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

Methodical Papers on Nonlinear Dynamics

Physics and intellectual development of personality

Review of the new book «About science, events in the history of the study of light, oscillations, waves, their researchers, as well as glosses and etymons» by Igor V. Izmailov and Boris N. Poizner is given.

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.

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.

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 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. 

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 ).

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.

Terms and definitions

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

Bogdanov–Takens bifurcation: from flows to discrete systems

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 semi-explicit Euler scheme is discussed. On the base of the continuous prototype the discrete model of Bogdanov– Takens oscillator is constructed. The analytical analysis of bifurcations of a codimension one and two for discrete model is realized.