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Oscillatory dynamics pervades the universe, appearing in systems of all scales. Whilst autonomous oscillatory dynamics has been extensively studied and is well understood, the very important problem of non-autonomous oscillatory dynamics is less well understood. Here, we provide a framework for non-autonomous oscillatory dynamics, within which we can define intermittent phenomena such as intermittent phase synchronisation. Moreover, we demonstrate this framework with a coupled pair of non-autonomous phase oscillators as well as a higher-dimensional system comprising of two interacting phase-oscillator networks. • A new framework for phase-dynamics phenomena in non-autonomous oscillatory systems. • Interactions between pairs of non-autonomous oscillators and networks are considered. • Intermittent synchronisation is formulated mathematically. • Relevant for many real-world problems. [ABSTRACT FROM AUTHOR]

A description in terms of phase and amplitude variables is given for nonlinear oscillators subject to white Gaussian noise described by Itô stochastic differential equations. The stochastic differential equations derived for the amplitude and the phase are rigorous, and their validity is not limited to the weak noise limit. It is shown that if Floquet vectors are used, then in the neighborhood of a limit cycle the phase variable coincides with the asymptotic phase defined through isochrons. Two techniques for the analysis of the phase and amplitude equations are discussed, that is, asymptotic expansion method and a phase reduction procedure based on projection operators technique. Formulas for the expected angular frequency, expected oscillation amplitude, and amplitude variance are derived using Itô calculus. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Synaptically coupled neurons show in-phase or antiphase synchrony depending on the chemical and dynamical nature of the synapse. Deterministic theory helps predict the phase differences between two phase-locked oscillators when the coupling is weak. In the presence of noise, however, deterministic theory faces difficulty when the coexistence of multiple stable oscillatory solutions occurs. We analyze the solution structure of two coupled neuronal oscillators for parameter values between a subcritical Hopf bifurcation point and a saddle node point of the periodic branch that bifurcates from the Hopf point, where a rich variety of coexisting solutions including asymmetric localized oscillations occurs. We construct these solutions via a multiscale analysis and explore the general bifurcation scenario using the λ-ω model. We show for both excitatory and inhibitory synapses that noise causes important changes in the phase and amplitude dynamics of such coupled neuronal oscillators when multiple oscillatory solutions coexist. Mixed-mode oscillations occur when distinct bistable solutions are randomly visited. The phase difference between the coupled oscillators in the localized solution, coexisting with in-phase or antiphase solutions, is clearly represented in the stochastic phase dynamics. [ABSTRACT FROM AUTHOR]

In this study, we investigate the role of stochastic parametric noise on the phase dynamics and the frequency characteristics of a vortex induced vibration (VIV) system, in the framework of synchronisation theory. In-phase, arbitrary-phase and anti-phase synchronisations exist during pre lock-in, lock-in and post lock-in regimes of the deterministic system, respectively. However the noise induced phase dynamics is found to be considerably different from the corresponding deterministic scenario and is largely dependent on the intensity of noise. Stochastic noise alters the conventional route to instability and is seen to induce new dynamical states such as Noise Induced Intermittency (NII) and quasi-periodicity in the structural response. It is also observed that noise triggers additional system frequencies in both structural and flow responses albeit not the same way. The birth of new dynamical states is strongly linked with these frequency characteristics of the structural and the flow oscillators. Phase can characterise the overall dynamics of coupled oscillators systematically and in the present study relative phase has been exploited in a stochastic framework in terms of both averaged and instantaneous quantities. The stochastic VIV system manifests asynchronous and synchronous phase dynamics as well as regions of phase slips during the transition boundaries of stochastic bifurcations. These are quantitatively and qualitatively monitored through phase coherence values in the desired parametric ranges. [ABSTRACT FROM AUTHOR]

Phase models are a powerful method to quantify the coupled dynamics of nonlinear oscillators from measured data. We use two phase modeling methods to quantify the dynamics of pairs of coupled electrochemical oscillators, based on the phases of the two oscillators independently and the phase difference, respectively. We discuss the benefits of the two-dimensional approach relative to the one-dimensional approach using phase difference. We quantify the dependence of the coupling functions on the coupling magnitude and coupling time delay. We show differences in synchronization predictions of the two models using a toy model. We show that the two-dimensional approach reveals behavior not detected by the one-dimensional model in a driven experimental oscillator. This approach is broadly applicable to quantify interactions between nonlinear oscillators, especially where intrinsic oscillator sensitivity and coupling evolve with time. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this manuscript is to propose a method for investigating the global dynamics of nonlinear oscillatory networks, with arbitrary couplings. The procedure is based on the assumption that each oscillator can be accurately described via its time-varying amplitude and phase variables. The proposed method allows one to derive a set of coupled nonlinear ordinary differential equations governing this couple of variables. By analyzing the evolution of amplitudes and phases, one can investigate the stability properties of the limit cycles for the whole system in a simpler way with respect to the latest available methodologies. Furthermore, it is proved that this technique also works for weakly connected oscillatory networks. Finally, as a case study, a chain of third-order oscillators (Chua’s circuits) is considered and the results are compared to those obtained via a numerical technique, entirely based on the harmonic balance (HB) approach. [Copyright &y& Elsevier]

