Landau Oscillators Research Articles

Dynamic Survivability Centrality in Nonlinear Oscillator Systems

In light of the fact that existing centrality indexes disregard the influence of dynamic characteristics and lack generalizability due to standard diversification, this study investigates dynamic survivability centrality, which enables quantification of oscillators’ capacity to impact the dynamic survivability of nonlinear oscillator systems. Taking an Erdős–Rényi random graph system consisting of Stuart–Landau oscillators as an illustrative example, the typical symmetry synchronization is considered as the key mission to be accomplished in light of the study and the dynamic survivability centrality value is found to be dependent on both the system size and connection density. Starting with a small scale system, the correctness of the theoretical results and the superiority in comparison to traditional indexes are verified. Further, we present the quantitative results by means of error analysis, distribution comparison of various indexes and relationship with system structure exploration, and give the position of the key oscillator. The results demonstrate a negligible error between the theoretical and numerical outcomes, and highlighting that the distribution of dynamic survivability centrality closely resembles the distribution of system state changes. The conclusions serve as evidence for the accuracy and validity of the proposed index. The findings provide an effective approach to protect systems to improve dynamic survivability.

Explosive synchronization in interacting star networks

We study the transition to phase synchronization in an ensemble of Stuart–Landau oscillators interacting on a star network. We observe that by introducing frequency-weighted coupling and timescale variations in the dynamics of nodes, the system exhibits a first-order explosive transition to phase synchrony. Further, we extend this study to understand the nature of synchronization in the case of two coupled star networks. If the coupled star networks are identical, we observe that with increasing inter-star coupling strength, the hysteresis width initially increases, reaches a maximum value, then decreases before saturating. If the interacting star networks are non-identical, we observe that the transition to the coherent state is preceded by the occurrence of intermittent in-phase and anti-phase synchrony for small inter-star coupling. However, for large values of coupling strengths, we observe that the intermittent state disappears and the hysteresis width changes as in coupled identical star networks. We characterize these transitions by plotting the Lyapunov exponents for the system and the master stability function.

Higher-Order Network Interactions Through Phase Reduction for Oscillators with Phase-Dependent Amplitude

Coupled oscillator networks provide mathematical models for interacting periodic processes. If the coupling is weak, phase reduction—the reduction of the dynamics onto an invariant torus—captures the emergence of collective dynamical phenomena, such as synchronization. While a first-order approximation of the dynamics on the torus may be appropriate in some situations, higher-order phase reductions become necessary, for example, when the coupling strength increases. However, these are generally hard to compute and thus they have only been derived in special cases: This includes globally coupled Stuart–Landau oscillators, where the limit cycle of the uncoupled nonlinear oscillator is circular as the amplitude is independent of the phase. We go beyond this restriction and derive second-order phase reductions for coupled oscillators for arbitrary networks of coupled nonlinear oscillators with phase-dependent amplitude, a scenario more reminiscent of real-world oscillations. We analyze how the deformation of the limit cycle affects the stability of important dynamical states, such as full synchrony and splay states. By identifying higher-order phase interaction terms with hyperedges of a hypergraph, we obtain natural classes of coupled phase oscillator dynamics on hypergraphs that adequately capture the dynamics of coupled limit cycle oscillators.

Synchronization of two indirectly coupled singly resonant optical parametric oscillators

We analyse a system of a singly resonant optical parametric oscillator for a second order nonlinear material. First, we show that the dynamics of the resonating cavity signal mode can be expressed by a Stuart–Landau oscillator, for a certain pumping powers close to the threshold. Second, we couple two optical parametric oscillators indirectly via a cold resonator. When the condition of a weak coupling is satisfied, the limit-cycle of each oscillator is unaltered, and the system is described by a coupled phase oscillator model (Kuramoto model), where a frequency synchronization of the two oscillators occurs at a critical coupling constant.

Effects of uncommon non-isochronicities on remote synchronization

Remote synchronization (RS) is referred to as the weak synchronization between indirectly coupled nodes (among leaf nodes) while asynchronous between the directed coupled nodes (i.e., hub and leaf nodes). The underlying mechanisms are related to (i) coupling terms indirectly connecting two nearly identical oscillators while having a much different mediator in between, (ii) non-isochronicity of the oscillators. Here we show the effects of uncommon non-isochronicity values of oscillators on the occurrence of RS. In addition, we show clear difference of using either the first order phase approximations of the Stuart–Landau oscillators or Kuramoto–Sakaguchi (KS) models to study RS. By both theoretical and numerical analysis, we show that the critical coupling for synchronization between the directly connected nodes is inversely proportional to the difference value of non-isochronicity. On the contrary, the critical coupling for RS between indirectly connected leaf nodes is determined by the sum of non-isochronicity. In addition, RS may experience fading out after the first appearance when the coupling is progressively increased, which is indeed time delayed phase synchronization.

