We study an ensembles of globally coupled or forced identical phase oscillators subject to independent white Cauchy noise. We demonstrate, that if the oscillators are forced in several harmonics, stationary synchronous regimes can be exactly described with a finite number of complex order parameters. The corresponding distribution of phases is a product of wrapped Cauchy distributions. For sinusoidal forcing, the Ott-Antonsen low-dimensional reduction is recovered.
Phase oscillator systems with global sine-coupling are known to exhibit low-dimensional dynamics. In this paper, such characteristics are extended to phase oscillator systems driven by Cauchy noise. The low-dimensional dynamics solution agreed well with the numerical simulations of noise-driven phase oscillators in the present study. The low-dimensional dynamics of identical oscillators with Cauchy noise coincided with those of heterogeneous oscillators with Cauchy-distributed natural frequencies. This allows for the study of noise-driven identical oscillator systems through heterogeneous osci…
We have investigated synchronized pattern in a network of Thomas oscillators coupled with sinusoidal nonlinear coupling. Pattern like chimera states are not only observed for many non-locally coupled oscillators but there is a signature of it even for locally coupled few oscillators. For certain range of intermediate coupling, clusters are also observed. These patterns do resemble with motion of real self propelled coupled dynamical systems.
We study the collective dynamics of identical phase oscillators on globally coupled networks whose interactions are asymmetric and mediated by positive and negative couplings. We split the set of oscillators into two interconnected subpopulations. In this setup, oscillators belonging to the same group interact via symmetric couplings while the interaction between subpopulations occurs in an asymmetric fashion. By employing the dimensional reduction scheme of the Ott-Antonsen (OA) theory, we verify the existence of traveling-wave and $π$-states, in addition to the classical fully synchronized a…
The Ott--Antonsen ansatz is a powerful tool to extract the behaviors of coupled phase oscillators, but it imposes a strong restriction on the initial condition. Herein, a systematic extension of the Ott--Antonsen ansatz is proposed to relax the restriction, enabling the systematic approximation of the behavior of a globally coupled phase oscillator system with an arbitrary initial condition. The proposed method is validated on the Kuramoto--Sakaguchi model of identical phase oscillators. The method yields cluster and chimera-like solutions that are not obtained by the conventional ansatz.
Nonautonomous driving of an oscillator has been shown to enlarge the Arnold tongue in parameter space, but little is known about the analogous effect for a network of oscillators. To test the hypothesis that deterministic nonautonomous perturbation is a good candidate for stabilising complex dynamics, we consider a network of identical phase oscillators driven by an oscillator with a slowly time-varying frequency. We investigate both the short- and long-term stability of the synchronous solutions of this nonautonomous system. For attractive couplings we show that the region of stability grows …
We show that a network of non-identical nodes, with excitable dynamics, pulse-coupled, with coupling delays depending on the Euclidean distance between nodes, is able to adapt the topology of its connections to obtain spike frequency synchronization. The adapted network exhibits remarkable properties: sparse, anti-cluster, necessary presence of a minimum of inhibitory nodes, predominance of connections from inhibitory nodes over those from excitatory nodes and finally spontaneous spatial structuring of the inhibitory projections: the furthest the most intense.
An outstanding problem in the study of networks of heterogeneous dynamical units concerns the development of rigorous methods to probe the stability of synchronous states when the differences between the units are not small. Here, we address this problem by presenting a generalization of the master stability formalism that can be applied to heterogeneous oscillators with large mismatches. Our approach is based on the simultaneous block diagonalization of the matrix terms in the variational equation, and it leads to dimension reduction that simplifies the original equation significantly. This n…
Stability of the synchronization manifold in nearest neighbors non identical van der Pol-like oscillators
我们研究了环和最近邻耦合自持系统的开放链中同步流形的稳定性，每个自持系统由多个极限环van der Pol振子组成.这种模型代表，例如，在生物系统中的相干振荡，通过在脑波模型中的铁电行为的酶底物反应的情况下。分别考虑全同振子和非全同振子的环和开环链。通过使用主稳定性函数方法（对于相同的情况）和复Kuramoto序参量（对于不相同的情况），我们得到了同步流形的稳定性边界。我们发现，同步发生在一个系统中的许多耦合的修改vanderPol振荡器，它是稳定的，即使在存在的传播参数。
Based on a local greedy numerical algorithm, we compute the topology of weighted, directed, and of unlimited extension networks of non identical Kuramoto oscillators which simultaneously satisfy 2 criteria: i) global frequency synchronisation and ii) minimal total mass of the connection weights. This problem has been the subject of many previous interesting studies, but from our best knowledge, this is the first time that no a priori constraint is imposed, either on the form or on the dynamics of the connections. The results are surprising: the optimal networks turn out to be strongly symmetri…
Synchronization in systems of coupled Kuramoto oscillators may depend on their natural frequencies, coupling, and underlying networks. In this paper, we reduce the alternatives to only one by considering identical oscillators where the only parameter that is allowed to change is the underlying network. While such a model was analyzed over the past few decades by studying the size of the basin of attraction of the synchronized state on restricted families of graphs, here we address a qualitative question on general graphs. In an analogy to resistive networks with current sources, we describe an…
