Emergence of synchronization in multiplex networks of mobile Rössler oscillators

Amplitude responses of swarmalators

Swarmalators are entities that swarm through space and sync in time and are potentially considered to replicate the complex dynamics of many real-world systems. So far, the internal dynamics of swarmalators have been taken as a phase oscillator inspired by the Kuramoto model. Here we examine the internal dynamics utilizing an amplitude oscillator capable of exhibiting periodic and chaotic behaviors. To incorporate the dual interplay between spatial and internal dynamics, we propose a general model that keeps the properties of swarmalators intact. This adaptation calls for a detailed study, which we present in this paper. We establish our study with the Rössler oscillator by taking parameters from both chaotic and periodic regions. While the periodic oscillator mimics most of the patterns in the previous phase oscillator model, the chaotic oscillator brings some fascinating states.

Directional synchrony among self-propelled particles under spatial influence

Synchronization is one of the emerging collective phenomena in interacting particle systems. Its ubiquitous presence in nature, science, and technology has fascinated the scientific community over the decades. Moreover, a great deal of research has been, and is still being, devoted to understand various physical aspects of the subject. In particular, the study of interacting active particles has led to exotic phase transitions in such systems which have opened up a new research front-line. Motivated by this line of work, in this paper, we study the directional synchrony among self-propelled particles. These particles move inside a bounded region, and crucially their directions are also coupled with spatial degrees of freedom. We assume that the directional coupling between two particles is influenced by the relative spatial distance which changes over time. Furthermore, the nature of the influence is considered to be both short and long-ranged. We explore the phase transition scenario in both the cases and propose an approximation technique which enables us to analytically find the critical transition point. The results are further supported with numerical simulations. Our results have potential importance in the study of active systems like bird flocks, fish schools, and swarming robots where spatial influence plays a pertinent role.

Swarmalators with delayed interactions

We investigate the effects of delayed interactions in a population of "swarmalators," generalizations of phase oscillators that both synchronize in time and swarm through space. We discover two steady collective states: a state in which swarmalators are essentially motionless in a disk arranged in a pseudocrystalline order, and a boiling state in which the swarmalators again form a disk, but now the swarmalators near the boundary perform boiling-like convective motions. These states are reminiscent of the beating clusters seen in photoactivated colloids and the living crystals of starfish embryos.

Flocking and swarming in a multi-agent dynamical system

Over the past few decades, the research community has been interested in the study of multi-agent systems and their emerging collective dynamics. These systems are all around us in nature, such as bacterial colonies, fish schools, and bird flocks, as well as in technology, such as microswimmers and robotics, to name a few. Flocking and swarming are two key components of the collective behaviors of multi-agent systems. In flocking, the agents coordinate their direction of motion, but in swarming, they congregate in space to organize their spatial position. We propose a minimal mathematical model of a locally interacting multi-agent system where the agents simultaneously swarm in space and exhibit flocking behavior. Various cluster structures are found depending on the interaction range. When the coupling strength value exceeds a crucial threshold, flocking behavior is observed. We do in-depth simulations and report the findings by changing the other parameters and with the incorporation of noise.

Antiphase synchronization in a population of swarmalators

Swarmalators are oscillatory systems endowed with a spatial component, whose spatial and phase dynamics affect each other. Such systems can demonstrate fascinating collective dynamics resembling many real-world processes. Through this work, we study a population of swarmalators where they are divided into different communities. The strengths of spatial attraction, repulsion, as well as phase interaction differ from one group to another. Also, they vary from intercommunity to intracommunity. We encounter, as a result of variation in the phase coupling strength, different routes to achieve the static synchronization state by choosing several parameter combinations. We observe that when the intercommunity phase coupling strength is sufficiently large, swarmalators settle in the static synchronization state. However, with a significant small phase coupling strength the state of antiphase synchronization as well as chimeralike coexistence of sync and async are realized. Apart from rigorous numerical results, we have been successful to provide semianalytical treatment for the existence and stability of global static sync and the antiphase sync states.

