Well-Posed and Regular Nonlocal Boundary-Value Problems for Partial Differential Equations

The present paper deals with the well-posedness and regularity of one class of one-dimensional time-dependent boundary-value problems with global boundary conditions on the entire time interval. We establish conditions for the well-posedness of boundary-value problems for partial differential equations in the class of bounded differentiable functions. A criterion for the regularity of the problem under consideration is also obtained.     

Dynamical analysis and multistability in autonomous hyperchaotic oscillator with experimental verification

Nonlinear Dynamics - In this contribution, a modified oscillator of Tamasevicius et al. (Electron Lett 33:542–544, 1997) (referred to as the mTCMNL oscillator hereafter) is introduced with...     

On the analysis of bipolar transistor based chaotic circuits: case of a two-stage colpitts oscillator

We perform a systematic analysis of a system consisting of a two-stage Colpitts oscillator. This well-known chaotic oscillator is a modification of the standard Colpitts oscillator obtained by adding an extra transistor and a capacitor to the basic circuit. The two-stage Colpitts oscillator exhibits better spectral characteristics compared to a classical single-stage Colpitts oscillator. This interesting feature is suitable for chaos-based secure communication applications. We derive a smooth mathematical model (i.e., sets of nonlinear ordinary differential equations) to describe the dynamics of the system. The stability of the equilibrium states is carried out and conditions for the occurrence of Hopf bifurcations are obtained. The numerical exploration reveals various bifurcation scenarios including period-doubling and interior crisis transitions to chaos. The connection between the system parameters and various dynamical regimes is established with particular emphasis on the role of both bias (i.e., power supply) and damping on the dynamics of the oscillator. Such an approach is particularly interesting as the results obtained are very useful for design engineers. The real physical implementation (i.e., use of electronic components) of the oscillator is considered to validate the theoretical analysis through several comparisons between experimental and numerical results.     

Stabilization of the Cauchy problem for integro-differential equations

In the present paper, we obtain a criterion for the stabilization of the Cauchy problem for an integro-differential equation in the class of functions of polynomial growth γ ≥ 0. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1571–1576, November, 2005.     

Transmission of solitary pulse in dissipative nonlinear transmission lines

A class of dissipative complex Ginzburg–Landau (DCGL) equations that govern the wave propagation in dissipative nonlinear transmission lines is solved exactly by means of the Hirota bilinear method. Two-soliton solutions of the DCGL equations, from which the one-soliton solutions are deduced, are obtained in analytical form. The modified Hirota method imposes some restrictions on the coefficients equations. Namely, the second-order dispersion must be real. The physical requirement of the solutions imposes complementary conditions on the combination of the dispersion and nonlinear gain/loss terms of the equation, as well as on the coefficient of the Kerr nonlinearity. The analytical solutions for one-solitary pulses are tested in direct simulations.     

Dynamical properties and chaos synchronization of improved Colpitts oscillators

In this paper, the dynamics and synchronization of improved Colpitts oscillators designed to operate in ultrahigh frequency range are considered. The model is described by a continuous time four-dimensional autonomous system with an exponential nonlinearity. The system is integrated numerically and various bifurcation diagrams and corresponding graphs of largest 1D Lyapunov exponent are plotted to summarize different scenarios leading to chaos. It is found that the oscillator moves from the state of fixed point motion to chaos via the usual paths of period-doubling, intermittency and interior crisis routes when monitoring the bias (i.e. power supply) in tiny ranges. In order to promote chaos-based synchronization designs of this type of oscillators, a synchronization strategy based upon the design of a nonlinear state observer is successfully adapted. The suggested approach enables synchronization to be achieved via a scalar transmitted signal which represents a suitable feature for communication applications. Numerical simulations are performed to demonstrate the effectiveness and feasibility of the proposed technique.     

Perturbation of a Two-Point Problem

We investigate the problem of the effect of integral terms in boundary conditions on the well-posedness of nonlocal boundary-value problems for partial differential equations.