Define the Poincaré index of a simple closed curve, not necessarily a trajectory, and the Poincaré index of an isolated fixed point $\mathbf{x}_{0}$ for a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{2}$. State the Poincaré index of a periodic orbit.

Consider the system

$$\begin{aligned} &\dot{x}=y+a x-b x^{3} \\ &\dot{y}=x^{3}-x \end{aligned}$$

where $a$ and $b$ are constants and $a \neq 0$.

(a) Find and classify the fixed points, and state their Poincaré indices.

(b) By considering a suitable function $H(x, y)$, show that any periodic orbit $\Gamma$ satisfies

$$\oint_{\Gamma}\left(x-x^{3}\right)\left(a x-b x^{3}\right) d t=0$$

where $x(t)$ is evaluated along the orbit.

(c) Deduce that if $b / a<1$ then the second-order differential equation

$$\ddot{x}-\left(a-3 b x^{2}\right) \dot{x}+x-x^{3}=0$$

has no periodic solutions.