Phase transitions on a multiplex of swarmalators

Dynamics of bidirectionally coupled swarmalators subject to attractive and repulsive couplings is analyzed. The probability of two elements in different layers being connected strongly depends on a defined vision range r_{c} which appears to lead both layers in different patterns while varying its values. Particularly, the interlayer static sync π has been found and its stability is proven. First-order transitions are observed when the repulsive coupling strength σ_{r} is very small for a fixed r_{c} and, moreover, in the absence of the repulsive coupling, they also appear for sufficiently large values of r_{c}. For σ_{r}=0 and for sufficiently small values of r_{c}, both layers achieve a second-order transition in a surprising two steps that are characterized by the drop of the energy of the internal phases while increasing the value of the interlayer attractive coupling σ_{a} and later a smooth jump, up to high energy value where synchronization is achieved. During these transitions, the internal phases present rotating waves with counterclockwise and later clockwise directions until synchronization, as σ_{a} increases. These results are supported by simulations and animations added as supplemental materials.

Oscillation suppression and chimera states in time-varying networks

Complex network theory has offered a powerful platform for the study of several natural dynamic scenarios, based on the synergy between the interaction topology and the dynamics of its constituents. With research in network theory being developed so fast, it has become extremely necessary to move from simple network topologies to more sophisticated and realistic descriptions of the connectivity patterns. In this context, there is a significant amount of recent works that have emerged with enormous evidence establishing the time-varying nature of the connections among the constituents in a large number of physical, biological, and social systems. The recent review article by Ghosh et al. [Phys. Rep. 949, 1-63 (2022)] demonstrates the significance of the analysis of collective dynamics arising in temporal networks. Specifically, the authors put forward a detailed excerpt of results on the origin and stability of synchronization in time-varying networked systems. However, among the complex collective dynamical behaviors, the study of the phenomenon of oscillation suppression and that of other diverse aspects of synchronization are also considered to be central to our perception of the dynamical processes over networks. Through this review, we discuss the principal findings from the research studies dedicated to the exploration of the two collective states, namely, oscillation suppression and chimera on top of time-varying networks of both static and mobile nodes. We delineate how temporality in interactions can suppress oscillation and induce chimeric patterns in networked dynamical systems, from effective analytical approaches to computational aspects, which is described while addressing these two phenomena. We further sketch promising directions for future research on these emerging collective behaviors in time-varying networks.

Dynamics on higher-order networks: a review

Network science has evolved into an indispensable platform for studying complex systems. But recent research has identified limits of classical networks, where links connect pairs of nodes, to comprehensively describe group interactions. Higher-order networks, where a link can connect more than two nodes, have therefore emerged as a new frontier in network science. Since group interactions are common in social, biological and technological systems, higher-order networks have recently led to important new discoveries across many fields of research. Here, we review these works, focusing in particular on the novel aspects of the dynamics that emerges on higher-order networks. We cover a variety of dynamical processes that have thus far been studied, including different synchronization phenomena, contagion processes, the evolution of cooperation and consensus formation. We also outline open challenges and promising directions for future research.

Two classes of functional connectivity in dynamical processes in networks

The relationship between network structure and dynamics is one of the most extensively investigated problems in the theory of complex systems of recent years. Understanding this relationship is of relevance to a range of disciplines-from neuroscience to geomorphology. A major strategy of investigating this relationship is the quantitative comparison of a representation of network architecture (structural connectivity, SC) with a (network) representation of the dynamics (functional connectivity, FC). Here, we show that one can distinguish two classes of functional connectivity-one based on simultaneous activity (co-activity) of nodes, the other based on sequential activity of nodes. We delineate these two classes in different categories of dynamical processes-excitations, regular and chaotic oscillators-and provide examples for SC/FC correlations of both classes in each of these models. We expand the theoretical view of the SC/FC relationships, with conceptual instances of the SC and the two classes of FC for various application scenarios in geomorphology, ecology, systems biology, neuroscience and socio-ecological systems. Seeing the organisation of dynamical processes in a network either as governed by co-activity or by sequential activity allows us to bring some order in the myriad of observations relating structure and function of complex networks.

