Modulation instability of an electromagnetic wave interacting with a counterpropagating electron beam under condition of cyclotron resonance absorption.

In this paper, transmission of a monochromatic wave through a counterpropagating electron beam under the condition of cyclotron resonance absorption is studied by theoretical analysis and numerical simulation. Conditions of the modulation instability (MI) are analyzed. The MI strongly affects the regimes of transmission. We also derive explicit periodic stationary solutions expressed in terms of elliptic Jacobi functions, as well as bright- and dark-soliton solutions. Analysis of these solutions allows obtaining threshold values of the driving power and frequency for the different regimes of transmission, such as cyclotron absorption, multifrequency self-modulation oscillations, and stationary single-frequency propagation. The theoretical predictions are verified by numerical simulation. In this way, we obtain the conditions at which a continuous-wave driving signal disintegrates into a close-to-periodic train of microwave soliton pulses.

Entrainment, stopping, and transmission of microwave solitons of self-induced transparency in counter-propagating magnetized electron beam.

Based on numerical simulations of a boundary problem, we study various scenarios of microwave soliton formation in the process of cyclotron resonance interaction of a short electromagnetic pulse with a counter-propagating initially rectilinear electron beam taking into account the relativistic dependence of the cyclotron frequency on the electrons' energy. When a certain threshold in the pulse energy is exceeded, the incident pulse can propagate without damping in the absorbing beam, similar to the effect of self-induced transparency in optics. However, mutual motion of the wave and electrons can lead to some novel effects. For relatively small energy of the incident pulse, the microwave soliton is entrained by the electron beam opposite to the direction of the wave's group velocity. With an increase in the pulse energy, soliton stopping occurs. This regime is characterized by the close-to-zero pulse velocity and can be interpreted as a variety of the "light stopping." High-energy microwave solitons propagate in the direction of the unperturbed group velocity. Their amplitude may exceed the amplitude of the incident pulse, i.e., nonlinear self-compression takes place. A further increase in the incident energy leads to the formation of additional high-order solitons whose behavior is similar to that of the first-order ones. The characteristics of each soliton (its amplitude and duration) correspond to analytical two-parametric soliton solutions that are to be found from consideration of the unbounded problem.

Bifurcation analysis of multistability of synchronous states in the system of two delay-coupled oscillators.

Systems of mutually coupled oscillators with delay coupling are of great interest for various applications in electronics, laser physics, biophysics, etc. Time delay usually originates from the finite speed of propagation of the coupling signal. In this paper, we present the results of detailed bifurcation analysis of two delay-coupled limit-cycle (Landau-Stuart) oscillators. First, we study the simplified case when the delay time is much smaller than the oscillation build-up time. When the coupling signal propagates between the two counterparts, it acquires a phase shift, which strongly affects the synchronization pattern. Depending on this phase shift, the system may demonstrate the behavior typical for either dissipative or conservative (reactive) coupling. We examine stability of the in-phase and anti-phase synchronous states and reveal the complicated pattern of the synchronization domains on the frequency mismatch-coupling strength parameter plane paying a special attention to the mechanisms of appearance and disappearance of the phase multistability. We demonstrate that taking into account reactive phase nonlinearity the coupling signal acquires an additional phase shift, which depends on the signal intensity. We also examine the more complicated case of finite delay time. The increase of the reactive nonlinearity parameter and the delay time leads to transformations of synchronization domains similar to those that occur when the phase shift increases. For the bifurcation analysis, we employ XPPAUT and DDEBifTool package and verify the results by direct numerical integration.

Phase locking of two limit cycle oscillators with delay coupling.

Mutual phase locking in the system of two limit cycle oscillators with delay coupling is studied. Conditions of phase locking are derived as a result of analysis of a generalized Adler equation. The analytical results are compared with numerical simulation. Depending on the phase shift of the coupling signal propagating between the two oscillators, either in-phase or anti-phase mode of synchronization may arise. The number of possible modes of synchronization increases with the delay time.

Hyperbolic chaos in the klystron-type microwave vacuum tube oscillator.

The ring-loop oscillator consisting of two coupled klystrons which is capable of generating hyperbolic chaotic signal in the microwave band is considered. The system of delayed-differential equations describing the dynamics of the oscillator is derived. This system is further reduced to the two-dimensional return map under the assumption of the instantaneous build-up of oscillations in the cavities. The results of detailed numerical simulation for both models are presented showing that there exists large enough range of control parameters where the sustained regime corresponds to the structurally stable hyperbolic chaos.

Nonlinear waves, chaos and patterns in microwave electronic devices.

We discuss some problems, concerning the application of nonlinear dynamics methods and ideas to vacuum microwave electronics. We consider such phenomena as solitons, deterministic chaos and pattern formation in different models of electron flows and devices. Our results reveal that microwave electronics is an interesting field of application of nonlinear dynamics. (c) 1996 American Institute of Physics.