Fluctuational Transitions through a Fractal Basin Boundary

Fluctuational transitions between two coexisting chaotic attractors, separated by a fractal basin boundary, are studied in a discrete dynamical system. It is shown that the transition mechanism is determined by a hierarchy of homoclinic points. The most probable escape path from a chaotic attractor to the fractal boundary is found using both statistical analyses of fluctuational trajectories and the Hamiltonian theory of fluctuations.

Figure 1

The coexisting chaotic attractors (solid black regions) and their basins of attraction represented by gray and white, respectively. The accessible boundary saddle points of period 3 are shown by the small black circles labeled S3. Their stable manifolds are shown as solid black lines. The saddle points of period 1 are shown by the crosses labeled S1. The saddle point at the origin is labeled O.

Figure 2

(a) The most probable escape path (dashed line) connecting the CA with the period-3 saddle cycle lying on the fractal boundary, obtained from the Monte Carlo simulations with $D={10}^{-5}$. The optimal path found by the solution of the boundary-value problem (see text) is shown as a solid line. The $x$ coordinate of the saddle point S1 is shown by the horizontal dashed line. (b) A two-dimensional plot of the paths presented in (a) where the results obtained by solution of the boundary-value problem [consecutively numbered points indicated by circles, corresponding to the numbered points in (a)] coincide almost perfectly with those obtained by numerical simulation (stars). Inset in (a): the optimal force as determined in the numerical simulation.

Figure 3

Probabilities of finding a fragment corresponding to the different period-$T$ original saddle cycle in the collected escape trajectories calculated for the different values of the noise intensity $D$.