We experimentally investigate the forced synchronization of a self-excited chaotic thermoacoustic oscillator with two natural frequencies, f1 and f2. On increasing the forcing amplitude, xf, at a fixed forcing frequency, ff, we find two different types of synchronization: (i) ff/f1 = 1: 1 or 2: 1 chaos-destroying synchronization (CDS), and (ii) phase synchronization of chaos (PSC). En route to 1: 1 CDS, the system transitions from an unforced chaotic state (CH1,2) to a forced chaotic state (CH1,2,f), then to a two-frequency quasiperiodic state where chaos is destroyed (T²2,f), and finally to a phase-locked period-1 state (P1f). The route to 2: 1 CDS is similar, but the quasiperiodic state hosts a doubled torus (2T²2,f) that transforms into a phase-locked period-2 orbit (P2f) when CDS occurs. En route to PSC, the system transitions to a forced chaotic state (CH1,2,f) followed by a phase-locked chaotic state, where f1, f2 and ff still coexist but their phase difference remains bounded. We find that the maximum reduction in thermoacoustic amplitude occurs near the onset of CDS, and that the critical xf required for the onset of CDS does not vary significantly with ff. We then use two unidirectionally coupled Anishchenko-Astakhov oscillators to phenomenologically model the experimental synchronization dynamics, including (i) the route to 1: 1 CDS, (ii) various phase dynamics, such as phase drifting, slipping and locking, and (iii) the thermoacoustic amplitude variations in the ff/f1-xf plane. This study extends the applicability of open-loop control further to a chaotic thermoacoustic system, demonstrating (i) the feasibility of using an existing actuation strategy to weaken aperiodic thermoacoustic oscillations, and (ii) the possibility of developing new active suppression strategies based on both established and emerging methods of chaos control. [ABSTRACT FROM AUTHOR]

This manuscript provides a comprehensive tutorial on the operating principles of a bio-inspired Cellular Nonlinear Network, leveraging the local activity of NbO x memristors to apply a spike-based computing paradigm, which is expected to deliver such a separation between the steady-state phases of its capacitively-coupled oscillators, relative to a reference cell, as to unveal the classification of the nodes of the associated graphs into the least number of groups, according to the rules of a non-deterministic polynomial-hard combinatorial optimization problem, known as vertex coloring. Besides providing the theoretical foundations of the bio-inspired signal-processing paradigm, implemented by the proposed Memristor Oscillatory Network, and presenting pedagogical examples, illustrating how the phase dynamics of the memristive computing engine enables to solve the graph coloring problem, the paper further presents strategies to compensate for an imbalance in the number of couplings per oscillator, to counteract the intrinsic variability observed in the electrical behaviours of memristor samples from the same batch, and to prevent the impasse appearing when the array attains a steady-state corresponding to a local minimum of the optimization goal. The proposed Memristor Cellular Nonlinear Network, endowed with ad hoc circuitry for the implementation of these control strategies, is found to classify the vertices of a wide set of graphs in a number of color groups lower than the cardinality of the set of colors identified by traditional either software or hardware competitor systems. Given that, under nominal operating conditions, a biological system, such as the brain, is naturally capable to optimise energy consumption in problem-solving activities, the capability of locally-active memristor nanotechnologies to enable the circuit implementation of bio-inspired signal processing paradigms is expected to pave the way toward electronics with higher time and energy efficiency than state-of-the-art purely-CMOS hardware. [ABSTRACT FROM AUTHOR]

We study a learning and recalling model of phase patterns in a two- or three-coupled BVP oscillators system with a time delay δ. The coupling strengths are modulated by the Hebbian learning rule. Assuming the first-order approximation, we calculate the optimal condition of δ for exact recall by applying the phase dynamics theory. When α = 0, where α represents the coupling from activator to inhibitor, the correlation between the learning and the retrieval phase depends on δ. When α = 1, exact recall is achieved independent of δ. The results can be explained by the phase dynamics theory. [ABSTRACT FROM AUTHOR]

The well-known method of estimating the delay time of coupling between self-oscillatory systems on the basis of their time series is based on empirical modeling of the observed phase dynamics by a system of first-order phase oscillators under the assumption that amplitude fluctuations can be ignored. We have studied the applicability of this method in the presence of significant amplitude fluctuations caused by an external noise or the chaotic dynamics of oscillators. Numerical experiments with systems of standard coupled oscillators show that the method under consideration gives incorrect estimations in some situations. A diagnostics of these situations is proposed, and modification of the estimator that eliminates errors is described and justified. [ABSTRACT FROM AUTHOR]

We present OIM (Oscillator Ising Machines), a new way to make Ising machines using networks of coupled self-sustaining nonlinear oscillators. OIM is theoretically rooted in a novel result that establishes that the phase dynamics of coupled oscillator systems, under the influence of subharmonic injection locking, are governed by a Lyapunov function that is closely related to the Ising Hamiltonian of the coupling graph. As a result, the dynamics of such oscillator networks evolve naturally to local minima of the Lyapunov function. Two simple additional steps (i.e., turning subharmonic locking on and off smoothly, and adding noise) enable the network to find excellent solutions of Ising problems. We demonstrate our method on Ising versions of the MAX-CUT and graph colouring problems, showing that it improves on previously published results on several problems in the G benchmark set. Using synthetic problems with known global minima, we also present initial scaling results. Our scheme, which is amenable to realisation using many kinds of oscillators from different physical domains, is particularly well suited for CMOS IC implementation, offering significant practical advantages over previous techniques for making Ising machines. We report working hardware prototypes using CMOS electronic oscillators. [ABSTRACT FROM AUTHOR]

Synchronization is common in both nature and human physiology, often used to illustrate nonlinear dynamics. Interestingly, one can initiate their comprehension of this phenomenon from pure linear systems. In this Tutorial, we begin with a theoretical exploration of coupled oscillators' dynamic behavior, enabling us to discern and contrast the unique attributes of dissipative coupling as opposed to commonly observed coherent coupling. We then examine synchronization in two dissipative coupled linear systems: one with two pendulums mutually linked via the Lenz effect and the other with two RLC oscillators coupled via a resistor. This Tutorial is designed to serve as a concise starting point for researchers interested in exploring synchronization phenomena using a simplified model driven solely by dissipative coupling. [ABSTRACT FROM AUTHOR]

