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Swarmalators are oscillators that swarm through space as they synchronize in time. Introduced a few years ago to model many systems that mix synchrony with self-assembly, they remain poorly understood theoretically. Here, we obtain the first analytic results on swarmalators moving in two spatial dimensions by introducing a simplified model where the swarmalators have no hard-shell interaction terms and move on a periodic plane. These simplifications allow expressions for order parameters, stabilities and bifurcations to be derived exactly. [ABSTRACT FROM AUTHOR]

ABSTRACT The lattice loop formed by adjacent coupling of tunnel diode LC oscillators has been shown to generate topological defects due to broken mirror symmetry, which rotate in the opposite direction to the rotating pulse. Leveraging the injection locking of the rotational dynamics of these topological defects can lead to a subharmonic injection locking scheme with a large division ratio. This paper aims to enhance the design by elucidating the relationship between loop size, the number of topological defects, and the division ratio. Furthermore, we perform bifurcation analysis of the injection locking system, demonstrating that when the external signal strength reaches a certain threshold, defect pinning by the external signal occurs, leading to an extension of the lock range. The dynamics of topological defects are well suited to the so‐called progressive multiphase injection locking. In this paper, we clarify the degree of lock range extension achieved by introducing this technique. [ABSTRACT FROM AUTHOR]

This research delves into simple Jerk-type dynamical networks, constructed using a specific bidirectional coupling scheme that influences the sub-oscillators and compound nonlinearity gradient functions that perturb each sub-unit of the network. The novelty of this work is to demonstrate a new pedagogical method able to stimulate the formation of higher-order multiscroll dynamics in the Jerk sub-oscillator. This is done by modeling a simple dynamic network built by applying a special bidirectional coupling between the Jerk sub-systems. Another novelty of this approach is that it allows us to study the collective dynamics of Jerk system networks, which has not yet been done in the literature. Multiscroll systems are exceedingly complex dynamically, making them valuable in chaos-based applications. The pedagogical approach used in this study is exceptional because it produces many additional equilibrium points (from 5 to 25 in dynamical network 1, and from 3 to 15 in dynamical network 2) in each Jerk sub-unit of the network. These equilibria elevate the complexity of Jerk systems by emulating higher-order multiscroll dynamics. The methodology used in this study is efficient and differs from those used in the literature, which mainly uses nonlinear multizero functions in dissipative systems. This research further explores the dynamic system characterization tools and conducts an experimental investigation on a microcontroller (ATMEGA2560) to confirm the predictions. [ABSTRACT FROM AUTHOR]

Synchronization of coupled oscillators is a universal phenomenon encountered across different scales and contexts, e.g., chemical wave patterns, superconductors, and the unison applause we witness in concert halls. The existence of common underlying coupling rules defines universality classes, revealing a fundamental sameness between seemingly distinct systems. Identifying rules of synchronization in any particular setting is hence of paramount relevance. Here, we address the coupling rules within an embryonic oscillator ensemble linked to vertebrate embryo body axis segmentation. In vertebrates, the periodic segmentation of the body axis involves synchronized signaling oscillations in cells within the presomitic mesoderm (PSM), from which somites, the prevertebrae, form. At the molecular level, it is known that intact Notch-signaling and cell-to-cell contact are required for synchronization between PSM cells. However, an understanding of the coupling rules is still lacking. To identify these, we develop an experimental assay that enables direct quantification of synchronization dynamics within mixtures of oscillating cell ensembles, for which the initial input frequency and phase distribution are known. Our results reveal a "winner-takes-it-all" synchronization outcome, i.e., the emerging collective rhythm matches one of the input rhythms. Using a combination of theory and experimental validation, we develop a coupling model, the "Rectified Kuramoto" (ReKu) model, characterized by a phase-dependent, nonreciprocal interaction in the coupling of oscillatory cells. Such nonreciprocal synchronization rules reveal fundamental similarities between embryonic oscillators and a class of collective behaviors seen in neurons and fireflies, where higher-level computations are performed and linked to nonreciprocal synchronization. [ABSTRACT FROM AUTHOR]

Most complex systems are nonlinear, relying on emergent behavior resulting from many interacting subsystems, which are often characterized by oscillatory dynamics. Having collective oscillatory behavior is an essential requirement for an appropriate functioning of various real-world systems. Complex networks have proven to be exceptionally efficient in elucidating the topological structures of both natural and artificial systems, as well as describing diverse processes taking place over them. Remarkable advancements have been achieved in recent years in comprehending the emergent dynamics atop complex networks. Specifically, among other processes, a large body of works intend to explore the dynamical robustness of complex networks, which is the networks' ability to withstand dynamical degradation in the network constituents while maintaining collective oscillatory dynamics. Indeed, various physical and biological systems are recognized to undergo a decline in their dynamic activities, whether occurring naturally or influenced by environmental factors. The impact of such damages on network performance can be significant, and the system's robustness is indicative of its capability to maintain fundamental functionality in the face of dynamic deteriorations, often called aging. This review offers a comprehensive excerpt of notable research endeavors that scrutinize how networks sustain global oscillation under a growing number of inactive dynamical units. We present the contemporary research dedicated to the theoretical understanding and the enhancement mechanisms of the dynamical robustness in complex networks. Our emphasis lies on various network topologies and coupling functions, elucidating the persistence of networked systems. We cover variants of system characteristics from heterogeneity in network connectivity to heterogeneity in the dynamical units. Finally we discuss challenges ahead in this potential field and open areas for future studies. [ABSTRACT FROM AUTHOR]

