Paper 2, Section II, A

Differential Equations | Part IA, 2021

The function θ=θ(t)\theta=\theta(t) takes values in the interval (π,π](-\pi, \pi] and satisfies the differential equation

d2θdt2+(λ2μ)sinθ+2μsinθ5+4cosθ=0\frac{\mathrm{d}^{2} \theta}{\mathrm{d} t^{2}}+(\lambda-2 \mu) \sin \theta+\frac{2 \mu \sin \theta}{\sqrt{5+4 \cos \theta}}=0

where λ\lambda and μ\mu are positive constants.

Let ω=θ˙\omega=\dot{\theta}. Express ()(*) in terms of a pair of first order differential equations in (θ,ω)(\theta, \omega). Show that if 3λ<4μ3 \lambda<4 \mu then there are three fixed points in the region 0θπ.0 \leqslant \theta \leqslant \pi .

Classify all the fixed points of the system in the case 3λ<4μ3 \lambda<4 \mu. Sketch the phase portrait in the case λ=1\lambda=1 and μ=3/2\mu=3 / 2.

Comment briefly on the case when 3λ>4μ3 \lambda>4 \mu.

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