Investigating the dynamics of a network of oscillatory systems is a timely and urgent topic. Phase synchronization has proven paradigmatic to study emergent collective behavior within a network. Defining the phase dynamics, however, is not a trivial task. The literature provides an arsenal of solutions, but results are scattered and their formulation is far from standardized. Here, we present, in a unified language, a catalogue of popular techniques for deriving the phase dynamics of coupled oscillators. Traditionally, approaches to phase reduction address the (weakly) perturbed dynamics of an oscillator. They fall into three classes. (i) Many phase reduction techniques start off with a Hopf normal form description, thereby providing mathematical rigor. There, the caveat is to first derive the proper normal form. We explicate several ways to do that, both analytically and (semi-)numerically. (ii) Other analytic techniques capitalize on time scale separation and/or averaging over cyclic variables. While appealing for their more intuitive implementation, they often lack accuracy. (iii) Direct numerical approaches help to identify oscillatory behavior but may limit an overarching view how the reduced phase dynamics depends on model parameters. After illustrating and reviewing the necessary mathematical details for single oscillators, we turn to networks of coupled oscillators as the central issue of this report. We show in detail how the concepts of phase reduction for single oscillators can be extended and applied to oscillator networks. Again, we distinguish between numerical and analytic phase reduction techniques. As the latter dwell on a network normal form, we also discuss associated reduction methods. To illustrate benefits and pitfalls of the different phase reduction techniques, we apply them point-by-point to two classic examples: networks of Brusselators and a more elaborate model of coupled Wilson–Cowan oscillators. The reduction of complex oscillatory systems is crucial for numerical analyses but more so for analytical estimates and model prediction. The most common reduction is towards phase oscillator networks that have proven successful in describing not only the transition between incoherence and global synchronization, but also in predicting the existence of less trivial network states. Many of these predictions have been confirmed in experiments. As we show, however, the phase dynamics depends to large extent on the employed phase reduction technique. In view of current and future trends, we also provide an overview of various methods for augmented phase reduction as well as for phase–amplitude reduction. Weindicate how these techniques can be extended to oscillator networks and, hence, may allow for an improved derivation of the phase dynamics of coupled oscillators. [ABSTRACT FROM AUTHOR]

Nonlinear dynamical systems such as coupled oscillators are being actively investigated as Ising machines for solving computationally hard problems in combinatorial optimization. Prior works have established the equivalence between the global minima of the cost function describing the coupled oscillator system and the ground state of the Ising Hamiltonian. However, the properties of the oscillator Ising machine (OIM) from a nonlinear control viewpoint, such as the stability of the OIM solutions, remain unexplored. Therefore, in this work, using nonlinear control-theoretic analysis, we (i) identify the conditions required to ensure the functionality of the coupled oscillators as an Ising machine, (ii) show that all globally optimal phase configurations may not always be stable, resulting in some configurations being more favored over others and, thus, creating a biased OIM, and (iii) elucidate the impact of the stability of locally optimal phase configurations on the quality of the solution computed by the system. Our work, fostered through the unique convergence between nonlinear control theory and analog systems for computing, provides a new toolbox for the design and implementation of dynamical system-based computing platforms. [ABSTRACT FROM AUTHOR]

Synchronization occurs in many natural and technological systems, from cardiac pacemaker cells to coupled lasers. In the synchronized state, the individual cells or lasers coordinate the timing of their oscillations, but they do not move through space. A complementary form of self-organization occurs among swarming insects, flocking birds, or schooling fish; now the individuals move through space, but without conspicuously altering their internal states. Here we explore systems in which both synchronization and swarming occur together. Specifically, we consider oscillators whose phase dynamics and spatial dynamics are coupled. We call them swarmalators, to highlight their dual character. A case study of a generalized Kuramoto model predicts five collective states as possible long-term modes of organization. These states may be observable in groups of sperm, Japanese tree frogs, colloidal suspensions of magnetic particles, and other biological and physical systems in which self-assembly and synchronization interact. [ABSTRACT FROM AUTHOR]

We investigate the phase dynamics of a closed-loop MEMS-based oscillator and demonstrate how one can exploit nonlinear behavior to improve oscillator phase noise characteristics by synchronizing the oscillator with a weak harmonic drive. Analytical predictions are based on an oscillator model that incorporates a resonator element with a weak cubic (i.e., Duffing type) nonlinearity, weak coupling to a clean external harmonic drive, and both thermal and frequency noise terms. The method of stochastic averaging is used to derive an expression for the rate of phase diffusion induced by the noises, and the results predict a remarkable phase noise reduction of three order of magnitude when the oscillator is synchronized to the weak external field. The predictions are experimentally demonstrated using a closed-loop oscillator with a double-anchored double-ended-tuning-fork MEMS resonator with coupling to a small external sinusoidal signal from a signal generator, demonstrating a 30 dB/Hz drop in phase noise. The results show how one can generate a clean high-power signal from a relative noisy oscillator by synchronizing it with a low-power external signal. The results also confirm recent studies showing that the parameter range of synchronization is significantly expanded when the resonator operates in its nonlinear regime. [2016-0067] [ABSTRACT FROM AUTHOR]

We analyze the features of the oscillations arising in forced inductor-capacitor (LC) oscillators when they operate in the periodic pulling mode, under the action of a weak injection signal. In radio frequency integrated circuits, both voltage-controlled oscillators subject to undesired couplings and injection-locked frequency dividers behave like forced LC oscillators. These are modeled as second-order driven oscillators working in the subharmonic (secondary) resonance regime. The analysis is based on the generalized Adler's equation, which we introduce to describe the phase dynamics of dividers of any division ratio and to derive closed-form expressions for the spectrum components of the system's oscillatory response. We show that the spectrum is double-sided and asymmetric, unlike the single-sided spectrum of systems with primary resonance. Numerical and experimental results are given to validate the presented results, which significantly generalize those available in the literature. [ABSTRACT FROM AUTHOR]