Decades have now passed since Colin Pittendrigh first proposed a model of a circadian clock composed of two coupled oscillators, individually responsive to the rising and setting sun, as a flexible solution to the challenge of behavioral and physiological adaptation to the changing seasons. The elegance and predictive power of this postulation has stimulated laboratories around the world in searches to identify and localize such hypothesized evening and morning oscillators, or sets of oscillators, in insects, rodents, and humans, with experimental designs and approaches keeping pace over the years with technological advances in biology and neuroscience. Here, we recount the conceptual origin and highlight the subsequent evolution of this dual oscillator model for the circadian clock in the mammalian suprachiasmatic nucleus; and how, despite our increasingly sophisticated view of this multicellular pacemaker, Pittendrigh's binary conception has remained influential in our clock models and metaphors. [ABSTRACT FROM AUTHOR]

Summary: This study provides design criteria of multiphase injection locking for an oscillator loop with multiple rotary pulses that can be utilized for an effective frequency multiplier. Coupled oscillator lattice loops can support phase waves. In the case where N pulses rotate in the loop, the oscillatory signal exhibits N times higher frequency than the rotation frequency of a single rotary pulse. Once an external signal succeeds in injection locking the single‐pulse rotation, the loop operates as an efficient N times frequency multiplier. As in a ring oscillator, the progressive multiphase injection locking is best employed, when the loop supports only a single rotary pulse. Because the phase of the external signal becomes different for each pulse, progressive phase assignment does not work well for multiple rotary pulses; therefore, a more efficient scheme of phase assignment is investigated and validated by bifurcation analyses and measurements. The most effective multiphase injection locking is expected for the phase difference between two neighboring oscillators to be set to −4π/N (4π/N), when N pulses rotate to the cell address increases (decreases) in an oscillator loop. [ABSTRACT FROM AUTHOR]

A ring of coupled oscillators with delayed feedback with various types of coupling between the oscillators is considered. For each type of coupling, the asymptotic behavior of the model solutions with respect to a large parameter is constructed for a wide variety of initial conditions. It is shown that the studying the behavior of solutions to the original infinite-dimensional models can be reduced to studying the dynamics of the constructed finite-dimensional mappings. High quality conclusions about the dynamics of the original systems are made. It is shown that the behavior of solutions significantly varies with variations in the type of coupling. Conditions on the system parameters are found under which the synchronization, two-cluster synchronization, and more complex modes are possible. [ABSTRACT FROM AUTHOR]

Plane waves have commonly been observed in recordings of human brains. These waves take the form of spatial phase gradients in the oscillatory potentials picked up by implanted electrodes. We first show that long but finite chains of nearest-neighbor coupled phase oscillators can produce an almost constant phase gradient when the edge effects interact with small heterogeneities in the local frequency. Next, we introduce a continuum model with nonlocal coupling and use singular perturbation methods to show similar interactions between the boundaries and small frequency differences. Finally, we show that networks of Wilson--Cowan equations can generate plane waves with the same mechanism. [ABSTRACT FROM AUTHOR]

We study the dynamics of a swarmalator model with higher harmonic phase coupling. We analyze stability, bifurcation, and structural properties of several novel attracting states, including the formation of spatial clusters with distinct phases, and single spatial clusters with a small number of distinct phases. We use mean-field (centroid) dynamics to analytically determine intercluster distance. We also find states with two large clusters along with a small number of swarmalators that are trapped between the two clusters and vacillate (waver) between them. In the case of a single vacillator we use a mean-field reduction to reduce the dynamics to two dimensions, which enables a detailed bifurcation analysis. We show excellent agreement between our reduced two-dimensional model and the dynamics and bifurcations of the full swarmalator model. [ABSTRACT FROM AUTHOR]

The synchronization of stochastic hybrid-impulses complex-valued multi-links coupled systems (SHCMCS) based on discrete-time state observations control is studied. Therein, hybrid impulses, multiple links and complex-valued factors are considered simultaneously, which make the model more common. By using the Lyapunov method and Kirchhoff's Matrix Tree Theorem, average impulsive interval and average impulsive gain, the synchronization is achieved. Moreover, synchronization criteria are obtained, whose sufficient conditions reflect that the realization of synchronization depends on coupled strength and stochastic disturbance strength. In addition, stochastic complex-valued multi-links coupled oscillators with hybrid impulses are studied. Finally, a numerical test is presented to illustrate the validity of results. [ABSTRACT FROM AUTHOR]

A comprehensive theory is developed for the chiral optical response of two configurations of the N-oscillator Born–Kuhn model (NOBK): the helically stacked and the corner stacked models. In the helical NOBK model, there is always a chiral response regardless of the value of N, whereas in the corner NOBK, only configurations with even N demonstrate a chiral response. Generally, the magnitudes of optical rotatory dispersion (ORD) and circular dichroism (CD) increase with N when the parameters of each oscillator are fixed. In cases of weak coupling, the spectral shapes of ORD and CD remain invariant, while strong coupling significantly alters the spectral shapes. For large damping, the spectral amplitude becomes smaller, and the spectral features become broader. In the presence of small damping, strong coupling introduces degeneracy in the coupled oscillator system, leading to multiple spectral features in both ORD and CD across the entire spectral region. This simple model can not only help in the design of tunable chiral metamaterials but also enhance our understanding of chiro-optical responses in structures with different configurations. [ABSTRACT FROM AUTHOR]

Abstract—The possibility of the existence of laminar chaos in coupled time-delayed feedback systems is investigated. The cases of unidirectional and mutual coupling of time-delay systems are considered. It is shown for the first time that laminar chaos can exist not only in a system with a variable delay time, but also in a system with a constant delay time, if it is coupled with a system in the regime of laminar chaos. [ABSTRACT FROM AUTHOR]

A system of three-coupled quantized fields is studied under the intrinsic decoherence scheme given by the Milburn equation. Taking the closed analytical form of this relation, without second-order approximation, we arrive at an explicit analytical solution for the wavefunction dynamics. We show the decaying behavior in the number of modes of each quantized field for suitable initial conditions. One of the modes goes coherently driven, and the others are vacuum states. We discuss the redistribution of the excitations. [ABSTRACT FROM AUTHOR]

