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The research of collective dynamics and cooperation mechanisms among coupled particles in various topological structures is highly significant across many fields. This study proposes a coupled system consisting of oscillators with frequency fluctuation under a framework of general network. We initially perform a theoretical analysis of system synchronization, from which we derive the first-order and second-order asymptotic stability conditions of the mean field. Under the first-order asymptotic stability condition, we derive the system stationary-state response and obtain the output amplitude amplification (OAA) to unveil the collective behaviors of coupled system. It is shown that there exist rich generalized stochastic resonance (GSR) phenomena. Through numerical simulations within scale-free complex networks, we validate the analytical results regarding collective behaviors. With the introduction of numerical definitions of synchronization probability and mean first synchronization time, we analyze the impact of different parameters on system synchronization. Our findings indicate that both synchronization probability and mean first synchronization time exhibit non-monotonous phenomena varying with network heterogeneity reflected in the power-law exponent. Moreover, it is also observed that the process of system synchronization can be induced by the synergism of noise and coupling in scale-free complex networks with different characteristics and scales. [ABSTRACT FROM AUTHOR]

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]

Noise can induce nontrivial behaviors in systems with multiple timescales. The dynamical reduction of fast-slow systems remains a challenging and intriguing problem. Self-induced stochastic resonance is a noise-induced coherent phenomenon that noise is introduced to the fast dynamics. By applying the embedding phase reduction approach, the synchronization behaviors of two non-identically coupled self-induced stochastic resonance oscillators are investigated. The first passage time distribution on the slow manifolds, which is crucial in the reduced phase equation, is well approximated by the Weibull distribution and is continuously estimated regarding the noise intensity by the polynomial interpolation. The effectiveness of the reduced phase equation is validated by the Monte Carlo simulations. The methods and results may motivate data-based investigations on other multiple timescale systems and experimental findings. [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]

The choice of which state variable to use in coupling network oscillators is a critical factor to reach synchronization. Nonetheless, few works have developed procedures to aid in such a choice, and in many studies, the coupling variable is chosen in an ad hoc fashion without any justification. To avoid a such situation, in this work, a procedure based on the Master Stability Function (MSF) is developed to rank the state variables according to their coupling ability to reach a Complete Synchronization (CS) in networks of identical oscillators. In some cases, the only information needed is the oscillator dynamics, in other cases, the ranking of coupling variables also requires information about the network topology. The developed criteria are related to features such as coupling strength and the speed with which disturbances are rejected. The results are validated via Monte Carlo simulation of the networks. [ABSTRACT FROM AUTHOR]

This study explores the dynamics of sequentially arranged coupled oscillator loops, focusing on inducing steadily rotating phase waves. Notably, when interconnected loops of differing sizes are analyzed, the resulting phase waves exhibit a synchronization pattern, either in-phase or out-of-phase, with pulse counts proportional to their size ratios. This synchronization is further investigated through the introduction of subharmonic injection locking, achieved by applying an external frequency signal matching one of the pulses within the rotating phase waves. Our research delineates the core characteristics of phase waves across oscillator loops of varying sizes, offering a novel approach to generate multiple rotary pulses and amplify the subharmonic response by fine-tuning the oscillators' bias voltage. This paper sheds light on the underlying mechanisms of phase wave synchronization and subharmonic injection locking, with potential applications in enhancing the functionality of coupled oscillator systems. [ABSTRACT FROM AUTHOR]

This paper presents the study of the opposition to the synchronization of bistable chaotic oscillator systems in basic motif configurations. The following configurations were analyzed: Driver-response oscillator systems coupling, two driver oscillator systems to one response oscillator, and a three-oscillator systems ring unidirectional configuration. The study was conducted using the differential equations representing the piecewise linear Rössler-like electronic circuits; the initial conditions were changed to achieve a bistable characteristic Homoclinic H-type or Rössler R-type attractor. Analyzing a sweep of the initial conditions, the basin attractor was obtained. It can be observed that each system has a preferred Homoclinic chaotic attractor with any perturbation or change in initial conditions. A similarity analysis based on the coupling factor was also performed and found that the system has a preferentially Homoclinic chaotic attractor.