Synchronization in microstructures is a widely explored domain due to its diverse dynamic traits and promising practical applications. Within synchronization analysis, the synchronization bandwidth serves as a pivotal metric. While current research predominantly focuses on symmetric evaluations of synchronization bandwidth, the investigation into potential asymmetries within nonlinear oscillators remains unexplored, carrying implications for sensor application performance. This paper conducts a comprehensive exploration employing straight and arch beams capable of demonstrating linear, hardening, and softening characteristics to thoroughly scrutinize potential asymmetry within the synchronization region. Through the introduction of weak harmonic forces to induce synchronization within the oscillator, we observe distinct asymmetry within its synchronization range. Additionally, we present a robust theoretical model capable of fully capturing the linear, hardening, and softening traits of resonators synchronized to external perturbation. Further investigation into the effects of feedback strength and phase delay on synchronization region asymmetry, conducted through analytical and experimental approaches, reveals a consistent alignment between theoretical predictions and experimental outcomes. These findings hold promise in providing crucial technical insights to enhance resonator performance and broaden the application landscape of MEMS (Micro-Electro-Mechanical Systems) technology. [ABSTRACT FROM AUTHOR]

A foremost challenge in modern network science is the inverse problem of reconstruction (inference) of coupling equations and network topology from the measurements of the network dynamics. Of particular interest are the methods that can operate on real (empirical) data without interfering with the system. One such earlier attempt (Tokuda et al. 2007 Phys. Rev. Lett. 99, 064101. (doi:10.1103/PhysRevLett.99.064101)) was a method suited for general limit-cycle oscillators, yielding both oscillators' natural frequencies and coupling functions between them (phase equations) from empirically measured time series. The present paper reviews the above method in a way comprehensive to domain-scientists other than physics. It also presents applications of the method to (i) detection of the network connectivity, (ii) inference of the phase sensitivity function, (iii) approximation of the interaction among phase-coherent chaotic oscillators, and (iv) experimental data from a forced Van der Pol electric circuit. This reaffirms the range of applicability of the method for reconstructing coupling functions and makes it accessible to a much wider scientific community. [ABSTRACT FROM AUTHOR]

Phase reduction is a general tool widely used to describe forced and interacting self-sustained oscillators. Here, we explore the phase coupling functions beyond the usual first-order approximation in the strength of the force. Taking the periodically forced Stuart-Landau oscillator as the paradigmatic model, we determine and numerically analyse the coupling functions up to the fourth order in the force strength. We show that the found nonlinear phase coupling functions can be used for predicting synchronization regions of the forced oscillator. [ABSTRACT FROM AUTHOR]

In the present work, we present a new algorithm for assessing causality in uni-directionally coupled chaotic oscillators with small frequency mismatch embedded in heavy white Gaussian noise. This method is based on the correlation between changes in the phase dynamics of the slave oscillator and the dynamics of the phase difference between the oscillators. To recover the phase at low signal-to-noise ratio, a nonlinear adaptive denoising algorithm based on finding sinusoidal fits to the local neighbourhood of the reconstructed phase space is used. Application of the proposed approach to master-slave Rössler systems showed that the new algorithm is well-suited for assessing the presence and direction of coupling in highly noisy uni-directionally coupled chaotic oscillators, especially in the case of weak and moderate coupling. [ABSTRACT FROM AUTHOR]

This research paper presents a circuit-based phase noise model of a cross-coupled LC tank oscillator. The effect of the Oscillator's noise is one of the most critical items in designing modern RF communication systems. The noise factor necessitates a more thorough comprehension of design tradeoffs in RF VCO IC implementations. This research paper introduces a new analytical derivation to produce a well-bounded third-order polynomial equation accurately representing the LC oscillator's phase noise behaviour. The third-order polynomial equation includes circuit parameters such as capacitance, inductance and quality factors to understand the phase noise characteristics. The proposed method determines the amplitude and phase noise and forms a link with the circuit operating point, and phase noise is produced as a direct result of the circuit parameters. The advantage of the proposed resulting equation is that it avoids the need for complex simulations to identify the noise characteristics in such a process. The proposed equation was validated by comparing the tank circuit simulation results. The simulation results verify the applicability of the third-order polynomial equation representing the noise characteristics of the LC oscillator. [ABSTRACT FROM AUTHOR]

The emergence of digital computers has profoundly reshaped our interactions with technology and the processing of information. Despite excelling in data processing and arithmetics, these computers face limitations in tackling complex nondeterministic-polynomial (NP) problems. In response, researchers have started searching for new computational paradigms that possess the natural tendency of solving these problems. Oscillator-based optimizers are one such paradigm, where the idea is to exploit the parallelism of oscillators networks in order to efficiently solve NP problems. This involves a process of mapping a given optimization task to a quadratic unconstrained binary optimization program and then mapping the resulting program onto an inter-oscillator coupling circuit encoding its coefficients. This paper presents a comprehensive approach to constructing oscillator-based optimizers, offering both the rationale for employing oscillator networks and formulas for linking optimization coefficients to inter-oscillator coupling. Here, we cover most aspects of oscillator-based optimization starting from the design of the network up to its technical implementation. Moreover, we provide a platform-independent wave digital algorithm, which allows for emulating our network's behavior in a highly parallel fashion. [ABSTRACT FROM AUTHOR]

Synchronization study allows a better understanding of the exchange of information among systems. In this work, we study experimental data recorded from a set of Rössler-like chaotic electronic oscillators arranged in a complex network, where the interactions between the oscillators are given in terms of a connectivity matrix, and their intensity is controlled by a global coupling parameter. We use the zero and one persistent homology groups to characterize the point clouds obtained from the signals recorded in pairs of oscillators. We show that the normalized persistent entropy (N P E) allows us to characterize the effective coupling between pairs of oscillators because it tends to increase with the coupling strength and to decrease with the distance between the oscillators. We also observed that pairs of oscillators that have similar degrees and are nearest neighbors tend to have higher N P E values than pairs with different degrees. However, large variability is found in the N P E values. Comparing the N P E behavior with that of the phase-locking value (P L V , commonly used to evaluate the synchronization of phase oscillators), we find that for large enough coupling, P L V only displays a monotonic increase, while N P E shows a richer behavior that captures variations in the behavior of the oscillators. This is due to the fact that P L V only captures coupling-induced phase changes, while N P E also captures amplitude changes. Moreover, when we consider the same network but with Kuramoto phase oscillators, we also find that N P E captures the transition to synchronization (as it increases with the coupling strength), and it also decreases with the distance between the oscillators. Therefore, we propose N P E as a data analysis technique to try to differentiate pairs of oscillators that have strong effective coupling because they are first or near neighbors, from those that have weaker coupling because they are distant neighbors. [ABSTRACT FROM AUTHOR]