Circadian rhythm, cell division and metabolic oscillations are rhythmic cellular behaviors that must be both robust but also to respond to changes in their environment. In this work, we study emergent behavior of coupled biochemical oscillators, modeled as repressilators. While more traditional approaches to oscillators synchronization often use phase oscillators, our approach uses switching systems that may be more appropriate for cellular networks dynamics governed by biochemical switches. We show that while one-directional coupling maintains stable oscillation of individual repressilators, there are well-characterized parameter regimes of mutually coupled repressilators, where oscillations stop. In other parameter regimes, joint oscillations continue. Our results may have implications for the understanding of condition-dependent coupling and un-coupling of regulatory networks. [ABSTRACT FROM AUTHOR]

Chimera dynamics, an intriguing phenomenon of coupled oscillators, is characterized by the coexistence of coherence and incoherence, arising from a symmetry-breaking mechanism. Extensive research has been performed in various systems, focusing on a system of Kuramoto–Sakaguchi (KS) phase oscillators. In recent developments, the system has been extended to the so-called generalized Kuramoto model, wherein an oscillator is situated on the surface of an M -dimensional unit sphere, rather than being confined to a unit circle. In this paper, we exploit the model introduced in Tanaka (2014 New. J. Phys. 16 023016) where the macroscopic dynamics of the system was studied using the extended Watanabe–Strogatz transformation both for real and complex spaces. Considering two-population networks of the generalized KS oscillators in 2D complex spaces, we demonstrate the existence of chimera states and elucidate different motions of the order parameter vectors depending on the strength of intra-population coupling. Similar to the KS model on the unit circle, stationary and breathing chimeras are observed for comparatively strong intra-population coupling. Here, the breathing chimera changes their motion upon decreasing intra-population coupling strength via a global bifurcation involving the completely incoherent state. Beyond that, the system exhibits periodic alternation of the two order parameters with weaker coupling strength. Moreover, we observe that the chimera state transitions into a componentwise aperiodic dynamics when the coupling strength weakens even further. The aperiodic chimera dynamics emerges due to the breaking of conserved quantities that are preserved in the stationary, breathing and alternating chimera states. We provide a detailed explanation of this scenario in both the thermodynamic limit and for finite-sized ensembles. Furthermore, we note that an ensemble in 4D real spaces demonstrates similar behavior. [ABSTRACT FROM AUTHOR]

A dual‐core voltage‐controlled oscillator (VCO) using harmonic resonance is presented in this letter. Without an additional switch capacitor bank, harmonic resonance implemented in VCO is realized by only a single control voltage, satisfying the requirement of frequency‐modulated continuous‐wave radar. Two pairs of varactors are added to the transformer‐based tank, which ensures a constant ratio between the two resonant frequencies of the tank in the entire tuning range. Robustly single‐frequency oscillation is obtained by increasing the coupling coefficient of the transformer and the coupling capacitor's capacitance. Implemented in a 28‐nm complementary metal–oxide–semiconductor, the VCO shows a 14.75% frequency tuning range covering 34.16–39.6 GHz. The measured results show it exhibits −102.13 dBc/Hz phase noise at a 1 MHz offset from the 34.16 GHz. The oscillator core draws 21–28 mA from a 0.9 V supply. [ABSTRACT FROM AUTHOR]

In this work, we analyze the Kuramoto model (KM) with inertia on a convergent family of graphs. It is assumed that the intrinsic frequencies of the individual oscillators are sampled from a probability distribution. In addition, a given graph, which may also be random, assigns network connectivity. As in the original KM, in the model with inertia, the weak coupling regime features mixing, the state of the network when the phases (but not velocities) of all oscillators are distributed uniformly around the unit circle. We study patterns, which emerge when mixing loses stability under the variation of the strength of coupling. We identify a pitchfork (PF) and an Andronov–Hopf (AH) bifurcations in the model with multimodal intrinsic frequency distributions. To this effect, we use a combination of the linear stability analysis and Penrose diagrams, a geometric technique for studying stability of mixing. We show that the type of a bifurcation and a nascent spatiotemporal pattern depend on the interplay of the qualitative properties of the intrinsic frequency distribution and network connectivity. [ABSTRACT FROM AUTHOR]

Agricultural pests are increasingly appreciated as subjects of ecology. One particular case, a pest in coffee production, is analyzed here using the conceptual framework of complex systems, increasingly acknowledged as having an obvious home in the field of ecology, notorious for its complex structures. The particular case analyzed here arguably falls under the control of the complexity of the ecological system rather than of a simple magic bullet of population regulation. The system, which has been under study in southern Mexico for the past quarter century, is analyzed through the lens of neutral oscillations of the classical nondissipative Lotka-Volterra system. Based on three consumer/resource pairs (populations of [1] an ant, [2] a scale insect, [3] a beetle predator of the scale insect, [4] a fungal pathogen of the scale insect, and [5] a fly parasitoid of the ant), this five-dimensional system is well known qualitatively. Coupling all agents through both direct effects and trait-mediated indirect effects, the behavior of the neutral oscillation form of the system reveals a complex set of behaviors, including harmonized invariant sets, chaos, and/or quasiperiodicity. Such behaviors are well-known subjects in the science of complex systems and, it is argued, are ultimately sufficient to effect a degree of regulation on the pest, independent of explicit density-dependent feedback. Control of the system is thus seen as arguably actuated through its complexity, independent of any classic dissipative force. [ABSTRACT FROM AUTHOR]

