Past exam of the differential equations and applications course of the University of Oxford / 2007 / A / 2 / Solution

For zero damping the conserved mechanical energy isFor $\gamma <0$, the potential grows to infinity in both directions, so every nonstationary finite-energy trajectory is trapped between two turning points and is periodic; the zero-energy solution is stationary. The case $\gamma =0$ is the harmonic oscillator. The negative-$\gamma$ comparison extends the parameter range initially stated in the source.

For $\gamma >0$, the origin lies in a local potential well, with barriers at $x=\pm {\gamma }^{-1/2}$ of height $1/(4\gamma )$. A trajectory launched from the origin oscillates precisely when its initial energy is below that barrier:At equality the trajectory approaches a barrier equilibrium asymptotically, rather than being periodic; for example the positive branch is $x={\gamma }^{-1/2}\tanh (t/\sqrt{2})$. Above the barrier the trajectory escapes the well. The source's inequality without the absolute value tacitly takes $v\ge 0$; for signed initial velocity the energy criterion requires $|v|$.

Next set $\beta =\gamma =0$ and assume $\alpha >0$. The energy $H=({\dot{x}}^2+x^2)/2$ satisfies $\dot{H}=-\alpha x^2{\dot{x}}^2\le 0$. This bounds the complete trajectory. The zero-dissipation set is the union of $x=0$ and $\dot{x}=0$, and its only invariant trajectory is the origin: a nonzero point on either axis immediately leaves it under the differential equation. The LaSalle invariance principle therefore gives $\boxed{x(t)\to 0,\dot{x}(t)\to 0}$. If $\alpha =0$ as well, damping is absent and that conclusion fails for nonzero $v$; the positive order-one damping assumption is essential.

For the weakly nonlinear periodic motion, write $\gamma =\alpha \kappa$, set $\tau =\omega t$, $\omega =1+\alpha {\omega }_1+\cdots$, and expand $x=A\cos \tau +\alpha x_1+\cdots$. The first correction obeysResonant forcing at the fundamental frequency would generate secular terms. Eliminating both coefficients yieldsConsequently the maximum displacement is $\boxed{2\sqrt{\beta }+O(\alpha )}$ for fixed positive $\beta$. A possible third-harmonic correction is $x_1=-A^3\sin 3\tau /32-\kappa A^3\cos 3\tau /32$, up to homogeneous amplitude and phase choices. The leading amplitude also balances averaged damping: its slow evolution is $\dot{A}=\alpha A(\beta /2-A^2/8)$, whose nonzero fixed point is attracting. The case $\beta =0$ is a degenerate zero-amplitude limit, rather than a nonzero limit cycle.