# delayed feedback

## Reconstructing the neuron-like oscillator equations modeled by a phase-locked system with delay from scalar time series

The purpose of this work is to develop the reconstruction technique for the neuron-like oscillator equations descibed by a phase-locked system model with delay from scalar time series. Methods. We reconstruct the state vector given a scalar series of only one variable corresponding to the transmembrane potential. The second variable is obtained by numerical differentiation with smoothing by a polynomial. The third variable is obtained by numerical integration using the Simpson method.

## Normalized boundary value problems in the model of optoelectronic oscillator delayed

Purpose of this work is reduction of differential-difference-model of optic-electronic oscillator to more simple normalized boundary value problems. We study the dynamics of an optoelectronic oscillator with delayed feedback in the vicinity of the zero equilibrium state. The differential-difference-model contains a small parameter with the derivative. It is shown that in a certain neighborhood of the bifurcation point, the number of roots of the characteristic equation that have a real part close to zero increases unlimitedly with decreasing small parameter.

## Numerical simulation of nonlinear dynamics in multiple cavity klystron oscillator with delayed feedback by the "partikle-in- cell» method

The 1.5 D code program of numerical simulation of nonlinear nonstationary processes in the klystron-type devices based on the nonstationary L.A. Vainshtein’s theory of cavity excitation and the «particle-in-cell» method for modeling of the electron beam dynamics is developed. The results of numerical simulation of the basic oscillation modes of the fourcavity klystron oscillator with the external delayed feedback are presented.

## Experiments with a source of chaos – a radioelectronic device with squarelaw phase modulator and interference amplification of quasiharmonic signal

A modified radioelectronic analog of the nonlinear ring cavity is realized in laboratory. The device represents a special class of oscillations or waves sources. An operation principle of the sources is based on interference amplification of feedback signal by an input signal. A laboratory experiments are performed, the likeness of their results and simulation data is shown. An intermittency, chaos, regular, static modes are detected. A thesis on controlled nonlinearity of dynamical systems is suggested.

## Chaos in radio device with squarelaw phase modulator and interference amplification of quasiharmonic signal: a model and simulation

The attempt is undertaken to define a class of oscillations or waves sources, the operation principle of which is based on interference amplification of feedback signal by an input signal. The precedent here is the optical Ikeda’s system. The radio-electronic analog of a nonlinear ring interferometer and it modification are offered, the block diagrams and mathematical models are constructed. The computer simulation is performed. An intermittency, chaos, regular, static modes are detected.

## Reflex klystron as an example of a self-oscillating delayed feedback system

Nonstationary theory of the reflex klystron oscillator based on differential equation with delay is developed. Analysis of self-excitation conditions, steady-state oscillation regimes and their stability is presented. Application of the developed theory for calculating of output characteristics of micromachined submillimetre-band reflex klystron is presented as well. Theoretical results are compared with the results of numerical simulation based on the particle-in-cell method.

## Controlling chaos in Ikeda system. Spatio–temporal model

The method for controlling chaos in a ring resonator filled with a medium with cubic phase nonlinearity (Ikeda system), suggested in [1], is investigated within the framework of a distributed spatio-temporal model described by a Nonlinear Schrodinger Equation with time-delayed boundary condition. Numerical results are presented which confirm the capability of the suggested method. For the case of weakly dispersive nonlinear medium, the results are in good agreement with the approximate theory based on the return map [1].

## Controlling chaos in Ikeda system. Symplified discrete map model

Method of controlling chaos in a ring cavity containing a media with cubic phase nonlinearity (Ikeda system) is considered. The proposed method is based on introduction of an additional feedback loop with parameters chosen so that the fundamental frequency components after passing through different feedback loops appear in phase, while the most unstable sidebands appear in antiphase, thus suppressing each other. In the weak dispersion limit a discrete map is derived that is a modification of the well-known Ikeda map.

## Automodulation and chaotic regimes of generation in a two-resonator gyroklystron with delayed feedback

Topic and aim. The dynamics of a double-resonator gyroklystron of the 93 GHz band with delayed feedback is studied. A comparative analysis of the dynamical regimes of amplifier generation obtained in the numerical experiment both on the basis of averaged equations and in the framework of direct numerical simulation by the «particle-in-cells» method using the KARAT code is carried out. Method.