Self-Modulation and Chaotic Regimes of Generation in a Relativistic Backward-Wave Oscillator with End Reflections

We study self-modulation and chaotic regimes of generation in a relativistic backward-wave oscillator in the presence of reflections of radiation from the boundaries of a slow-wave structure. The onset of self-modulation is examined in detail in the cases of weak and strong reflections. A numerical simulation of transition-to-chaos scenarios is performed over a wide range of parameters. The relation of bifurcation transitions between different regimes to the formation of spatio-temporal structures in the electron beam is discussed.     

Observation of chaotic dynamics in a powerful backward-wave oscillator

Self-modulation regimes of generation in a powerful 10-micros X-band backward-wave oscillator were studied theoretically and experimentally. The sequence of the self-modulation patterns and corresponding bifurcation values observed as the current was increased were in good agreement with the results of simulations. It was found that at a current of 120 A chaotic self-modulation set in at a power of 2 MW and a relative spectral width of 4%.     

Study of the Nonlinear Dynamics of a Traveling-Wave-Tube Oscillator with Delayed Feedback

We present the results of numerical simulations of the nonlinear dynamics of a traveling-wave-tube (TWT) oscillator with delayed feedback. Basic properties of stationary single-frequency oscillation regimes are considered, and the onset of self-modulation is studied in detail. Various route-to-chaos scenarios corresponding to successively increasing values of the beam current are simulated numerically. It is shown that the basic scenario is a quasi-periodic route to chaos, while the beam deceleration in strongly nonlinear regimes causes transitions via intermittency to regimes based on modes with higher frequencies. Competition between these two scenarios leads to a complex picture of regular and chaotic self-modulation regimes in the parameter space. Such a behavior is typical of distributed electron–wave self-excited oscillators with delayed feedback.     

Complex dynamics of a simple distributed self-oscillatory model system with delay

A simple model for a distributed self-oscillatory system with cubic nonlinearity and delay is presented. Conditions for oscillation self-excitation and stationary oscillation conditions, as well as the stability of the oscillations, are analyzed. Nonstationary self-modulation regimes (including conditions of complex dynamics and chaos) are simulated numerically over a wide range of control parameters. As the factor of nonequilibrium grows, regular and chaotic regimes alternate in a complex manner. The transitions to chaos may follow all scenarios known for finite-dimensional systems. The model suggested is somewhat akin to a number of earlier finite-dimensional models aimed at studying mode competition in resonance electron masers.     

Transitions to chaos in intracavity parametric wave mixing

The complex dynamics of intracavity three-wave mixing are discussed. Detailed results obtained by numerical simulation of routes to chaos are presented for a wide range of parameters. As the pump intensity increases, a complex sequence of alternating regular and chaotic self-modulation regimes is observed. The relationship between these regimes and soliton formation and propagation is analyzed.     

Observation of the chaotic spin-wave soliton trains in magnetic films

The transition from stationary to chaotic spin-wave soliton trains has been observed. The experiment utilized cw excitation of envelope solitons through self-modulation instability of spin waves. By increasing the spin-wave power, the secondary self-modulation instability succeeded the primary modulation instability, resulting in after-modulation of the soliton train amplitude. Further increase of the spin-wave power led to development of the higher-order instabilities, resulting in formation of the chaotic soliton train.     

Chaos and hyperchaos in a gyrotron

We study oscillation in a gyrotron with allowance for reflections from an output horn. Regions with different system behaviors, such as stationary oscillation, self-modulation, and complex-dynamics regimes are found in the parameter plane. The scenarios of appearance of chaotic oscillations are considered. It is shown that they can emerge via either a sequence of period-doubling bifurcations or destruction of quasiperiodic motion. For chaotic attractors, Lyapunov exponents are calculated and their dimensions are estimated on the basis of the Kaplan-Yorke formula. The dimension values turn out to be anomalously large, which is stipulated by the presence of a large number of high-Q eigenmodes in the gyrotron cavity due to operation near the cutoff frequency of an electrodynamic system. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 49, No. 10, pp. 887–899, October 2006.     