The investigated parametrically coupled electromechanical structure is composed of a mechanical Duffing oscillator whose mass sits on a moving belt surface. The driving electrical network is a van der Pol oscillator whose aim is to actuate the attached DC motor to provide some rotatry unbalances and parametric coupling in the vibrating structure. The coupled oscillator is applied to energy harvesting and overcomes the limitation of low energy generation associated with a single oscillator of this kind. The system was solved analytically and validated by numerical methods. The global dynamics of the structure were investigated, and nonlinear phenomena such as Neimark–Sacker bifurcation, discontinuity-induced bifurcation, grazing–sliding, and bifurcation to multiple tori were identified. These nonlinear behaviors affect the harvested energy at bifurcation points, resulting in jumps from one energy level to another. In addition to harnessing the highest energy under hard parametric coupling, the coupling ensures that higher and more useful energy is harvested over a wider range of belt speeds. Finally, the qualitative validation of the numerical concept by experimental setup verifies the workings of the model. [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]

Physiological networks are usually made of a large number of biological oscillators evolving on a multitude of different timescales. Phase oscillators are particularly useful in the modelling of the synchronization dynamics of such systems. If the coupling is strong enough compared to the heterogeneity of the internal parameters, synchronized states might emerge where phase oscillators start to behave coherently. Here, we focus on the case where synchronized oscillators are divided into a fast and a slow component so that the two subsets evolve on separated timescales. We assess the resilience of the slow component by, first, reducing the dynamics of the fast one using Mori-Zwanzig formalism. Second, we evaluate the variance of the phase deviations when the oscillators in the two components are subject to noise with possibly distinct correlation times. From the general expression for the variance, we consider specific network structures and show how the noise transmission between the fast and slow components is affected. Interestingly, we find that oscillators that are among the most robust when there is only a single timescale, might become the most vulnerable when the system undergoes a timescale separation. We also find that layered networks seem to be insensitive to such timescale separations. [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]

We discuss an analog simulation of the Kuramoto model of coupled phase oscillators that can be constructed in the laboratory by coupling a set of light-emitting diodes (LEDs) through a microcontroller. The components of the circuit, namely a WS2812B LED strip, Arduino UNO R3, and a standard power supply, are inexpensive and can be sourced locally. By increasing the strength of the coupling, the ensemble of LEDs can be made to flash in synchrony, and the extent of synchronization can be quantified and measured. [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]

Coordinated fluctuations in female reproductive physiology and thermoregulatory output have been reported for over a century. These changes occur rhythmically at the hourly (ultradian), daily (circadian), and multi-day (ovulatory) timescales, are critical for reproductive function, and have led to the use of temperature patterns as a proxy for female reproductive state. The mechanisms underlying coupling between reproductive and thermoregulatory systems are not fully established, hindering the expansion of inferences that body temperature can provide about female reproductive status. At present, numerous digital tools rely on temperature to infer the timing of ovulation and additional applications (e.g., monitoring ovulatory irregularities and progression of puberty, pregnancy, and menopause are developed based on the assumption that reproductive-thermoregulatory coupling occurs across timescales and life stages. However, without clear understanding of the mechanisms and degree of coupling among the neural substrates regulating temperature and the reproductive axis, whether such approaches will bear fruit in particular domains is uncertain. In this overview, we present evidence supporting broad coupling among the central circuits governing reproduction, thermoregulation, and broader systemic physiology, focusing on timing at ultradian frequencies. Future work characterizing the dynamics of reproductive-thermoregulatory coupling across the lifespan, and of conditions that may decouple these circuits (e.g., circadian disruption, metabolic disease) and compromise female reproductive health, will aid in the development of strategies for early detection of reproductive irregularities and monitoring the efficacy of fertility treatments.

