User:Peterpan98/sandbox

Example
Consider the following system of differential equations in polar coordinates, $$(\theta, r)\in \mathbb{S}^1\times \mathbb{R}^+ $$:

\begin{cases} \dot{\theta} = 1\\ \dot{r} = (1-r^2)r \end{cases} $$ The flow of the system can be obtained by integrating the equation: for the $$\theta$$ component we simply have $$ \theta(t) = \theta_0 + t $$ while for the $$r$$ component we need to separate the variables and integrate:

\int \frac{1}{(1-r^2)r} dr = \int dt \Longrightarrow \log\left(\frac{r}{\sqrt(1-r^2)}\right) = t+c $$ Inverting last expression gives

r(t) = \sqrt{\frac{e^{2(t+c)}}{1+e^{2(t+c)}}} $$ and since

r(0)=\sqrt{\frac{e^{2c}}{1+e^{2c}}} $$ we find

r(t) = \sqrt{\frac{e^{2t}r_0^2}{1+r_0^2(e^{2t}-1)}} = \sqrt{\frac{1}{1+e^{-2t}\left(\frac{1}{r_0^2}-1\right)}} $$ The flow of the system is therefore

\Phi_t(\theta, r) = \left(\theta+ t, \sqrt{\frac{1}{1+e^{-2t}\left(\frac{1}{r^2}-1\right)}}\right) $$ The behaviour of the flow is the following: Therefore, the solution with initial data $$(\theta_0, r_0\neq 1)$$ draws a spiral that tends towards the radius 1 circle.
 * The angle $$\theta$$ increases monotonically and at constant rate.
 * The radius $$r$$ tends to the equilibrium $$\bar{r}=1$$ for every value.

We can take as Poincaré section for this flow the positive horizontal axis, namely $$ \Sigma = \{(\theta, r) \ : \ \theta =0 \} $$: obviously we can use $$r$$ as coordinate on the section. Every point in $$\Sigma$$ returns to the section after a time $$t=2\pi$$ (this can be understood by looking at the evolution of the angle): we can take as Poicaré map the restriction of $$\Phi$$ to the section $$\Sigma$$ computed at the time $$2\pi$$, $$\Phi_{2\pi}|_{\Sigma}$$. The Poincaré map is therefore :$$\Psi(r) = \sqrt{\frac{1}{1+e^{-4\pi}\left(\frac{1}{r^2}-1\right)}}$$

The behaviour of the orbits of the discrete dynamical system $$ (\Sigma, \mathbb{Z}, \Psi) $$ is the following:
 * The point $$r=1$$ is fixed, so $$\Psi^n(1)=1$$ for every $$n$$.
 * Every other point tends monotonically to the equilibrium, $$\Psi^n(z) \to 1$$ for $$n\to \pm \infty$$.