Controlling chaotic transport in Hamiltonian systems

With the aid of an original reformulation of the KAM theory, it is shown that a relevant control of Hamiltonian chaos is possible through suitable small perturbations whose form can be explicitly computed. In particular, it is shown that it is possible to control (reduce) the chaotic diffusion in the phase space of a 1.5 degrees of freedom Hamiltonian which models the diffusion of charged test particles in “turbulent” electric fields across the confining magnetic field in controlled thermonuclear fusion devices. Though still far from practical applications, this result suggests that some strategy to control turbulent transport in magnetized plasmas, in particular tokamaks, is conceivable. It is well known that anomalous, noncollisional, losses of energy and particles in magnetic confinement devices of tokamak type still represent a serious obstacle to the attainement of the feasibility proof of controlled thermonuclear fusion, [1]. Anomalous transport, being of noncollisional origin, is currently attributed to the presence of turbulent fluctuations of mainly electric field in fusion plasmas. Several years ago, it has been shown that the E × B modeling of the guiding centres motions of charged test particles provides a natural explanation of the diffusion across the confining magnetic field B. In fact the intrinsic chaoticity of the dynamics provides a source of strong diffusion [2]. Moreover, even though somewhat too idealized, these models yield chaotic diffusion coefficients in a fairly good agreement with their experimental counterparts [3]. Now, both the empirically found states of improved confinement in tokamaks, and the possibility of reducing and even suppressing chaos with open-loop parametric perturbations of dissipative systems [4, 5], suggest to investigate the possibility of devising a strategy of control of anomalouschaotic transport through some smart perturbation acting at the microscopic level of charged particles motions. The mentioned models, however, are Hamiltonian and controlling chaos in these models is rather problematic because no attracting sets exist in their phase space. e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] [email protected] e-mail: [email protected] 38 Physics AUC, vol. 15 (part I), 38-44 (2005) PHYSICS AUC However, as it is shown in the present paper, also chaotic Hamiltonian dynamics can be controlled. The central idea and meaning of “control” is that one aims at inducing a relevant change in the dynamics (for example reducing or suppressing chaos) by means of a small perturbation (either openor closed-loop) so that the original structure of the system under investigation is substantially kept unaltered. In the case of dissipative systems, an efficient strategy of control works by stabilizing unstable periodic orbits where the dynamics is eventually attracted, whereas – at present – for Hamiltonian systems the only hope seems that of looking for a small perturbation, if any, making the system integrable or closer to integrability. In what follows we show that this is actually possible. First we briefly describe the 1.5 degrees of freedom Hamiltonian modeling the E×B motions of charged particles in a “spatially turbulent” electric field, then we sketch a new formulation of the KAM theory due to one of us ([6]) which is to be used to work out a controlling perturbation. Finally, we report the numerical evidence of the effectiveness of the method. Let us begin by describing the model whose dynamics we want to control. In the guiding centres approximation, the equations of motion of charged particles in presence of a strong toroidal magnetic field and of a nonstationary electric field are