Introduction: In order to understand human hearing, it helps to understand how the ears of lower vertebrates, like, for instance, lizards, function. A key feature in common is that the ears of both humans and lizards emit faint, pure tones known as spontaneous otoacoustic emissions (SOAEs). More than four decades after their discovery, the mechanism underlying these emissions is still imperfectly understood, although it is known that they are important for improving the sensitivity and sharpness of hearing. In both humans and lizards, the frequencies of SOAEs change by a few percent when static pressure is applied to the tympanic membrane. For the human ear, this observation is normally explained by a so-called global oscillator model (such as with Shera's coherent reflection model), in which the emissions result from standing waves, and external pressure changes the boundary conditions – the stiffness of the oval and round windows – which then has a global effect on the SOAE frequencies. Methods: Here we investigate how changing parameters of an earlier developed local oscillator model for the lizard ear can change the frequencies of the SOAEs. A major feature of the model is that each oscillator is coupled only to its immediate neighbours. The oscillators then cluster into groups of identical frequency, and each of these so-called frequency plateaus can be taken to represent an SOAE. Results: Even though the natural (unperturbed) frequencies of all the oscillators remain fixed, here we find for several model parameters that by slightly changing their value the frequency plateaus – the SOAEs – shift by a few percent. Plots of how these changes alter SOAE frequencies are given, and their magnitude corresponds well with observations of SOAE changes in lizards. Discussion: Investigation of the influence of the change of parameters in an earlier developed local oscillator model for the lizard ear shows that a local oscillator model can explain small SOAE frequency changes as well as a global oscillator model. [ABSTRACT FROM AUTHOR]

Summary: Coupled tunnel‐diode oscillator lattice loops are investigated to characterize the mutual synchronization of nonlinear rotary pulses. In a properly biased lattice loop, a pulsed phase wave autonomously rotates. When two loops are coupled, these pulses are supposed to be mutually synchronized to start rotating with a fixed phase difference. To quantify the synchronization properties of both the in‐phase and out‐of‐phase rotary pulses, two five‐cell loops that are point‐coupled by a capacitor are studied using bifurcation analyses with respect to the bias voltage and coupling capacitance. In‐phase pulses are allowed irrespective of the coupling capacitance, whereas relatively small coupling is favorable for out‐of‐phase pulses. Moreover, a solution exists whose phase difference continuously varies with the bias voltage, which bifurcates from the out‐of‐phase pulses through pitchfork bifurcation. This study presents the result from the bifurcation analyses. For validation of the rotary pulse in the coupled loops, we measured a breadboarded test circuit in both the frequency and time domains. The in‐phase, out‐of‐phase, and symmetry‐broken rotary pulses were detected successfully. [ABSTRACT FROM AUTHOR]

Undoubtedly, multistability represents one of the most followed venues for researchers working in the field of nonlinear science. Multistability refers to the situation where a combination of two or more attractors occurs for the same rank of parameters. However, to the best of our knowledge, the situation encountered in the relevant literature is never one where periodicity, chaos and hyperchaos coexist. In this article, we study a fourth-order autonomous dynamical system composing of a Duffing oscillator coupled to a van der Pol oscillator. Coupling consists in disturbing the amplitude of one oscillator with a signal proportional to the amplitude of the other. We exploit analytical and numerical methods (bifurcation diagrams, phase portraits, basins of attraction) to shed light on the plethora of bifurcation modes exhibited by the coupled system. Several ranks of parameters are revealed where the coupled system exhibits two or more competing behaviors. In addition to the transient dynamics, the most gratifying behavior reported in this article concerns the coexistence of four attractors consisting of a limit cycle of period-n, a pair of chaotic attractors and a hyperchaotic attractor. The impact of a fractional-order derivative is also examined. A physical implementation of the coupled oscillator system is performed and the PSpice simulations confirm the predictions of the theoretical study conducted in this work. [ABSTRACT FROM AUTHOR]

We study dynamics of a unidirectional ring of three Rulkov neurons coupled by chemical synapses. We consider both deterministic and stochastic models. In the deterministic case, the neural dynamics transforms from a stable equilibrium into complex oscillatory regimes (periodic or chaotic) when the coupling strength is increased. The coexistence of complete synchronization, phase synchronization, and partial synchronization is observed. In the partial synchronization state either two neurons are synchronized and the third is in antiphase, or more complex combinations of synchronous and asynchronous interaction occur. In the stochastic model, we observe noise-induced destruction of complete synchronization leading to multistate intermittency between synchronous and asynchronous modes. We show that even small noise can transform the system from the regime of regular complete synchronization into the regime of asynchronous chaotic oscillations. [ABSTRACT FROM AUTHOR]

• The effect of the modulation signal on the dynamics of coupled Erbium-Doped Fiber Lasers is studied. • Two distinct clusters are formed in the non-modulated systems, which are lag-synchronized. • When the modulation signal is considered, the coupled systems can reach a chimera state. Studying the dynamics of coupled Erbium-Doped Fiber Lasers helps in optimizing the performance of these systems in different applications. An important factor in these systems is the modulation signal which can affect the dynamics of systems, leading to complex behaviors including diverse periodic and chaotic dynamics. This paper investigates the dynamics of a ring network of Erbium-Doped Fiber Lasers and compares its behaviors in the presence and absence of the modulation. The results reveal that although complete synchronization is not reachable, certain synchronization patterns exist during the periodic oscillations. In the non-modulated network, two distinct clusters are formed which can achieve lag synchronization under specific coupling intensities. While in the coupled modulated systems, the chimera state may form. These findings represent the complex relationship among modulation, coupling intensity, and network dynamics in EDFL systems. [ABSTRACT FROM AUTHOR]

This paper is concerned with periodic solutions of coupled oscillators. A systematic method of existence and stability of periodic solutions for coupled oscillators is provided in a combined method of graph theory, coincidence degree theory and Lyapunov method. Sufficient conditions for periodicity are obtained, which can be easily checked in the real applications. Finally, a numerical example is also provided to illustrate the correctness of theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we investigate nonlinear systems of second order ODEs describing the dynamics of two coupled nonlinear oscillators of a mechanical system. We obtain, under certain assumptions, some stability results for the null solution. Also, we show that in the presence of a time-dependent external force, every solution starting from sufficiently small initial data and its derivative are bounded or go to zero as the time tends to +8, provided that suitable conditions are satisfied. Our theoretical results are illustrated with numerical simulations. [ABSTRACT FROM AUTHOR]

