that any trajectory that begins within $\delta_\epsilon$ of the origin remains within $\epsilon$ of the origin.

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More related questions

accounting

Garcia Corporation recently hired a new accountant with extensive experience in accounting for partnerships. Because of the pressure of the new job, the accountant was unable to review what he had learned earlier about corporation accounting. During the first month, he made the following entries for the corporation’s capital stock.

May 2       Cash                                            104,000
              Capital Stock                                             104,000
                (Issued 8,000 shares of $10 par value
                common stock at$13 per share)
                
   10       Cash                                            530,000
              Capital Stock                                             530,000
                (Issued 10,000 shares of $20 par value
                preferred stock at$53 per share)
 
   15       Capital Stock                                     7,200
              Cash                                                        7,200
                (Purchased 600 shares of common stock
                for the treasury at $12 per share)

Instructions

On the basis of the explanation for each entry, prepare the entries that should have been made for the capital stock transactions.

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accounting

May 2       Cash                                            104,000
              Capital Stock                                             104,000
                (Issued 8,000 shares of $10 par value
                common stock at$13 per share)
                
   10       Cash                                            530,000
              Capital Stock                                             530,000
                (Issued 10,000 shares of $20 par value
                preferred stock at$53 per share)
 
   15       Capital Stock                                     7,200
              Cash                                                        7,200
                (Purchased 600 shares of common stock
                for the treasury at $12 per share)

Instructions

On the basis of the explanation for each entry, prepare the entries that should have been made for the capital stock transactions.

1/7

Related questions with answers

Either by hand or using a computer software package, sketch the phase plane diagrams for the given system. Discuss the stability of each limit cycle and critical point. Here $x=r \cos \theta$ and $y=r \sin \theta$ .
$\frac{d x}{d t}=y+x \sin (1 / r)$ ,
$\frac{d y}{d t}=-x+y \sin (1 / r)$

Create a free account to view solutions

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