(a) An autonomous dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{2}$ has a periodic orbit $\mathbf{x}=\mathbf{X}(t)$ with period $T$. The linearized evolution of a small perturbation $\mathbf{x}=\mathbf{X}(t)+\boldsymbol{\eta}(t)$ is given by $\eta_{i}(t)=\Phi_{i j}(t) \eta_{j}(0)$. Obtain the differential equation and initial condition satisfied by the matrix $\Phi(t)$.

Define the Floquet multipliers of the orbit. Explain why one of the multipliers is always unity and give a brief argument to show that the other is given by

$$\exp \left(\int_{0}^{T} \nabla \cdot \mathbf{f}(\mathbf{X}(t)) d t\right)$$

(b) Use the energy-balance method for nearly Hamiltonian systems to find leading-order approximations to the two limit cycles of the equation

$$\ddot{x}+\epsilon\left(2 \dot{x}^{3}+2 x^{3}-4 x^{4} \dot{x}-\dot{x}\right)+x=0$$

where $0<\epsilon \ll 1$.

Determine the stability of each limit cycle, giving reasoning where necessary.

[You may assume that $\int_{0}^{2 \pi} \cos ^{4} \theta d \theta=3 \pi / 4$ and $\int_{0}^{2 \pi} \cos ^{6} \theta d \theta=5 \pi / 8$.]