Fluctuation theorems for the undriven Duffing oscillator

Undamped autonomous nonlinear systems with one degree of freedom are described by a Hamiltonian }[p^2+V(x)]/2m. When their damping is of the form  gamma p, we prove by two methods that H(x,p)/gamma is a global Lyapunov function. We exploit this result to verify the Jarzynski relation, past the pitchfork bifurcation of the undriven Duffing oscillator, to within an error of order 10^{-7}. Along the first method, we also obtain the explicit form of a detailed fluctuation theorem, analog to T Delta S= Delta U. We verify numerically this  theorem for the undriven Duffing oscillator, also within an error of order 10^{-7}.