A criterion for mixed dynamics in two-dimensional reversible maps.

We give conditions for non-conservative dynamics in reversible maps with transverse and non-transverse homoclinic orbits.

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 to 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 Painlevé II equation.

Equilibration of energy in slow-fast systems.

Ergodicity is a fundamental requirement for a dynamical system to reach a state of statistical equilibrium. However, in systems with several characteristic timescales, the ergodicity of the fast subsystem impedes the equilibration of the whole system because of the presence of an adiabatic invariant. In this paper, we show that violation of ergodicity in the fast dynamics can drive the whole system to equilibrium. To show this principle, we investigate the dynamics of springy billiards, which are mechanical systems composed of a small particle bouncing elastically in a bounded domain, where one of the boundary walls has finite mass and is attached to a linear spring. Numerical simulations show that the springy billiard systems approach equilibrium at an exponential rate. However, in the limit of vanishing particle-to-wall mass ratio, the equilibration rates remain strictly positive only when the fast particle dynamics reveal two or more ergodic components for a range of wall positions. For this case, we show that the slow dynamics of the moving wall can be modeled by a random process. Numerical simulations of the corresponding springy billiards and their random models show equilibration with similar positive rates.

Exponential energy growth due to slow parameter oscillations in quantum mechanical systems.

It is shown that a periodic emergence and destruction of an additional quantum number leads to an exponential growth of energy of a quantum mechanical system subjected to a slow periodic variation of parameters. The main example is given by systems (e.g., quantum billiards and quantum graphs) with periodically divided configuration space. In special cases, the process can also lead to a long period of cooling that precedes the acceleration, and to the desertion of the states with a particular value of the quantum number.

Leaky Fermi accelerators.

A Fermi accelerator is a billiard with oscillating walls. A leaky accelerator interacts with an environment of an ideal gas at equilibrium by exchange of particles through a small hole on its boundary. Such interaction may heat the gas: we estimate the net energy flow through the hole under the assumption that the particles inside the billiard do not collide with each other and remain in the accelerator for a sufficiently long time. The heat production is found to depend strongly on the type of Fermi accelerator. An ergodic accelerator, i.e., one that has a single ergodic component, produces a weaker energy flow than a multicomponent accelerator. Specifically, in the ergodic case the energy gain is independent of the hole size, whereas in the multicomponent case the energy flow may be significantly increased by shrinking the hole size.

Exponential energy growth in adiabatically changing Hamiltonian systems.

We show that the mixed phase space dynamics of a typical smooth Hamiltonian system universally leads to a sustained exponential growth of energy at a slow periodic variation of parameters. We build a model for this process in terms of geometric Brownian motion with a positive drift, and relate it to the steady entropy increase after each period of the parameters variation.

Robust exponential acceleration in time-dependent billiards.

A class of nonrelativistic particle accelerators in which the majority of particles gain energy at an exponential rate is constructed. The class includes ergodic billiards with a piston that moves adiabatically and is removed adiabatically in a periodic fashion. The phenomenon is robust: deformations that keep the chaotic character of the billiard retain the exponential energy growth. The growth rate is found analytically and is, thus, controllable. Numerical simulations corroborate the analytic predictions with good precision. The acceleration mechanism has a natural thermodynamical interpretation and is applied to a hot dilute gas of repelling particles.

Exponential energy growth in a Fermi accelerator.

An unbounded energy growth of particles bouncing off two-dimensional (2D) smoothly oscillating polygons is observed. Notably, such billiards have zero Lyapunov exponents in the static case. For a special 2D polygon geometry--a rectangle with a vertically oscillating horizontal bar--we show that this energy growth is not only unbounded but also exponential in time. For the energy averaged over an ensemble of initial conditions, we derive an a priori expression for the rate of the exponential growth as a function of the geometry and the ensemble type. We demonstrate numerically that the ensemble averaged energy indeed grows exponentially, at a close to the analytically predicted rate-namely, the process is controllable.

Chaotic soliton walk in periodically modulated media.

We show that a weak transverse spatial modulation in (2+1) nonlinear Schrödinger-type equation can result in nontrivial dynamics of a radially symmetric soliton. We provide examples of chaotic soliton motion in periodic media both for conservative and dissipative cases. We show that complex dynamics can persist even for soliton sizes greater than the modulation period.

Chaotic bound state of localized structures in the complex Ginzburg-Landau equation.

Stable dynamic bound states of dissipative localized structures are found. It is characterized by chaotic oscillations of distance between the localized structures, their phase difference, and the center of mass velocity.

Delay differential equations for mode-locked semiconductor lasers.

We propose a new model for passive mode locking that is a set of ordinary delay differential equations. We assume a ring-cavity geometry and Lorentzian spectral filtering of the pulses but do not use small gain and loss and weak saturation approximations. By means of a continuation method, we study mode-locking solutions and their stability. We find that stable mode locking can exist even when the nonlasing state between pulses becomes unstable.