Exercise (10 equally weighted items)
This could have been an exam (exercise).
But the exam will be about \(\dot x=f(x)\), \(\dot x=f(x,r)\), and a bit of \(\dot x=f(x,r,h)\) perhaps.
Periodicity in \(x\) will not occur.
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With \[x=r\cos\theta,\quad y=r\sin\theta\] the linear system \[\dot{x} = ax-by; \] \[\dot{y} = cx+dy \] transforms into \[\frac{\dot r}r=a\cos^2\theta+(c-b)\cos\theta\sin\theta+d\sin^2\theta;\] \[\dot\theta=c\cos^2\theta+(d-a)\cos\theta\sin\theta+b\sin^2\theta.\] The right hand sides are \(\pi\)-periodic in \(\theta\).

For this exam we forget about the \(r\)-equation and take \(a=5,\,b=c=d=1\) for a start. Then \[\dot\theta=\cos^2\theta-4\cos\theta\sin\theta+\sin^2\theta=1-2\sin2\theta.\]

1. Let \(\bar\theta>0\) be the smallest positive steady state for the \(\theta\)-equation. Explain why it is stable.
2. Determine the smallest unstable stable positive steady state and explain why it is unstable.
3. Draw a horizontal \(\theta\)-line to exhibit the stationary states and the flow of the \(\theta\)-equation.
4. Derive an equation for \(S=\tan\theta\).
5. Explain why it has two steady states, both positive, and why the smallest one is stable.
6. For which \(a\) and \(d\) in the orginal \(\theta\)-equation is this also the case if \(b=c\)?


Now take \[a=2,b=c=d=1,\] and let \(\theta(t)\) be the solution with \(\theta(0)=0\) for what follows.


7. Explain why \(\theta(t)\to\infty\) as \(t\to\infty\) and \(\theta(t)\to-\infty\) as \(t\to-\infty\).
8. Suppose that \(T\) is a solution of \(\theta(T)=\pi\). Explain why \[T=\int_0^\pi\frac{d\theta}{1-\frac12\sin(2\theta)}.\]
9. What is the equation for \(S=\tan\theta\) in this case?
10. Explain why every positive solution \(S(t)\) blows up both in forward and negative finite time.