Asymmetry-induced order in multilayer networks

Delayed-feedback oscillators replicate the dynamics of multiplex networks: Wavefront propagation and stochastic resonance

The widespread development and use of neural networks have significantly enriched a wide range of computer algorithms and promise higher speed at lower cost. However, the imitation of neural networks by means of modern computing substrates is highly inefficient, whereas physical realization of large scale networks remains challenging. Fortunately, delayed-feedback oscillators, being much easier to realize experimentally, represent promising candidates for the empirical implementation of neural networks and next generation computing architectures. In the current research, we demonstrate that coupled bistable delayed-feedback oscillators emulate a multilayer network, where one single-layer network is connected to another single-layer network through coupling between replica nodes, i.e. the multiplex network. We show that all the aspects of the multiplexing impact on wavefront propagation and stochastic resonance identified in multilayer networks of bistable oscillators are entirely reproduced in the dynamics of time-delay oscillators. In particular, varying the coupling strength allows suppressing and enhancing the effect of stochastic resonance, as well as controlling the speed and direction of both deterministic and stochastic wavefront propagation. All the considered effects are studied in numerical simulations and confirmed in physical experiments, showing an excellent correspondence and disclosing thereby the robustness of the observed phenomena.

Transients versus network interactions give rise to multistability through trapping mechanism

In networked systems, the interplay between the dynamics of individual subsystems and their network interactions has been found to generate multistability in various contexts. Despite its ubiquity, the specific mechanisms and ingredients that give rise to multistability from such interplay remain poorly understood. In a network of coupled excitable units, we demonstrate that this interplay generating multistability occurs through a competition between the units' transient dynamics and their coupling. Specifically, the diffusive coupling between the units reinjects them into the excitability region of their individual state space, effectively trapping them there. We show that this trapping mechanism leads to the coexistence of multiple types of oscillations: periodic, quasi-periodic, and even chaotic, although the units separately do not oscillate. Interestingly, we find that the attractors emerge through different types of bifurcations-in particular, the periodic attractors emerge through either saddle-node of limit cycles bifurcations or homoclinic bifurcations-but in all cases, the reinjection mechanism is present.

Shrimp structure as a test bed for ordinal pattern measures

Identifying complex periodic windows surrounded by chaos in the two or higher dimensional parameter space of certain dynamical systems is a challenging task for time series analysis based on complex network approaches. This holds particularly true for the case of shrimp structures, where different bifurcations occur when crossing different domain boundaries. The corresponding dynamics often exhibit either period-doubling when crossing the inner boundaries or, respectively, intermittency for outer boundaries. Numerically characterizing especially the period-doubling route to chaos is difficult for most existing complex network based time series analysis approaches. Here, we propose to use ordinal pattern transition networks (OPTNs) to characterize shrimp structures, making use of the fact that the transition behavior between ordinal patterns encodes additional dynamical information that is not captured by traditional ordinal measures such as permutation entropy. In particular, we compare three measures based on ordinal patterns: traditional permutation entropy εO, average amplitude fluctuations of ordinal patterns ⟨σ⟩, and OPTN out-link transition entropy εE. Our results demonstrate that among those three measures, εE performs best in distinguishing chaotic from periodic time series in terms of classification accuracy. Therefore, we conclude that transition frequencies between ordinal patterns encoded in the OPTN link weights provide complementary perspectives going beyond traditional methods of ordinal time series analysis that are solely based on pattern occurrence frequencies.

Multi-type synchronization for coupled van der Pol oscillator systems with multiple coupling modes

In this paper, we investigate synchronous solutions of coupled van der Pol oscillator systems with multiple coupling modes using the theory of rotating periodic solutions. Multiple coupling modes refer to two or three types of coupling modes in van der Pol oscillator networks, namely, position, velocity, and acceleration. Rotating periodic solutions can represent various types of synchronous solutions corresponding to different phase differences of coupled oscillators. When matrices representing the topology of different coupling modes have symmetry, the overall symmetry of the oscillator system depends on the intersection of the symmetries of the different topologies, determining the type of synchronous solutions for the coupled oscillator network. When matrices representing the topology of different coupling modes lack symmetry, if the adjacency matrices representing different coupling modes can be simplified into structurally identical quotient graphs (where weights can be proportional) through the same external equitable partition, the symmetry of the quotient graph determines the synchronization type of the original system. All these results are consistent with multi-layer networks where connections between different layers are one-to-one.

The complementary contribution of each order topology into the synchronization of multi-order networks

Higher-order interactions improve our capability to model real-world complex systems ranging from physics and neuroscience to economics and social sciences. There is great interest nowadays in understanding the contribution of higher-order terms to the collective behavior of the network. In this work, we investigate the stability of complete synchronization of complex networks with higher-order structures. We demonstrate that the synchronization level of a network composed of nodes interacting simultaneously via multiple orders is maintained regardless of the intensity of coupling strength across different orders. We articulate that lower-order and higher-order topologies work together complementarily to provide the optimal stable configuration, challenging previous conclusions that higher-order interactions promote the stability of synchronization. Furthermore, we find that simply adding higher-order interactions based on existing connections, as in simple complexes, does not have a significant impact on synchronization. The universal applicability of our work lies in the comprehensive analysis of different network topologies, including hypergraphs and simplicial complexes, and the utilization of appropriate rescaling to assess the impact of higher-order interactions on synchronization stability.

