Fluctuational transitions across different kinds of fractal basin boundaries

We study fluctuational transitions in discrete and continuous dynamical systems that have two coexisting attractors in phase space, separated by a fractal basin boundary which may be either locally disconnected or locally connected. Theoretical and numerical evidence is given to show that, in each case, the transition occurs via a unique accessible point on the boundary, both in discrete systems and in flows. The complicated structure of the escape paths inside the locally disconnected fractal basin boundary is determined by a hierarchy of homoclinic points. The interrelation between the mechanism of transitions and the hierarchy is illustrated by consideration of fluctuational transitions in dynamical systems demonstrating “fractal-fractal” basin boundary metamorphosis at some value of a control parameter. The most probable escape path from an attractor, which can be either regular or chaotic, is found for each type of boundary using both statistical analysis of fluctuational trajectories and the Hamiltonian theory of fluctuations.

Figure 1

(a) (Color online) One of the coexisting CAs is indicated by the centrally placed filled curve. Its basin of attraction is shown in white, whereas points belonging to the other basin are shaded in gray. The most probable escape path connecting the CA with the period-3 saddle cycle $S3$ lying on the fractal boundary is shown by the sequentially numbered small circles (boundary value problem) and stars (Monte Carlo simulations), which practically coincide. The remaining unnumbered circles show how the MPEP evolves, just outside the basin of attraction of the initial state. (b) The $x$ coordinates of the optimal escape paths shown in (a) which were obtained both from the Monte Carlo simulations with $D={10}^{-5}$ (dashed line) and from the solution of the boundary value problem (solid line). The abscissa of the saddle point $S1$ is shown by the thin-dashed line. (c) The optimal fluctuational force moving the system (1) to the LD FBB. All units are dimensionless.

Figure 2

Details of the escape process: $x$ is a point inside the accessible basin of attraction $\Omega 1$, $f\left(x\right)$ is the noise-free image of $x$, and $y$ is the point on the boundary $d\Omega$ that minimizes the “energy” cost; $y^{*}$ is another (nonoptimal) point in $\Omega 2$ (the set of all points in the space is inaccessible from $\Omega 1$). The units are dimensionless.

Figure 3

Illustration of the hierarchical heteroclinic intersections between the stable and unstable manifolds of the original saddles. The units are dimensionless.

Figure 4

(Color online) (a) The LD FBB separating the stable point of period 2 (labeled as $C2$) from the attractor at infinity, at $A=1.38$, before the “fractal-fractal” basin boundary metamorphosis (dimensionless units). The unstable manifold of the saddle point of period 3 $\left(S3\right)$ and the stable manifold of the accessible boundary point of period 4 $\left(S4\right)$, indicated by the solid lines, do not intersect. (b) The LD FBB at $A=1.405$ after the “fractal-fractal” boundary crisis. The saddle point $S3$ has become the accessible boundary point, and $S4$ lies inside the LD FBB which has jumped inside the open neighborhood containing the stable point $C2$.

Figure 5

(a) (Color online) The LD FBB of the system (10) before the boundary crisis $A=1.38$. The stable manifold of the saddle point $S3$ (indicated by crosses) shown by the solid lines. The optimal escape path obtained from the Monte Carlo simulations with $D=1.3\times {10}^{-3}$ is shown by the thick solid line, while points belonging to the MPEP are shown by open circles. (b) The $x$ coordinates of the optimal escape paths before (solid line) and after (dashed line) the “fractal-fractal” boundary crisis at $A=1.405$. (c) The optimal fluctuational forces calculated before (black filled squares) and after (open circles) the crisis. All units are dimensionless.

Figure 6

(Color online) (a) Stroboscopic $(\Omega t=0,2\pi ∕\Omega ,4\pi ∕\Omega ,\dots )$ section of the system (12) before the “fractal-fractal” basin boundary crisis at $\Gamma =1$, $\alpha =10$, $\beta =100$, $\Omega =3.76$, and $A=0.9$ (dimensionless units). Two coexisting stable points of period 2 are indicated by filled triangles. Their basins of attraction are colored in white and gray, respectively. The stable manifolds of the saddle points $S3$ and $S4$ (indicated by filled circles) are shown by the solid thin lines. The optimal escape path obtained from the Monte Carlo simulations with $D=3\times {10}^{-5}$ is shown by the thick solid line. (b) The same stroboscopic section after the boundary crisis at $A=0.915$. All other parameters have the same values as in the previous figure. The optimal escape path obtained from the simulations with $D=2\times {10}^{-5}$ is shown by the thick solid line.

Figure 7

(a) The locally connected FBB (solid closed curve), unstable node of period 9 (crosses), and points on the optimal escape path obtained from the Monte Carlo simulations (filled circles) with $D=5\times {10}^{-3}$. Points of the MPEP are shown by empty squares. (b) The $x$ (solid line) and $y$ (dashed line) components of the optimal fluctuational force. All units are dimensionless.