Networks Of Nonlinear Oscillators Research Articles

Transition of a network of nonlinear interactions into a regime of strong nonlinear fluctuations: A paradigm for the edge localized mode onset

We study how a network of nonlinear oscillators transits into a regime of strong nonlinear fluctuations when perturbed by a triad. In this regime, most of the potential energy contained in the waves is made available to the system through strong nonlinear fluctuations. This analysis is motivated by recent experimental observations [Dominski and Diallo, Plasma Phys. Control. Fusion 62, 095011 (2020)] where it was found that magnetic fluctuations trigger the onset of edge localized modes by suddenly exciting a network of nonlinear interactions. In our study, we consider the simplest system of many harmonic oscillators that are organized in a network of nonlinear triads. We model and simulate the sudden transition of this network of triads into a regime of strong nonlinear fluctuations—reminiscent of the onset of edge localized modes in tokamaks. This transition is triggered by the activation of a nonlinear perturbation. An abrupt rise of the system's disorder (an entropy-like quantity) is measured during the transition. This transition from weak to strong nonlinear fluctuations is even more abrupt when these fluctuations are chaotic, i.e., when the timescale of the nonlinear interaction is comparable to the timescale of the wave oscillations.

Studying the Effect of Metre Perception on Rhythm and Melody Modelling with LSTMs

In this paper we take a connectionist machine learning approach to the problem of metre perception and melody learning in musical signals. We present a two-layered network consisting of a nonlinear oscillator network and a recurrent neural network. The oscillator network acts as an entrained resonant filter to the musical signal. It `perceives' metre by resonating nonlinearly to the inherent periodicities within the signal, creating a hierarchy of strong and weak periods. The neural network learns the long-term temporal structures present in this signal. We show that this network outperforms our previous approach of a single layer recurrent neural network in a melody and rhythm prediction task. We hypothesise that our system is enabled to make use of the relatively long temporal resonance in the oscillator network output, and therefore model more coherent long-term structures. A system such as this could be used in a multitude of analytic and generative scenarios, including live performance applications.

Dynamical consistency in networks of nonlinear oscillators

We investigate, numerically and experimentally, the phenomenon of dynamical consistency in a network of nonlinear oscillators. Dynamical consistency is defined as the ability of a dynamical system to respond in the same way when it is perturbed by the same external signal. This property, fulfilled by specific dynamical systems, has been mainly studied in single oscillators (e.g., lasers or electronic circuits), but just a few works have focused on dynamical systems connected through a non-regular network. Using an electronic array, we can connect a series of nonlinear (analog) electronic circuits to create complex networks, where nodes are coupled between them in a given structure, and an external signal can also be introduced into each particular circuit. Furthermore, the strength of the connections is re-configurable. Next, we analyze the consequences of varying the network’s coupling strength and investigate how synchronization and consistency of the whole ensemble arise. Variations in both coupling strengths, i.e. (i) between oscillators and (ii) with the external signal, show the existence of a region where electronic oscillators can respond in the same way to an external perturbation even though they are not completely synchronized with the external signal. We use the Pearson correlation coefficient to measure the consistency of the network, which is characterized by a high correlation between the network output signals and, at the same time, a low correlation with the input signal.

Synchronizing Chaos with Imperfections.

Previous research on nonlinear oscillator networks has shown that chaos synchronization is attainable for identical oscillators but deteriorates in the presence of parameter mismatches. Here, we identify regimes for which the opposite occurs and show that oscillator heterogeneity can synchronize chaos for conditions under which identical oscillators cannot. This effect is not limited to small mismatches and is observed for random oscillator heterogeneity on both homogeneous and heterogeneous network structures. The results are demonstrated experimentally using networks of Chua's oscillators and are further supported by numerical simulations and theoretical analysis. In particular, we propose a general mechanism based on heterogeneity-induced mode mixing that provides insights into the observed phenomenon. Since individual differences are ubiquitous and often unavoidable in real systems, it follows that such imperfections can be an unexpected source of synchronization stability.

Collective Synchronous Spiking in a Brain Network of Coupled Nonlinear Oscillators.

A network of propagating nonlinear oscillatory modes (waves) in the human brain is shown to generate collectively synchronized spiking activity (hypersynchronous spiking) when both amplitude and phase coupling between modes are taken into account. The nonlinear behavior of the modes participating in the network are the result of the nonresonant dynamics of weakly evanescent cortical waves that, as shown recently, adhere to an inverse frequency-wave number dispersion relation when propagating through an inhomogeneous anisotropic media characteristic of the brain cortex. This description provides a missing link between simplistic models of synchronization in networks of small amplitude phase coupled oscillators and in networks built with various empirically fitted models of pulse or amplitude coupled spiking neurons. Overall the phase-amplitude coupling mechanism presented in the Letter shows significantly more efficient synchronization compared to current standard approaches and demonstrates an emergence of collective synchronized spiking from subthreshold oscillations that neither phase nor amplitude coupling alone are capable of explaining.