Emergent behavior in complex networks can be predicted and analyzed via the mechanism of spontaneous symmetry-breaking bifurcation, in which solutions of related bifurcation problems lose symmetry as some parameters are varied, even though the equations that such solutions satisfy retain the full symmetry of the system. A less common mechanism is that of forced symmetry-breaking, in which either a bifurcation problem has symmetry on both the state variables and the parameters, or one where the equations have less symmetry when a certain parameter is varied. In this manuscript, it is shown that in certain networks with parameter mismatches the governing equations remain unchanged when the group of symmetries acts on both the state variables and the parameter space. Based on this observation we study the existence and stability of collective patterns in symmetric networks with parameters mismatches from the point of view of forced symmetry-breaking bifurcations. Treating the parameters as state variables, we perform center manifold reductions, which allow us to understand how the disorder in parameters affects the bifurcation points as well as the stability properties of the ensuing patterns. Theoretical results are validated with numerical simulations. [ABSTRACT FROM AUTHOR]

Synchronization is an omnipresent collective phenomenon in nature and technology, whose understanding is still elusive for real-world systems in particular. We study the synchronization transition in a phase oscillator system with two nonvanishing Fourier-modes in the interaction function, hence going beyond the Kuramoto paradigm. We show that the transition scenarios crucially depend on the interplay of the two coupling modes. We describe the multistability induced by the presence of a second coupling mode. By extending the collective coordinate approach, we describe the emergence of various states observed in the transition from incoherence to coherence. Remarkably, our analysis suggests that, in essence, the two-mode coupling gives rise to states characterized by two independent but interacting groups of oscillators. We believe that these findings will stimulate future research on dynamical systems, including complex interaction functions beyond the Kuramoto-type. [ABSTRACT FROM AUTHOR]

We study the dynamics of multistable coexisting rotating waves that propagate along a unidirectional ring consisting of coupled double-well Duffing oscillators with different numbers of oscillators. By employing time series analysis, phase portraits, bifurcation diagrams, and basins of attraction, we provide evidence of multistability on the route from coexisting stable equilibria to hyperchaos via a sequence of bifurcations, including the Hopf bifurcation, torus bifurcations, and crisis bifurcations, as the coupling strength is increased. The specific bifurcation route depends on whether the ring comprises an even or odd number of oscillators. In the case of an even number of oscillators, we observe the existence of up to 32 coexisting stable fixed points at relatively weak coupling strengths, while a ring with an odd number of oscillators exhibits 20 coexisting stable equilibria. As the coupling strength increases, a hidden amplitude death attractor is born in an inverse supercritical pitchfork bifurcation in the ring with an even number of oscillators, coexisting with various homoclinic and heteroclinic orbits. Additionally, for stronger coupling, amplitude death coexists with chaos. Notably, the rotating wave speed of all coexisting limit cycles remains approximately constant and undergoes an exponential decrease as the coupling strength is increased. At the same time, the wave frequency varies among different coexisting orbits, exhibiting an almost linear growth with the coupling strength. It is worth mentioning that orbits originating from stronger coupling strengths possess higher frequencies. [ABSTRACT FROM AUTHOR]

Synchronization of two or more self-sustained oscillators is a well-known and studied phenomenon, appearing both in natural and designed systems. In some cases, the synchronized state is undesired, and the aim is to destroy synchrony by external intervention. In this paper, we focus on desynchronizing two self-sustained oscillators by short pulses delivered to the system in a phase-specific manner. We analyze a non-trivial case when we cannot access both oscillators but stimulate only one. The following restriction is that we can monitor only one unit, be it a stimulated or non-stimulated one. First, we use a system of two coupled Rayleigh oscillators to demonstrate how a loss of synchrony can be induced by stimulating a unit once per period at a specific phase and detected by observing consecutive inter-pulse durations. Next, we exploit the phase approximation to develop a rigorous theory formulating the problem in terms of a map. We derive exact expressions for the phase–isostable coordinates of this coupled system and show a relation between the phase and isostable response curves to the phase response curve of the uncoupled oscillator. Finally, we demonstrate how to obtain phase response information from the system using time series and discuss the differences between observing the stimulated and unstimulated oscillator. [ABSTRACT FROM AUTHOR]

Adaptive interactions are an important property of many real-word network systems. A feature of such networks is the change in their connectivity depending on the current states of the interacting elements. In this work, we study the question of how the heterogeneous character of adaptive couplings influences the emergence of new scenarios in the collective behavior of networks. Within the framework of a two-population network of coupled phase oscillators, we analyze the role of various factors of heterogeneous interaction, such as the rules of coupling adaptation and the rate of their change in the formation of various types of coherent behavior of the network. We show that various schemes of heterogeneous adaptation lead to the formation of transient phase clusters of various types. [ABSTRACT FROM AUTHOR]

In this work a master-slave configuration to obtain synchronization between the double-well Duffing-van der Pol (master system) and the three-well Φ6 Duffing oscillators (slave system) is studied. For this configuration, we analyze the system when the dissipative and one that combines the elastic and dissipative couplings are used. Whenever the dissipative coupling is used, we observed a vertical shift synchronization in the slave system. However, when the combination of the elastic and dissipative couplings is used, the vertical shift disappears obtaining complete synchronization. We resort to perturbation method to corroborate analytically these kind of synchronization, when the master system provides an harmonic function to the slave system. These synchronizations are numerically corroborated. [ABSTRACT FROM AUTHOR]