In this investigation, two different models for two coupled asymmetrical oscillators, known as, coupled forced damped Duffing oscillators (FDDOs) are reported. The first model of coupled FDDOs consists of a nonlinear forced damped Duffing oscillator (FDDO) with a linear oscillator, while the second model is composed of two nonlinear FDDOs. The Krylov–Bogoliubov–Mitropolsky (KBM) method, is carried out for analyzing the coupled FDDOs for any model. To do that, the coupled FDDOs are reduced to a decoupled system of two individual FDDOs using a suitable linear transformation. After that, the KBM method is implemented to find some approximations for both unforced and forced damped Duffing oscillators (DDOs). Furthermore, the KBM analytical approximations are compared with the fourth-order Runge–Kutta (RK4) numerical approximations to check the accuracy of all obtained approximations. Moreover, the RK4 numerical approximations to both coupling and decoupling systems of FDDOs are compared with each other. [ABSTRACT FROM AUTHOR]

The functionalization of internal resonance (IR) is theoretically and experimentally demonstrated on a nonlinear hybrid vibration energy harvester (HVEH) based on piezoelectric (PE) and electromagnetic (EM) transductions. This nonlinear phenomenon is tuned by adjusting the gaps between the moving magnets of the structure, enabling 1:1 and 2:1 IR. The experimental results prove that the activation of 2:1 IR with a realistic excitation amplitude allows the improvement of both the frequency bandwidth (BW) and the harvested power (HP) by 300 % and 100 % , respectively compared to the case away from IR. These remarkable results open the way towards a very large scale integration of coupled resonators with simultaneous internal resonances. [ABSTRACT FROM AUTHOR]

The effect of topological disorder on the dynamic behaviour of topologically constrained oscillator chains is investigated through a normal mode analysis. A measure for mode localization based on the skewness of the density of energy states of normal modes is proposed, which correlates well with main configuration characteristics. Appropriate configuration sampling is achieved by employing the Ising model together with the Metropolis algorithm. [ABSTRACT FROM AUTHOR]

A resonant pulse transformer with shock excitation is compared with a system of coupled pendulums with different masses tuned to resonance. For the Tesla transformer and coupled pendulums, conditions for complete energy transfer during one-half of the beat cycle were obtained. The requirements for the system parameters and initial conditions necessary for complete transfer of energy in the case of two spring-coupled pendulums with different masses and of the same length were first determined theoretically and then verified experimentally. The dependence of the minimum time for complete transfer of energy from one pendulum to the other on the system parameters is obtained. [ABSTRACT FROM AUTHOR]

• Fractional-order Stuart-Landau oscillators with nonisochronicity parameter are investigated. • Nonisochronicity parameter can induce the swing of synchronization mediated by chimera state. • The effect of the fractional derivatives on synchronization transition is revealed. In this letter, we investigate the various dynamical behaviors in fractional-order coupled Stuart-Landau oscillator networks with the nonisochronicity parameter. We find that the nonisochronicity parameter can induce the swing of synchronization mediated by the imperfect synchronized state and the amplitude cluster state, and we verify the accuracy of the emergence of the swing of synchronization by the strength of incoherence and stability theory. Moreover, we discuss the effects of different parameters on the swing of synchronization mediated by the chimera state. First, as the nonisochronicity parameter increases, the range of the swing increases. Second, the decreasing of fractional-order derivative will lead to the reduction in the range of swing intervals. Third, the system exhibits the swing of synchronization mediated by the imperfect synchronized state when different initial conditions are employed. In particular, the system can show the imperfect breathing chimera state under symmetric initial conditions. [ABSTRACT FROM AUTHOR]

We consider a ring of phase oscillators interacting with distance-dependent time delays in general forms. It is found that the distance-dependent interaction delay is an essential driving mechanism for the twisted state to occur as a generic state. We carry out rigorous stability analysis for the existence and stability of the twisted states along the Ott–Antonsen invariant manifold and derive the exact characteristic equations for the stability conditions for arbitrarily chosen distance-dependent delay function. The complete stability diagrams are illustrated for two types of distance-dependent delay schemes, the propagational and step-wise interaction delays. Our theoretical results are verified using the direct numerical simulations of the model system. • Low-dimensional dynamics of oscillator system with distance-dependent time delays is found along the Ott–Antonsen invariant manifold. • Exact characteristic equations for the stability conditions for arbitrary distance-dependent delays are derived. • Complete stability diagrams are illustrated for two types of distance-dependent delay schemes. [ABSTRACT FROM AUTHOR]