Topology-Dependent Excitation Response of Networks of Linear and Nonlinear Oscillators

Abstract This paper investigates the near-resonance response to exogenous excitation of a class of networks of coupled linear and nonlinear oscillators with emphasis on the dependence on network topology, distribution of nonlinearities, and damping ratios. The analysis shows a qualitative transition between the behaviors associated with the extreme cases of all linear and all nonlinear oscillators, respectively, even allowing for such a transition under continuous variations in the damping ratios but for fixed topology. Theoretical predictions for arbitrary members of the network class using the multiple-scales perturbation method are validated against numerical results obtained using parameter continuation techniques. The latter include the tracking of families of quasi-periodic invariant tori emanating from saddle-node and Hopf bifurcations of periodic orbits. In networks in the class of interest with special topology, 1:1 and 1:3 internal resonances couple modes of oscillation, and the conditions to suppress the influence of these resonances are explored.

On the Scaling Law of Phase Drift in Coupled Nonlinear Oscillators for Precision Timing

Computational and experimental works reveal that the coupling of similar crystal oscillators leads to a variety of collective patterns, mainly various forms of discrete rotating waves and synchronization patterns, which have the potential for developing precision timing devices through phase drift reduction. Among all observed patterns, the standard traveling wave, in which consecutive crystals oscillate out of phase by [Formula: see text], where [Formula: see text] is the network size, leads to optimal phase drift error that scales down as [Formula: see text] as opposed to [Formula: see text] for an uncoupled ensemble. In this manuscript, we provide an analytical proof of the scaling laws, for uncoupled and coupled symmetric networks, and show that [Formula: see text] is the fundamental limit of phase-error reduction that one can obtain with a symmetric network of nonlinear oscillators of any type, not just crystals.

Identifiability of structural networks of nonlinear electronic oscillators

The interplay between structure and function is critical in the understanding of complex systems, their dynamics and their behavior. We investigated the interplay between structural and functional networks by means of the differential identifiability framework, which here quantifies the ability of identifying a particular network structure based on (1) the observation of its functional network and (2) the comparison with a prior observation under different initial conditions. We carried out an experiment consisting of the construction of M=20 different structural networks composed of N=28 nonlinear electronic circuits and studied the regions where network structures are identifiable. Specifically, we analyzed how differential identifiability is related to the coupling strength between dynamical units (modifying the level of synchronization) and what are the consequences of increasing the amount of noise existing in the functional networks. We observed that differential identifiability reaches its highest value for low to intermediate coupling strengths. Furthermore, it is possible to increase the identifiability parameter by including a principal component analysis in the comparison of functional networks, being especially beneficial for scenarios where noise reaches intermediate levels. Finally, we showed that the regime of the parameter space where differential identifiability is the highest is highly overlapped with the region where structural and functional networks correlate the most.

Anormal diffusion enhancement of resonant responses for coupled oscillator networks to weak signals.

The normal diffusion effect is introduced as a new regulating factor into the established diffusive coupling model for bistable oscillator networks. We find that the response of the system to the weak signal is substantially enhanced by the anormal diffusion, which is termed anormal-diffusion-induced resonance. We also reveal that the diffusive coupling-induced transition, which changes the system from a bistable to a monostable state, is of fundamental importance for the occurrence of resonance. The proposed approach is validated using simulation studies and theoretical analyses. Our results suggest that diffusion induced resonance can be more easily observed in nonlinear oscillator networks.

Stability analysis of nonlinear oscillator networks based on the mechanism of cascading failures

Abstract Traditional methods for resisting cascading failures mainly focus on optimizing the topology of the network. Recent research indicates that the exceptional dynamics of the nodes, which caused by an intentional attack, can also induce cascading failures of nonlinear oscillator networks. This implies that the tolerance of the network to cascading failures can be enhanced by stabilizing the exceptional nodes after the attack. In this paper, we have constructed a framework to identify and control the exceptional nodes of the network to avoid further cascading failures. First, the node with the largest load is removed to simulate the situation that the network is intentionally attacked and that will cause the exceptional dynamics of other nodes. Then, the exceptional nodes which can induce cascading failures are identified by the small disturbance analysis method. Finally, the specific external control will be applied in the exceptional nodes respectively to stabilize them based on the phase differences analysis of their dynamic equations. Results show that the number of exceptional nodes in the Sci-Grid and the Spanish power grid can be reduced by more than 50% via increasing the coupling strength and the unstable network can be stabilized by controlling the remaining exceptional nodes (usually no more than 10% of the total).

