Control of escapes in two-degree-of-freedom open Hamiltonian systems

We investigate the possibility of avoiding the escape of chaotic scattering trajectories in two-degree-of-freedom Hamiltonian systems. We develop a continuous control technique based on the introduction of coupling forces between the chaotic trajectories and some periodic orbits of the system. The main results are shown through numerical simulations, which confirm that all trajectories starting near the stable manifold of the chaotic saddle can be controlled. We also show that it is possible to jump between different unstable periodic orbits until reaching a stable periodic orbit belonging to a Kolmogorov–Arnold–Moser island. Control of chaos in Hamiltonian systems is a relevant topic in physics that has aroused much attention in recent decades. However, mainly, discrete control schemes have been implemented. In this manuscript, we show that a continuous control inspired by the Pyragas method can be used to avoid the escape of trajectories in two-degree-of-freedom Hamiltonian systems, which are quite common in chaotic scattering problems. The control scheme starts with a systematic search of symmetric periodic orbits (SPOs) and their crossings with a predefined control axis. Once an arbitrary trajectory crosses this axis, a coupling force between the trajectory and a symmetric periodic orbit is activated. In the presence of the control force, the trajectory approaches the desired periodic orbit, thereby avoiding the escape. After some time, the control force becomes almost negligible and the system recovers its original energy. We test the effectiveness of the method by applying the control to a huge number of initial conditions, distributed along representative Poincaré sections. The numerical simulations show that a significant percentage of escaping initial conditions can be kept within the scattering region by using weak coupling forces and a relatively small amount of symmetric periodic orbits. The initial conditions that are not possible to control are those located far from the stable manifold of the chaotic saddle. In the final part of the work, we show that once a trajectory is stabilized into some periodic orbit, it is possible to move it to different periodic orbits by changing the coupling force at an appropriate time. The goal of this extension of the method is not only to avoid the escape but also to choose in which particular periodic orbit we want to stabilize the system. We hope this work will be useful in research fields such as fusion plasmas, celestial mechanics, conservative flows, and laser-driven reactions