Neuronal synchronization in long-range time-varying networks

We study synchronization in neuronal ensembles subject to long-range electrical gap junctions which are time-varying. As a representative example, we consider Hindmarsh-Rose neurons interacting based upon temporal long-range connections through electrical couplings. In particular, we adopt the connections associated with the direct 1-path network to form a small-world network and follow-up with the corresponding long-range network. Further, the underlying direct small-world network is allowed to temporally change; hence, all long-range connections are also temporal, which makes the model much more realistic from the neurological perspective. This time-varying long-range network is formed by rewiring each link of the underlying 1-path network stochastically with a characteristic rewiring probability p_r, and accordingly all indirect k(>1)-path networks become temporal. The critical interaction strength to reach complete neuronal synchrony is much lower when we take up rapidly switching long-range interactions. We employ the master stability function formalism in order to characterize the local stability of the state of synchronization. The analytically derived stability condition for the complete synchrony state agrees well with the numerical results. Our work strengthens the understanding of time-varying long-range interactions in neuronal ensembles.

Invariance and stability conditions of interlayer synchronization manifold

We investigate interlayer synchronization in a stochastic multiplex hypernetwork which is defined by the two types of connections, one is the intralayer connection in each layer with hypernetwork structure and the other is the interlayer connection between the layers. Here all types of interactions within and between the layers are allowed to vary with a certain rewiring probability. We address the question about the invariance and stability of the interlayer synchronization state in this stochastic multiplex hypernetwork. For the invariance of interlayer synchronization manifold, the adjacency matrices corresponding to each tier in each layer should be equal and the interlayer connection should be either bidirectional or the interlayer coupling function should vanish after achieving the interlayer synchronization state. We analytically derive a necessary-sufficient condition for local stability of the interlayer synchronization state using master stability function approach and a sufficient condition for global stability by constructing a suitable Lyapunov function. Moreover, we analytically derive that intralayer synchronization is unattainable for this network architecture due to stochastic interlayer connections. Remarkably, our derived invariance and stability conditions (both local and global) are valid for any rewiring probabilities, whereas most of the previous stability conditions are only based on a fast switching approximation.

Prediction of bifurcations by varying critical parameters of COVID-19

Coronavirus disease 2019 is a recent strong challenge for the world. In this paper, an epidemiology model is investigated as a model for the development of COVID-19. The propagation of COVID-19 through various sub-groups of society is studied. Some critical parameters, such as the background of mortality without considering the disease state and the speed of moving people from infected to resistance, affect the conditions of society. In this paper, early warning indicators are used to predict the bifurcation points in the system. In the interaction of various sub-groups of society, each sub-group can have various parameters. Six cases of the sub-groups interactions are studied. By coupling these sub-groups, various dynamics of the whole society are investigated.

Enhancing synchrony in multiplex network due to rewiring frequency

Most of the previous studies on synchrony in multiplex networks have been investigated using different types of intralayer network architectures which are either static or temporal. Effect of a temporal layer on intralayer synchrony in a multilayered network still remains elusive. In this paper, we discuss intralayer synchrony in a multiplex network consisting of static and temporal layers and how a temporal layer influences other static layers to enhance synchrony simultaneously. We analytically derive local stability conditions for intralayer synchrony based on the master stability function approach. The analytically derived results are illustrated by numerical simulations on up to five-layers multiplex networks with the paradigmatic Lorenz system as the node dynamics in each individual layer.

Multistability and basin stability in coupled pendulum clocks

In this paper, we investigate the phenomenon of multistability and the concept of basin stability in two coupled pendula with escapement mechanisms, suspended on horizontally oscillating beam. The dynamics of a single pendulum clock is studied and described, showing possible responses of the unit. The basin stability maps are discussed in two-parameters plane, where we vary both the system's stiffness as well as the damping. The possible attractors for the investigated clocks are discussed, showing that different patterns of synchronization and desynchronization can occur. The oscillators may completely synchronize in one of the three possible combinations (including inphase and antiphase ones), practically synchronize with some fluctuations or stay in the irregular pattern, which includes chaotic motion. The transitions between solutions are studied, uncovering that the road from one type of dynamics into another may become very complex. Moreover, we examine the multistability property of our model using the bifurcation diagrams and the basins of attraction maps, discussing possible scenarios in which the states co-exist. The analysis of attractors' basins uncovers complicated structure of the latter ones, exhibiting that the final behavior of investigated model may be hard to determine and trace. Our results are discussed for the cases of identical and nonidentical pendula, as well as light and heavy beam, showing that depending on considered scenario, various patterns of behaviors and transitions may be observed. The research described in this paper proves that the mechanical properties of the system's suspension may play a crucial role in the possibility of the appearance of different types of attractors and that the basin stabilities of states strictly depend on the values of considered parameters.