Polymorphisms and adiabatic chaos

At the end of the last century Vershik introduced some dynamical systems, called polymorphisms. Systems of this kind are multivalued self-maps of an interval, where (roughly speaking) each branch has some probability. By definition, the standard Lebesgue measure should be invariant. Unexpectedly, some class of polymorphisms appeared in the problem of destruction of an adiabatic invariant after a multiple passage through a separatrix. We discuss ergodic properties of polymorphisms from this class

On the phase change for perturbations of Hamiltonian systems (non-parametric case)

We consider perturbations of Hamiltonian systems with one degree of freedom such that the evolution leads to separatrix crossings. Such crossings are described by a parameter called the pseudo-phase. We prove a formula for the dependence of the pseudo-phase on the initial conditions

Violation of adiabaticity in magnetic billiards due to separatrix crossings

We consider dynamics of magnetic billiards with curved boundaries and strong inhomogeneous magnetic field. We investigate a violation of adiabaticity of charged particle motion in this system. The destruction of adiabatic invariance is due to the change of type of the particle trajectory: particles can drift along the boundary reflecting from it or rotate around the magnetic field at some distance from the boundary without collisions with it. Trajectories of these two types are demarcated in the phase space by a separatrix. Crossings of the separatrix result in jumps of the adiabatic invariant. We derive an asymptotic formula for such a jump and demonstrate that an accumulation of these jumps leads to the destruction of the adiabatic invariance

Directed transport in a spatially periodic harmonic potential under periodic nonbiased forcing

Transport of a particle in a spatially periodic harmonic potential under the influence of a slowly timedependent unbiased periodic external force is studied. The equations of motion are the same as in the problem of a slowly forced nonlinear pendulum. Using methods of the adiabatic perturbation theory we show that for a periodic external force of a general kind the system demonstrates directed ratchet transport in the chaotic domain on very long time intervals and obtain a formula for the average velocity of this transport. Two cases are studied: The case of the external force of small amplitude, and the case of the external force with amplitude of order one. The obtained formulas can also be used in case of a nonharmonic periodic potential

Remarkable charged particle dynamics near magnetic field null lines

The study of charged-particle motion in electromagnetic fields is a rich source of problems, models, and new phenomena for nonlinear dynamics. The case of a strong magnetic field is well studied in the framework of a guiding center theory, which is based on conservation of an adiabatic invariant – the magnetic moment. This theory ceases to work near a line on which the magnetic field vanishes – the magnetic field null line. In this paper we show that the existence of these lines leads to remarkable phenomena which are new both for nonlinear dynamics in general and for the theory of charged-particle motion. We consider the planar motion of a charged particle in a strong stationary perpendicular magnetic field with a null line and a strong electric field. We show that particle dynamics switch between a slow guiding center motion and the fast traverse along a segment of the magnetic field null line. This segment is the same (in the principal approximation) for all particles with the same total energy. During the phase of a guiding center motion, the magnetic moment of particle’s Larmor rotation stays approximately constant, i.e., it is an adiabatic invariant. However, upon each traversing of the null-line, the magnetic moment changes in a random fashion, causing the particle choose a new trajectory of the guiding center motion. This results in a stationary distribution of the magnetic moment, which only depends on the particle’s total energy. The jumps in the adiabatic invariant are described by Painleve II equation

Mapping for nonlinear electron interaction with whistler-mode waves

The resonant interaction of relativistic electrons and whistler waves is an important mechanism of electron acceleration and scattering in the Earth radiation belts and other space plasma systems. For low amplitude waves, such an interaction is well described by the quasi-linear di?usion theory, whereas nonlinear resonant e?ects induced by high-amplitude waves are mostly investigated (analytically and numerically) using the test particle approach. In this paper, we develop a mapping technique for the description of this nonlinearresonant interaction. Using the Hamiltonian theory for resonant systems, we derive the main characteristics of electron transport in the phase space and combine these characteristics to construct the map. This map can be considered as a generalization of the classical Chirikov map for systems with nondi?usive particle transport and allows us to model the long-term evolution of the electron distribution function.</div

Kinetic equation for nonlinear wave-particle interaction: solution properties and asymptotic dynamics

We consider a kinetic equation describing evolution of the particle distribution function in a system with nonlinear wave-particle interactions (trappings into resonance and nonlinear scatterings). We study properties of its solutions and show that the only stationary solution is a constant, and that all solutions with smooth initial conditions tend to a constant as time grows. The resulting flattening of the distribution function in the domain of nonlinear interactions is similar to one described by the quasi-linear plasma theory, but the distribution evolves much faster. The results are confirmed numerically for a model problem

Charged particle nonlinear resonance with localized electrostatic wave-packets

A resonant wave-particle interaction, in particular a nonlinear resonance characterized by particle phase trapping, is an important process determining charged particle energization in many space and laboratory plasma systems. Although an individual charged particle motion in the nonlinear resonance is well described theoretically, the kinetic equation modeling the long-term evolution of the resonant particle ensemble has been developed only recently. This study is devoted to generalization of this equation for systems with localized wave packets propagating with the wave group velocity different from the wave phase velocity. We limit our consideration to the Landau resonance of electrons and waves propagating in an inhomogeneous magnetic field. Electrons resonate with the wave field-aligned electric fields associated with gradients of wave electrostatic potential or variations of the field-aligned component of the wave vector potential. We demonstrate how wave-packet properties determine the efficiency of resonant particle acceleration and derive the nonlocal integral operator acting on the resonant particle distribution. This operator describes particle distribution variations due to interaction with one wave-packet. We solve kinetic equation with this operator for many wave-packets and show that solutions coincide with the results of the numerical integration of test particle trajectories. To demonstrate the range of possible applications of the proposed approach, we consider the electron evolution induced by the Landau resonances with packets of kinetic Alfven waves, electron acoustic waves, and very oblique whistler waves in the near-Earth space plasma

Lagrangian tori near resonances of near-integrable Hamiltonian systems

We study families of Lagrangian tori that appear in a neighbourhood of a resonance of a near-integrable Hamiltonian system. Such families disappear in the 'integrable' limit ε → 0. Dynamics on these tori are oscillatory in the direction of the resonance phases and rotating with respect to the other (non-resonant) phases. We also show that, if multiplicity of the resonance equals one, generically these tori occupy a set of a large relative measure in the resonant domains in the sense that the relative measure of the remaining 'chaotic' set is of the order $\sqrt \varepsilon$ . Therefore, for small ε > 0 a random initial condition in a $\sqrt \varepsilon$ -neighbourhood of a single resonance occurs inside this set (and therefore generates a quasi-periodic motion) with a probability much larger than in the 'chaotic' set. We present results of numerical simulations and discuss the form of projection of such tori to the action space. At the end of section 4 we discuss the relationship of our results and a conjecture that tori (in a near-integrable Hamiltonian systems) occupy all the phase space except a set of measure ~ε

Classical and quantum dynamics of a particle in a narrow angle

We consider the 2D Schr¨odinger equation with variable potential in the narrow domain diffeomorphic to the wedge with the Dirichlet boundary condition. The corresponding classical problem is the billiard in this domain. In general, the corresponding dynamical system is not integrable. The small angle is a small parameter which allows one to make the averaging and reduce the classical dynamical system to an integrable one modulo exponential small correction. We use the quantum adiabatic approximation (operator separation of variables) to construct the asymptotic eigenfunctions (quasimodes) of the Schrodinger operator. We discuss the relation between classical averaging and constructed quasimodes. The behavior of quasimodes in the neighborhood of the cusp is studied. We also discuss the relation between Bessel and Airy functions that follows from different representations of asymptotics near the cusp