Hyperchaotic attractors of three-dimensional maps and scenarios of their appearance

We study bifurcation mechanisms of the appearance of hyperchaotic attractors in three-dimensional maps. We consider, in some sense, the simplest cases when such attractors are homoclinic, i.e. they contain only one saddle fixed point and entirely its unstable manifold. We assume that this manifold is two-dimensional, which gives, formally, a possibility to obtain two positive Lyapunov exponents for typical orbits on the attractor (hyperchaos). For realization of this possibility, we propose several bifurcation scenarios of the onset of homoclinic hyperchaos 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 three-dimensional Mir\'a map.Comment: 40 pages, 24 figure

Self-oscillating systems with controlled phase of external force

The purpose of this work is to study self-oscillatory systems under adaptive external action. This refers to the situation when the phase of the external action additionally depends on the dynamical variable of the oscillator. In a review plan, the results are presented for the case of a linear damped oscillator. Two cases of self-oscillatory systems are studied: the van der Pol oscillator and an autonomous quasi-periodic generator with three-dimensional phase space. Methods. Methods of charts of dynamical regimes and charts of Lyapunov exponents are used, as well as the construction of phase portraits and stroboscopic sections. Results. In a review plan, the results are presented for the case of a linear damped oscillator. Two cases of self-oscillatory systems are studied: the van der Pol oscillator and an autonomous quasi-periodic generator with a three-dimensional phase space. The pictures of characteristic dynamical regimes are described. Scenarios for the development of multidimensional chaos are described. Illustrations are given of the influence of the control parameter, which is responsible for the degree of dependence of the phase on the oscillator variable, on the dynamics of the system at different frequencies of action. Conclusion. The taling into account of the dependence of the phase on a dynamical variable leads to an extension of the tongues of subharmonic resonances, which are weakly expressed in the classical van der Pol oscillator. This is especially noticeable for even resonances of periods 2 and 4. For the generator of quasi-periodic oscillations in the non-autonomous case, three-frequency tori are observed, their regions begin to dominate with an increase in the adaptivity parameter, displacing the tongues of resonant two-frequency tori. A variety of multidimensional chaos characterized by an additional Lyapunov exponent close to zero is discovered, the possibility of developing hyperchaos as a result of destruction is shown

Hidden and self-excited attractors in radiophysical and biophysical models

One of the central tasks of investigation of dynamical systems is the problem of analysis of the steady (limiting) behavior of the system after the completion of transient processes, i.e., the problem of localization and analysis of attractors (bounded sets of states of the system to which the system tends after transient processes from close initial states). Transition of the system with initial conditions from the vicinity of stationary state to an attractor corresponds to the case of a self-excited attractor. However, there exist attractors of another type: hidden attractors are attractors with the basin of attraction which does not have intersection with a small neighborhoods of any equilibrium points. Classification "hidden vs self-excited" attractors was introduced by Leonov and Kuznetsov. Discovery of the hidden chaotic attractor has shown the need for further study of the scenarios concerned with the appearance and properties of hidden attractors, since the appearance of such attractors in the system can lead to a qualitative change in the dynamics of the system. In the present work two directions have been chosen, for which the possibility of the appearance of hidden attractors can be critical: radiophysics and biophysics. The features of radiophysical generators which can be used for systems of secure communication based on the dynamical chaos are considered in detail. Using the Chua circuit as an example, we investigate the problem of synchronization between two coupled generators in case when the observed regimes are represented by hidden and self-excited attractors. This example shows that in case of hidden attractors under certain initial conditions desynchronization of the coupled subsystems is possible, and the system of secure communication becomes inoperative. Alternative new radiophysical generators with self-excited attractors are also proposed. In such generators, the dynamical chaos is stable to the variation of parameters, initial conditions. In the context of the biophysics problems, a simplified model describing the dynamics of beta-cells based on the Hodgkin-Huxley formalism is presented. It has a typical for such systems bursting attractor which became hidden. This model can be used for the description of various pathological states of cells formation

Stochastic switching in systems with rare and hidden attractors

Complex biochemical networks are commonly characterised by the coexistence of multiple stable attractors. This endows living systems with plasticity in responses under changing external conditions, thereby enhancing their probability for survival. However, the type of such attractors as well as their positioning can hinder the likelihood to randomly visit these areas in phase space, thereby effectively decreasing the level of multistability in the system. Using a model based on the Hodgkin–Huxley formalism with bistability between a silent state, which is a rare attractor, and oscillatory bursting attractor, we demonstrate that the noise-induced switching between these two stable attractors depends on the structure of the phase space and the disposition of the coexisting attractors to each other

Mixed-mode oscillation-incrementing bifurcations and a devil’s staircase from a nonautonomous, constrained Bonhoeffer-van der Pol oscillator

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