The idea that a ‘magnetized’ charged particle in interaction with ‘resonant’ photons operates from an energy level to another higher one by a stochastic acceleration effect suggests that such effects may represent a phenomenological physical mechanism which explains how an electron jumps to higher atomic orbits when it absorbs resonant photons. If we increase the number of iterations of the corresponding nonlinear system of equations, we obtain a Bohr image of an atom. Such (quantum-transition) jumps, their duration and physical mechanism have never been explained by the quantum theory of atoms. We thus offer through such a cascade of chaotic kicked (stochastic acceleration) effects a physical explanation of the quantum model of absorption of energy by an atom. The proposed equations can model a circuit biased with a traveling electromagnetic wave. Such a circuit can also simulate a stochastic acceleration and a chaotic atom.

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.

We obtain the classical spiral orbits from a chaotic behaviour of a particle- field nonlinear interaction. The accelerating spiral motion of a charged particle in a combination of electromagnet ic and static fields represents the classical (relativistic) analogue of quantum absorption. For the first time we propose a physical mechanism to explain a quantum transition, namely the chaotic gun effect that arises in the process of stochastic acceleration of particles and damping of waves. The results suggest that classical chaos, in conjunction with semiclassical assumptions, are meaningful in quantum theory. In essence, we obtain a wave chaos (an analog of quantum chaos) that deals with the classical wave properties of electrodynami cs and may find important practical applications in accelerator physics and electrical engineering. We generalise this idea to the case of a chaotic (turbulent) field and we define a (gravito) magnetic chaos with chaotic field-line orbits. As a new practical application of classical and quantum chaos we propose a (super)sensor based on the Josephson butterfly effect in order to extract energy from a linear gravitational field.