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]

Synchronous dynamics play a pivotal role in various cognitive processes. Previous studies extensively investigate noise-induced synchrony in coupled neural oscillators, with a focus on scenarios featuring uniform noise and equal coupling strengths between neurons. However, real-world or experimental settings frequently exhibit heterogeneity, including deviations from uniformity in coupling and noise patterns. This study investigates noise-induced synchrony in a pair of coupled excitable neurons operating in a heterogeneous environment, where both noise intensity and coupling strength can vary independently. Each neuron is an excitable oscillator, represented by the normal form of Hopf bifurcation (HB). In the absence of stimulus, these neurons remain quiescent but can be triggered by perturbations, such as noise. Typically, noise and coupling exert opposing influences on neural dynamics, with noise diminishing coherence and coupling promoting synchrony. Our results illustrate the ability of asymmetric noise to induce synchronization in such coupled neural oscillators, with synchronization becoming increasingly pronounced as the system approaches the excitation threshold (i.e., HB). Additionally, we find that uneven coupling strengths and noise asymmetries are factors that can promote in-phase synchrony. Notably, we identify an optimal synchronization state when the absolute difference in coupling strengths ismaximized, regardless of the specific coupling strengths chosen. Furthermore, we establish a robust relationship between coupling asymmetry and the noise intensity required to maximize synchronization. Specifically, when one oscillator (receiver neuron) receives a strong input from the other oscillator (source neuron) and the source neuron receives significantly weaker or no input from the receiver neuron, synchrony is maximized when the noise applied to the receiver neuron is much weaker than that applied to the source neuron. These findings reveal the significant connection between uneven coupling and asymmetric noise in coupled neuronal oscillators, shedding light on the enhanced propensity for in-phase synchronization in two-neuron motifs with one-way connections compared to those with two-way connections. This research contributes to a deeper understanding of the functional roles of network motifs that may serve within neuronal dynamics. [ABSTRACT FROM AUTHOR]

Electro-mechanical systems are key elements in engineering. They are designed to convert electrical signals and power into mechanical motion and vice-versa. As the number of networked systems grows, the corresponding mathematical models become more and more complex, and novel sophisticated techniques for their analysis and design are required. We present a novel methodology for the analysis and design of electro-mechanical systems subject to random external inputs. The method is based on the joint application of a model order reduction technique, by which the original electro-mechanical variables are projected onto a lower dimensional space, and of a stochastic averaging technique, which allows the determination of the stationary probability distribution of the system mechanical energy. The probability distribution can be exploited to assess the system performance and for system optimization and design. As examples of application, we apply the method to power factor correction for the optimization of a vibration energy harvester, and to analyse a system composed by two coupled electro-mechanical resonators for sensing applications. [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]

Physiological networks are usually made of a large number of biological oscillators evolving on a multitude of different timescales. Phase oscillators are particularly useful in the modelling of the synchronization dynamics of such systems. If the coupling is strong enough compared to the heterogeneity of the internal parameters, synchronized states might emerge where phase oscillators start to behave coherently. Here, we focus on the case where synchronized oscillators are divided into a fast and a slow component so that the two subsets evolve on separated timescales. We assess the resilience of the slow component by, first, reducing the dynamics of the fast one using Mori-Zwanzig formalism. Second, we evaluate the variance of the phase deviations when the oscillators in the two components are subject to noise with possibly distinct correlation times. From the general expression for the variance, we consider specific network structures and show how the noise transmission between the fast and slow components is affected. Interestingly, we find that oscillators that are among the most robust when there is only a single timescale, might become the most vulnerable when the system undergoes a timescale separation. We also find that layered networks seem to be insensitive to such timescale separations.

This article compares two SiGe Colpitts quadrature voltage-controlled oscillators (QVCO) with different coupling techniques in the low E-Band, intended to be used as signal sources for push-push frequency doublers. The first QVCO is based on a cross-coupled tail-current topology, while the second is based on a fundamental active coupling network. The cross-coupled QVCO has a center frequency of 64.3 GHz and a bandwidth of 2.5 GHz. This circuit realization provides up to 12.2 dBm output power per channel and has a power consumption of 385 mW, resulting in a dc-to-RF efficiency of 8.6%. The phase noise of this oscillator at 1 MHz offset frequency is as low as −105 dBc/Hz. The fundamentally coupled QVCO has a center frequency of 67 GHz with a bandwidth of 3.9 GHz. It provides 13.1 dBm output power per channel while consuming 410 mW of power, resulting in a dc-to-RF efficiency of 9.9%. The oscillator's phase noise at 1 MHz offset frequency is as low as −105.2 dBc/Hz. In addition to the presented circuits, this article introduces a method to measure the relative phase error of quadrature signals utilizing a vector network analyzer. This method is verified with measurements of the developed QVCOs.

