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In this work, we investigate the impact of mixed coupling on synchronization in a multiplex oscillatory network. The network mimics the neural–glial systems by incorporating interacting slow (“glial”) and fast (“neural”) oscillatory layers. Connections between the “glial” elements form a regular periodic structure, in which each element is connected to the eight other neighbor elements, whereas connections among “neural” elements are represented by Watts–Strogatz networks (from regular and small-world to random Erdos–Renyi graph) with a matching mean node degree. We find that the random rewiring toward small-world topology readily yields the dynamics close to that exhibited for a completely random graph, in particular, leading to coarse-graining of dynamics, suppressing multi-stability of synchronization regimes, and the onset of Kuramoto-type synchrony in both layers. The duration of transient dynamics in the system measured by relaxation times is minimized with the increase of random connections in the neural layer, remaining substantial only close to synchronization–desynchronization transitions. “Inhibitory” interactions in the “neural” subnetwork layer undermine synchronization; however, the strong coupling with the “glial” layer overcomes this effect.

Nonlinear dynamics makes use of attactors to describe asymptotic behavior of complex systems. However, in real life can be unattainable, as achieving them might require time much exceeding all relevant timescales of a system. Therefore, there is an increasing interest in quasi-stationary states, where the system rapidly converges to and remains for a long time, before getting into an absorbing (asymptotic) state. Exemplifying in the famous Dawkins‘ Battle of the Sexes game, we demonstrate that quasi-stationary distributions can produce not simply different, but a much more complex behavior, then the asymptotic ones, that is transient self-sustained oscillations of player numbers and the corresponding non-unimodal probability distribution. We find that parameters of the quasi-stationary limit cycle depend on the population size. The work was supported by the Russian Foundation forBasic Research grant 18-32-20221.

Recently, it has been demonstrated that asymptotic states of open quantum system can undergo qualitative changes resembling pitchfork, saddle-node, and period doubling classical bifurcations. Here, making use of the periodically modulated open quantum dimer model, we report and investigate a quantum Neimark-Sacker bifurcation. Its classical counterpart is the birth of a torus (an invariant curve in the Poincaré section) due to instability of a limit cycle (fixed point of the Poincaré map). The quantum system exhibits a transition from unimodal to bagel shaped stroboscopic distributions, as for Husimi representation, as for observables. The spectral properties of Floquet map experience changes reminiscent of the classical case, a pair of complex conjugated eigenvalues approaching a unit circle. Quantum Monte-Carlo wave function unraveling of the Lindblad master equation yields dynamics of single trajectories on “quantumtorus” and allows for quantifying it by rotation number. The bifurcation is sensitive to the number of quantum particles that can also be regarded as a control parameter.

Inspired by the recent interest in collective dynamics of biological neural networks immersed in the glial cell medium, we investigate the frequency and phase order, i.e., Kuramoto type of synchronization in a multiplex two-layer network of phase oscillators of different time scales and topologies. One of them has a long-range connectivity, exemplified by the Erdős-Renyi random network, and supports both kinds of synchrony. The other is a locally coupled two-dimensional lattice that can reach frequency synchronization but lacks phase order. Drastically different layer frequencies disentangle intra- and interlayer synchronization. We find that an indirect but sufficiently strong coupling through the regular layer can induce both phase order in the originally nonsynchronized random layer and global order, even when an isolated regular layer does not manifest it in principle. At the same time, the route to global synchronization is complex: an initial onset of (partial) synchrony in the regular layer, when its intra- and interlayer coupling is increased, provokes the loss of synchrony even in the originally synchronized random layer. Ultimately, a developed asynchronous dynamics in both layers is abruptly taken over by the global synchrony of both kinds.

