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}.
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