arXiv subjects
Publications and source records attributed to Nataliya Stankevich.
Using the example of three-dimensional Mira map, it is shown that the destruction of a multi-turn invariant curve can occur through the appearance of local multiple bends. It was found that, depending on the precision of machine arithmetic, a complication of the multi-turn invariant curve can be observed, which is a numerical artifact. Artifact can be avoided with increasing of machine arithmetic accuracy.
We study bifurcation mechanisms for the appearance of hyperchaotic attractors in three-dimensional diffeomorphisms, i.e., such attractors whose orbits have two positive Lyapunov exponents in numerical experiments. In order to possess this property periodic orbits belonging to the attractor should have two-dimensional unstable invariant manifolds. For realization of this possibility, we propose several bifurcation scenarios that include cascades of both supercritical period-doubling bifurcations with saddle periodic orbits and supercritical Neimark-Sacker bifurcations with stable periodic orbits, as well as various combinations of these cascades. In the paper, these scenarios are illustrated by an example of the three-dimensional Mirá map.
Characterizing accurately chaotic behaviors is not a trivial problem and must allow to determine the properties that two given chaotic invariant sets share or not. The underlying problem is the classification of chaotic regimes, and their labelling. Addressing these problems correspond to the development of a dynamical taxonomy, exhibiting the key properties discriminating the variety of chaotic behaviors discussed in the abundant literature. Starting from the hierarchy of chaos initially proposed by one of us, we systematized the description of chaotic regimes observed in three- and four-dimensional spaces, which cover a large variety of known (and less known) examples of chaotic systems. Starting with the spectrum of Lyapunov exponents as the first taxonomic ranks, we extended the description to higher ranks with some concepts inherited from topology (bounding torus, surface of section, first-return map...). By treating extensively the Rössler and the Lorenz attractors, we extended the description of branched manifold -- the highest taxonomic rank for classifying chaotic attractor -- by a linking matrix (or linker) to multi-component attractors (bounded by a torus whose genus g <= 3
The dynamics of a multiplex heterogeneous network of oscillators is studied. Two types of similar models based on the Hodgkin-Huxley formalism are used as the basic elements of the networks. The first type model demonstrates bursting oscillations. The second model demonstrates bistability between bursting attractor and stable steady state. Basin of attraction of the stable equilibrium in the model is very small. Bistabilty is a result taking into account an additional ion channel, which has a non-monotonic characteristic and can be interpreted as a channel with a communication defect. Suggested multiplex networks assumed more active communication between models with a defect as a result in such networks it is enough to have one element with a communication defect in the subnetworks in order to stabilize the state of equilibrium in the entire network.
The dynamics of a non-autonomous oscillator in which the phase and frequency of the external force depend on the dynamical variable is studied. Such a control of the phase and frequency of the external force leads to the appearance of complex chaotic dynamics in the behavior of oscillator. A hierarchy of various periodic and chaotic oscillations is observed. The paper studies the structure of the space of control parameters. It is shown that in the dynamics of the system there are oscillatory modes similar to those of a non-autonomous oscillator with a potential in the form of a periodic function, but there are also significant differences.
The cooperative dynamics of cellular populations emerging from the underlying interactions determines cellular functions, and thereby their identity in tissues. Global deviations from this dynamics on the other hand reflects pathological conditions. However, how these conditions are stabilized from dis-regulation on the level of the single entities is still unclear. Here we tackle this question using generic Hodgkin-Huxley type of models that describe physiological bursting dynamics of pancreatic beta-cells, and introduce channel dis-function to mimic pathological silent dynamics. The probability for pathological behavior in beta-cell populations is $\sim100\%$ when all cell have these defects, despite the negligible size of the silent state basin of attraction for single cells. In stark contrast, in a more realistic scenario for a heterogeneous population, stabilization of the pathological state depends on the size of the sub-population which acquired the defects. However, the probability to exhibit stable pathological dynamics in this case is less than $\sim10\%$. These results therefore suggest that the physiological bursting dynamics of a population of beta-cells is cooperatively regulated, even under intercellular communication defects induced by dis-functional channels of single-cells.
In the present paper the result of numerical simulations of the model based on the Hodgkin-Huxley formalism with bistability between hidden and rare attractors in the presence of noise are studied.
The dynamics of the Chua circuit is studied. Analysis of equilibrium states was revealed. Parameter plane of the picewise linear voltage-current was obtained. For this system was shown sequence of bifurcations of symmetry broken.
The dynamics of two coupled generators of quasiperiodic oscillations is studied. The opportunity of complete and phase synchronization of generators in the regime of quasiperiodic oscillations is obtained. The features of structure of parameter plane is researched using charts of dynamical regimes and charts of Lyapunov exponents, in which typical structures as resonance Arnold web were revealed. The possible quasiperiodic bifurctions in the system are discussed.