We investigate the asymptotic state of a periodically driven many-body quantum system which is weakly coupled to an environment. The combined action of the modulations and the environment steers the system towards a state being characterized by a time-periodic density operator. To resolve this asymptotic non-equilibrium state at stroboscopic instants of time, we introduce the dissipative Floquet map, evaluate the stroboscopic density operator as its eigen-element and elucidate how particle interactions affect properties of the density operator. We illustrate the idea with a periodically modulated Bose-Hubbard dimer and discuss the relations between the interaction-induced bifurcations in a mean-field dynamics and changes in the characteristics of the genuine quantum many-body state. We argue that Floquet maps provide insight into the system relaxation towards its asymptotic state and may help to understand whether it is possible (or not) to construct a stroboscopic time-independent generator mimicking the action of the original time-dependent one., 12 pages, 4 figures

We propose and study models of two distributed synthetic gene circuits, toggle-switch and oscillator, each split between two cell strains and coupled via quorum-sensing signals. The distributed toggle switch relies on mutual repression of the two strains, and oscillator is comprised of two strains, one of which acts as an activator for another that in turn acts as a repressor. Distributed toggle switch can exhibit mobile fronts, switching the system from the weaker to the stronger spatially homogeneous state. The circuit can also act as a biosensor, with the switching front dynamics determined by the properties of an external signal. Distributed oscillator system displays another biosensor functionality: oscillations emerge once a small amount of one cell strain appears amid the other, present in abundance. Distribution of synthetic gene circuits among multiple strains allows one to reduce crosstalk among different parts of the overall system and also decrease the energetic burden of the synthetic circuit per cell, which may allow for enhanced functionality and viability of engineered cells.

In an isolated single-particle quantum system a spatial disorder can induce Anderson localization. Being a result of interference, this phenomenon is expected to be fragile in the face of dissipation. Here we show that dissipation can drive a disordered system into a steady state with tunable localization properties. This can be achieved with a set of identical dissipative operators, each one acting non-trivially only on a pair of neighboring sites. Operators are parametrized by a uniform phase, which controls selection of Anderson modes contributing to the state. On the microscopic level, quantum trajectories of a system in a localized steady regime exhibit intermittent dynamics consisting of long-time sticking events near selected modes interrupted by jumps between them., Comment: 5 pages, 5 figures

We present computational data on the thermal conductivity of nonlinear waves in disordered chains. Disorder induces Anderson localization for linear waves and results in a vanishing conductivity. Cubic nonlinearity restores normal conductivity, but with a strongly temperature-dependent conductivity $\kappa(T)$. We find indications for an asymptotic low-temperature $\kappa \sim T^4$ and intermediate temperature $\kappa \sim T^2$ laws. These findings are in accord with theoretical studies of wave packet spreading, where a regime of strong chaos is found to be intermediate, followed by an asymptotic regime of weak chaos (EPL 91 (2010) 30001)., Comment: 8 pages, 3 figures

Recently q-breathers - time-periodic solutions which localize in the space of normal modes and maximize the energy density for some mode vector q0 - were obtained for finite nonlinear lattices. We scale these solutions to arbitrarily large lattices in various lattice dimensions. We study the scaling consequence for previously obtained analytical estimates of the localization length of q-breathers for β-FPU and α-FPU lattices. The first finding is that the degree of localization depends only on intensive quantities and is size independent. Secondly, a critical wave vector km is identified, which depends on one effective nonlinearity parameter, q-breathers minimize the localization length at k0 = km and completely delocalize in the limit k0 → 0, π.

In dissipationless linear media, spatial disorder induces Anderson localization of matter, light, and sound waves. The addition of nonlinearity causes interaction between the eigenmodes, which results in a slow wave diffusion. We go beyond the dissipationless limit of Anderson arrays and consider nonlinear disordered systems that are subjected to the dissipative losses and energy pumping. We show that the Anderson modes of the disordered Ginsburg-Landau lattice possess specific excitation thresholds with respect to the pumping strength. When pumping is increased above the threshold for the band-edge modes, the lattice dynamics yields an attractor in the form of a stable multi-peak pattern. The Anderson attractor is the result of a joint action by the pumping-induced mode excitation, nonlinearity-induced mode interactions, and dissipative stabilization. The regimes of Anderson attractors can be potentially realized with polariton condensates lattices, active waveguide or cavity-QED arrays., 11 pages, 4 figures

