Nonlinear Sciences

Locusts are significant agricultural pests. Under favorable environmental conditions flightless juveniles may aggregate into coherent, aligned swarms referred to as hopper bands. These bands are often observed as a propagating wave having a dense front with rapidly decreasing density in the wake. A tantalizing and common observation is that these fronts slow and steepen in the presence of green vegetation. This suggests the collective motion of the band is mediated by resource consumption. Our goal is to model and quantify this effect. We focus on the Australian plague locust, for which excellent field and experimental data is available. Exploiting the alignment of locusts in hopper bands, we concentrate solely on the density variation perpendicular to the front. We develop two models in tandem; an agent-based model that tracks the position of individuals and a partial differential equation model that describes locust density. In both these models, locust are either stationary (and feeding) or moving. Resources decrease with feeding. The rate at which locusts transition between moving and stationary (and vice versa) is enhanced (diminished) by resource abundance. This effect proves essential to the formation, shape, and speed of locust hopper bands in our models. From the biological literature we estimate ranges for the ten input parameters of our models. Sobol sensitivity analysis yields insight into how the band's collective characteristics vary with changes in the input parameters. By examining 4.4 million parameter combinations, we identify biologically consistent parameters that reproduce field observations. We thus demonstrate that resource-dependent behavior can explain the density distribution observed in locust hopper bands. This work suggests that feeding behaviors should be an intrinsic part of future modeling efforts.

The role of counter-rotating oscillators in an ensemble of coexisting co- and counter-rotating oscillators is examined by increasing the proportion of the latter. The phenomenon of aging transition was identified at a critical value of the ratio of the counter-rotating oscillators, which was otherwise realized only by increasing the number of inactive oscillators to a large extent. The effect of the mean-field feedback strength in the symmetry preserving coupling is also explored. The parameter space of aging transition was increased abruptly even for a feeble decrease in the feedback strength and subsequently, the aging transition was observed at a critical value of the feedback strength surprisingly without any counter-rotating oscillators. Further, the study was extended to symmetry breaking coupling using conjugate variables and it was observed that the symmetry breaking coupling can facilitating the onset of aging transition even in the absence of counter-rotating oscillators and for the unit value of the feedback strength. In general, the parameter space of aging transition was found to increase by increasing the frequency of oscillators and by increasing the proportion of the counter-rotating oscillators in both the symmetry preserving and symmetry breaking couplings. Further, the transition from oscillatory to aging transition occurs via a Hopf bifurcation, while the transition from aging transition to oscillation death state emerges via Pitchfork bifurcation. Analytical expressions for the critical ratio of the counter- rotating oscillators are deduced to find the stable boundaries of the aging transition.

In this article, we report the development of an emerging dynamical state, namely, the alternating chimera, in a network of identical neuronal systems induced by an external electromagnetic field. Owing to this interaction scenario, the nonlinear neuronal oscillators are coupled indirectly via electromagnetic induction with magnetic flux, through which neurons communicate in spite of the absent physical connections among them. The evolution of each neuron, here, is described by the three-dimensional Hindmarsh-Rose dynamics. We demonstrate that the presence of such non-locally and globally interacting external environments induce a stationary alternating chimera pattern in the ensemble of neurons, whereas in the local coupling limit the network exhibits transient chimera state whenever the local dynamics of the neurons is of chaotic square-wave bursting type. For periodic square-wave bursting of the neurons, similar qualitative phenomenon has been witnessed with the exception of the disappearance of cluster states for non-local and global interactions. Besides these observations, we advance our work while providing confirmation of the findings for neuronal ensembles exhibiting plateau bursting dynamics. These results may deliver better interpretations for different aspects of synchronization appearing in network of neurons through field coupling that also relaxes the prerequisite of synaptic connectivity for realizing chimera state in neuronal networks.

We report the emergence of stable amplitude chimeras and chimera death in a two-layer network where one layer has an ensemble of identical nonlinear oscillators interacting directly through local coupling and indirectly through dynamic agents that form the second layer. The nonlocality in the interaction among the dynamical agents in the second layer induces different types of chimera related dynamical states in the first layer. The amplitude chimeras developed in them are found to be extremely stable, while chimera death states are prevalent for increased coupling strengths. The results presented are for a system of coupled Stuart-Landau oscillators and can in general represent systems with short-range interactions coupled to another set of systems with long range interactions. In this case, by tuning the range of interactions among the oscillators or the coupling strength between the two types of systems, we can control the nature of chimera states and the system can be restored to homogeneous steady states. The dynamic agents interacting nonlocally with long-range interactions can be considered as a dynamic environment or medium interacting with the system. We indicate how the second layer can act as a reinforcement mechanism on the first layer under various possible interactions for desirable effects.

We study the death and restoration of collective oscillations in networks of oscillators coupled through random-walk diffusion. Differently than the usual diffusion coupling used to model chemical reactions, here the equilibria of the uncoupled unit is not a solution of the coupled ensemble. Instead, the connectivity modifies both, the original unstable fixed point and the stable limit-cycle, making them node-dependent. Using numerical simulations in random networks we show that, in some cases, this diffusion induced heterogeneity stabilizes the initially unstable fixed point via a Hopf bifurcation. Further increasing the coupling strength the oscillations can be restored as well. Upon numerical analysis of the stability properties we conclude that this is a novel case of amplitude death. Finally we use a heterogeneous mean-field reduction of the system in order to proof the robustness of this phenomena upon increasing the system size.