In this work, we present a parsimonious set of ordinary differential equations (ODEs), describing with good precision and over a wide range of frequencies the linear and nonlinear dynamics of different types of fully premixed, turbulent, swirl-stabilized flames. This phenomenological model comprises eight ODEs and can physically be interpreted as a superposition of two mass–spring–damper systems. The model is characterized by 11 parameters and one nonlinear term of structure β x 2 x ̇ in particular, with position x and rate of displacement x ̇ of a given oscillator mass. The model can be optimized in frequency domain or in time domain. The former is suitable for situations where experimental data of the flame transfer function (FTF) and flame describing function (FDF) are available. The latter is suitable for situations where time series (input and output) from numerical simulations or experiments are available. In the present work, large eddy simulation (LES) data is used. Choosing a model comprising ODEs enables us to describe the dynamics of the flame using linear and nonlinear input-to-output transformations via the states. This is a sharp contrast to commonly applied finite impulse response (FIR) models, which can only directly employ nonlinear functions acting on the input or output — not on derivatives or other states of the underlying dynamical system. We demonstrate the simplicity of implementing our model by investigating the coupling of a time-domain acoustic solver with the set of ODEs proposed, all within the graphical interface of Simulink. The x 2 x ̇ nonlinearity plays a crucial role in achieving good agreement in the nonlinear regime, as observed in the FDF, as well as in both the amplitude and frequency of the investigated limit cycle, which exhibits a simple period-one oscillation. However, there is still room for improvement in the nonlinear model if more complex dynamics need to be modeled, thanks to the flexible ODE structure proposed in this work. Novelty and significance A phenomenological model, characterized by a parsimonious set of ordinary differential equations is presented. This model is able to describe the dynamic flame response of typical turbulent, swirl-stabilized flames to acoustic disturbances in their linear and nonlinear regime. The model can be calibrated from broadband time series data or reference frequency domain data. After calibration, the model can be coupled to an acoustic network to predict the occurrence of limit cycles and (if they are present) their frequency as well as amplitude. • We propose a parsimonious system of ODEs for flame response modeling • The system of ODEs describes the flame response of typical turbulent swirled flames • Model parameters are calibrated using frequency domain data or time series data • In the linear regime the model performs comparably to established FIR techniques • In the nonlinear regime, the model qualitatively recovers the FDF • We demonstrate how to couple the nonlinear model to an acoustic network • Using a combination of model and acoustic network, simple limit cycles are recovered • The proposed model can be understood as the superposition of two subsystems • Each subsystem characterizes the flame's response to swirl or axial fluctuations [ABSTRACT FROM AUTHOR]

Biological processes are governed by the expression of proteins, and for some proteins, their level of expression can fluctuate periodically over time (i.e., they oscillate). Many oscillatory proteins (e.g., cell cycle proteins and those from the HES family of transcription factors) are connected in complex ways, often within large networks. This complexity can be elucidated by developing intuitive mathematical models that describe the underlying critical aspects of the relationships between these processes. Here, we provide a mathematical explanation of a recently discovered biological phenomenon: the phasic position of the gene Hes1's oscillatory expression at the beginning of the cell cycle of an individual human breast cancer stem cell can have a predictive value on how long that cell will take to complete a cell cycle. We use a two-component model of coupled oscillators to represent Hes1 and the cell cycle in the same cell with minimal assumptions. Inputting only the initial phase angles, we show that this model is capable of predicting the dynamic mitosis to mitosis behaviour of Hes1 and predicting cell cycle length patterns as found in real-world experimental data. Moreover, we discover that bidirectional coupling between Hes1 and the cell cycle is critical within the system for the data to be reproduced and that nonfixed asymmetry in the interactions between the oscillators is required. The phase dynamics we present here capture the complex interplay between Hes1 and the cell cycle, helping to explain nongenetic cell cycle variability, which has critical implications in cancer treatment contexts. [ABSTRACT FROM AUTHOR]

In this paper we investigate the stability of the null solution of a system of ODEs describing the motion of two coupled damped nonlinear oscillators. We also show that for any solution (x; y) of the system we have limt!+1 x (t) = limt!+1 x_ (t) = limt!+1 y (t) = limt!+1 y_ (t) = 0; for small initial data in the case when the uniqueness of solutions is not guaranteed. Our proofs are mainly based on a generalized form of Schauder-Tychonoff fixed point theorem. The theoretical results are illustrated with numerical simulations. [ABSTRACT FROM AUTHOR]

Background: An earlier oscillator model for the generation of spontaneous otoacoustic emissions (SOAEs) from the lizard ear is extended with a connection of the oscillators to the basilar papilla, to make it possible that these SOAEs can be transported to the tympanic membrane, to be emitted. Material and methods: The generators of spontaneous otoacoustic emissions are modelled as a one-dimensional array of Hopf-resonators. The resonators (or oscillators) are coupled to their neighbours, and to the basilar papilla. The papilla is modelled as a rigid structure, that is flexibly connected to its surroundings. Results: Frequency spectra are given for different sets of coupling parameters, both for nearest neighbour coupling of the oscillators, and for coupling to the papilla, and also after the introduction of irregularities in the damping of the oscillators. Waterfall and density plots show clustering of the oscillators in frequency plateaus, and entrainment of a cluster of oscillators by an externally applied sinusoidal force. All these model outcomes correspond with characteristics of SOAEs emitted by real lizard ears. Conclusions: The present model is a useful extension of an earlier model. Because its characteristics differ from that of a model that is used to describe the generation of SOAEs by mammalian ears, it revives the discussion whether different models are needed for SOAE generation in different animal species [ABSTRACT FROM AUTHOR]

Embryonic development hinges on effective coordination of molecular events across space and time. Waves have recently emerged as constituting an ubiquitous mechanism that ensures rapid spreading of regulatory signals across embryos, as well as reliable control of their patterning, namely, for the emergence of body plan structures. In this article, we review a selection of recent quantitative work on signaling waves and present an overview of the theory of waves. Our aim is to provide a succinct yet comprehensive guiding reference for the theoretical frameworks by which signaling waves can arise in embryos. We start, then, from reaction–diffusion systems, both static and time dependent; move to excitable dynamics; and conclude with systems of coupled oscillators. We link these theoretical models to molecular mechanisms recently elucidated for the control of mitotic waves in early embryos, patterning of the vertebrate body axis, micropattern cultures, and bone regeneration. Our goal is to inspire experimental work that will advance theory in development and connect its predictions to quantitative biological observations. [ABSTRACT FROM AUTHOR]