Universal transition to inactivity in global mixed coupling

The collective dynamics of a network of nonlinear oscillators can be represented in terms of activity level of the network. We have studied a universal transition from activity to inactivity in a globally coupled network of identical oscillators. We consider mixed coupling, where some of the network elements interact through the similar variables while others with dissimilar variables. The coupling strength at which the network become inactive is inversely proportional to the fraction of oscillators coupled through dissimilar variables. Results are presented for the network of various globally coupled limit-cycle oscillators such as Stuart-Landau oscillators, MacArthur prey-predator model as well as for the chaotic Rössller oscillators. The analytical condition for the onset of inactivity in the system is calculated using linear stability analysis which is found to be in good agreement with the numerical results.

Impact of mixed measurements in detecting phase synchronization in networks using multivariate singular spectrum analysis

Multivariate singular spectrum analysis (M-SSA), with a varimax rotation of eigenvectors, was recently proposed to provide detailed information about phase synchronization in networks of nonlinear oscillators without any a priori need for phase estimation. The discriminatory power of M-SSA is often enhanced by using only the time series of the variable that provides the best observability of the node dynamics. In practice, however, diverse factors could prevent one to have access to this variable in some nodes and other variables should be used, resulting in a mixed set of variables. In the present work, the impact of this mixed measurement approach on the M-SSA is numerically investigated in networks of R\"ossler systems and cord oscillators. The results are threefold. First, a node measured by a poor variable, in terms of observability, becomes virtually invisible to the technique. Second, a side effect of using a poor variable is that the characterization of phase synchronization clustering of the {\it other}\, nodes is hindered by a small amount. This suggests that, given a network, synchronization analysis with M-SSA could be more reliable by not measuring those nodes that are accessible only through poor variables. Third, global phase synchronization could be detected even using only poor variables, given enough of them are measured. These insights could be useful in defining measurement strategies for both experimental design and real world applications for use with M-SSA.

Role of time scales and topology on the dynamics of complex networks.

The interplay between time scales and structural properties of complex networks of nonlinear oscillators can generate many interesting phenomena, like amplitude death, cluster synchronization, frequency synchronization, etc. We study the emergence of such phenomena and their transitions by considering a complex network of dynamical systems in which a fraction of systems evolves on a slower time scale on the network. We report the transition to amplitude death for the whole network and the scaling near the transitions as the connectivity pattern changes. We also discuss the suppression and recovery of oscillations and the crossover behavior as the number of slow systems increases. By considering a scale free network of systems with multiple time scales, we study the role of heterogeneity in link structure on dynamical properties and the consequent critical behaviors. In this case with hubs made slow, our main results are the escape time statistics for loss of complete synchrony as the slowness spreads on the network and the self-organization of the whole network to a new frequency synchronized state. Our results have potential applications in biological, physical, and engineering networks consisting of heterogeneous oscillators.

Mode locking in periodically forced gradient frequency neural networks.

We study mode locking in a canonical model of gradient frequency neural networks under periodic forcing. The canonical model is a generic mathematical model for a network of nonlinear oscillators tuned to a range of distinct frequencies. It is mathematically more tractable than biological neuron models and allows close analysis of mode-locking behaviors. Here we analyze individual modes of synchronization for a periodically forced canonical model and present a complete set of driven behaviors for all parameter regimes available in the model. Using a closed-form approximation, we show that the Arnold tongue (i.e., locking region) for k:m synchronization gets narrower as k and m increase. We find that numerical simulations of the canonical model closely follow the analysis of individual modes when forcing is weak, but they deviate at high forcing amplitudes for which oscillator dynamics are simultaneously influenced by multiple modes of synchronization.

Stochastic Thermodynamics of Oscillators' Networks.

We apply the stochastic thermodynamics formalism to describe the dynamics of systems of complex Langevin and Fokker-Planck equations. We provide in particular a simple and general recipe to calculate thermodynamical currents, dissipated and propagating heat for networks of nonlinear oscillators. By using the Hodge decomposition of thermodynamical forces and fluxes, we derive a formula for entropy production that generalises the notion of non-potential forces and makes transparent the breaking of detailed balance and of time reversal symmetry for states arbitrarily far from equilibrium. Our formalism is then applied to describe the off-equilibrium thermodynamics of a few examples, notably a continuum ferromagnet, a network of classical spin-oscillators and the Frenkel-Kontorova model of nano friction.

