Solution of the boundary value problem for optimal escape in continuous stochastic systems and maps

Topologies of invariant manifolds and optimal trajectories are investigated in stochastic continuous systems and maps. A topological method is introduced that simplifies the solution of boundary value problems: The activation energy is calculated as a function of a set of parameters characterizing the initial conditions of the escape path. The method is applied explicitly to compute the optimal escape path and the activation energy for a variety of dynamical systems and maps.

Figure 1

Structure of the escape trajectories for the IVDP system. Here the origin is a stable point for the noise-free dynamics and the cycle (bold line) is an unstable limit cycle. The MPEP is shown as a thick line. It reaches the limit cycle asymptotically with no momentum. Some trajectories that are small perturbations of the MPEP are also shown (thin lines). They may either cross the cycle or be reflected, moving back to the initial state. The MPEP separates these two families of trajectories.

Figure 2

(Color online) (a) Points in the vicinity of a period-3 orbit are described using the integer $i∊[1\cdots 3]$ and a real vector $\delta x$ describing the displacement; (b) the region of coordinate space in the neighborhood of a limit cycle is described using a real parameter $\gamma$ and a vector $\delta x$ measuring the displacement.

Figure 3

(Color online) Comparison between numerical simulation (small dots) and theory (black thin line) for the system (46). The noise intensity is $ϵ=0.1$, and parameter $a=0.4$. The numerical value of $W=0.842255$ and the theoretical value is $W=1-a^2=0.840000$. The theory and the numerics almost completely overlap. Similar agreement has been obtained for a wide range of values of $a$ and noise intensities $ϵ$.

Figure 4

(Color online) (a) Possible way to parametrize the unstable manifold of a stable point in a two-dimensional system. The point $P$ is chosen on a small circle (radius $⪡\sqrt{ϵ}$); the angular position is the parameter we use to describe the set of escape trajectories on the manifold. (b) The parametrization of the trajectories for a two-dimensional map. The initial conditions are chosen inside an annulus limited by radii $r_1$ and $r_2$, whose values are chosen so as to cover all possible trajectories on the manifold.

Figure 5

(Color online) Three different classes of trajectories are possible for a system with a saddle cycle. ${\gamma }_0$ is the heteroclinic path: It reaches the cycle (dashed line) asymptotically for $t\to +\infty$; ${\gamma }_1$ has a bigger momentum and crosses the cycle; ${\gamma }_2$ has insufficient momentum to reach the cycle and it is reflected back.

Figure 6

(a) The action plot for the IVDP system. The typical hills-peaks structure is evident. Arrows (1) and (2) show two discontinuity points in the first hill. Arrows (3) and (4) show other discontinuities in the second hill. (b) A zoom of the first peak [area included between the points marked by the arrows (1) and (2)] reveals the self-repeating structure of the action plot. The points marked with (5) and (6) are other discontinuity points in the peak.

Figure 7

Interesting paths (solid lines) in the pattern of optimal trajectories and caustics (dashed line) for the IVDP system. A caustic present in the system. “a” is the optimal escape path: It is the heteroclinic trajectory with the least action. “b” is the trajectory that hits the cycle in the same point of the external branch of the caustic. “c” is a nonoptimal heteroclinic path.

Figure 8

The activation energy for escape in the inverted van der Pol oscillator as a function of the $\eta$ parameter. The results of Monte Carlo simulations (crosses) are compared with the action calculated using the action plot technique (full curve). Error bars are shown for some of the points.

Figure 9

(Color online) The optimal escape trajectory for the overdamped Duffing oscillator. Here the frequency of the harmonic driving is set to $\omega =1$ and the amplitude to $A=0.1$. The optimal path is shown by a full line. It is clearly a heteroclinic trajectory. The dashed line is the switching line for the system. It is clear from the figure that the switching line and MPEP never cross, but they asymptotically touch when they converge to the cycle as $t\to \infty$.

Figure 10

(Color online) Comparison of theory and numerical simulations for the escape problem in the Henon map (parameters are $a=1.1$ and $b=-0.3$). The dots indicate the basin boundary of the stable orbit (the boundary is, of course, continuous; the dots appear due to finite resolution). The theoretical MPEP (dashed line) and numerical (full line) escape paths are in excellent agreement. Escape takes place through the saddle point in $(-1.88;-1.88)$

Figure 11

(Color online) Comparison of theory and experiment for the escape problem in the Henon map (parameters are $a=1.405$ and $b=-0.3$). The solid line is a numerical realization of the escape; the dashed line is the theoretical trajectory found using the actionplot. Agreement between them inside the domain of attraction is excellent. On the boundary, the numerical dynamics diverges from theory due to the finiteness of the noise.

Figure 12

(Color online) The theoretical MPEP (dashed line) for the Henon map compared with an escape path (solid line) found by numerical simulation in the iteration-coordinate domain: $n$ represents the number of iterations (the origin $n=0$ is chosen arbitrarily) and $x$ is the coordinate. The parameters are $a=1.405$ and $b=-0.3$. Agreement between experiment and theory is excellent.

Figure 13

(Color online) Comparison of theory and numerical simulations for the escape problem in the Julia map. The theoretical MPEP (dashed line) and numerical (full line) escape paths agree well inside the basin of attraction. Escape takes place through the saddle cycle of period 9.

Figure 14

(Color online) Comparison of theory and numerical experiment for the escape problem in the Julia map in the coordinate space $x-y$. The theoretical MPEP (dashed line) and numerical (full line) escape paths agree well. The dynamics of the MPEP on the boundary follows a saddle cycle of period 9.