Effect of colored noise on heteroclinic orbits

The dynamics of a weakly dissipative Hamiltonian system submitted to stochastic perturbations has been investigated by means of asymptotic methods. The probability of noise-induced separatrix crossing, which drastically changes the fate of the system, is derived analytically in the case where noise is an additive Kubo-Anderson process. This theory shows how the geometry of the separatrix, as well as the noise intensity and correlation time, affect the statistics of crossing. Results can be applied to a wide variety of systems, and are valid in the limit where the noise correlation time scale is much smaller than the time scale of the undisturbed Hamiltonian dynamics.

Figure 1

Sketch of the undisturbed dynamics with a heteroclinic trajectory $AB$ [solid line, ($\mathrm{\Lambda }=0$ and ${\varepsilon }_i=0$)]. The long-dashed line is a trajectory under weak dissipative perturbation ($\mathrm{\Lambda }>0$ and ${\varepsilon }_i=0$). The thin-dashed line is a realization of the weakly dissipative stochastic system ($\mathrm{\Lambda }>0$ and ${\varepsilon }_i>0$).

Figure 2

Sketch of the plane $({\varepsilon }_1,{\varepsilon }_2)$, together with the triangular zone where condition (20) is fulfilled and the system is sheltered from noise.

Figure 3

Histograms of normalized noise amplitudes ${\xi }_k$ for the mechanical pendulum [(a) uniformly distributed noise, (c) dichotomous noise]. Durations $\delta {\tau }_k$ follow an exponential distribution with average $\delta \tau =0.005$. Graphs on the right column are the histograms of the corresponding Hamiltonian jump [(b) uniformly distributed noise, (d) dichotomous noise]. Solid lines in (b) and (d) are the normal distribution with average and variance taken from the theoretical result (44).

Figure 4

Probability of noise-induced escape of the mechanical pendulum, for ${\omega }_0=1$ and various friction coefficients $\mathrm{\Lambda }$. Symbols are the percentage of escape obtained from numerical simulations, by counting the number of pendulums for which $\theta >\pi$ during the run. Circles: uniformly distributed noise intensities. Triangles: dichotomous noise. Solid lines: theoretical result (45). Inset (ii) shows the same data, but in terms of the renormalized variable $X={\varepsilon }_2\sqrt{\delta \tau }/\mathrm{\Lambda }{\sqrt{\omega }}_0$. Inset (i) shows a typical cloud of particles in the phase space $(q,p)$, as it reaches stagnation point $B$.

Figure 5

Probability of noise-induced escape of the mechanical pendulum, for $\mathrm{\Lambda }=0.002,{\varepsilon }_2=0.06,\delta \tau =0.05$, and ${\omega }_0$ in the range [1,30]. Circles: uniformly distributed noise intensities. Triangles: dichotomous noise. For the dichotomous noise, the abrupt drop of the probability is clearly visible as ${\omega }_0$ approaches ${\omega }_0^c=23.6$.