Complex dynamics of a double-cavity delayed feedback Klystron oscillator

The complex dynamics of a model double-cavity delayed feedback klystron oscillator is considered. The self-oscillation and stationary oscillation conditions are analyzed theoretically. The results of numerical simulation of the self-modulation and chaotic regimes are presented, and routes to chaos at the center and boundaries of the oscillation zone are studied in detail. The effect of space charge forces on the oscillator dynamics is discussed.     

Chaotic generation in a megawatt backward-wave tube

Experimental and theoretical studies on the self-modulation mode of generation in a high-power BWT with the electrodynamic system representing a slightly corrugated waveguide are presented. The BWT is fed by an electron beam with the energy 150 keV and the current 150 A. The system operates at the E ₀₁ mode with the mean frequency 8.7 GHz. Dynamic chaos is obtained by a three-fold increase in the length of the interaction space in comparison with the prototype exhibiting stationary generation. The stationary generation was changed to periodic sinusoidal self-modulation and then to chaotic self-modulation as the current increases from 6 to 60 A. The generation mode is simplified when the current ranges from 70 to 90 A and becomes complicated again for the current exceeding 100 A. Experimental observations are in good agreement with the results of simulation predicting a certain simplification of the self-modulation mode at the currents 70–90 A owing to the effect of high-frequency space charge. Under the conditions of chaotic generation, the mean power was as high as 2 MW at the relative spectral width of the signal 4% and the total duration of the microwave pulse 10 μs.     

Transition to fully developed chaos in a system of two unidirectionally coupled backward-wave oscillators

The chaotic dynamics of a system of two unidirectionally coupled backward-wave oscillators (BWOs) is studied in the case when a signal from the driving BWO in (periodic or chaotic) self-modulation mode is applied to the driven oscillator, which exhibits strong periodic self-modulation in the autonomous case. The oscillation evolution with the amount of coupling is traced. The use of a chain of coupled BWOs is shown to significantly reduce the threshold of transition to the regime of wide-band chaotic oscillations with a uniform continuous spectrum (so-called fully developed chaos), which is of interest for applications.     

Bright matter-wave soliton collisions in a harmonic trap: regular and chaotic dynamics

Collisions between bright solitary waves in the 1D Gross-Pitaevskii equation with a harmonic potential, which models a trapped atomic Bose-Einstein condensate, are investigated theoretically. A particle analogy for the solitary waves is formulated and shown to be integrable for a two-particle system. The extension to three particles is shown to support chaotic regimes. Good agreement is found between the particle model and simulations of the full wave dynamics, suggesting that the dynamics can be described in terms of solitons both in regular and chaotic regimes, presenting a paradigm for chaos in wave mechanics.     

Chaos and Hyperchaos in a Backward-Wave Oscillator

Based on a numerical solution of the equations of the nonstationary nonlinear theory, we study chaotic self-oscillation regimes in a backward-wave oscillator. For “weak” chaos, arising via a period-doubling cascade of self-modulation for moderate values of the normalized-length parameter, and for developed chaos, which corresponds to large values of this parameter, we present the temporal dependences of the output-signal amplitude, the phase portraits, and the statistical parameters of the dynamics. It is shown that developed chaos is characterized by the presence of more than one positive Lyapunov exponent (hyperchaos). We also present estimates of the Kolmogorov–Sinai entropy, the Lyapunov dimension, and the correlation dimension obtained from the Grassberger–Procaccia algorithm. The results confirm that a finite-dimensional strange attractor is responsible for the chaotic regimes in a backward-wave oscillator.     

Self-organization of temporal structures in a multiequilibrium self-excited oscillator system with frequency control

An analysis is made of the nonlinear dynamics of a model of a self-excited oscillator system with automatic fine tuning of the frequency and possessing more than one equilibrium state. It is shown that, depending on the delay parameters of the control loop and the initial frequency detuning, various periodic and stochastic temporal structures may be formed, accompanied by the generation of various limit cycles and chaotic attractors in phase space. Reasons for the onset of self-modulation are set forth, and the position of the different oscillation regions is established. The main bifurcations are studied together with scenarios for the transformation of the oscillator self-modulation modes as a function of the parameters. Zh. Tekh. Fiz. 67, 1–8 (March 1997)     

Some special features of the transition to chaos in the self-modulation of surface spin waves

