Zhivko D. Georgiev

Analysis and synthesis of perturbed Duffing oscillators

SUMMARY Analysis and synthesis of perturbed Dung oscillators have been presented. The oscillations in such systems are regarded as limit cycles in perturbed Hamiltonian systems under polynomial perturbations of sixth degree and are analysed by using the Melnikov function. It has been proved that there exists a polynomial perturbation depending on the zeros of the Melnikov function so that the system considered can have either two simple limit cycles, or one limit cycle of multiplicity 2, or one simple limit cycle. A synthesis of such oscillators based on the Melnikovs theory has been proposed. Copyright ? 2006 John Wiley & Sons, Ltd.

Time shift in the presence of linear and nonlinear gain, spectral filtering, third-order of dispersion and self-steepening effect

We study the soliton time shift in the presence of linear and nonlinear gain, saturation of the nonlinear refractive index, spectral filtering, third-order of dispersion and self-steepening effect. The applied model generalizes the complex cubicquintic Ginzburg-Landau equation (CCQGLE) with the basic higher-order effects in fibers: the intrapulse Raman scattering (IRS), third-order of dispersion (TOD) and self-steepening effect. Soliton perturbation theory (SPT) is derived with which the influence of the saturation of the nonlinear refraction index, self-steepening and TOD on the appearance of the Poincare-Andronov-Hopf bifurcation is analyzed. It has been shown that TOD and self-steepening effect can lead to reduction in the time shift of the pulse. This prediction has been verified by numerical solution of generalized CCQGLE.

Localized Pulsating Solutions of the Generalized Complex Cubic-Quintic Ginzburg-Landau Equation

We study the dynamics of the localized pulsating solutions of generalized complex cubic-quintic Ginzburg-Landau equation (CCQGLE) in the presence of intrapulse Raman scattering (IRS). We present an approach for identification of periodic attractors of the generalized CCQGLE. Using ansatz of the travelling wave and fixing some relations between the material parameters, we derive the strongly nonlinear Lienard-Van der Pol equation for the amplitude of the nonlinear wave. Next, we apply the Melnikov method to this equation to analyze the possibility of existence of limit cycles. For a set of fixed parameters we show the existence of limit cycle that arises around a closed phase trajectory of the unperturbed system and prove its stability. We apply the Melnikov method also to the equation of Duffing-Van der Pol oscillator used for the investigation of the influence of the IRS on the bandwidth limited amplification. We prove the existence and stability of a limit cycle that arises in a neighborhood of a homoclinic trajectory of the corresponding unperturbed system. The condition of existence of the limit cycle derived here coincides with the relation between the critical value of velocity and the amplitude of the solitary wave solution (Uzunov, 2011).

Analysis and synthesis of oscillator systems described by perturbed double hump Duffing equations

This paper presents an analysis of oscillator systems described by double hump Duffing equations under polynomial perturbations of fourth degree. It has been proved that such a system can have unique hyperbolic limit cycle whose properties depend on the perturbation coefficients. The analytical condition for the arising of a limit cycle has been derived. Moreover, a method for the synthesis of oscillator systems of the considered type, having preliminarily assigned properties, is proposed. The synthesis consists of an appropriate choice of the perturbation coefficients in such a way that the oscillator equation is to have in advance assigned limit cycle. Both the analysis and the synthesis are performed with the aid of the Melnikov function. Copyright

Influence of intrapulse Raman scattering on the stationary pulses in the presence of linear and nonlinear gain as well as spectral filtering

In this paper we present numerical investigation of the influence of intrapulse Raman scattering (IRS) on the stable stationary pulses. Our basic equation, namely cubic-quintic Ginzburg-Landau equation describes the propagation of ultra-short optical pulses under the effect of IRS in the presence of linear and nonlinear gain as well as spectral filtering. Our aim is to examine numerically the influence of IRS, on the stable stationary pulses in the presence of constant linear and nonlinear gain as well as spectral filtering. Numerical solution of our basic equation is performed by means of the “fourth-order Runge-Kutta method in the interaction picture method” method. We found that the small change of the value of the parameter which describes IRS leads to qualitatively different behavior of the evolution of pulse amplitudes. In order to study the observed strong dependence on the IRS, the perturbation method of conserved quantities of the nonlinear Schrodinger equation is applied. The numerical analysis of the derived nonlinear system of ordinary differential equations has shown that our numerical findings are related to the existence of the Poincare-Andronov-Hopf bifurcation.

Influence of the intrapulse Raman scattering on the localized pulsating solutions of generalized complex-quintic Ginzburg-Landau equation

We study the dynamics of the localized pulsating solutions of generalized complex cubic– quintic Ginzburg-Landau equation (CCQGLE) in the presence of intrapulse Raman scattering (IRS). We present an approach for identification of periodic attractors of the generalized CCQGLE. At first using ansatz of the travelling wave, and fixing some relations between the material parameters, we derive the strongly nonlinear Lienard - Van der Pol equation for the amplitude of the nonlinear wave. Next, we apply the Melnikov method to this equation to analyze the possibility of existence of limit cycles. For a set of fixed material parameters we show the existence of limit cycle that arises around a closed phase trajectory of the unperturbed system and prove its stability.

Synthesis and experimental verification of sinusoidal oscillator based on the modified Van der Pol equation

The article presents a method for synthesis of sinusoidal oscillators based on the modified Van der Pol equation. The synthesis applies the Melnikov theory, which allows obtaining a differential equation with amplitude and frequency assigned in advance. Formulas for determination of basic parameters of the oscillations are presented. The modified Van der Pol equation is practically modelled by a Field Programmable Analogue Array AN221E04 of Anadigm Inc. The obtained experimental results demonstrate the basic theoretical relations.