Biased van der Pol relaxation period ($\mathcal{T}\sim k[3+({\lambda }^2-1)\log ((4-{\lambda }^2)/(1-{\lambda }^2))]$)

For $\ddot{x}+k(x^2-1)\dot{x}+x=\lambda$, with fixed $|\lambda |<1$ and $k\to \infty$, the Liénard coordinate $y=\dot{x}/k+x^3/3-x$ has fast horizontal jumps and slow motion on $y=x^3/3-x$. The slow time is $t/k$. Integrating $(x^2-1)dx/dT=\lambda -x$ along $[-2,-1]$ and $[2,1]$ gives the displayed period. The two jumps take negligible time at this scale. For fixed $\lambda >1$, the right attracting branch contains the stable equilibrium and the large relaxation cycle disappears.