(i) Define the Poincaré index of a curve $\mathcal{C}$ for a vector field $\mathbf{f}(\mathbf{x}), \mathbf{x} \in \mathbb{R}^{2}$. Explain why the index is uniquely given by the sum of the indices for small curves around each fixed point within $\mathcal{C}$. Write down the indices for a saddle point and for a focus (spiral) or node, and show that the index of a periodic solution of $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ has index unity.

A particular system has a periodic orbit containing five fixed points, and two further periodic orbits. Sketch the possible arrangements of these orbits, assuming there are no degeneracies.

(ii) A dynamical system in $\mathbb{R}^{2}$ depending on a parameter $\mu$ has a homoclinic orbit when $\mu=\mu_{0}$. Explain how to determine the stability of this orbit, and sketch the different behaviours for $\mu<\mu_{0}$ and $\mu>\mu_{0}$ in the case that the orbit is stable.

Now consider the system

$$\dot{x}=y, \quad \dot{y}=x-x^{2}+y(\alpha+\beta x)$$

where $\alpha, \beta$ are constants. Show that the origin is a saddle point, and that if there is an orbit homoclinic to the origin then $\alpha, \beta$ are related by

$$\oint y^{2}(\alpha+\beta x) d t=0$$

where the integral is taken round the orbit. Evaluate this integral for small $\alpha, \beta$ by approximating $y$ by its form when $\alpha=\beta=0$. Hence give conditions on (small) $\alpha, \beta$ that lead to a stable homoclinic orbit at the origin. [Note that $y d t=d x$.]