Open system control of dynamical transitions under the generalized Kruskal-Neishtadt-Henrard theorem

Useful dynamical processes often begin through barrier-crossing dynamical transitions; engineering system dynamics in order to make such transitions reliable is therefore an important task for biological or artificial microscopic machinery. Here, we first show by example that adding even a small amount of back-reaction to a control parameter, so that it responds to the system's evolution, can significantly increase the fraction of trajectories that cross a separatrix. We then explain how a post-adiabatic theorem due to Neishtadt can quantitatively describe this kind of enhancement without having to solve the equations of motion, allowing systematic understanding and design of a class of self-controlling dynamical systems.

Figure 1

Instantaneous energy contours in phase space for $\lambda =0$, when the separatrix is at its largest size. Shown are orbits with energies $H(0,\pi /2,0)$ (orange), $H(0,\pi ,0)$ (the separatrix, blue), and $H(0,\pi ,0.5)$ (red) for $g={(\pi /4)}^2/9,\alpha =1$.

Figure 2

Time evolution of $q$ (a) and $p$ (b) for trajectories of the responsive case transitioning to different phases of motion. The same qualitative behavior can be seen for the unresponsive case, $q$ (c) and $p$ (d). Slight differences between the orange curves in (b) and (d) can be seen around $t_{\mathrm{pc}}$, and also at late times in the decreasing amplitude of the orange-curve oscillations in (b), with no corresponding amplitude decrease in (d). The constants are $g={(\pi /4)}^2/9,\alpha =1,\gamma =1$, and $\varepsilon =0.002$.

Figure 3

Time evolution of the control parameter $\lambda$ for the same initial conditions as in Fig. 2. The orange curve is the responsive ${\lambda }_{\mathrm{R}}\left(t\right)$ for the $q,p$ trajectory which successfully enters the separatrix, while the blue curve shows the responsive ${\lambda }_{\mathrm{R}}\left(t\right)$ for the trajectory which fails to cross the separatrix. The red dotted curve is the unresponsive ${\lambda }_{\mathrm{U}}\left(t\right)$, which is the same for both kinds of system trajectory. (b) depicts the same evolutions zoomed in around ${\lambda }_{\mathrm{pc}}$. The three curves all remain close until after the system encounters the separatrix at $t_{\mathrm{pc}}$, showing that the success or failure of the control task depends sensitively on the precise time dependence of the control parameter.

Figure 4

Probabilities of success in the dragging control task, as predicted post-adiabatically by the KNH (blue dashed) and GKNH (orange) formulas, with the parameters from Fig. 2 for different ${\lambda }_{\mathrm{pc}}$. To calculate each probabilities we evolved $N={10}^4$ trajectories numerically for different random initial conditions and determined the probability by counting the trajectories entering the separatrix, $N_{\mathrm{sep}}$, and then taking the fraction $N_{\mathrm{sep}}/N$. Blue dots without borders mark the probabilities for the unresponsive system, while orange dots with black borders represent the responsive system. What appear to be blue dots with black borders, towards the sides of the plot, are in fact borderless blue dots overlying bordered orange dots, because the probabilities in the two cases are indistinguishable.

Figure 5

Evolution for $p$ and $\dot{\lambda }$ in both the responsive (orange) and unresponsive (red) case for the same initial conditions with the same parameters as before. Note how the $p\left(t\right)$ oscillations are very similar for the two cases up until $t_{\mathrm{pc}}$, but the oscillations in the responsive orange ${\dot{\lambda }}_{\mathrm{R}}\left(t\right)$ closely follow the oscillations in $p\left(t\right)$ for $t<t_{\mathrm{pc}}$, while the unresponsive red curve for $\dot{\lambda }\left(t\right)\to {\dot{\lambda }}_{\mathrm{U}}\left(t\right)$ does not. This difference in the responsive ${\dot{\lambda }}_{\mathrm{R}}\left(t\right)$ behavior is small, but systematically cooperative in triggering capture.

Figure 6

Change of the energy interval $[0,\delta h_{⌢}]$ (upper bound blue, lower bound orange) over time, during the evolution along upper and lower arc of the separatrix. Along the upper arc the energy $\delta h_{⌢}$ gets lost, yielding interval $[-\delta h_{⌢},0]$. Along the lower arc, the energy $-\delta h_{⌣}$ is gained; orbits which thereby rise above zero energy again are not returning to their initial phase space region above the separatrix, however, but are rather proceeding below the lower arc of the separatrix, where energy is also positive. These blue-shaded orbits will therefore not cycle around the separatrix again and be captured; instead they have successfully crossed the whole negative energy region, climbing back up to positive energy, and will never be captured. The captured fraction is what remains below the separatrix energy, colored in orange.