A ring of unidirectionally coupled phase oscillators is simple and easy to implement but not suitable for application to central pattern generators (CPGs) owing to the presence of coexisting stable equilibria corresponding to a gait and useless pattern. In this study, we propose a novel approach to applying a ring of unidirectionally coupled phase oscillators to CPGs by incorporating additional circuitry that alters state transition timing. This circuitry comprises a linear-feedback shift register and comparator. Our proposed model successfully generated typical hexapod gait patterns, such as wave and tripod gait patterns, as well as transition patterns between them. The projected Poincarè map was numerically derived to reveal that the proposed model possesses a unique stable equilibrium corresponding to these desired patterns. Furthermore, we implemented the proposed model on a field-programmable gate array (FPGA) to experimentally validate its effectiveness in generating gaits for a hexapod robot. Finally, the proposed model is demonstrated to require fewer FPGA resources compared with conventional and state-of-the-art CPG models.

Biology is suffused with rhythmic behaviour, and interacting biological oscillators often synchronize their rhythms with one another. Colonies of some ant species are able to synchronize their activity to fall into coherent bursts, but models of this phenomenon have neglected the potential effects of intrinsic noise and interspecific differences in individual-level behaviour. We investigated the individual and collective activity patterns of two Leptothorax ant species. We show that in one species (Leptothorax sp. W), ants converge onto rhythmic cycles of synchronized collective activity with a period of about 20 min. A second species (Leptothorax crassipilis) exhibits more complex collective dynamics, where dominant collective cycle periods range from 16 min to 2.8 h. Recordings that last 35 h reveal that, in both species, the same colony can exhibit multiple oscillation frequencies. We observe that workers of both species can be stimulated by nest-mates to become active after a refractory resting period, but the durations of refractory periods differ between the species and can be highly variable. We model the emergence of synchronized rhythms using an agent-based model informed by our empirical data. This simple model successfully generates synchronized group oscillations despite the addition of noise to ants' refractory periods. We also find that adding noise reduces the likelihood that the model will spontaneously switch between distinct collective cycle frequencies.

Synchronous dynamics play a pivotal role in various cognitive processes. Previous studies extensively investigate noise-induced synchrony in coupled neural oscillators, with a focus on scenarios featuring uniform noise and equal coupling strengths between neurons. However, real-world or experimental settings frequently exhibit heterogeneity, including deviations from uniformity in coupling and noise patterns. This study investigates noise-induced synchrony in a pair of coupled excitable neurons operating in a heterogeneous environment, where both noise intensity and coupling strength can vary independently. Each neuron is an excitable oscillator, represented by the normal form of Hopf bifurcation (HB). In the absence of stimulus, these neurons remain quiescent but can be triggered by perturbations, such as noise. Typically, noise and coupling exert opposing influences on neural dynamics, with noise diminishing coherence and coupling promoting synchrony. Our results illustrate the ability of asymmetric noise to induce synchronization in such coupled neural oscillators, with synchronization becoming increasingly pronounced as the system approaches the excitation threshold (i.e., HB). Additionally, we find that uneven coupling strengths and noise asymmetries are factors that can promote in-phase synchrony. Notably, we identify an optimal synchronization state when the absolute difference in coupling strengths is maximized, regardless of the specific coupling strengths chosen. Furthermore, we establish a robust relationship between coupling asymmetry and the noise intensity required to maximize synchronization. Specifically, when one oscillator (receiver neuron) receives a strong input from the other oscillator (source neuron) and the source neuron receives significantly weaker or no input from the receiver neuron, synchrony is maximized when the noise applied to the receiver neuron is much weaker than that applied to the source neuron. These findings reveal the significant connection between uneven coupling and asymmetric noise in coupled neuronal oscillators, shedding light on the enhanced propensity for in-phase synchronization in two-neuron motifs with one-way connections compared to those with two-way connections. This research contributes to a deeper understanding of the functional roles of network motifs that may serve within neuronal dynamics.

