Sergei Rybalko

MODEL OF CARDIAC TISSUE AS A CONDUCTIVE SYSTEM WITH INTERACTING PACEMAKERS AND REFRACTORY TIME

The model of the cardiac tissue as a conductive system with two interacting pacemakers and a refractory time is proposed. In the parametric space of the model the phase locking areas are investigated in detail. The obtained results make possible to predict the behavior of excitable systems with two pacemakers, depending on the type and intensity of their interaction and the initial phase. Comparison of the described phenomena with intrinsic pathologies of cardiac rhythms is given.

Suppression of chaos by cyclic parametric excitation in two-dimensional maps

We study the qualitative change of the dynamics of a generalized two-dimensional quadratic map under the influence of parametric perturbations which operate in the chaotic parameter set. It is shown that such perturbations can lead to the suppression of chaos and appearance of a regular (periodic) behaviour. Numerically we can argue that the suppression of chaos due to the parametric excitation is caused by a shift of the windows of periodic behaviour in the bifurcation diagram.

Intermittency as a universal characteristic of the complete chromosome DNA sequences of eukaryotes: From protozoa to human genomes

Large-scale dynamical properties of complete chromosome DNA sequences of eukaryotes are considered. Using the proposed deterministic models with intermittency and symbolic dynamics we describe a wide spectrum of large-scale patterns inherent in these sequences, such as segmental duplications, tandem repeats, and other complex sequence structures. It is shown that the recently discovered gene number balance on the strands is not of a random nature, and certain subsystems of a complete chromosome DNA sequence exhibit the properties of deterministic chaos.

A GENERALIZED MODEL OF ACTIVE MEDIA WITH A SET OF INTERACTING PACEMAKERS: APPLICATION TO THE HEART BEAT ANALYSIS

We propose a fairly general model of active media by considering of the interaction between pacemakers via their phase response curves. This model describes a network of pulse oscillators coupled by their response to the internal depolarization of mutual stimulations. First, a macroscopic level corresponding to an arbitrary large number of oscillatory elements coupled globally is considered. As a specific and important case of the proposed model, the bidirectional interaction of two cardiac nodes is described. This case is generalized by means of an additional pacemaker, which can be expounded as an external stimulator. The behavior of such a system is analyzed. Second, the microscopic level corresponding to the representation of cardiac nodes by two-dimensional lattices of pulse oscillators coupled via the nearest neighbors is described. The model is a universal one in the sense that on its basis one can easily construct discrete distributed media of active elements, which interact via phase response curves.

Information encoding by stabilized cycles of dynamical systems

A new method is proposed for masking transferred data with the aid of chaotic maps. The cryptographic stability is analyzed by the method of total probing. A correlation analysis of the obtained codes is performed and predictability of the code sequence is evaluated. A network application is developed, which allows legal users to exchange messages protected by the proposed method.

Dynamics of Inhomogeneous Chains of Coupled Quadratic Maps

A new effective local analysis method is elaborated for coupled map dynamics. In contrast to the previously suggested methods, it allows visually investigating the evolution of synchronization and complex-behavior domains for a distributed medium described by a set of maps. The efficiency of the method is demonstrated with examples of ring and flow models of diffusively coupled quadratic maps. An analysis of a ring chain in the presence of space defects reveals some new global-behavior phenomena.

MODEL OF A SPATIALLY INHOMOGENEOUS ONE-DIMENSIONAL ACTIVE MEDIUM

We investigate the dynamics of one-dimensional discrete models of a one-component active medium analytically. The models represent spatially inhomogeneous diffusively concatenated systems of one-dimensional piecewise-continuous maps. The discontinuities (the defects) are interpreted as the differences in the parameters of the maps constituting the model. Two classes of defects are considered: spatially periodic defects and localized defects. The area of regular dynamics in the space of the parameters is estimated analytically. For the model with a periodic inhomogeneity, an exact analytic partition into domains with regular and with chaotic types of behavior is found. Numerical results are obtained for the model with a single defect. The possibility of the occurrence of each behavior type for the system as a whole is investigated.

Parametric Perturbations and Dynamic System Control

The article analyzes dynamical systems with externally applied periodic perturbations in a general setting. We provide a rigorous justification of an approach that reduces such systems to autonomous systems and thus simplifies the analysis. The behavior of families of quadratic one-dimensional maps and circle maps in the presence of parametric perturbations is studied in detail. We prove the existence of periodic perturbations acting strictly on a chaotic subset that stabilize the dynamics and induce the emergence of stable cycles in initially chaotic maps. The analytical results are supplemented with numerical data. It is shown that chaos may be suppressed by a sufficiently complex periodic perturbation.

Mechanisms of non-feedback controlling chaos and suppression of chaotic motion

We describe a rigorous approach to the investigation of qualitative changes in the behaviour of chaotic dynamical systems under external periodic perturbations and propose an analytical key to find such perturbations. It is proven that through a simple periodic perturbation one can stabilize the chosen periodic orbits of any unimodal maps. As an example the quadratic family maps is investigated. Also, it is proven that for piecewise linear family maps and for a two-dimensional map having a hyperbolic attractor there are feedback-free perturbations which lead to suppression of chaos and stabilization of certain periodic orbits.

Modeling and controlling the heart conductive system

A model of the cardiac tissue as a conductive excitable system with a pacemaker under external stimuli is proposed. At certain conditions this model is reduced to the standard circle map. It is analytically shown that 2-periodic perturbations of the constructed map lead to the stabilization of the prescribed orbits and thus, allow us to realize the control of complex cardiac rhythms and remove the heart behaviour to the required dynamical regime.