## 4 Numerical Algorithm and Results
### Example
As we can see from Figures 3d and 4, for a fixed \(T\), the peak trajectory of the nonstationary PPD coincides with the NOP only if the noise intensity is small enough, while for a fixed \(\varepsilon\), the deviation of the peak trajectory from the NOP increases as \(T\) increases. The larger \(T\) is, the smaller the \(\varepsilon\) needed to ensure the matching effect between the two paths. The numerical results are in perfect agreement with our theoretical predictions. Next, we explore how the \(\varepsilon\)-independent parameter \(\kappa\) evolves when \(T\) and \(x_{f}\) vary. The NOPs for differential target points \(x_{f}\) and \(T\) are shown in Figure 5, with their corresponding dispersion parameters displayed in Figure 6. This shows that \(\kappa\) varies approximately linearly near the initial and the final points, and continuously increases to reach its maximum. For a short time span \(T=1\), \(\kappa\) has a single peak, while if we choose \(x_{0}=1\) and increase \(T\), two peaks of \(\kappa\) will emerge. Moreover, the value of the dispersion parameter grows dramatically when \(T\) increases, as shown in Figure 7. Since the dispersion of the PPD reflects the sensitivity to fluctuations along the NOP, a larger \(\kappa\) implies that stochastic trajectories are more likely to be biased from the optimal one and, thus, a smaller \(\varepsilon\) is needed to ensure the convergence. This is consistent with our previous analysis. In particular, it is worth pointing out that \(\kappa(t^{*})=1/2\) at the moment where the optimal path reaches  
Figure 2: **(a)** The vector field of the Hamiltonian system. The red and green curves denote the OP from \(x_{sl}\) to \(x_{sr}\) and its reverse, respectively. The purple curve is a typical example of the NOP from \(x_{sl}\) to \(x_{sr}\). **(b)** The time required for a Hamiltonian orbit to hit the line \(x=x_{sr}\) from the initial point \((x_{xl},p_{0})\). the unstable fixed point \(x_{nop}(t^{*})=0\). This is because the NOP approaches the stationary one as \(T\) becomes large enough. As a result, the value of \(\kappa\) near the unstable fixed point can be approximately described by the quasi-potential of the reversed process--that is, \(\kappa(t^{*})\approx-1/(2U^{\prime}(0))\). Figure 4: Comparison of the NOP obtained by the MAM and the peak trajectories of the PPDs with different noise intensities: (**a**) \(T=5\). (**b**) \(T=7\). Figure 3: The PPDs and their peak trajectories; here, \(x_{0}=-1\), \(x_{f}=1\) and \(T=10\). (**a**) \(\epsilon=0.001\). (**b**) \(\epsilon=0.01\). (**c**) \(\epsilon=0.1\). (**d**) Comparison of the NOP obtained by the MAM and the peak trajectories of the PPDs with different noise intensities. Figure 5: The NOP for different target positions \(x_{f}\). (**a**) \(T=1\), \(\varepsilon=0.01\). (**b**) \(T=4\), \(\varepsilon=0.001\). Figure 6: The noise-independent dispersion parameter \(\kappa\) as a function of \(t\) and \(x\) for different final positions \(x_{f}\). (**a,b**) \(T=1\), \(\varepsilon=0.01\); (**c,d**) \(T=4\), \(\varepsilon=0.001\). In the end, we show the multiple-peaks effect of the PPD. Here, we can assume that \(-1<x_{0}<x_{f}<0\). In this case, the circular phase trajectories in the interior of the heteroclinic orbits now serve as the optimal phase orbits of our optimization problem. The time required for an NOP to reach the target point is illustrated in Figure 8a. We can divide them into five scenarios, each of which has a typical example displayed in Figure 8b. This shows that, as \(T\to\infty\), there are only two principal phase orbits as candidates for the NOP: case B and case C (case A must have a larger action). Which one dominates the PPD depends on the value of the action. Since they approach the heteroclinic orbits, their action could be approximately estimated by \(V_{B}=2U(x_{f})-2U(x_{sl})\) and \(V_{\text{C}}=2U(x_{u})-2U(x_{0})\), respectively.