Generalized Lyapunov exponents of the random harmonic oscillator: Cumulant expansion approach

The cumulant expansion is used to estimate generalized Lyapunov exponents of the random-frequency harmonic oscillator. Three stochastic processes are considered: Gaussian white noise, Ornstein-Uhlenbeck, and Poisson shot noise. In some cases, nontrivial numerical difficulties arise. These are mostly solved by implementing an appropriate importance-sampling Monte Carlo scheme. We analyze the relation between random-frequency oscillators and many-particle systems with pairwise interactions like the Lennard-Jones gas.

Figure 1

Lyapunov exponents versus noise strength for the harmonic oscillator with random frequency. Symbols correspond to numerical results for $\lambda$ (small circles) and ${\lambda }_2$ (large circles: importance-sampling; triangles: simple Gaussian sampling). Two values of the damping constant were used: $\alpha =0$ (hollow symbols) and $\alpha =1$ (filled symbols). In both cases, ${\kappa }_0=1$. In all cases, we averaged over ${10}^3$ trajectories. Resampling time was set to $t_{\mathrm{res}}=1.0$. Lines correspond to exact theoretical expressions.

Figure 2

Lyapunov exponents vs noise strength for the harmonic oscillator with random damping. Lines represent exact theoretical results. Symbols correspond to numerical calculations for ${\lambda }_2$ (large circles) and $\lambda$ (small circles). Importance-sampling Monte Carlo was used in the case of ${\lambda }_2$. We chose two values for the average damping coefficient: ${\alpha }_0=1$ (top panel), ${\alpha }_0=4$ (bottom panel). In both cases, $\kappa =1$. In all cases, we averaged over ${10}^3$ trajectories. Resampling time was set to $t_{\mathrm{res}}=0.2$.

Figure 3

Harmonic oscillator with correlated random frequency. Symbols indicate numerical results for ${\lambda }_2$ (full circles) and $\lambda$ (hollow circles) as a function of the noise amplitude $\Delta$ (averages over ${10}^4$ trajectories). Parameters are $\alpha =0$, ${\kappa }_0=0$, and $\tau =1$. Resampling time was set to $t_{\mathrm{res}}=20$. Solid lines correspond to theoretical results. For ${\lambda }_2$ we used the cumulant expansion blue (middle solid line): truncation at the second cumulant; dark blue (top solid line): including the fourth cumulant. An approximate analytical expression for $\lambda$ (red) is also shown.

Figure 4

Harmonic oscillator with Poisson-shot-noise frequency. We show the Lyapunov exponents $\lambda$, ${\lambda }_2$, and ${\lambda }_4$ as a function of collision frequency $\rho$. Red (top solid line) and blue (middle solid line) indicate theoretical estimates for ${\lambda }_4$ and ${\lambda }_2$, respectively, and the corresponding symbols stand for numerical calculations using either simple-Gaussian sampling (open symbols) or importance sampling (full symbols). Also shown is the theoretical result for $2{\lambda }_2-{\lambda }_4$ (black line), which is an estimate of the standard Lyapunov exponent $\lambda$ (circles, numerical).