ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


Методические заметки по нелинейной динамике

Хаотическая динамика кольцевой цепочки маятников с вибрирующим подвесом

Показана возможность реализации хаоса, обусловленного присутствием грубого гиперболического аттрактора, в кольцевой цепочке связанных маятников с диссипацией, когда частота вертикальных колебаний подвеса периодически переключается так, что обеспечивается попеременное параметрическое возбуждение одной или другой колебательной моды. Возникновение аттрактора в виде соленоида Смейла – Вильямса обусловлено тем, что для образующихся на кольцевой цепочке стоячих волн пространственная фаза за период модуляции накачки трансформируется в соответствии с растягивающим отображением окружности.

An electronic device implementing a strange nonchaotic Hunt–Ott attractor

Topic and aim. The aim of the article is to propose an electronic device representing a non-autonomous dynamical system with a strange nonchaotic attractor insensitive to variation of parameters (with the only limitation that the ratio of the frequencies of the components of the external control driving remains unchanged being equal to a fixed irrational number). Investigated model.

One more on universality of oscillatory and wave processes. Foundations for construction of mathematical models

Nonlinear systems with random sources are considered. As a rule, such systems cannot be solved both analytically and numerically. But due to the universality of the oscillation theory we can use simple models and obtain qualitative results.

Bifurcations of three­ and four­dimensional maps: universal properties

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.

On modelling the dynamics of coupled self-oscillators using the simplest phase maps

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 di?erences 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 di?erent model systems.

Electronic circuits manifesting hyperbolic chaos and simulation of their dynamics using software package multisim

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.

Phenomenon of lotka–volterra mathematical model and similar models

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.

Аnalytical research of nonlinear properties of ferroelectrics

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 ).

Terms and definitions

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

Reflex klystron as an example of a self­oscillating delayed feedback system

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.

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