In this study, we introduce a design of a feedback-type fluidic oscillator with elastic structures surrounding its feedback channel. By employing phase reduction theory, we extract the phase sensitivity function of the complex fluid–structure coupled system, which represents the system's oscillatory characteristics. We show that the frequency of the oscillating flow inside the fluidic oscillator can be modulated by inducing synchronization with the weak periodic forcing from the elastic structure vibration. This design approach adds controllability to the fluidic oscillator, where conventionally, the intrinsic oscillatory characteristics of such device were highly determined by its geometry. The synchronization-induced control also changes the physical characteristics of the oscillatory fluid flow, which can be beneficial for practical applications, such as promoting better fluid mixing without changing the overall geometry of the device. Furthermore, by analyzing the phase sensitivity function, we demonstrate how the use of phase reduction theory gives good estimation of the synchronization condition with minimal number of experiments, allowing for a more efficient control design process. Finally, we show how an optimal control signal can be designed to reach the fastest time to synchronization. [ABSTRACT FROM AUTHOR]

A detailed study of the characteristics of the time-delay signature of a single loop optoelectronic oscillator incorporating a nonlinear amplifier of sine hyperbolic type is presented. We demonstrate that with appropriate values of the system parameters, the unique time-delay of the system cannot be traced in its dynamics. Such an advantage is exploited in synchronization using a scheme that confirms the use of the hidden time-delay as a key. Furthermore, a new tunable parameter which may serve as a key in a secure communication scheme arises as a result of the introduction of the nonlinear amplifier. We expect this system to emulate a general benefit in the physical equipment, and reduce their cost and congestion. [ABSTRACT FROM AUTHOR]

Ising Machines (IMs) have the potential to outperform conventional Von-Neuman architectures in notoriously difficult optimization problems. Various IM implementations have been proposed based on quantum, optical, digital and analog CMOS, as well as emerging technologies. Networks of coupled electronic oscillators have recently been shown to exhibit characteristics required for implementing IMs. However, for this approach to successfully solve complex optimization problems, a highly reconfigurable implementation is needed. In this work, the possibility of implementing highly reconfigurable oscillator-based IMs is explored. An implementation based on quasiperiodically modulated coupling strength through a common medium is proposed and its potential is demonstrated through numerical simulations. Moreover, a proof-of-concept implementation based on CMOS coupled ring oscillators is proposed and its functionality is demonstrated. Simulation results show that our proposed architecture can consistently find the Max-Cut solution and demonstrate the potential to greatly simplify the physical implementation of highly reconfigurable oscillator-based IMs. [ABSTRACT FROM AUTHOR]

The classical wake oscillator model is capable of predicting the vortex-induced vibration response of a cylinder at high mass-damping ratios, but it fails to perform satisfactorily at low mass-damping ratios. A modified wake oscillator model is presented in this paper. The modification method involves analyzing the variation law of the add mass coefficient of the cylinder versus reduced velocity and expressing the reference lift coefficient CL0 as a function of the add mass coefficient. The modified wake oscillator model has been demonstrated to have better accuracy in capturing maximum amplitudes and flow velocity at low mass-damping ratios. However, the modified model at present form is unable to accurately predict the vortex-induced vibration response at high damping ratios. The purpose of this paper is to propose a new modification idea. In order to achieve better results when applying this modification idea to particular objects, it may be necessary to first understand the response law of these kinds of objects. [ABSTRACT FROM AUTHOR]

We present a theoretical study of resonant frequencies and spatial correlations of Josephson phases in frustrated arrays of Josephson junctions. Two types of one-dimensional arrays, namely, the diamond and sawtooth chains, are discussed in detail. For these arrays in the linear regime the Josephson phase dynamics is characterized by multiband dispersion relation , and the lowest band becomes completely flat at a critical value of frustration, f   =  f c. In a strongly nonlinear regime such critical value of frustration determines the crossover from non-frustrated (0  <  f   <  f c) to frustrated (f c  <  f   <  1) regimes. The crossover is characterized by the thermodynamic spatial correlation functions of phases on vertices, , i.e. displaying the transition from long- to short-range spatial correlations. We find that higher-order correlations functions, e.g. p   =  2 and p   =  3, restore the long-range behavior deeply in the frustrated regime, . Monte-Carlo simulations of the thermodynamics of frustrated arrays of Josephson junctions are in good agreement with analytical results. We also outline the extension of our results to the case of kagome lattice, prototypical 2D frustrated lattice, and other higher dimensional lattices. [ABSTRACT FROM AUTHOR]

Even after about 50 years of intensive research, the dynamics of oscillator populations remain one of the most popular topics in nonlinear science. This Focus Issue brings together studies on such diverse aspects of the problem as low-dimensional description, effects of noise and disorder on synchronization transition, control of synchrony, the emergence of chimera states and chaotic regimes, stability of power grids, etc. [ABSTRACT FROM AUTHOR]

Combinatorial optimization problems are difficult to solve with conventional algorithms. Here we explore networks of nonlinear electronic oscillators evolving dynamically towards the solution to such problems. We show that when driven into subharmonic response, such oscillator networks can minimize the Ising Hamiltonian on non-trivial antiferromagnetically-coupled 3-regular graphs. In this context, the spin-up and spin-down states of the Ising machine are represented by the oscillators' response at the even or odd driving cycles. Our experimental setting of driven nonlinear oscillators coupled via a programmable switch matrix leads to a unique energy minimizer when one exists, and probes frustration where appropriate. Theoretical modeling of the electronic oscillators and their couplings allows us to accurately reproduce the qualitative features of the experimental results and extends the results to larger graphs. This suggests the promise of this setup as a prototypical one for exploring the capabilities of such an unconventional computing platform. Combinatorial optimization problems, such as the max-cut problem, present efficiency challenges for traditional algorithms but can be solved using unconventional computing platforms. Here we demonstrate that electrical-oscillator networks can suitably function as a computing architecture to minimize Ising-spin Hamiltonians for various 3-regular graphs. [ABSTRACT FROM AUTHOR]

