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Recent experiments in one and two-dimensional microfluidic arrays of droplets containing Belousov -Zhabotinsky reactants show a rich variety of spatial patterns [J. Phys. Chem. Lett. 1, 1241-1246 (2010)]. The dominant coupling between these…
We study spectral densities for systems on lattices, which, at a phase transition display, power-law spatial correlations. Constructing the spatial correlation matrix we prove that its eigenvalue density shows a power law that can be…
We demonstrate the emergence of self-organized structures in the course of the relaxation of an initially excited, dissipative and finite chain of interacting particles in a periodic potential towards its many particle equilibrium…
We design a system of phase oscillators that is able to produce temporally periodic sequences of patterns. Patterns are cluster partitions which encode information as phase differences between phase oscillators. The architecture of our…
We generalize our recent approach to reconstruction of phase dynamics of coupled oscillators from data [B. Kralemann et al., Phys. Rev. E, 77, 066205 (2008)] to cover the case of small networks of coupled periodic units. Starting from the…
We study phase entrainment of Kuramoto oscillators under different conditions on the interaction range and the natural frequencies. In the first part the oscillators are entrained by a pacemaker acting like an impurity or a defect. We…
The oscillatory dynamics of natural and man-made systems can be disrupted by their time-varying interactions, leading to oscillation quenching phenomena in which the oscillations are suppressed. We introduce a framework for analyzing,…
The formation of regular patterns is a well-known phenomenon in condensed matter physics. Systems that exhibit pattern formation are typically driven and dissipative with pattern formation occurring in the weakly non-linear regime and…
The behaviour of a space-modulated, so-called "argumental" oscillator is studied, which is represented by a model having an even-parity space-modulating function. Analytic expressions of a stability criterion and of discrete energy levels…
We describe a simple nearest-neighbor Ising model that is capable of supporting a gas, liquid, crystal, in characteristic relationship to each other. As the parameters of the model are varied one obtains characteristic patterns of phase…
The phase reduction technique is essential for studying rhythmic phenomena across various scientific fields. It allows the complex dynamics of high-dimensional oscillatory systems to be expressed by a single phase variable. This paper…
An overview is given on two representative methods of dynamical reduction known as center-manifold reduction and phase reduction. These theories are presented in a somewhat more unified fashion than the theories in the past. The target…
We introduce simple models of genetic regulatory networks and we proceed to the mathematical analysis of their dynamics. The models are discrete time dynamical systems generated by piecewise affine contracting mappings whose variables…
Many natural systems including the brain comprise coupled non-uniformly stimulated elements. In this paper we show that heterogeneously driven networks of excitatory-inhibitory units exhibit striking collective phenomena, including…
Coupled oscillator networks provide mathematical models for interacting periodic processes. If the coupling is weak, phase reduction -- the reduction of the dynamics onto an invariant torus -- captures the emergence of collective dynamical…
There are few examples of non-autonomous vector fields exhibiting complex dynamics that may be proven analytically. We analyse a family of periodic perturbations of a weakly attracting robust heteroclinic network defined on the two-sphere.…
We study the dynamical regimes demonstrated by a pair of identical 3-element ring oscillators (reduced version of synthetic 3-gene genetic Repressilator) coupled using the design of the "quorum sensing (QS)" process natural for…
We study the oscillations of neutrinos in a model in which the neutrino is coupled to a localized, idealized source and detector. By varying the spatial and temporal resolution of the source and detector we are able to model the full range…
Study of network dynamics is very active area in biological and social sciences. However, the relationship between the network structure and the attractors of the dynamics has not been fully understood yet. In this study, we numerically…
Identification of attractors, that is, stable states and sustained oscillations, is an important step in the analysis of Boolean models and exploration of potential variants. We describe an approach to the search for asynchronous cyclic…