Modeling reservoir computing with the discrete nonlinear Schrödinger equation

We formulate, using the discrete nonlinear Schroedinger equation (DNLS), a general approach to encode and process information based on reservoir computing. Reservoir computing is a promising avenue for realizing neuromorphic computing devices. In such computing systems, training is performed only at the output level, by adjusting the output from the reservoir with respect to a target signal. In our formulation, the reservoir can be an arbitrary physical system, driven out of thermal equilibrium by an external driving. The DNLS is a general oscillator model with broad application in physics and we argue that our approach is completely general and does not depend on the physical realisation of the reservoir. The driving, which encodes the object to be recognised, acts as a thermodynamical force, one for each node in the reservoir. Currents associated to these thermodynamical forces in turn encode the output signal from the reservoir. As an example, we consider numerically the problem of supervised learning for pattern recognition, using as reservoir a network of nonlinear oscillators.

Fast and exact search for the partition with minimal information loss.

In analysis of multi-component complex systems, such as neural systems, identifying groups of units that share similar functionality will aid understanding of the underlying structures of the system. To find such a grouping, it is useful to evaluate to what extent the units of the system are separable. Separability or inseparability can be evaluated by quantifying how much information would be lost if the system were partitioned into subsystems, and the interactions between the subsystems were hypothetically removed. A system of two independent subsystems are completely separable without any loss of information while a system of strongly interacted subsystems cannot be separated without a large loss of information. Among all the possible partitions of a system, the partition that minimizes the loss of information, called the Minimum Information Partition (MIP), can be considered as the optimal partition for characterizing the underlying structures of the system. Although the MIP would reveal novel characteristics of the neural system, an exhaustive search for the MIP is numerically intractable due to the combinatorial explosion of possible partitions. Here, we propose a computationally efficient search to precisely identify the MIP among all possible partitions by exploiting the submodularity of the measure of information loss, when the measure of information loss is submodular. Submodularity is a mathematical property of set functions which is analogous to convexity in continuous functions. Mutual information is one such submodular information loss function, and is a natural choice for measuring the degree of statistical dependence between paired sets of random variables. By using mutual information as a loss function, we show that the search for MIP can be performed in a practical order of computational time for a reasonably large system (N = 100 ∼ 1000). We also demonstrate that MIP search allows for the detection of underlying global structures in a network of nonlinear oscillators.

Synergistic control of a multi-segments vertebral column robot based on tensegrity for postural balance

ABSTRACTWe present a neuronal architecture to control a compliant robotic model of the human vertebral column for postural balance. The robotic structure is designed using the principle of tensegrity that ensures to be lightweight, auto-replicative with multi-degrees of freedom, flexible and also robust to perturbations. We model the central pattern generators of the spinal cords with a network of nonlinear Kuramoto oscillators coupled internally and externally to the structure and error-driven by a proportional derivative (PD) controller using an accelerometer for feedback. This coupling between the two controllers is original and we show it serves to generate controlled rhythmical patterns. We observe for certain coupling parameters some intervals of synchronization and of resonance of the neural units to the tensile structure to permit smooth control and balance. We show that the top-down PD control of the oscillators flexibly absorbs external shocks proportionally to the perturbation and converges to steady state behaviors. We discuss then about our neural architecture to model motor synergies for compliance control and also about tensegrity structures for soft robotics. The 3D printed model is provided as well as a movie at the addresshttps://sites.google.com/site/embodiedai/current-research/tensegrityrobots.

Boltzmann sampling from the Ising model using quantum heating of coupled nonlinear oscillators

A network of Kerr-nonlinear parametric oscillators without dissipation has recently been proposed for solving combinatorial optimization problems via quantum adiabatic evolution through its bifurcation point. Here we investigate the behavior of the quantum bifurcation machine (QbM) in the presence of dissipation. Our numerical study suggests that the output probability distribution of the dissipative QbM is Boltzmann-like, where the energy in the Boltzmann distribution corresponds to the cost function of the optimization problem. We explain the Boltzmann distribution by generalizing the concept of quantum heating in a single nonlinear oscillator to the case of multiple coupled nonlinear oscillators. The present result also suggests that such driven dissipative nonlinear oscillator networks can be applied to Boltzmann sampling, which is used, e.g., for Boltzmann machine learning in the field of artificial intelligence.

Synchronization of Chimera States in Coupled Networks of Nonlinear Chaotic Oscillators

Synchronization of Chimera States in Coupled Networks of Nonlinear Chaotic Oscillators