Dynamics of oscillator populations with disorder in the coupling phase shifts

We study populations of oscillators, all-to-all coupled by means of quenched disordered phase shifts. While there is no traditional synchronization transition with a nonvanishing Kuramoto order parameter, the system demonstrates a specific order as the coupling strength increases. This order is characterized by partial phase locking, which is put into evidence by the introduced novel correlation order parameter, which is shown to grow monotonically with the coupling strength, and via frequency entrainment by following concentration of the oscillators frequencies. Simulations with phase oscillators, Stuart–Landau oscillators, and chaotic Roessler oscillators demonstrate similar scaling of the correlation order parameter with the coupling and the system size and also similar behavior of the frequencies with maximal entrainment (at which the standard deviation of the frequencies is reduced by a factor close to four) at some finite coupling.

Emergent behavior of conjugate-coupled Stuart–Landau oscillators in directed star networks

Emergent behavior of conjugate-coupled Stuart–Landau oscillators in directed star networks

Modeling of Main Parameters for Near Wake from Pair of Side-by-Side not too Close Cylinders

A one-dimensional model of the near wake produced by a pair of not-too-close cylinders in a viscous incompressible fluid is developed on the basis of the Stuart–Landau model and perturbation theory. The whole wake is considered as two interacting partial von Kàrmàn vortex streets or two coupled Stuart–Landau oscillators, when the nonlinear nature of the interaction of these streets is taken into account. The observable spectrum of the global modes of the wake is obtained, the calculated and experimental eigenfrequencies of the oscillation modes being in agreement with respect to the modes.

Quantum synchronization effects induced by strong nonlinearities

A paradigm for quantum synchronization is the quantum analog of the Stuart--Landau oscillator, which corresponds to a van der Pol oscillator in the limit of weak (i.e. vanishingly small) nonlinearity. Due to this limitation, the quantum Stuart--Landau oscillator fails to capture interesting nonlinearity-induced phenomena such as relaxation oscillations. To overcome this deficiency we propose an alternative model which approximates the van der Pol oscillator to finitely large nonlinearities while remaining numerically tractable. This allows us to uncover interesting phenomena in the deep-quantum strongly-nonlinear regime with no classical analog, such as the persistence of amplitude death on resonance. We also report nonlinearity-induced position correlations in reactively coupled quantum oscillators. Such coupled oscillations become more and more correlated with increasing nonlinearity before reaching some maximum. Again, this behavior is absent classically. We also show how strong nonlinearity can enlarge the synchronization bandwidth in both single and coupled oscillators. This effect can be harnessed to induce mutual synchronization between two oscillators initially in amplitude death.

Topological insulators in relativistic quantum mechanics

By employing supersymmetric quantum mechanics, the analog of relativistic quantum mechanics for topological insulators is considered. The procedure determines the general structure of the spin–orbit coupling even though it constrains the coupling constant. Through analogies with nonelativistic quantum mechanics, we construct relativistic Hamiltonians that share properties of topological insulators. As an application of our results, we study in detail the nonabelian extension of the Dirac–Landau oscillator and relativistic scattering with a spin–orbit coupling term.

Spiral wave chimeras in populations of oscillators coupled to a slowly varying diffusive environment

Chimera states are firstly discovered in nonlocally coupled oscillator systems. Such a nonlocal coupling arises typically as oscillators are coupled via an external environment whose characteristic time scale τ is so small (i.e., τ → o) that it could be eliminated adiabatically. Nevertheless, whether the chimera states still exist in the opposite situation (i.e., τ ≫ 1) is unknown. Here, by coupling large populations of Stuart—Landau oscillators to a diffusive environment, we demonstrate that spiral wave chimeras do exist in this oscillator-environment coupling system even when τ is very large. Various transitions such as from spiral wave chimeras to spiral waves or unstable spiral wave chimeras as functions of the system parameters are explored. A physical picture for explaining the formation of spiral wave chimeras is also provided. The existence of spiral wave chimeras is further confirmed in ensembles of FitzHugh—Nagumo oscillators with the similar oscillator-environment coupling mechanism. Our results provide an affirmative answer to the observation of spiral wave chimeras in populations of oscillators mediated via a slowly changing environment and give important hints to generate chimera patterns in both laboratory and realistic chemical or biological systems.

