Methodical Papers on Nonlinear Dynamics

To describe the problem of the random nonlinear waves in fluid, we must know, exactly or approximately, how occurs the process of the vortex separation. For this it is conveniently to use models based on physical considerations and (or) some experimental data. The main attention in this review will be attended to random waves, emerging, for example, at stall flutter. Such waves often appear in fluid, and they are the main cause of many disasters is seas and oceans.

The Galerkin method together with the method of small parameter is applied for study of Routh­like equations describing the dynamics of two­dimensional inviscid incom­pressible fluid flows. A set of simple models for some vortical flows of such fluid in rectangular area has been derived and analysed.

The methodically important bifurcation – Bogdanov–Takens bifurcation – is discussed. For the primary model its bifurcations and evolution of phase portraits are described. The examples of nonlinear systems with such bifurcation are presented. The method of discrete models of construction that is founded on semi­explicit Euler scheme is discussed. On the base of the continuous prototype the discrete model of Bogdanov–Takens oscillator is constructed.

Nonstationary theory of the reflex klystron oscillator based on differential equation with delay is developed. Analysis of self­excitation conditions, steady­state oscillation regimes and their stability is presented. Application of the developed theory for calculating of output characteristics of micromachined submillimetre­band reflex klystron is presented as well. Theoretical results are compared with the results of numerical simulation based on the particle­in­cell method.

The paper is a proposal to discuss any terms and definitions of experiments in textbooks, scientific and methodical publications.

This paper describes a method to obtain an analytical expression for the nonlinear dependence of the ferroelectric polarization on the external electric field. To find analytical dependence, used a special Kml-function of second order. Listed structure built of series and groups  of series representing Kml-function at any point in the complex plane. Analyzed domain of  convergence of series , received as a boundary condition. To show the application of the method, as  an example used a ferroelectric polymer polyvinylidene fluoride (PVDF–TrFE ).

Lotka–Volterra mathematical model (often called «predator–prey» model) is applicable for different processes description in biology, ecology, medicine, in sociology investigations, in history, radiophysics, ets. Variants of this model is considered methodologicaly in this review.

We consider several electronic circuits, which are represented dynamical systems with hyperbolic chaotic attractors, such as Smale–Williams and Plykin attractors, and present results of their simulation using the software package NI Multisim 10. The approach developed is useful as an intermediate step of constructing real electronic devices with structurally stable hyperbolic chaos, which may be applicable in systems of secure communication, noise radar, for cryptographic systems, for random number generators.

The problem of describing the dynamics of coupled self-oscillators using discrete time systems on the torus is considered. We discuss the methodology for constructing such maps as a simple formal models, as well as physically motivated systems. We discuss the differences between the cases of the dissipative and inertial coupling. Using the method of Lyapunov exponents charts we identify the areas of two- and three-frequency quasiperiodicity and chaos. Arrangement of the Arnold resonance web is investigated and compared for different model systems.

The approach, in which the picture of bifurcations of discrete maps is considered in the space of invariants of perturbation matrix (Jacobi matrix), is extended to the case of three and four dimensions. In those cases the structure of surfaces, lines and points for bifurcations, that is universal for all maps, is revealed. We present the examples of maps, whose parameters are governed directly by invariants of the Jacobian matrix.