I'm struggling with exercise 4.1.9 of Strogatz's Nonlinear Dynamics and Chaos.
The questions asks: "In exercises 2.6.2 and 2.7.7 you were asked to give two analytical proofs that periodic solutions are impossible for vector fields on the line. Review these arguments and explain why they don't carry over to vector fields on the circle. Specifically which parts fail?"
It's the first part I'm stuck on.
First, a review of the case on the line. We have $\dot{x} = f(x)$. We assume $f(x)$ to be nontrivial, and we assume that there exists $T>0$ such that $x(t) = x(t+T)$ and $x(t) \neq x(t+s)$ for $0<s<T$. We then consider the integral:
$I = \displaystyle\int^{t+T}_t f(x)\frac{dx}{dt}dt = \int^{x(t+T)}_{x(t)}f(x)dx = F(x(t+T)) - F(x(t)) = 0$,
since $x(t)=x(t+T)$. However:
$I = \displaystyle\int^{t+T}_{t}[f(x)]^2dt > 0$,
since $f(x)$ is nontrivial and $T>0$. Hence our assumptions must be incorrect.
Now, for the circle, we have $\dot{\theta} = f(\theta)$, where $\theta = \theta(t)$ is periodic with period $T$, and $f(\theta) = f(\theta+ 2\pi)$ i.e periodic with period $2\pi$. Then considering
$\displaystyle I_\theta = \int^{t+T}_t f(\theta)\frac{d\theta}{dt}dt = 0$
by the argument above. (This step seems to me to hold, unless I'm missing something about the anti-derivative of periodic functions).
Now, where I expect the argument to fail is when we look at the second part:
$\displaystyle I_\theta = \displaystyle\int^{t+T}_{t}[f(\theta)]^2dt = \int^{t+T}_{t}\left[\frac{d\theta}{dt}\right]^2dt$.
For this to vanish, we need $\frac{d\theta}{dt} = 0$ between the two limits. Wouldn't this again imply that $\theta$ is just constant?