Amplitude death in multiplex networks with competing attractive and repulsive interactions

Amplitude death (AD) phenomenon is one of the most striking and ubiquitous collective behaviors of coupled dynamical networks, in which all the dynamical units will cease their oscillations totally when the coupling occurs. Various appealing works have been devoted to the study of AD in the single-layer networks from theoretical and experimental perspectives in the last decades. However, it is still unclear under which conditions AD can be induced in the multiplex networks. In this work, we investigate the onset of AD of coupled systems connected via multiplex network architectures in the presence of competing attractive and repulsive intralayer and interlayer interactions. We first theoretically derive the critical condition of AD for a class of generalized multiplex network models. Then we apply it to calculate the stable AD regions for the case of a single-layer network of Stuart–Landau oscillators and observe that in the single-layer ring network, AD only can occur for small networks regardless of how to tune the proportion of repulsive coupling. Whereas, for the single-layer all-to-all network, AD always can appear, and the larger the network size or the proportion of repulsive coupling, the smaller the required coupling strength for AD is. Finally, we show in the three-layer multiplex network of Stuart–Landau oscillators that the stable AD regions can be concluded to four types based on the intralayer and interlayer network topologies. In particular, even if AD does not exist separately in each layer network, it can be reached by choosing appropriately interlayer interaction in the three-layer multiplex network. Our study sheds a new insight into the generation of AD phenomenon and gives a general framework to analyze the steady state behaviors of coupled systems in the multiplex networks.

Transition from inhomogeneous limit cycles to oscillation death in nonlinear oscillators with similarity-dependent coupling.

We consider a system of coupled nonlinear oscillators in which the interaction is modulated by a measure of the similarity between the oscillators. Such a coupling is common in treating spatially mobile dynamical systems where the interaction is distance dependent or in resonance-enhanced interactions, for instance. For a system of Stuart-Landau oscillators coupled in this manner, we observe a novel route to oscillation death via a Hopf bifurcation. The individual oscillators are confined to inhomogeneous limit cycles initially and are damped to different fixed points after the bifurcation. Analytical and numerical results are presented for this case, while numerical results are presented for coupled Rössler and Sprott oscillators.

Phase frustration induced remote synchronization.

Remote synchronization (RS) may take an important role in brain functioning and its study has attracted much attention in recent years. So far, most studies of RS are focused on the Stuart-Landau oscillators with mean-field coupling. However, realistic cases may have more complicated couplings and behaviors, such as the brain networks. To make the study of RS a substantial progress toward realistic situations, we here present a model of RS with phase frustration and show that RS can be induced for those systems where no RS exists when there is no phase frustration. By numerical simulations on both the Stuart-Landau and Kuramoto oscillators, we find that the optimal range of RS depends on the match of phase frustrations between the hub and leaf nodes and a fixed relationship of this match is figured out. While for the non-optimal range of RS, we find that RS exists only in a linear band between the phase frustrations of the hub and leaf nodes. A brief theoretical analysis is provided to explain these results.

Publisher's Note: "Mean-field analysis of Stuart-Landau oscillator networks with symmetric coupling and dynamical noise" [Chaos 32, 063114 (2022)

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Stability in star networks of identical Stuart–Landau oscillators with asymmetric coupling

In this paper, we systematically investigate the stability issues of different kinds of collective behaviors, including the equilibrium point, complete synchronization, quenched hub incoherence, anti-phase remote synchronization and regular remote synchronization, which arise in identical Stuart–Landau oscillators asymmetrically coupled in a star network. We have analytically obtained their rigorous stability conditions and stable basins, which are well verified by numerical results. Additionally, we surprisingly find that, for measuring the remote synchronization, the two traditional order parameters rdirect and rindirect are not well-defined metrics. All these results may help us to further improve our understanding of nonlinear systems’ complete, partial and cluster synchronization behaviors in real or man-made complex networks.

Mean-field analysis of Stuart-Landau oscillator networks with symmetric coupling and dynamical noise.