About two decades ago it was discovered that systems of nonlocally coupled oscillators can exhibit unusual symmetry-breaking patterns composed of coherent and incoherent regions. Since then such patterns, called chimera states, have been the subject of intensive study but mostly in the stationary case when the coarse-grained system dynamics remains unchanged over time. Nonstationary coherence–incoherence patterns, in particular periodically breathing chimera states, were also reported, however not investigated systematically because of their complexity. In this paper we suggest a semi-analytic solution to the above problem providing a mathematical framework for the analysis of breathing chimera states in a ring of nonlocally coupled phase oscillators. Our approach relies on the consideration of an integro-differential equation describing the long-term coarse-grained dynamics of the oscillator system. For this equation we specify a class of solutions relevant to breathing chimera states. We derive a self-consistency equation for these solutions and carry out their stability analysis. We show that our approach correctly predicts macroscopic features of breathing chimera states. Moreover, we point out its potential application to other models which can be studied using the Ott–Antonsen reduction technique. [ABSTRACT FROM AUTHOR]

In weakly coupled neural oscillator networks describing brain dynamics, the coupling delay is often distributed. We present a theoretical framework to calculate the phase response curve of distributed-delay induced limit cycles with infinite-dimensional phase space. Extending previous works, in which non-delayed or discrete-delay systems were investigated, we develop analytical results for phase response curves of oscillatory systems with distributed delay using Gaussian and log-normal delay distributions. We determine the scalar product and normalization condition for the linearized adjoint of the system required for the calculation of the phase response curve. As a paradigmatic example, we apply our technique to the Wilson–Cowan oscillator model of excitatory and inhibitory neuronal populations under the two delay distributions. We calculate and compare the phase response curves for the Gaussian and log-normal delay distributions. The phase response curves obtained from our adjoint calculations match those compiled by the direct perturbation method, thereby proving that the theory of weakly coupled oscillators can be applied successfully for distributed-delay-induced limit cycles. We further use the obtained phase response curves to derive phase interaction functions and determine the possible phase locked states of multiple inter-coupled populations to illuminate different synchronization scenarios. In numerical simulations, we show that the coupling delay distribution can impact the stability of the synchronization between inter-coupled gamma-oscillatory networks. [ABSTRACT FROM AUTHOR]

In this work we present a parsimonious set of ordinary differential equations (ODEs) that describes with satisfactory precision the linear and non-linear dynamics of a typical laminar premixed flame in time and frequency domain. The proposed model is characterized by two ODEs of second-order that can be interpreted as two coupled mass-spring-damper oscillators with a symmetric, nonlinear damping term. This non-linear term is identified as function of the rate of displacement following x 2 x ˙ . The model requires only four constants to be calibrated. This is achieved by carrying out an optimization procedure on one input and one output broadband signal obtained from high-fidelity numerical simulations (CFD). Note that the Transfer Function (FTF) or describing function (FDF) of the flame under investigation are not known a-priori, and therefore not used in the optimization procedure. Once the model is trained on CFD input and output time series, it is capable of recovering with quantitative accuracy the impulse response of the laminar flame under investigation and, hence, the corresponding frequency response (FTF). If fed with harmonic signals of different frequency and amplitude, the trained model is capable of retrieving with qualitative precision the flame describing function (FDF) of the studied flame. We show that the non-linear term x 2 x ˙ is essential for capturing the gain saturation for high amplitudes of the input signal. All results are validated against CFD data. [ABSTRACT FROM AUTHOR]

Thermoacoustic instabilities in can-annular combus-tors of stationary gas turbines lead to unstable Bloch modes which appear as rotating acoustic pressure waves along the turbine annulus. The multiscale, multiphysical nature of the full problem makes a detailed analysis challenging. In this work, we derive a low-order, coupled oscillators model of an idealized can-annular combustor. The unimodal projection of the Helmholtz equation for the can acoustics is combined with the Rayleigh conductivity, which describes the aeroacoustic coupling between neighbouring cans. Using a Bloch-wave ansatz, the resulting system is reduced to a single equation for the frequency spectrum. A linear stability analysis is then performed to study the perturbation of the spectrum by the can-to-can interaction. It is observed that the acoustic coupling can suppress or amplify thermoacoustic instabilities, raising the potential for instabilities in nominally stable systems. [ABSTRACT FROM AUTHOR]

Measurements of interaction intensity are generally achieved by observing responses to perturbations. In biological and chemical systems, external stimuli tend to deteriorate their inherent nature, and thus, it is necessary to develop noninvasive inference methods. In this paper, we propose theoretical methods to infer coupling strength and noise intensity simultaneously in two well-synchronized noisy oscillators through observations of spontaneously fluctuating events such as neural spikes. A phase oscillator model is applied to derive formulae relating each of the parameters to spike time statistics. Using these formulae, each parameter is inferred from a specific set of statistics. We verify these methods using the FitzHugh-Nagumo model as well as the phase model. Our methods do not require external perturbations and thus can be applied to various experimental systems. [ABSTRACT FROM AUTHOR]

In Fernández-García and Vidal [Physica D401 (2020) 132129], the authors have analyzed the synchronization features between two identical 3D slow-fast oscillators, symmetrically coupled, and built as an extension of the FitzHugh—Nagumo dynamics generating Mixed-Mode Oscillations. The third variable in each oscillator aims at representing the time-varying intracellular calcium concentration in neurons. The global model is therefore six-dimensional with two fast variables and four slow variables with strong symmetry properties. In the present article, we consider an extension of this model in two different directions. First, we consider heterogeneity among cells and analyze the coupling of two oscillators with different values for one parameter which tunes the intrinsic frequency of the output. We therefore identify new patterns of antiphasic synchronization, with non trivial signatures and that exhibit a Devil's Staircase phenomenon in signature transitions when varying the coupling gain parameter value. Second, we introduce a network of N cells divided into two clusters: the coupling between neurons in each cluster is excitatory, while the coupling between the two clusters is inhibitory. Such system aims at modelling the interactions between neurons tending to synchronization in each of two subpopulations that inhibit each other, like ipsi- and contra-lateral motoneurons assemblies. To perform the numerical simulations in this case when N is large, as an initial step towards the network analysis, we consider Reduced Order Models in order to save computational costs. We present the numerical reduction results in a network of 100 cells. For the sake of validation of the numerical reduction method, we both compare the outputs and CPU times obtained with the original and the reduced models in different cases of network coupling structures. [ABSTRACT FROM AUTHOR]

