Define the terms homoclinic orbit, heteroclinic orbit and heteroclinic loop. In the case of a dynamical system that possesses a homoclinic orbit, explain, without detailed calculation, how to calculate its stability.

A second order dynamical system depends on two parameters $\mu_{1}$ and $\mu_{2}$. When $\mu_{1}=\mu_{2}=0$ there is a heteroclinic loop between the points $P_{1}, P_{2}$ as in the diagram.

When $\mu_{1}, \mu_{2}$ are small there are trajectories that pass close to the fixed points $P_{1}, P_{2}$ :

By adapting the method used above for trajectories near homoclinic orbits, show that the distances $y_{n}, y_{n+1}$ to the stable manifold at $P_{1}$ on successive returns are related to $z_{n}$, $z_{n+1}$, the corresponding distances near $P_{2}$, by coupled equations of the form

$$\left.\begin{array}{rl} z_{n} & =\left(y_{n}\right)^{\gamma_{1}}+\mu_{1}, \\ y_{n+1} & =\left(z_{n}\right)^{\gamma_{2}}+\mu_{2}, \end{array}\right\}$$

where any arbitrary constants have been removed by rescaling, and $\gamma_{1}, \gamma_{2}$ depend on conditions near $P_{1}, P_{2}$. Show from these equations that there is a stable heteroclinic orbit $\left(\mu_{1}=\mu_{2}=0\right)$ if $\gamma_{1} \gamma_{2}>1$. Show also that in the marginal situation $\gamma_{1}=2, \gamma_{2}=\frac{1}{2}$ there can be a stable fixed point for small positive $y, z$ if $\mu_{2}<0, \mu_{2}^{2}<\mu_{1}$. Explain carefully the form of the orbit of the original dynamical system represented by the solution of the above map when $\mu_{2}^{2}=\mu_{1}$.