Mechanism of solitary state appearance in an ensemble of nonlocally coupled Lozi maps

We study the peculiarities of the solitary state appearance in the ensemble of nonlocally coupled chaotic maps. We show that nonlocal coupling and features of the partial elements lead to arising of multistability in the system. The existence of solitary state is caused by formation of two attractive sets with different basins of attraction. Their basins are analysed depending on coupling parameters.Comment: 7 figures and 11 page

The role of coupling, noise and harmonic impact in oscillatory activity of an excitable FitzHugh–Nagumo oscillator network

Background and Objectives: The dynamics of a separate small ensemble and coupled small ensembles of excitable FitzHugh–Nagumo oscillators is studied. Different topologies and types of coupling between elements, as well as external noise and harmonic impact are considered. Models and Methods: The main model is a ring of five locally coupled excitable FitzHugh–Nagumo neurons, into which additional connections and external disturbances are introduced. Also, two such systems are connected via a hub, represented by a single FitzHugh–Nagumo neuron. To assess the influence of various system parameters on the neuronal spike activity, maps of the average firing frequency are constructed in the plane of control parameters, and the critical values of the parameters necessary for the occurrence of spikes are found. Results: It has been shown that a repulsive local coupling can excite spike activity in a network of excitable oscillators without external impact, and the addition of remote coupling expands the range of parameters in which firings are observed. Besides, by introducing anomalous Lévy noise, it is possible to excite oscillations in the system at lower values of the coupling strength between neurons than by utilising normal Gaussian noise. Also, in a system of two ensembles of neurons connected through a common hub, the interlayer coupling leads not only to synchronisation of the firing frequencies of these ensembles, but also to a transition to the spike activity mode even when no firing was observed in individual ensembles. By changing the parameters of the external harmonic impact and the coupling coefficients of the two ensembles with a common hub, it is possible to influence the average firing frequency

Analysing Dynamical Behavior of Cellular Networks via Stochastic Bifurcations

The dynamical structure of genetic networks determines the occurrence of various biological mechanisms, such as cellular differentiation. However, the question of how cellular diversity evolves in relation to the inherent stochasticity and intercellular communication remains still to be understood. Here, we define a concept of stochastic bifurcations suitable to investigate the dynamical structure of genetic networks, and show that under stochastic influence, the expression of given proteins of interest is defined via the probability distribution of the phase variable, representing one of the genes constituting the system. Moreover, we show that under changing stochastic conditions, the probabilities of expressing certain concentration values are different, leading to different functionality of the cells, and thus to differentiation of the cells in the various types

Coherence resonance and stochastic synchronization in a nonlinear circuit near a subcritical Hopf bifurcation

We analyze noise-induced phenomena in nonlinear dynamical systems near a subcritical Hopf bifurcation. We investigate qualitative changes of probability distributions (stochastic bifurcations), coherence resonance, and stochastic synchronization. These effects are studied in dynamical systems for which a subcritical Hopf bifurcation occurs. We perform analytical calculations, numerical simulations and experiments on an electronic circuit. For the generalized Van der Pol model we uncover the similarities between the behavior of a self-sustained oscillator characterized by a subcritical Hopf bifurcation and an excitable system. The analogy is manifested through coherence resonance and stochastic synchronization. In particular, we show both experimentally and numerically that stochastic oscillations that appear due to noise in a system with hard excitation, can be partially synchronized even outside the oscillatory regime of the deterministic system

Deterministic nonlinear systems: a short course

This text is a short yet complete course on nonlinear dynamics of deterministic systems. Conceived as a modular set of 15 concise lectures it reflects the many years of teaching experience by the authors. The lectures treat in turn the fundamental aspects of the theory of dynamical systems, aspects of stability and bifurcations, the theory of deterministic chaos and attractor dimensions, as well as the elements of the theory of Poincare recurrences.Particular attention is paid to the analysis of the generation of periodic, quasiperiodic and chaotic self-sustained oscillations and to the issue of synchronization in such systems.  This book is aimed at graduate students and non-specialist researchers with a background in physics, applied mathematics and engineering wishing to enter this exciting field of research

Deterministic Nonlinear SystemsA Short Course /

XIV, 294 p. 172 illus., 2 illus. in color.online

Phase portraits of a single oscillator for and different values of the noise intensity : (A) and (B) and for and (C) and (D) .

<p> and indicate the areas in which the stochastic trajectories spend most of the time.</p

The power spectra for a single oscillator in the presence of noise and external harmonic forcing for , and different values of the external force amplitude (indicated in the figure).

<p>The eigenfrequency is marked as . The frequency of the external forcing is . For convenience of the comparison of the results, the spectral power density is not normalized here.</p