The Jarzynski equality in van der Pol and Rayleigh oscillators

We have studied the Jarzynski equality (JE) in van der Pol and Rayleigh oscillators which are typical deterministic non-Hamiltonian models, but not expected to rigorously satisfy the JE because they are not microscopically reversible. Our simulations that calculate the contribution to the work $W$ of an applied ramp force with a duration $\tau$ show that the JE approximately holds for a fairly wide range of $\tau$ including $\tau \rightarrow 0$ and $\tau \rightarrow \infty$, except for $\tau \sim T$ where $T$ denotes the period of relaxation oscillations in the limit cycle. The work distribution function (WDF) is shown to be non-Gaussian with the $U$-shaped structure for a strong damping parameter. The $\tau$ dependence of $R$ $(=- k_B T \ln<e^{-\beta W}>)$ obtained by our simulations is semi-quantitatively elucidated with the use of a simple expression for limit-cycle oscillations, where the bracket $<\cdot>$ expresses an average over the WDF. The result obtained in self-excited oscillators is in contrast with the fact that the JE holds in the Nos\'{e}-Hoover oscillator which also belongs to deterministic non-Hamiltonian models.