We investigate a system of four nearest neighbour bidirectional coupled phase oscillators of dissimilar initial frequencies in a ring at the changeover into a synchronizing state. There are twenty four permutations upon assigning the initial frequencies to the oscillators. The local interaction between the adjacent coupled phase oscillators introduces details to the synchronization features. Therefore, for the four unalike local coupled oscillators, we classify all possible arrangements into three classes, where each class contains eight configurations. The synchronized state appears at the di…
We demonstrate that strongly asymmetric limit cycles can be observed in the system of three identical ring oscillators (3-gene networks known as Repressilators) globally coupled by signal molecule diffusion added to the model in a way like the known bacterial "quorum-sensing" mechanism. These cycles are stable over a wide interval of the coupling strengths where they expel the dominant hyperchaotic regime existing in three Repressilators in very large areas of parameters. The bifurcations of the inhomogeneous limit cycle, with a high-amplitude orbit for one oscillator and two low-amplitude ide…
Previous research on nonlinear oscillator networks has shown that chaos synchronization is attainable for identical oscillators but deteriorates in the presence of parameter mismatches. Here, we identify regimes for which the opposite occurs and show that oscillator heterogeneity can synchronize chaos for conditions under which identical oscillators cannot. This effect is not limited to small mismatches and is observed for random oscillator heterogeneity on both homogeneous and heterogeneous network structures. The results are demonstrated experimentally using networks of Chua's oscillators an…
The simplest ring oscillator is made from three strongly nonlinear elements repressing each other unidirectionally resulting in the emergence of a limit cycle. A popular implementation of this scheme uses repressive genes in bacteria creating the synthetic genetic oscillator known as the Repressilator. Here, we consider the main collective modes produced when two identical Repressilators are mean-field coupled via the quorum sensing (QS) mechanism which is realized via production of diffusive signal molecules. Using the rate of the repressor's production and the value of coupling strength as t…
Frequency plays a crucial role in exhibiting various collective dynamics in the coexisting co- and counter-rotating (CR) systems. To illustrate the impact of CR frequencies, we consider a network of non-identical and globally coupled Stuart-Landau oscillators with additional perturbation. Primarily, we investigate the dynamical transitions in the absence of perturbation, demonstrating that the transition from desynchronized state to cluster oscillatory state occurs through an interesting partial synchronization state. Followed by this, the system dynamics transits to amplitude death and oscill…
The emergence of rich dynamical phenomena in coupled self-sustained oscillators, primarily synchronization and amplitude death, has attracted considerable interest in several fields of science and engineering. Here, we present a comprehensive theoretical study on the manifestation of these exquisite phenomena in a reduced-order model of two coupled Rijke tube oscillators, which are prototypical thermoacoustic oscillators. We characterize the dynamical behaviors of two such identical and non-identical oscillators by varying both system parameters (such as the uncoupled amplitudes and the natura…
A long-standing expectation is that two repulsively coupled oscillators tend to oscillate in opposite directions. It has been difficult to achieve complete synchrony in coupled identical oscillators with purely repulsive coupling. Here, we introduce a general coupling condition based on the linear matrix of dynamical systems for the emergence of the complete synchronization in pure repulsively coupled oscillators. The proposed coupling profiles (coupling matrices) define a bidirectional cross-coupling link that plays the role of indicator for the onset of complete synchrony between identical o…
Twisted states with non-zero winding numbers composed of sinusoidally coupled identical oscillators have been observed in a ring. The phase of each oscillator in these states constantly shifts, following its preceding neighbor in a clockwise direction, and the summation of such phase shifts around the ring over $2π$ characterizes the winding number of each state. In this work, we consider finite-sized $d$-dimensional hypercubic lattices, namely square ($d=2$) and cubic ($d=3$) lattices with periodic boundary conditions. For identical oscillators, we observe new states in which the oscillators …
We propose Moebius maps as a tool to model synchronization phenomena in coupled phase oscillators. Not only does the map provide fast computation of phase synchronization, it also reflects the underlying group structure of the sinusoidally coupled continuous phase dynamics. We study map versions of various known continuous-time collective dynamics, such as the synchronization transition in the Kuramoto-Sakaguchi model of non-identical oscillators, chimeras in two coupled populations of identical phase oscillators, and Kuramoto-Battogtokh chimeras on a ring, and demonstrate similarities and dif…
We consider the Kuramoto-Sakaguchi model of identical coupled phase oscillators with a common noisy forcing. While common noise always tends to synchronize the oscillators, a strong repulsive coupling prevents the fully synchronous state and leads to a nontrivial distribution of oscillator phases. In previous numerical simulations, a formation of stable multicluster states has been observed in this regime. However we argue here, that because identical phase oscillators in the Kuramoto-Sakaguchi model form a partially integrable system according to the Watanabe-Strogatz theory, the formation of…
In this work, we experimentally investigate the dynamics of pairs of opto-thermally driven, mechanically coupled, doubly clamped, silicon micromechanical oscillators, and numerically investigate the dynamics of the corresponding lumped-parameter model. Coupled limit cycle oscillators exhibit striking nonlinear dynamics and bifurcations in response to variations in system parameters. We show that the input laser power influences the frequency detuning between two non-identical oscillators. As the laser power is varied, different regimes of oscillations such as the synchronized state, the drift …
