Variational principle for the determination of unstable periodic orbits and instanton trajectories at saddle points

The complexity of arbitrary dynamical systems and chemical reactions, in particular, can often be resolved if only the appropriate periodic orbit—in the form of a limit cycle, dividing surface, instanton trajectories, or some other related structure—can be uncovered. Determining such a periodic orbit, no matter how beguilingly simple it appears, is often very challenging. We present a method for the direct construction of unstable periodic orbits and instanton trajectories at saddle points by means of Lagrangian descriptors. Such structures result from the minimization of a scalar-valued phase-space function without the need for any additional constraints or knowledge. We illustrate the approach for two-degree of freedom systems at a rank-1 saddle point of the underlying potential-energy surface by constructing both periodic orbits at energies above the saddle point as well as instanton trajectories below the saddle-point energy.

Figure 1

Visualization of five POs (solid lines) at different energies above the activation energy. Panel (a) shows a LD cut through phase space according to Eq. (1) with $\tau =10$ and a selection of POs in $(x,y,v_x)$ space (analogous results are also obtained for $v_y$), and the LD is calculated for initial conditions $v_x=v_y=0$. The LD exhibits a minimum valley (dashed black and white lines) whose phase-space coordinates belong to the periodic orbit at the respective energy. Panel (b) presents the configuration space projection of the POs together with the underlying potential. The POs presented have periods in the range of $2.19\lesssim T\lesssim 2.45$ so that the integration time $\tau$ is sufficient to provide a high-resolution minimum LD valley.

Figure 2

Complete time-dependent phase-space representation of a periodic orbit (solid line) and the corresponding LD (1). The comparison shows that the PO's coordinates are equivalent to the LD minimum throughout. The panels (a)–(f) show the time evolution of both these objects for the different two-variable combinations of the variables $x,y,v_x,$ and $v_y$. The values of the LD in each time slice are shaded in the same dimensionless units as shown in the legend of Fig. 1.

Figure 3

Visualization of two POs (solid lines) at different energies below the activation energy on the inverted potential $-V$ of Eq. (6): (a) A colored contour surface of LD values obtained at $\tau =10$ with POs cutting through the minimum values. (b) Configuration space projection of the instanton trajectories drawn on the potential.