We study collective dynamics of interacting centers of exciton-polariton condensation in presence of spatial inhomogeneity, as modeled by diatomic active oscillator lattices. The mode formalism is developed and employed to derive existence and stability criteria of plane wave solutions. It is demonstrated that $k_0=0$ wave number mode with the binary elementary cell on a diatomic lattice possesses superior existence and stability properties. Decreasing net on-site losses (balance of dissipation and pumping) or conservative nonlinearity favors multistability of modes, while increasing frequency mismatch between adjacent oscillators detriments it. On the other hand, spatial inhomogeneity may recover stability of modes at high nonlinearities. Entering the region where all single-mode solutions are unstable we discover subsequent transitions between localized quasiperiodic, chaotic and global chaotic dynamics in the mode space, as nonlinearity increases. Importantly, the last transition evokes the loss of synchronization. These effects may determine lasing dynamics of interacting exciton-polariton condensation centers., 9 pages, 3 figures

In dissipationless linear lattices, spatial disorder or quasiperiodic modulations in on-site potentials induce localization of the eigenstates and block the spreading of wave packets. Quasiperiodic inhomogeneities allow for the metal–insulator transition at a finite modulation amplitude already in one dimension. We go beyond the dissipationless limit and consider nonlinear quasi-periodic arrays that are additionally subjected to dissipative losses and energy pumping. We find finite excitation thresholds for oscillatory phases in both metallic and insulating regimes. In contrast to disordered arrays, the transition in the metallic and weakly insulating regimes display features of the second order phase transition accompanied by a large-scale cluster synchronization. In the limit of strong localization, we find the existence of globally stable asymptotic states consisting of several localized modes. These localization attractors and chaotic synchronization effects can be potentially implemented with polariton condensate lattices and cavity-QED arrays.

Single particle states in a chain with quasiperiodic potential show a metal-insulator transition upon the change of the potential strength. We consider two particles with local interaction in the single particle insulating regime. The two particle states change from being localized to delocalized upon an increase of the interaction strength to a nonperturbative finite value. At even larger interaction strength the states become localized again. This transition of two particle bound states into a correlated metal is due to a resonant mixing of the noninteracting two particle eigenstates. In the discovered correlated metal states two particles move coherently together through the whole chain, therefore contributing to a finite conductivity., 4 pages, 4 figures

We present experimental and numerical evidence of synchronization of burst events in two different modulated ${\mathrm{CO}}_{2}$ lasers. Bursts appear randomly in each laser as trains of large amplitude spikes intercalated by a small amplitude chaotic regime. Experimental data and model show the frequency locking of bursts in a suitable interval of coupling strength. We explain the mechanism of this phenomenon and demonstrate the inhibitory properties of the implemented coupling.

We study phase synchronization effects of chaotic oscillators with a type-I intermittency behavior. The external and mutual locking of the average length of the laminar stage for coupled discrete and continuous in time systems is shown and the mechanism of this synchronization is explained. We demonstrate that this phenomenon can be described by using results of the parametric resonance theory and that this correspondence enables one to predict and derive all zones of synchronization.

We study the effects of mutual and external chaotic phase synchronization in ensembles of bursting oscillators. These oscillators (used for modeling neuronal dynamics) are essentially multiple time scale systems. We show that a transition to mutual phase synchronization takes place on the bursting time scale of globally coupled oscillators, while on the spiking time scale, they behave asynchronously. We also demonstrate the effect of the onset of external chaotic phase synchronization of the bursting behavior in the studied ensemble by a periodic driving applied to one arbitrarily taken neuron. We also propose an explanation of the mechanism behind this effect. We infer that the demonstrated phenomenon can be used efficiently for controlling bursting activity in neural ensembles.

We study the effect of noncoherence on the onset of phase synchronization of two coupled chaotic oscillators. Depending on the coherence properties of oscillations characterized by the phase diffusion, three types of transitions to phase synchronization are found. For phase-coherent attractors this transition occurs shortly after one of the zero Lyapunov exponents becomes negative. At rather strong phase diffusion, phase locking manifests a strong degree of generalized synchronization, and occurs only after one positive Lyapunov exponent becomes negative. For intermediate phase diffusion, phase synchronization sets in via an interior crises of the hyperchaotic set.