Experimental observation of multifrequency patterns in arrays of coupled nonlinear oscillators

Frequency-related oscillations in coupled oscillator systems, in which one or more oscillators oscillate at different frequencies than the other oscillators, have been studied using group theoretical methods by Armbruster and Chossat [Phys. Lett. A 254, 269 (1999)] and more recently by Golubitsky and Stewart [in Geometry, Mechanics, and Dynamics, edited by P. Newton, P. Holmes, and A. Weinstein (Springer, New York, 2002), p. 243]. We demonstrate, experimentally, via electronic circuits, the existence of frequency-related oscillations in a network of two arrays of N oscillators, per array, coupled to one another. Under certain conditions, one of the arrays can be induced to oscillate at N times the frequency of the other array. This type of behavior is different from the one observed in a driven system because it is dictated mainly by the symmetry of the coupled system.     

Parametric generated high-efficiency oscillations in acoustoelectric oscillators

Diagnostic experiments are presented indicating that the r-? emission of acoustoelectric oscillators includes important range of components due to parametric interactions. The process of conversion is characterised by the emission of a transitory frequency spectrum.     

Remarks on multi-frequency oscillations in symmetrically coupled oscillators

Coupled nonlinear oscillators that lead to oscillations where one oscillator oscillates with frequencies that are integer multiples of all other oscillators are analyzed. It is shown that oscillations with multiple frequencies < n occur in systems of n identical and symmetrically coupled oscillators (S_n symmetry). Solutions with n-fold frequencies occur for systems of n identical oscillators symmetrically coupled to each other and to one additional different oscillator.     

Chaotic oscillations in a coupled system of bistable oscillators

The influence of the asymmetry of the nonlinear element characteristic on the chaotic oscillations of Chua’s bistable oscillator is studied. It is shown that such asymmetry causes asymmetry of a chaotic attractor that maps the switching of motions between two basins of attraction up to the concentration of oscillations in one basin. Oscillation control in a bistable chaotic self-oscillating system (two coupled Chua’s oscillators) is considered. It is demonstrated that oscillations excited in two basins of attraction may pass to one of them and that oscillations may build up in two basins when they are autonomously excited in different basins. It is also found that chaotic oscillations in a coupled system may be excited at parameter values for which the autonomous chaotic oscillations of partial oscillators are absent. The influence of external noiselike oscillations is investigated.     

Synchronized oscillations in a system of two coupled van-der-pol-duffing oscillators

The results of analysis of the periodic solutions obtained within the framework of complete and truncated equations for a system of identical Van-der-Pol-Duffing oscillators with nonlinear coupling are compared. This work was presented at the Summer Workshop “Dynamic Days” (Nizhny Novgorod, June 30–July 2, 1998). Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 12, pp. 1531–1536, December, 1998.     

Stochastic phase dynamics and noise-induced mixed-mode oscillations in coupled oscillators

Synaptically coupled neurons show in-phase or antiphase synchrony depending on the chemical and dynamical nature of the synapse. Deterministic theory helps predict the phase differences between two phase-locked oscillators when the coupling is weak. In the presence of noise, however, deterministic theory faces difficulty when the coexistence of multiple stable oscillatory solutions occurs. We analyze the solution structure of two coupled neuronal oscillators for parameter values between a subcritical Hopf bifurcation point and a saddle node point of the periodic branch that bifurcates from the Hopf point, where a rich variety of coexisting solutions including asymmetric localized oscillations occurs. We construct these solutions via a multiscale analysis and explore the general bifurcation scenario using the lambda-omega model. We show for both excitatory and inhibitory synapses that noise causes important changes in the phase and amplitude dynamics of such coupled neuronal oscillators when multiple oscillatory solutions coexist. Mixed-mode oscillations occur when distinct bistable solutions are randomly visited. The phase difference between the coupled oscillators in the localized solution, coexisting with in-phase or antiphase solutions, is clearly represented in the stochastic phase dynamics.     

Periodic orbits in glycolytic oscillators: From elliptic orbits to relaxation oscillations

We consider the Sel’kov model of glycolytic oscillator for a quantitative study of the limit cycle oscillations in the system. We identify a region of parameter space where perturbation theory holds and use both Linstedt Poincaré technique and harmonic balance to obtain the shape and frequency of the limit cycle. The agreement with the numerically obtained result is excellent. We also find a different extreme, where the limit cycle is of the relaxation oscillator variety, has a large time period and it is seen that, as a particular parameter in the model is varied, the time period increases indefinitely. We characterize this divergence numerically. A calculational method is devised to capture the divergence approximately.     

Mutual synchronization in a system of two bistable oscillators executing regular and chaotic oscillations

A numerical analysis of a new model describing two coupled modified Chua??s oscillators is conducted. Equations of a partial oscillator differ from classical equations in that the former contain additional delayed feedback in another writing of dimensionless time. Changeover from regular oscillations in the absence of additional feedback to additional-feedback-induced (switchable) chaotic oscillations is studied. It is shown that, when normal regular oscillations, as well as additional-feedback-induced chaotic oscillations, are synchronized, difference oscillations are left. They are absent only when the control parameters of partial oscillators are identical. The application of a harmonic signal allows one to control the oscillations of a chaotic system of coupled modified bistable oscillators.     

A multifrequency gyrotron

A theory of a gyrotron that generates at frequencies that are multiples of the cyclotron frequency is considered. In the steady-state regime, this radiation appears as a nonsinusoidal electromagnetic oscillation whose waveform depends on the amplitudes of its harmonics. The theory is developed for the weakly relativistic case and is based on known transverse momentum equations for electrons moving in an electromagnetic field. Under optimal conditions, the single-harmonic emission of a multifrequency gyrotron is more efficient than that of a single-frequency device.     

Autodyne effect in multifrequency osciliators

The variations of the individual-oscillator amplitudes, generalized phase differences of the generated harmonics, and self-oscillation frequency of a multifrequency autodyne oscillator under the influence of reflected radiation for each of the generated harmonics are studied theoretically. Basic relations are obtained for analysis of the autodyne response of a multifrequency oscillator in synchronous and asynchronous operation. General conclusions are drawn concerning the self-oscillation characteristics of such autodynes that make it possible to improve the performance and expand the functional possibilities of autodyne systems for short-range radar.