Locating a regular needle in a chaotic haystack using Lyapunov Weighted Dynamics
Lai-Sang Young 2013 J. Phys. A: Math. Theor. 46 254001
Trajectories that matter, from stable planetary orbits to non-linear rogue waves, are often exceptional; locating them is of great importance, but difficult in the case of systems with many degrees of freedom.
We used a Monte-Carlo algorithm, developed by some of us a few years ago , to single out trajectories of a dynamical system according to their Lyapunov exponents — an observable that measures sensitivity to initial conditions and hence chaoticity.
The idea of the algorithm is to evolve a population of copies of a system, called clones, and to copy and kill them according to the rate at which they contribute to the Lyapunov exponents. By doing so in a controlled way, we nudge the clone population to efficiently sample regions of the trajectory space that would never be visited by brute-force sampling.
This can be used in both low and high dimensions, so that potential applications range from celestial mechanics to statistical physics.
For instance, we studied the restricted gravitational three-body problem in which a small element of negligible mass evolves in the gravitational field of two larger interacting bodies of rescaled masses μ and 1-μ. By increasing μ or the eccentricity e of the larger body trajectories, one can destabilize the regular islands surrounding the Lagrange points L4 and L5 (which are stationary points for the smallest body when e=0).
Figure 1a. Stability region (in grey) of the Lagrange points in terms of eccentricity and mass ratio.
Using Lyapunov Weighted Dynamics (LWD), we were able to track the region in the (e, μ) plane where these regular islands survive and confirm a linear stability result by Danby  (figure 1a). Furthermore, we were able to isolate the periodic orbits at the center of these islands which form the patterns repeated quasi-periodically by the...