The results of experimental investigations of the transition from regular to stochastic self-modulation of intense surface spin waves are presented. It is found that the transition to chaos follows the scenario of a sequence of period-doubling bifurcations. The fractal dimensions and the Kolmogorov entropy are determined for different regimes. The experiments are performed on an apparatus consisting of a microwave oscillator with a spin-wave delay line in the feedback circuit. Pis’ma Zh. éksp. Teor. Fiz. 66, No. 4, 243–246 (25 August 1997)     

Dynamical study and control of drift waves in a magnetized laboratory plasma

The various dynamical regimes of collisional drift waves in a magnetized plasma column are experimentally studied. These unstable low-frequency electrostatic waves are related with strong modulations of the ion and electron density. The angular velocity of the rotating plasma column is the control parameter of the dynamics: regular, chaotic and turbulent regimes are obtained. The spatial extension of the system allows for the occurrence of spatiotemporal chaos. The time-delay auto-synchronization method of controlling chaos [K. Pyragas, Phys. Lett. A 170, 421 (1992)] though purely temporal is successfully applied. A numerical study using coupled nonlinear oscillators exhibiting chaos is compared to the experimental results. The control method is tested on this model.     

Self modulation phenomena in an external-cavity semiconductor laser

We report the observed self-modulation phenomena in an external-cavity semiconductor laser, in which a slant ridge-loading waveguide was formed for the gain medium. A region beside the ridge at a cleaved end of the waveguide served as a saturable absorber for self-modulation. Under different conditions of injection current and cavity alignment, we observed three laser operation regimes, including pure cw oscillation, constant-amplitude pulse-like signal on the top of a cw component, and amplitude-modulated pulse-like signal coexistent with a cw component. The amplitude modulation phenomenon is different from the previously reported concurrence of mode-locking and self-pulsation. A model was proposed to explain the observed results.     

Stochastic simulation of unidirectional intense waves in deep water applied to rogue waves

The results of numerical and laboratory simulation of ensembles of quasi-random unidirectional intense surface gravity waves in deep water have been summarized. The role of nonlinear self-modulation of the waves applied to the problem of ocean rogue waves, as well as the appearance, dynamics, and manifestation of non-linear wave packets in stochastic wave fields, is discussed.     

Nonstationary processes in a diffraction-output orotron

The nonlinear dynamics of an orotron with diffraction output is studied. The evolution of the longitudinal field distribution is described by means of a parabolic equation subject to the condition of zero reflection at the collector end of the interaction space. For stationary regimes, the regions of high efficiency are delineated on the parameter plane, with the parameters being the departure from the critical frequency and the reduced length of the interaction space. From numerical results, the parameter plane is divided into the regions of stationary operation, periodic self-modulation, and stochastic self-modulation. It is demonstrated that self-modulation oscillating modes can be obtained most easily if the effective bounce frequency slightly exceeds the cutoff frequency, with the former being the blinking frequency of the dipole comprising an electron and its reflection in the regularly corrugated slow-wave guide.     

On the theory of self-modulation instability in an FEL amplifier due to stimulated scattering of counterpropagating waves

We present the results of numerical simulation of the self-modulation processes in an amplifier based on the effect of stimulated scattering of two counterpropagating transverse electromagnetic waves by a relativistic electron beam (FEL amplifier). Two models of the studied system are considered. One model allows for the effects of overbunching of electrons and is based on the modified method of macroparticles. The other is a simplified wave model obtained in the approximation that the amplitude of a combination wave is small if the nonlinearity of the electron-beam processes is negligibly small. The mechanisms of self-modulation are studied. The scenarios of transition to chaos, observed with increase in the input-signal intensity and system length, are examined. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 50, No. 6, pp. 471–484, June 2007.     

Self-modulation lasing regimes in a fast-flow CO<Subscript>2</Subscript> laser

Numerical simulation of self-modulation lasing regimes has been carried out for a CO₂ laser with transverse flow of the active medium through an unstable resonator with inhomogeneous internal pumping. Two types of steady-state self-modulation oscillation differing in the feedback mechanism were observed. The type of lasing regime and the conditions of its realization depend both upon the pumping profile and upon the composition and pressure of the working mixture, which makes it generally possible to control temporal characteristics of lasing.