Summary: Model frameworks, based on Floquet theory, have been shown to produce effective tools for accurately predicting phase‐noise response of single (free‐running) oscillator systems. This method of approach, referred to herein as macro‐modeling, has been discussed in several highly influential papers and now constitutes an established branch of modern circuit theory. The increased application of, for example, injection‐locked oscillators and oscillator arrays in modern communication systems has subsequently exposed the demand for similar rigorous analysis tools aimed at coupled oscillating systems. This paper presents a novel solution in terms of a macro‐model characterizing the phase‐response of synchronized coupled oscillator circuits and systems perturbed by weak noise sources. The framework is generalized and hence applicable to all circuit configurations and coupling topologies generating a synchronized steady‐state. It advances and replaces the phenomenological descriptions currently found in the published literature pertaining to this topic and, as such, represents a significant breakthrough w.r.t. coupled oscillator noise modeling. The proposed model is readily implemented numerically using standard routines. [ABSTRACT FROM AUTHOR]

Phase reduction is a well-established technique used to analyze the timing of oscillations in response to weak external inputs. In the preceding decades, a wide variety of results have been obtained for weakly perturbed oscillators that place restrictive limits on the magnitude of the inputs or on the magnitude of the time derivatives of the inputs. By contrast, no general reduction techniques currently exist to analyze oscillatory dynamics in response to arbitrary, large magnitude inputs, and comparatively very little is understood about these strongly perturbed limit cycle oscillators. In this work, the theory of isostable reduction is leveraged to develop an adaptive phase-amplitude transformation that does not place any restrictions on the allowable input. Additionally, provided some of the Floquet multipliers of the underlying periodic orbits are near zero, the proposed method yields a reduction in dimension comparable to that of other phase-amplitude reduction frameworks. Numerical illustrations show that the proposed method accurately reflects synchronization and entrainment of coupled oscillators in regimes where a variety of other phase-amplitude reductions fail. [ABSTRACT FROM AUTHOR]

Many animals benefit from synchronizing their daily activities with conspecifics. In this hybrid paper, we first review recent literature supporting and extending earlier evidence for a lack of clear relationship between the level of sociality and social entrainment of circadian rhythms. Social entrainment is specifically potent in social animals that live in constant environments in which some or all individuals do not experience the ambient day-night cycles. We next focus on highly social honeybees in which there is good evidence that social cues entrain the circadian clocks of nest bees and can override the influence of conflicting light-dark cycles. The current understanding of social synchronization in honeybees is consistent with self-organization models in which surrogates of forager activity, such as substrate-borne vibrations and colony volatiles, entrain the circadian clocks of bees dwelling in the dark cavity of the nest. Finally, we present original findings showing that social synchronization is effective even in an array of individually caged callow bees placed on the same substrate and is improved for bees in connected cages. These findings reveal remarkable sensitivity to social time-giving cues and show that bees with attenuated rhythms (weak oscillators) can nevertheless be socially synchronized to a common phase of activity. This article is part of the theme issue 'Synchrony and rhythm interaction: from the brain to behavioural ecology'. [ABSTRACT FROM AUTHOR]

Multiplex networks are networks composed of multiple layers such that the number of nodes in all layers is the same and the adjacency matrices between the layers are diagonal. We consider the special class of multiplex networks where the adjacency matrices for each layer are simultaneously triagonalizable. For such networks, we derive the relation between the spectrum of the multiplex network and the eigenvalues of the individual layers. As an application, we propose a generalized master stability approach that allows for a simplified, low-dimensional description of the stability of synchronized solutions in multiplex networks. We illustrate our result with a duplex network of FitzHugh--Nagumo oscillators. In particular, we show how interlayer interaction can lead to stabilization or destabilization of the synchronous state. Finally, we give explicit conditions for the stability of synchronous solutions in duplex networks of linear diffusive systems. [ABSTRACT FROM AUTHOR]

We report on a stripe-core mixed spiral chimera in a system of nonlocally coupled phase oscillators, located on the spherical surface, where the spiral wave consisting of phase-locked oscillators is separated by a stripe-type region of incoherent oscillators into two parts. We analyze the existence and stability of the stripe-core mixed spiral chimera state rigorously, on the basis of the Ott–Antonsen reduction theory. The stability diagram for the stationary states including the spiral chimeras as well as incoherent state is presented. Our stability analysis reveals that the stripe-core mixed spiral chimera state emerges as a unique attractor and loses its stability via the Hopf bifurcation. We verify our theoretical results using direct numerical simulations of the model system. [ABSTRACT FROM AUTHOR]

Motion-induced oscillations of the flexible single link and its payload at the tip have negative impact on the anticipated performance of the flexible manipulators and thus should be suppressed to achieve tip positioning accuracy and high-speed operation. Because of the structural flexibility, the dynamics of the flexible manipulator can be described by coupled Duffing oscillators when considering the inherent structural nonlinearity of the flexible link into the dynamic modeling. However, little research has been focused on addressing the dynamic coupling issue in the nonlinear modeling of flexible-link manipulators using coupled Duffing oscillators. This article presents coupled Duffing oscillators for the nonlinear modeling of flexible single-link manipulators and then proposes a control method for suppressing the nonlinear vibrations of the coupled Duffing oscillators. Simulated and experimental results obtained from a flexible single-link manipulator test bench are in good agreement with the proposed nonlinear modeling and also demonstrate the effectiveness of the proposed control techniques for vibration suppression of the flexible manipulator. [ABSTRACT FROM AUTHOR]