(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 $\boldsymbol{\Phi}(t)$.

Define the Floquet multipliers of the orbit. Explain why one of the multipliers is always unity and 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 a leadingorder approximation to the amplitude of the limit cycle of the equation

$$\ddot{x}+\epsilon\left(\alpha x^{2}+\beta \dot{x}^{2}-\gamma\right) \dot{x}+x=0,$$

where $0<\epsilon \ll 1$ and $(\alpha+3 \beta) \gamma>0$.

Compute a leading-order approximation to the nontrivial Floquet multiplier of the limit cycle and hence determine its stability.

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