Dynamics of multilayer networks with amplification

Complete synchronization of three-layer Rulkov neuron network coupled by electrical and chemical synapses

This paper analyzes the complete synchronization of a three-layer Rulkov neuron network model connected by electrical synapses in the same layers and chemical synapses between adjacent layers. The outer coupling matrix of the network is not Laplacian as in linear coupling networks. We develop the master stability function method, in which the invariant manifold of the master stability equations (MSEs) does not correspond to the zero eigenvalues of the connection matrix. After giving the existence conditions of the synchronization manifold about the nonlinear chemical coupling, we investigate the dynamics of the synchronization manifold, which will be identical to that of a synchronous network by fixing the same parameters and initial values. The waveforms show that the transient chaotic windows and the transient approximate periodic windows with increased or decreased periods occur alternatively before asymptotic behaviors. Furthermore, the Lyapunov exponents of the MSEs indicate that the network with a periodic synchronization manifold can achieve complete synchronization, while the network with a chaotic synchronization manifold can not. Finally, we simulate the effects of small perturbations on the asymptotic regimes and the evolution routes for the synchronous periodic and the non-synchronous chaotic network.

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.

Faster network disruption from layered oscillatory dynamics

Nonlinear complex network-coupled systems typically have multiple stable equilibrium states. Following perturbations or due to ambient noise, the system is pushed away from its initial equilibrium, and, depending on the direction and the amplitude of the excursion, it might undergo a transition to another equilibrium. It was recently demonstrated [M. Tyloo, J. Phys. Complex. 3 03LT01 (2022)] that layered complex networks may exhibit amplified fluctuations. Here, I investigate how noise with system-specific correlations impacts the first escape time of nonlinearly coupled oscillators. Interestingly, I show that, not only the strong amplification of the fluctuations is a threat to the good functioning of the network but also the spatial and temporal correlations of the noise along the lowest-lying eigenmodes of the Laplacian matrix. I analyze first escape times on synthetic networks and compare noise originating from layered dynamics to uncorrelated noise.

Synchronization in a multilevel network using the Hamilton-Jacobi-Bellman (HJB) technique

This paper presents the optimal control and synchronization problem of a multilevel network of Rössler chaotic oscillators. Using the Hamilton-Jacobi-Bellman technique, the optimal control law with a three-state variable feedback is designed such that the trajectories of all the Rössler oscillators in the network are optimally synchronized at each level. Furthermore, we provide numerical simulations to demonstrate the effectiveness of the proposed approach for the cases of one and three networks. A perfect correlation between the MATLAB and PSpice results was obtained, thus allowing the experimental validation of our designed controller and shows the effectiveness of the theoretical results.

Emergent Dynamics and Spatio Temporal Patterns on Multiplex Neuronal Networks

We present a study on the emergence of a variety of spatio temporal patterns among neurons that are connected in a multiplex framework, with neurons on two layers with different functional couplings. With the Hindmarsh-Rose model for the dynamics of single neurons, we analyze the possible patterns of dynamics in each layer separately and report emergent patterns of activity like in-phase synchronized oscillations and amplitude death (AD) for excitatory coupling and anti-phase mixed-mode oscillations (MMO) in multi-clusters with phase regularities when the connections are inhibitory. When they are multiplexed, with neurons of one layer coupled with excitatory synaptic coupling and neurons of the other layer coupled with inhibitory synaptic coupling, we observe the transfer or selection of interesting patterns of collective behavior between the layers. While the revival of oscillations occurs in the layer with excitatory coupling, the transition from anti-phase to in-phase and vice versa is observed in the other layer with inhibitory synaptic coupling. We also discuss how the selection of these spatio temporal patterns can be controlled by tuning the intralayer or interlayer coupling strengths or increasing the range of non-local coupling. With one layer having electrical coupling while the other synaptic coupling of excitatory(inhibitory)type, we find in-phase(anti-phase) synchronized patterns of activity among neurons in both layers.

Synchronizability of double-layer dumbbell networks

Synchronization of multiplex networks has been a topical issue in network science. Dumbbell networks are very typical structures in complex networks which are distinguished from both regular star networks and general community structures, whereas the synchronous dynamics of a double-layer dumbbell network relies on the interlink patterns between layers. In this paper, two kinds of double-layer dumbbell networks are defined according to different interlayer coupling patterns: one with the single-link coupling pattern between layers and the other with the two-link coupling pattern between layers. Furthermore, the largest and smallest nonzero eigenvalues of the Laplacian matrix are calculated analytically and numerically for the single-link coupling pattern and also obtained numerically for the two-link coupling pattern so as to characterize the synchronizability of double-layer dumbbell networks. It is shown that interlayer coupling patterns have a significant impact on the synchronizability of multiplex systems. Finally, a numerical example is provided to verify the effectiveness of theoretical analysis. Our findings can facilitate company managers to select optimal interlayer coupling patterns and to assign proper parameters in terms of improving the efficiency and reducing losses of the whole team.