This paper presents analyses of networks composed of homogeneous Stuart-Landau oscillators with symmetric linear coupling and dynamical Gaussian noise. With a simple mean-field approximation, the original system is transformed into a surrogate system that describes uncorrelated oscillation/fluctuation modes of the original system. The steady-state probability distribution for these modes is described using an exponential family, and the dynamics of the system are mainly determined by the eigenvalue spectrum of the coupling matrix and the noise level. The variances of the modes can be expressed as functions of the eigenvalues and noise level, yielding the relation between the covariance matrix and the coupling matrix of the oscillators. With decreasing noise, the leading mode changes from fluctuation to oscillation, generating apparent synchrony of the coupled oscillators, and the condition for such a transition is derived. Finally, the approximate analyses are examined via numerical simulation of the oscillator networks with weak coupling to verify the utility of the approximation in outlining the basic properties of the considered coupled oscillator networks. These results are potentially useful for the modeling and analysis of indirectly measured data of neurodynamics, e.g., via functional magnetic resonance imaging and electroencephalography, as a counterpart of the frequently used Ising model.

Swarmalators under competitive time-varying phase interactions

Swarmalators are entities with the simultaneous presence of swarming and synchronization that reveal emergent collective behavior due to the fascinating bidirectional interplay between phase and spatial dynamics. Although different coupling topologies have already been considered, here we introduce time-varying competitive phase interaction among swarmalators where the underlying connectivity for attractive and repulsive coupling varies depending on the vision (sensing) radius. Apart from investigating some fundamental properties like conservation of center of position and collision avoidance, we also scrutinize the cases of extreme limits of vision radius. The concurrence of attractive–repulsive competitive phase coupling allows the exploration of diverse asymptotic states, like static π, and mixed phase wave states, and we explore the feasible routes of those states through a detailed numerical analysis. In sole presence of attractive local coupling, we reveal the occurrence of static cluster synchronization where the number of clusters depends crucially on the initial distribution of positions and phases of each swarmalator. In addition, we analytically calculate the sufficient condition for the emergence of the static synchronization state. We further report the appearance of the static ring phase wave state and evaluate its radius theoretically. Finally, we validate our findings using Stuart–Landau oscillators to describe the phase dynamics of swarmalators subject to attractive local coupling.

On the role of nonlinear correlations in reduced-order modelling

This work investigates nonlinear dimensionality reduction as a means of improving the accuracy and stability of reduced-order models of advection-dominated flows. Nonlinear correlations between temporal proper orthogonal decomposition (POD) coefficients can be exploited to identify latent low-dimensional structure, approximating the attractor with a minimal set of driving modes and a manifold equation for the remaining modes. By viewing these nonlinear correlations as an invariant manifold reduction, this least-order representation can be used to stabilize POD–Galerkin models or as a state space for data-driven model identification. In the latter case, we use sparse polynomial regression to learn a compact, interpretable dynamical system model from the time series of the active modal coefficients. We demonstrate this perspective on a quasiperiodic shear-driven cavity flow and show that the dynamics evolves on a torus generated by two independent Stuart–Landau oscillators. The specific approach to nonlinear correlations analysis used in this work is applicable to periodic and quasiperiodic flows, and cannot be applied to chaotic or turbulent flows. However, the results illustrate the limitations of linear modal representations of advection-dominated flows and motivate the use of nonlinear dimensionality reduction more broadly for exploiting underlying structure in reduced-order models.

System Identification of Two-Dimensional Transonic Buffet

When modeled within the unsteady Reynolds-averaged Navier–Stokes framework, the shock wave dynamics on a two-dimensional airfoil at transonic buffet conditions is characterized by time-periodic oscillations. Given the time series of the lift coefficient at different angles of attack for the OAT15A supercritical profile, the Sparse Identification of Nonlinear Dynamics (SINDy) technique is used to extract a parameterized, interpretable, and minimal-order description of this dynamics. For all the operating conditions considered, SINDy infers that the dynamics in the lift-coefficient time series can be modeled by a simple parameterized Stuart–Landau oscillator, reducing the computation time from hundreds of core hours to seconds. The identified models are then supplemented with equally parameterized low-dimensional dynamic mode decomposition based state observers enabling real-time estimation of the whole flowfield from the predicted lift. The value of this simple model is in the possibility of enriching sparse data sets in the buffet regime for new geometries. Only a few unsteady simulations need to be performed to adjust the coefficients of the identified model and then supplement the data set with new cases. Simplicity, accuracy, and interpretability make the identified model an attractive tool toward the construction of real-time systems to be used during the design, certification, and operational phases of the aircraft life cycle.