Chaos in dissipative relativistic standard maps

The relativistic generalization of the dissipative standard map is introduced, based on the problem of acceleration and heating (or cooling) of charged particles in the electric field of an electromagnetic wave packet. The question arises as to how the relativistic effects change the nonlinear dynamics described by a dissipative standard map. It is shown that the dissipation modifies the positions of the fixed points, but the origin (the central point) remains identical with that of the corresponding Hamiltonian system. However, the phase-space structure around the origin is drastically modified even if a small dissipation is present. The formation of an “ordered” stochastic structure which is not washed out (in the stochastic sea) for longer times shows that the phase mixing is weak and the nonuniformity of the stochastic acceleration increases because of the dissipation. A new type of stochastic attractor of a higher order is found by numerical simulations. In the context of a scaling-law hypothesis (or renormalization group approach), the transition stochastic sea (high acceleration of relativistic particles)–stochastic attractor (low acceleration) is similar to a Bose–Einstein condensation (or, simply, a condensation gas–liquid) at low temperatures, the dissipative parameter being the control parameter for such a transition. The dissipation parameter can also be considered as a time (aging) parameter of the system, and this may have some applications in biological systems. A Frenkel–Kontorova model of the dissipative relativistic standard map (DRSM) and possible applications to “incommensurate fractals” and lattice dynamics of thermoelectric materials are also considered.