Question

3. Consider the nonlinear oscillator equation for x(t) given by ?13 ?

x ̈+ε 3x ̇ −x ̇ +x=0, x(0)=0, x ̇(0)=2a

where a is a positive constant. If ε = 0 this is a simple harmonic oscillator with frequency 1. With non-zero ε this oscillator has a limit cycle, a sort of nonlinear center toward which all trajectories evolve: if you start with a small amplitude, it grows; if you start with a large amplitude, it decays.

For ε ≪ 1, find an approximate solution to the equation that is valid for all time. If you try a regular perturbation expansion, you’ll find a resonant term appears in the first order equation, which means the approximate solution blows up. So instead, introduce a slow time T = εt and use an asymptotic expansion x(t, ε) = x0(t, T ) + εx1(t, T ) + O(ε2). I recommend reading Strogatz Examples 7.6.1 and 7.6.2, and writing the solution to the O(1) part as x0 = r(T ) cos(t + φ(T )) (which is equivalent to the usual linear combination of sin t and cos t). Show that as t → ∞, the solution is approximately x ∼ 2 sin t for any initial amplitude a.

Answer 1