Summary: For wide application of the synchronization properties of nonlinear oscillators in control and signal processing engineering field, a common design method for nonlinear oscillator models is required. The output response of a nonlinear oscillator excited by a periodic signal implicitly includes information on both the instantaneous phase and the amplitude of the input signal. The design of a nonlinear oscillator model that can synchronize with an arbitrary cyclic phenomenon can be enabled by explicitly expressing the dynamics of both phase and amplitude with the phase reduction method and organizing them by the number of events in one period. Assuming the cyclic phenomenon to be a single periodic signal, the properties of the generalized nonlinear oscillator model depend on the phase resolution of the signal, which is associated with the phase‐shift of fractional calculus. Thus, this study verifies the validity of fractional calculus through the input‐output characteristic of the generalized nonlinear oscillator model. Numerical simulation using a fractional differentiator with backward‐difference demonstrated that increasing the phase resolution of a single periodic signal improves the estimation accuracy of the phase and amplitude of the signal by employing a generalized nonlinear oscillator model. [ABSTRACT FROM AUTHOR]

Conventional and modern methods for beam steering in antennas and radar systems have one goal in common: to manipulate the phase shift between oscillating components, so that the direction of a radiating intensity pattern can be controlled. Typical components include arrays of nonlinear oscillators connected in a chain configuration. Modern methods for beam steering take advantage of the inherent nonlinearities of the individual components and of the collective dynamics of the array of oscillators to successfully manipulate phase shift. None of those methods include, however, signal amplification. In this manuscript, we introduce a novel array configuration: a feedforward network, which allows, simultaneously, for beam steering and signal amplification capabilities. We show that the amplitudes of certain bifurcating solutions in a feedforward network of nonlinear oscillators exhibit growth rates that are significantly larger than in other type of networks that operate near the onset of a Hopf bifurcation. Those branches of oscillations have the potential to lead to significant signal amplification of a radiating beam. [ABSTRACT FROM AUTHOR]

We proposed a new optomechanical system (OMS) based on parallel suspended one-dimensional optomechanical crystal (1D-OMC) nanobeam cavities for optomechanical synchronization. The optomechanical oscillators (OMOs) were spaced apart by an air-slot gap and coupled through optical radiation fields. The numerical simulation showed that the evolution process of 1D-OMC nanobeam cavities to mechanical synchronization could be divided into three clear stages. The synchronization of two mechanical breathing modes at 5.8846 GHz was achieved by using a single laser source. Finally, we investigated the relationship between the threshold power and detuning of an input laser for self-sustaining and synchronization states. Such chip-based structures hold great potential for large-scale synchronized oscillator networks. [ABSTRACT FROM AUTHOR]

Virtual oscillator control (VOC) is a time-domain strategy that leverages the dynamics of nonlinear oscillator circuits for synchronizing and regulating grid-forming inverters. In this article, we examine a class of second-order circuits composed of a harmonic oscillator and nonlinear state-dependent damping that has found extensive interest in the context of VOC. We center our analysis on the Van der Pol, Dead-zone, and Andronov-Hopf oscillators; these are characterized by several distinguishing attributes but they all share the common structure noted above. Analytical methods based on averaging and perturbation theory are outlined to derive several performance metrics related to harmonic and dynamic properties of these oscillators under a unified framework. Our results reveal that the Andronov-Hopf oscillator is well suited for grid-forming inverter applications since it can yield harmonics-free waveforms without compromising dynamic performance. Analytical results are validated with numerical simulations and experiments, and a multiinverter hardware setup is used to illustrate a practical use case. [ABSTRACT FROM AUTHOR]

Networks of weakly coupled oscillators had a profound impact on our understanding of complex systems. Studies on model reconstruction from data have shown prevalent contributions from hypernetworks with triplet and higher interactions among oscillators, in spite that such models were originally defined as oscillator networks with pairwise interactions. Here, we show that hypernetworks can spontaneously emerge even in the presence of pairwise albeit nonlinear coupling given certain triplet frequency resonance conditions. The results are demonstrated in experiments with electrochemical oscillators and in simulations with integrate-and-fire neurons. By developing a comprehensive theory, we uncover the mechanism for emergent hypernetworks by identifying appearing and forbidden frequency resonant conditions. Furthermore, it is shown that microscopic linear (difference) coupling among units results in coupled mean fields, which have sufficient nonlinearity to facilitate hypernetworks. Our findings shed light on the apparent abundance of hypernetworks and provide a constructive way to predict and engineer their emergence. Networks with higher-order interactions are relevant to variety of real-world applications, they can be good description of data even if the system has only pairwise interactions. The authors uncover the hypernetwork emergence in coupled nonlinear oscillators and electrochemical experiments. [ABSTRACT FROM AUTHOR]

In this work, we present a method of generating a class of nonlinear ordinary differential equations (ODEs), representing the dynamics of appropriate nonlinear oscillators, that have the characteristics of either amplitude independent frequency of oscillations or amplitude-dependent frequency of oscillations from the integrals of the simple harmonic oscillator equation. To achieve this, we consider the case where the integrals are in the same form both for the linear and the nonlinear oscillators in either of the cases. We also discuss the method of deriving the associated integrals and the general solution in harmonic form for both the types. We demonstrate the applicability of this method up to 2N coupled first-order nonlinear ODEs in both the cases. Further, we illustrate the theory with an example in each case. [ABSTRACT FROM AUTHOR]

