Connection between nonlinear energy optimization and instantons

How systems transit between different stable states under external perturbation is an important practical issue. We discuss here how a recently developed energy optimization method for identifying the minimal disturbance necessary to reach the basin boundary of a stable state is connected to the instanton trajectory from large deviation theory of noisy systems. In the context of the one-dimensional Swift–Hohenberg equation, which has multiple stable equilibria, we first show how the energy optimization method can be straightforwardly used to identify minimal disturbances—minimal seeds—for transition to specific attractors from the ground state. Then, after generalizing the technique to consider multiple, equally spaced-in-time perturbations, it is shown that the instanton trajectory is indeed the solution of the energy optimization method in the limit of infinitely many perturbations provided a specific norm is used to measure the set of discrete perturbations. Importantly, we find that the key features of the instanton can be captured by a low number of discrete perturbations (typically one perturbation per basin of attraction crossed). This suggests a promising new diagnostic for systems for which it may be impractical to calculate the instanton.

Figure 1

The nonlinear solutions to SH [Eq. (1)] with $a=-0.3$, shown in black (stable) or gray (unstable). $U_{1.5}$, $U_{2.5}$, and $U_{3.5}$ are unstable edge states with only a single unstable eigenvector and are not $\mathcal{Z}$-symmetric. $U_2$, $U_3$, $U_4$, and $U_5$ are $\mathcal{Z}$-symmetric unstable solutions with two unstable eigenvectors. We plot the sum of the solution and $\mathcal{Z}$-symmetric eigenvectors (with some small amplitude) in red dashed lines, and the sum of the solution and eigenvectors without $\mathcal{Z}$ symmetry in blue dotted lines.

Figure 2

Schematic diagram of the $\mathcal{Z}$-symmetric manifold. The dashed lines show basin boundaries, and the unstable states are drawn with their unstable manifold. The unstable states are edge states of the $\mathcal{Z}$-symmetric dynamics. Left: The trajectories of the minimal seeds for each stable state are shown in different colors. The dotted line corresponds to a perturbation, and the solid colored line corresponds to the evolution of SH. The minimal seed is the closest point on the basin boundary to $O$. Right: The trajectory of the optimal set of two perturbations (green) and the instanton (orange). Because we fix the time between the two perturbations, the first perturbation for the green curve does not go to $M_2$.

Figure 3

Schematic diagram of the phase space around $U_2$ (left) and $U_4$ (right). Both states have two unstable eigenvectors—one tangential to the $\mathcal{Z}$-symmetric manifold, and one directed out of the manifold. The arrows are all in the two-dimensional unstable manifold of $U_2$ and $U_4$, and the colors correspond to the basin of attraction of the different stable states within the unstable manifold.

Figure 4

Minimum energy perturbations to $O$ to the three other stable states, $S_2$, $S_3$, and $P$ (shown with black stars). The trajectories are also plotted, showing that in each case, the minimal seed is on the stable manifold of one of the $\mathcal{Z}$-symmetric unstable states, $U_2$, $U_3$, or $U_4$. Tick marks are placed on the trajectories every 5 time units. See Fig. 5 for each of the perturbations.

Figure 5

The minimal seeds leading to stable states $S_2$, $S_3$, and $P$. They each evolve toward one of the $\mathcal{Z}$-symmetric unstable states ($U_2$, $U_3$, or $U_4$) before reaching the desired stable state.

Figure 6

Instanton trajectory (yellow to black line), and trajectories for optimal set of two and five perturbations to transition from $O$ to $P$. The instanton trajectory's color corresponds to the size of the perturbation $\delta u_I\left(t\right)$ at that position in the trajectory. The perturbations $\delta u_{2P,i}$ and $\delta u_{5P,i}$ are shown in dotted lines. Tick marks are placed on each trajectory every five time units. Long tick marks denote states and perturbations which are plotted in Fig. 8. After they reach $U_4$, all three trajectories are identical, so they are all denoted with the black line. The trajectories associated with the optimal set of two and five perturbations are very close to each other, but are different from the instanton trajectory, or the minimal seed trajectory $M_P$ (Fig. 4).

Figure 7

The amplitude of the instanton's perturbation $\delta u_I\left(t\right)$ (black line), and each of the optimal perturbations $\delta u_{2P}$ and $\delta u_{5P}$ at the times of the perturbation. Also shown is the amplitude of the sum of $\delta u_I$ between $t=10(i-1)$ and $10i$ (orange circles). The largest perturbations in all cases are near $t=0$ and near $t=25$. This corresponds to perturbing the system toward $U_2$ and then subsequently perturbing the system toward $U_4$.

Figure 8

The solution along the instanton trajectory ($I$) and along the trajectory associated with the optimal set of two and five perturbations ($2P$ and $5P$). $I$ is shown every 10 time units in black, with the dashed orange line showing the solution plus the sum of the perturbations over the next 10 time units. For $2P$ and $5P$, we show the solution right before each perturbation (in black), as well as right after each perturbation (in green or pink; dashed). In all cases, initially the system develops two central large amplitude maxima, followed by two outer medium amplitude maxima.

Figure 9

The instanton Hamiltonian (normalized by the instanton Lagrangian) as a function of time. The perturbations only act until $t_f=50$, so ${\mathcal{H}}_I$ is identically zero at later times. The typical size of the terms in the Hamiltonian are given by ${\mathcal{L}}_I$, but they largely cancel out. Thus, the Hamiltonian is very nearly constant, showing that the associated trajectory is an instanton.