Subtle escaping modes and subset of patterns from a nonhyperbolic chaotic attractor

Noise-induced escape from the domain of attraction of a nonhyperbolic chaotic attractor in a periodically excited nonlinear oscillator is further investigated. Deviations are found to be amplified at the primary homoclinic tangency from which the optimal force begins to fluctuate dramatically. Escaping trajectories turn out to possess several modes to pass through the saddle cycle on the basin boundary, and each mode corresponds to a certain type of value of the action plot, respectively. A subset of the pattern of fluctuational paths from the chaotic attractor is obtained, showing the existence of complicated singularities.

Figure 1

(a) The Lagrangian manifold is traced out by trajectories of the auxiliary Hamiltonian system. The Lagrangian manifold has folds and cusps, and a trajectory on it is plotted by a curve with arrows. (b) Extreme paths are projections onto the coordinate space. The caustics are projections of folds in (a), and a trajectory going over the fold is described by an extreme path reflected by the caustic.

Figure 2

Global results of (1) without noise in the Poincaré section $q_3=0\left(\mathrm{mod}2\pi \right)$. (a) The basin of attraction of the chaotic attractor indicated by the cyan shaded regions. The magenta curves are the stable manifolds of UC1, and the green curves are its unstable manifolds. (b) A local refinement of the region near the attractor indicated by the blue dots. The red dots denote the saddle cycle of period 9 (UC9 for short), and the black ones denote the saddle cycle of period 3 (UC3 for short). The magenta dots at the bottom represent UC1. The cell size is $0.0009375\times 0.0009375$.

Figure 3

(a) The action plot for our system. The $x$ axis is the angular position to parametrize the escape trajectories, ranging from $\left[0,2\pi \right)$. Some ${10}^6$ points are taken in this range. (b)–(d) Comparisons between the optimal path of the action plot (blue solid lines) and the result of statistical analysis (red, green, and yellow lines). (b)$q_1$. (c) $q_2$. (d) $p_2$.

Figure 4

The optimal path in the section $q_3=0\left(\mathrm{mod}2\pi \right)$, plotted by blue stars with numbers to indicate the orders. The green lines are the stable manifolds of UC1, and the magenta lines are one branch of the unstable manifold of UC1 (left) and the unstable manifold of UC9 (right), respectively. The label −10 corresponds to the same point in Fig. 3.

Figure 5

(a) A local amplification of the action plot. Finer and nested structures are evident, and infinite many local minima between each pair of peaks can be seen. (b) Escape path corresponding to arrow 1 in (a) where a discontinuity is present. (c) Escape path corresponding to arrow 2 in (a). (d) Escape path corresponding to arrow 3 in (a).

Figure 6

Patterns of the fluctuational paths of (4), the initial point of which is chosen as point 0 of UC9. The dashed squares inside indicate where singularities exist.

Figure 7

(a) Fluctuational paths passing through UC1 are plotted with the cyan dotted curves. The red solid curve is obtained by statistical analysis in Fig. 3, and the green dashed curve is the optimal path. (b) Actions calculated along various paths. The minimum is taken by the optimal path, corresponding to the red solid line and the green dashed line.