In this work, we experimentally investigate the dynamics of pairs of opto-thermally driven, mechanically coupled, doubly clamped, silicon micromechanical oscillators, and numerically investigate the dynamics of the corresponding lumped-parameter model. Coupled limit cycle oscillators exhibit striking nonlinear dynamics and bifurcations in response to variations in system parameters. We show that the input laser power influences the frequency detuning between two non-identical oscillators. As the laser power is varied, different regimes of oscillations such as the synchronized state, the drift state, and the quasi-periodic state are mapped at minimal and high coupling strengths. For non-identical oscillators, coexistence of two states, the synchronized state and the quasi-periodic state, is demonstrated at high coupling and high laser power. Experimentally, this bistability manifests as irregular oscillations as the system rapidly switches between the two states due to the system’s sensitive dependence on initial conditions in the presence of noise. We provide a qualitative comparison of the experimental and numerical results to elucidate the behavior of the system. [2022-0005] [ABSTRACT FROM AUTHOR]

A coupled model of a top tensioned riser (TTR) under combined excitations from vortices and time-varying tension is proposed. And the van der Pol oscillator is introduced to simulate the time-varying characteristics of vortex shedding. The first three mode approximations as a six degree-of-freedom (DOF) system of vibrations of the fluid-structure interaction system are derived. Two cases of resonances are analyzed by using the numerical method. One case is on the vortex induced vibration (VIV) of the six DOF system, where the 1-to-1 internal resonance (also named "lock-in" phenomenon) between frequencies of the first modes of structure and the wake oscillator are investigated. Bifurcation curves between dynamic responses in terms of the shedding frequency are derived. Comparing to the vibrations of the first order mode, the "lock-in" phenomenon for vibrations of the higher order modes can occur in the wider parameter range with large values. The second case is on the combined resonance between the 1-to-1 internal and 1-to-2 parametric resonance. Response curves versus the shedding frequency under different amplitude ratios are calculated by the Runge–Kutta method. Effects of system parameters including the amplitude ratio of the parametric excitation, shedding frequency on the response curves are investigated. Comparing to the case when only considering the 1-to-1 internal resonance in the VIV, results of the combined resonance show those large-amplitude responses of the structure and wake oscillator can be induced. Responses will increase rapidly as the amplitude of the varying tension increases. Unstable responses can occur within a large frequency band of nondimensional shedding frequency. Furthermore, the dynamic characteristics of the responses of the six DOF system including the bifurcation diagrams, time histories, phase-plane projection and Poincaré sections are computed for different amplitude ratios. Results indicate that complex dynamics such as the quasi-periodic and chaotic solutions can occur for different amplitude ratios. The analysis can be helpful for the parameter design of deep-water risers. [ABSTRACT FROM AUTHOR]

Using a stochastic nonlinear phase oscillator model, we study the effect of event-triggered feedback on the statistics of interevent intervals. Events are associated with the entering of a new cycle. The feedback is modeled by an instantaneous increase (positive feedback) or decrease (negative feedback) of the oscillator frequency whenever an event occurs followed by an exponential decay on a slow time scale. In addition to the known excitable and oscillatory regimes, which are separated by a saddle node on invariant circle bifurcation, positive feedback can lead to bistable dynamics md a change of the system's excitability. The feedback has also a strong effect on noise-induced phenomena like coherence resonance or anticoherence resonance. Both positive and negative feedback can lead to more regular output for particular noise strengths. Finally, we investigate serial correlations in the sequence of interevent intervals that occur due to the additional slow dynamics. We derive approximations for the serial correlation coefficient and show that positive feedback results in extended positive interval correlations, whereas negative feedback yields short-ranging negative correlations. Investigating the interplay of feedback and the nonlinear phase dynamics close to the bifurcation, we find that correlations are most pronounced for optimal feedback strengths. [ABSTRACT FROM AUTHOR]

We analyze the effect of adding a weak, localized, inhomogeneity to a two dimensional array of oscillators with nonlocal coupling. We propose and also justify a model for the phase dynamics in this system. Our model is a generalization of a viscous eikonal equation that is known to describe the phase modulation of traveling waves in reaction–diffusion systems. We show the existence of a branch of target pattern solutions that bifurcates from the spatially homogeneous state when , the strength of the inhomogeneity, is nonzero and we also show that these target patterns have an asymptotic wavenumber that is small beyond all orders in . The strategy of our proof is to pose a good ansatz for an approximate form of the solution and use the implicit function theorem to prove the existence of a solution in its vicinity. The analysis presents two challenges. First, the linearization about the homogeneous state is a convolution operator of diffusive type and hence not invertible on the usual Sobolev spaces. Second, a regular perturbation expansion in does not provide a good ansatz for applying the implicit function theorem since the nonlinearities play a major role in determining the relevant approximation, which also needs to be ‘correct’ to all orders in . We overcome these two points by proving Fredholm properties for the linearization in appropriate Kondratiev spaces and using a refined ansatz for the approximate solution which was obtained using matched asymptotics. [ABSTRACT FROM AUTHOR]

Dynamical systems theory has emerged as an interdisciplinary area of research to characterize the complex dynamical transitions in real-world systems. Various nonlinear dynamical phenomena and bifurcations have been discovered over the decades using different reduced-order models of oscillators. Different measures and methodologies have been developed theoretically to detect, control, or suppress the nonlinear oscillations. However, obtaining such phenomena experimentally is often challenging, time-consuming, and risky mainly due to the limited control of certain parameters during experiments. With this review, we aim to introduce a paradigmatic and easily configurable Rijke tube oscillator to the dynamical systems community. The Rijke tube is commonly used by the combustion community as a prototype to investigate the detrimental phenomena of thermoacoustic instability. Recent investigations in such Rijke tubes have utilized various methodologies from dynamical systems theory to better understand the occurrence of thermoacoustic oscillations and their prediction and mitigation, both experimentally and theoretically. The existence of various dynamical behaviors has been reported in single and coupled Rijke tube oscillators. These behaviors include bifurcations, routes to chaos, noise-induced transitions, synchronization, and suppression of oscillations. Various early warning measures have been established to predict thermoacoustic instabilities. Therefore, this review article consolidates the usefulness of a Rijke tube oscillator in terms of experimentally discovering and modeling different nonlinear phenomena observed in physics, thus transcending the boundaries between the physics and the engineering communities. [ABSTRACT FROM AUTHOR]