Composed solutions of synchronized patterns in multiplex networks of Kuramoto oscillators

Networks with different levels of interactions, including multilayer and multiplex networks, can display a rich diversity of dynamical behaviors and can be used to model and study a wide range of systems. Despite numerous efforts to investigate these networks, obtaining mathematical descriptions for the dynamics of multilayer and multiplex systems is still an open problem. Here, we combine ideas and concepts from linear algebra and graph theory with nonlinear dynamics to offer a novel approach to study multiplex networks of Kuramoto oscillators. Our approach allows us to study the dynamics of a large, multiplex network by decomposing it into two smaller systems: one representing the connection scheme within layers (intra-layer), and the other representing the connections between layers (inter-layer). Particularly, we use this approach to compose solutions for multiplex networks of Kuramoto oscillators. These solutions are given by a combination of solutions for the smaller systems given by the intra- and inter-layer systems, and in addition, our approach allows us to study the linear stability of these solutions.

Dynamics of a two-layer neuronal network with asymmetry in coupling

Investigating the effect of changes in neuronal connectivity on the brain's behavior is of interest in neuroscience studies. Complex network theory is one of the most capable tools to study the effects of these changes on collective brain behavior. By using complex networks, the neural structure, function, and dynamics can be analyzed. In this context, various frameworks can be used to mimic neural networks, among which multi-layer networks are a proper one. Compared to single-layer models, multi-layer networks can provide a more realistic model of the brain due to their high complexity and dimensionality. This paper examines the effect of changes in asymmetry coupling on the behaviors of a multi-layer neuronal network. To this aim, a two-layer network is considered as a minimum model of left and right cerebral hemispheres communicated with the corpus callosum. The chaotic model of Hindmarsh-Rose is taken as the dynamics of the nodes. Only two neurons of each layer connect two layers of the network. In this model, it is assumed that the layers have different coupling strengths, so the effect of each coupling change on network behavior can be analyzed. As a result, the projection of the nodes is plotted for several coupling strengths to investigate how the asymmetry coupling influences the network behaviors. It is observed that although no coexisting attractor is present in the Hindmarsh-Rose model, an asymmetry in couplings causes the emergence of different attractors. The bifurcation diagrams of one node of each layer are presented to show the variation of the dynamics due to coupling changes. For further analysis, the network synchronization is investigated by computing intra-layer and inter-layer errors. Calculating these errors shows that the network can be synchronized only for large enough symmetric coupling.

Multiplexing-based control of stochastic resonance

We show that multiplexing (Here, the term "multiplexing" means a special network topology where a one-layer network is connected to another one-layer networks through coupling between replica nodes. In the present paper, this term does not refer to the signal processing issues and telecommunications.) allows us to control noise-induced dynamics of multilayer networks in the regime of stochastic resonance. We illustrate this effect on an example of two- and multi-layer networks of bistable overdamped oscillators. In particular, we demonstrate that multiplexing suppresses the effect of stochastic resonance if the periodic forcing is present in only one layer. In contrast, multiplexing allows us to enhance the stochastic resonance if the periodic forcing and noise are present in all the interacting layers. In such a case, the impact of multiplexing has a resonant character: the most pronounced effect of stochastic resonance is achieved for an appropriate intermediate value of coupling strength between the layers. Moreover, multiplexing-induced enhancement of the stochastic resonance can become more pronounced for the increasing number of coupled layers. To visualize the revealed phenomena, we use the evolution of the dependence of the signal-to-noise ratio on the noise intensity for varying strength of coupling between the layers.

Emergent microrobotic oscillators via asymmetry-induced order

Spontaneous oscillations on the order of several hertz are the drivers of many crucial processes in nature. From bacterial swimming to mammal gaits, converting static energy inputs into slowly oscillating power is key to the autonomy of organisms across scales. However, the fabrication of slow micrometre-scale oscillators remains a major roadblock towards fully-autonomous microrobots. Here, we study a low-frequency oscillator that emerges from a collective of active microparticles at the air-liquid interface of a hydrogen peroxide drop. Their interactions transduce ambient chemical energy into periodic mechanical motion and on-board electrical currents. Surprisingly, these oscillations persist at larger ensemble sizes only when a particle with modified reactivity is added to intentionally break permutation symmetry. We explain such emergent order through the discovery of a thermodynamic mechanism for asymmetry-induced order. The on-board power harvested from the stabilised oscillations enables the use of electronic components, which we demonstrate by cyclically and synchronously driving a microrobotic arm. This work highlights a new strategy for achieving low-frequency oscillations at the microscale, paving the way for future microrobotic autonomy.