Random kicks leave a non-Gaussian oscillator fingerprint

A stable underdamped oscillator is kicked at Poisson arrival times \(t_j\): \[ m\ddot x+m\Gamma\dot x+m\omega_0^2x =\sum_jJ_j\delta(t-t_j). \] The arrivals have rate \(\lambda\), and the independent impulse amplitudes \(J_j\) are also independent of the arrival process. Work in the stationary regime extending from the remote past. Derive the causal displacement response to one unit impulse and express \(x(t)\) as a shot-noise sum. For general kick moments, derive the mean, the continuous two-sided force spectrum after removing any DC mean, and the displacement spectrum. State and use the compound-Poisson cumulant rule \[ \kappa_n[x(t)]=\lambda\langle J^n\rangle \int_0^\infty h(s)^n\,ds. \] Evaluate the second and third equal-time cumulants explicitly. Explain why a Gaussian white force with the same spectrum reproduces the variance but not the third cumulant. Take \(m=0.500\,\mathrm{kg}\), \(\Gamma=1.00\,\mathrm{s^{-1}}\), \(\omega_0=5.00\,\mathrm{s^{-1}}\), \(\lambda=4.00\,\mathrm{s^{-1}}\), and let \(J=2J_0\) with probability \(1/3\), while \(J=-J_0\) with probability \(2/3\), where \(J_0=0.0200\,\mathrm{N\,s}\). Compute \(\omega_d\), the kick moments through order three, displacement variance, RMS displacement, third cumulant, standardized skewness, and \(S_x(0)\).

For one force impulse, \(h(t)=\Theta(t)e^{-\Gamma t/2}\sin(\omega_dt)/(m\omega_d)\).