Nonlinear Sciences

From fireflies to heart cells, many systems in Nature show the remarkable ability to spontaneously fall into synchrony. By imitating Nature's success at self-synchronizing, scientists have designed cost-effective methods to achieve synchrony in the lab, with applications ranging from wireless sensor networks to radio transmission. A similar story has occurred in the study of swarms, where inspiration from the behavior flocks of birds and schools of fish has led to 'low-footprint' algorithms for multi-robot systems. Here, we continue this 'bio-inspired' tradition, by speculating on the technological benefit of fusing swarming with synchronization. The subject of recent theoretical work, minimal models of so-called 'swarmalator' systems exhibit rich spatiotemporal patterns, hinting at utility in 'bottom-up' robotic swarms. We review the theoretical work on swarmalators, identify possible realizations in Nature, and discuss their potential applications in technology.

The linear response is studied in globally coupled oscillator systems including the Kuramoto model. We develop a linear response theory which can be applied to systems whose coupling functions are generic. Based on the theory, we examine the role of asymmetry introduced to the natural frequency distribution, the coupling function, or the coupling constants. A remarkable difference appears in coexistence of the divergence of susceptibility at the critical point and a nonzero phase gap between the order parameter and the applied external force. The coexistence is not allowed by the asymmetry in the natural frequency distribution but can be realized by the other two types of asymmetry. This theoretical prediction and the coupling-constant dependence of the susceptibility are numerically verified by performing simulations in N -body systems and in reduced systems obtained with the aid of the Ott-Antonsen ansatz.

This paper deals with an optimal control problem associated to the Kuramoto model describing the dynamical behavior of a network of coupled oscillators. Our aim is to design a suitable control function allowing us to steer the system to a synchronized configuration in which all the oscillators are aligned on the same phase. This control is computed via the minimization of a given cost functional associated with the dynamics considered. For this minimization, we propose a novel approach based on the combination of a standard Gradient Descent (GD) methodology with the recently-developed Random Batch Method (RBM) for the efficient numerical approximation of collective dynamics. Our simulations show that the employment of RBM improves the performances of the GD algorithm, reducing the computational complexity of the minimization process and allowing for a more efficient control calculation.

We consider the stochastic phase models for the community effect of cardiac muscle cells. The model is the extension of the stochastic integrate-and-fire model in which we incorporate the irreversibility after beating, induced beating and refractory. We focus on investigating the expectation and variance of (synchronized) beating interval. In particular, for the single-isolated cell, we obtain the closed-form expectation and variance of the beating interval, and we discover that the coefficient of variance (CV) has upper limit 2/3 − − − √ . For two-coupled cells, we derive the partial differential equations (PDEs) for the expected synchronized beating intervals and the distribution density of phase. Moreover, we also consider the conventional Kuramoto model for both two- and N -cells models, where we establish a new analysis using stochastic calculus to obtain the CV of the ''synchronized'' beating interval, and make some improvement to the literature work.

In this article, the dynamics and complexity of a noise induced blood flow system have been investigated. Changes in the dynamics have been recognized by measuring the periodicity over significant parameters. Chaotic as well as non-chaotic regimes have also been classified. Further, dynamical complexity has been studied by phase space based weighted entropy. Numerical results show a strong correlation between the dynamics and complexity of the noise induced system. The correlation has been confirmed by a cross-correlation analysis.

Recent attempts at cooperating on climate change mitigation highlight the limited efficacy of large-scale agreements, when commitment to mitigation is costly and initially rare. Bottom-up approaches using region-specific mitigation agreements promise greater success, at the cost of slowing global adoption. Here, we show that a well-timed switch from regional to global negotiations dramatically accelerates climate mitigation compared to using only local, only global, or both agreement types simultaneously. This highlights the scale-specific roles of mitigation incentives: local incentives capitalize on regional differences (e.g., where recent disasters incentivize mitigation) by committing early-adopting regions, after which global agreements draw in late-adopting regions. We conclude that global agreements are key to overcoming the expenses of mitigation and economic rivalry among regions but should be attempted once regional agreements are common. Gradually up-scaling efforts could likewise accelerate mitigation at smaller scales, for instance when costly ecosystem restoration initially faces limited public and legislative support.

Collective behavior in large ensembles of dynamical units with non-pairwise interactions may play an important role in several systems ranging from brain function to social networks. Despite recent work pointing to simplicial structure, i.e., higher-order interactions between three or more units at a time, their dynamical characteristics remain poorly understood. Here we present an analysis of the collective dynamics of such a simplicial system, namely coupled phase oscillators with three-way interactions. The simplicial structure gives rise to a number of novel phenomena, most notably a continuum of abrupt desynchronization transitions with no abrupt synchronization transition counterpart, as well as, extensive multistability whereby infinitely many stable partially synchronized states exist. Our analysis sheds light on the complexity that can arise in physical systems with simplicial interactions like the human brain and the role that simplicial interactions play in storing information.

In this paper, we discuss distributed adaptive algorithms for synchronization of complex networks, consensus of multi-agents with or without pinning controller. The dynamics of individual node is governed by generalized QUAD condition. We design new algorithms, which can keep the left eigenvector of the adaptive coupling matrix corresponding to the zero eigenvalue invariant. Based on this invariance, various distributive adaptive synchronization, consensus, anti-synchronization models are given.

Many complex systems occurring in the natural or social sciences or economics are frequently described on a microscopic level, e.g., by lattice- or agent-based models. To analyze the states of such systems and their bifurcation structure on the level of macroscopic observables, one has to rely on equation-free methods like stochastic continuation. Here, we investigate how to improve stochastic continuation techniques by adaptively choosing the parameters of the algorithm. This allows one to obtain bifurcation diagrams quite accurately, especially near bifurcation points. We introduce lifting techniques which generate microscopic states with a naturally grown structure, which can be crucial for a reliable evaluation of macroscopic quantities. We show how to calculate fixed points of fluctuating functions by employing suitable linear fits. This procedure offers a simple measure of the statistical error. We demonstrate these improvements by applying the approach in analyses of (i) the Ising model in two dimensions, (ii) an active Ising model, and (iii) a stochastic Swift-Hohenberg model. We conclude by discussing the abilities and remaining problems of the technique.

We consider a model of three interacting sets of decision-making agents, labeled Blue, Green and Red, represented as coupled phased oscillators subject to frustrated synchronisation dynamics. The agents are coupled on three networks of differing topologies, with interactions modulated by different cross-population frustrations, internal and cross-network couplings. The intent of the dynamic model is to examine the degree to which two of the groups of decision-makers, Blue and Red, are able to realise a strategy of being ahead of each others' decision-making cycle while internally seeking synchronisation of this process -- all in the context of further interactions with the third population, Green. To enable this analysis, we perform a significant dimensional reduction approximation and stability analysis. We compare this to a numerical solution for a range of internal and cross-network coupling parameters to investigate various synchronisation regimes and critical thresholds. The comparison reveals good agreement for appropriate parameter ranges. Performing parameter sweeps, we reveal that Blue's pursuit of a strategy of staying too-far ahead of Red's decision cycles triggers a second-order effect of the Green population being ahead of Blue's cycles. This behaviour has implications for the dynamics of multiple interacting social groups with both cooperative and competitive processes.