Collective decision making processes lie at the heart of many social, political and economic challenges. The classical voter model is a well-established conceptual model to study such processes. In this work, we define a new form of adaptive (or co-evolutionary) voter model posed on a simplicial complex, i.e., on a certain class of hypernetworks/hypergraphs. We use the persuasion rule along edges of the classical voter model and the recently studied re-wiring rule of edges towards like-minded nodes, and introduce a new peer pressure rule applied to three nodes connected via a 2-simplex. This simplicial adaptive voter model is studied via numerical simulation. We show that adding the effect of peer pressure to an adaptive voter model leaves its fragmentation transition, i.e., the transition upon varying the re-wiring rate from a single majority state into to a fragmented state of two different opinion subgraphs, intact. Yet, above and below the fragmentation transition, we observe that the peer pressure has substantial quantitative effects. It accelerates the transition to a single-opinion state below the transition and also speeds up the system dynamics towards fragmentation above the transition. Furthermore, we quantify that there is a multiscale hierarchy in the model leading to the depletion of 2-simplices, before the depletion of active edges. This leads to the conjecture that many other dynamic network models on simplicial complexes may show a similar behaviour with respect to the sequential evolution of simplicies of different dimensions.

Starting from recent experimental observations of starlings and jackdaws, we propose a minimal agent-based mathematical model for bird flocks based on a system of second-order delayed stochastic differential equations with discontinuous (both in space and time) right-hand side. The model is specifically designed to reproduce self-organized spontaneous sudden changes of direction, not caused by external stimuli like predator's attacks. The main novelty of the model is that every bird is a potential turn initiator, thus leadership is formed in a group of indistinguishable agents. We investigate some theoretical properties of the model and we show the numerical results. Biological insights are also discussed.

Chemical and electrical synapses shape the dynamics of neuronal networks. Numerous theoretical studies have investigated how each of these types of synapses contributes to the generation of neuronal oscillations, but their combined effect is less understood. This limitation is further magnified by the impossibility of traditional neuronal mean-field models ---also known as firing rate models, or firing rate equations--- to account for electrical synapses. Here we introduce a novel firing rate model that exactly describes the mean field dynamics of heterogeneous populations of quadratic integrate-and-fire (QIF) neurons with both chemical and electrical synapses. The mathematical analysis of the firing rate model reveals a well-established bifurcation scenario for networks with chemical synapses, characterized by a codimension-2 Cusp point and persistent states for strong recurrent excitatory coupling. The inclusion of electrical coupling generally implies neuronal synchrony by virtue of a supercritical Hopf bifurcation. This transforms the Cusp scenario into a bifurcation scenario characterized by three codimension-2 points (Cusp, Takens-Bogdanov, and Saddle-Node Separatrix Loop), which greatly reduces the possibility for persistent states. This is generic for heterogeneous QIF networks with both chemical and electrical coupling. Our results agree with several numerical studies on the dynamics of large networks of heterogeneous spiking neurons with electrical and chemical coupling.

Transfer entropy in information theory was recently demonstrated [Phys. Rev. E 102, 012404 (2020)] to enable us to elucidate the interaction domain among interacting elements solely from an ensemble of trajectories. There, only pairs of elements whose distances are shorter than some distance variable, termed cutoff distance, are taken into account in the computation of transfer entropies. The prediction performance in capturing the underlying interaction domain is subject to noise level exerted on the elements and the sufficiency of statistics of the interaction events. In this paper, the dependence of the prediction performance is scrutinized systematically on noise level and the length of trajectories by using a modified Vicsek model. The larger the noise level and the shorter the time length of trajectories, the more the derivative of average transfer entropy fluctuates, which makes it difficult to identify the interaction domain in terms of the position of global minimum of the derivative of average transfer entropy. A measure to quantify the degree of strong convexity at coarse-grained level is proposed. It is shown that the convexity score scheme can identify the interaction distance fairly well even while the position of global minimum of the derivative of average transfer entropy does not. We also derive an analytical model to explain the relationship between the interaction domain and the change of transfer entropy that supports our cutoff distance technique to elucidate the underlying interaction domain from trajectories.

Self-organisation lies at the core of fundamental but still unresolved scientific questions, and holds the promise of de-centralised paradigms crucial for future technological developments. While self-organising processes have been traditionally explained by the tendency of dynamical systems to evolve towards specific configurations, or attractors, we see self-organisation as a consequence of the interdependencies that those attractors induce. Building on this intuition, in this work we develop a theoretical framework for understanding and quantifying self-organisation based on coupled dynamical systems and multivariate information theory. We propose a metric of global structural strength that identifies when self-organisation appears, and a multi-layered decomposition that explains the emergent structure in terms of redundant and synergistic interdependencies. We illustrate our framework on elementary cellular automata, showing how it can detect and characterise the emergence of complex structures.