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]

Environmental rhythms such as the daily light-dark cycle selected for endogenous clocks. These clocks predict regular environmental changes and provide the basis for well-timed adaptive homeostasis in physiology and behavior of organisms. Endogenous clocks are oscillators that are based on positive feedforward and negative feedback loops. They generate stable rhythms even under constant conditions. Since even weak interactions between oscillators allow for autonomous synchronization, coupling/synchronization of oscillators provides the basis of self-organized physiological timing. Amongst the most thoroughly researched clocks are the endogenous circadian clock neurons in mammals and insects. They comprise nuclear clockworks of transcriptional/translational feedback loops (TTFL) that generate ~24 h rhythms in clock gene expression entrained to the environmental day-night cycle. It is generally assumed that this TTFL clockwork drives all circadian oscillations within and between clock cells, being the basis of any circadian rhythm in physiology and behavior of organisms. Instead of the current gene-based hierarchical clock model we provide here a systems view of timing. We suggest that a coupled system of autonomous TTFL and posttranslational feedback loop (PTFL) oscillators/clocks that run at multiple timescales governs adaptive, dynamic homeostasis of physiology and behavior. We focus on mammalian and insect neurons as endogenous oscillators at multiple timescales. We suggest that neuronal plasma membrane-associated signalosomes constitute specific autonomous PTFL clocks that generate localized but interlinked oscillations of membrane potential and intracellular messengers with specific endogenous frequencies. In each clock neuron multiscale interactions of TTFL and PTFL oscillators/clocks form a temporally structured oscillatory network with a common complex frequency-band comprising superimposed multiscale oscillations. Coupling between oscillator/clock neurons provides the next level of complexity of an oscillatory network. This systemic dynamic network of molecular and cellular oscillators/clocks is suggested to form the basis of any physiological homeostasis that cycles through dynamic homeostatic setpoints with a characteristic frequency-band as hallmark. We propose that mechanisms of homeostatic plasticity maintain the stability of these dynamic setpoints, whereas Hebbian plasticity enables switching between setpoints via coupling factors, like biogenic amines and/or neuropeptides. They reprogram the network to a new common frequency, a new dynamic setpoint. Our novel hypothesis is up for experimental challenge. [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]

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 $+\infty$, provided that suitable conditions are satisfied. Our theoretical results are illustrated with numerical simulations.

Environmental rhythms such as the daily light-dark cycle selected for endogenous clocks. These clocks predict regular environmental changes and provide the basis for well-timed adaptive homeostasis in physiology and behavior of organisms. Endogenous clocks are oscillators that are based on positive feedforward and negative feedback loops. They generate stable rhythms even under constant conditions. Since even weak interactions between oscillators allow for autonomous synchronization, coupling/synchronization of oscillators provides the basis of self-organized physiological timing. Amongst the most thoroughly researched clocks are the endogenous circadian clock neurons in mammals and insects. They comprise nuclear clockworks of transcriptional/translational feedback loops (TTFL) that generate ∼24 h rhythms in clock gene expression entrained to the environmental day-night cycle. It is generally assumed that this TTFL clockwork drives all circadian oscillations within and between clock cells, being the basis of any circadian rhythm in physiology and behavior of organisms. Instead of the current gene-based hierarchical clock model we provide here a systems view of timing. We suggest that a coupled system of autonomous TTFL and posttranslational feedback loop (PTFL) oscillators/clocks that run at multiple timescales governs adaptive, dynamic homeostasis of physiology and behavior. We focus on mammalian and insect neurons as endogenous oscillators at multiple timescales. We suggest that neuronal plasma membrane-associated signalosomes constitute specific autonomous PTFL clocks that generate localized but interlinked oscillations of membrane potential and intracellular messengers with specific endogenous frequencies. In each clock neuron multiscale interactions of TTFL and PTFL oscillators/clocks form a temporally structured oscillatory network with a common complex frequency-band comprising superimposed multiscale oscillations. Coupling between oscillator/clock neurons provides the next level of complexity of an oscillatory network. This systemic dynamic network of molecular and cellular oscillators/clocks is suggested to form the basis of any physiological homeostasis that cycles through dynamic homeostatic setpoints with a characteristic frequency-band as hallmark. We propose that mechanisms of homeostatic plasticity maintain the stability of these dynamic setpoints, whereas Hebbian plasticity enables switching between setpoints via coupling factors, like biogenic amines and/or neuropeptides. They reprogram the network to a new common frequency, a new dynamic setpoint. Our novel hypothesis is up for experimental challenge.

Chronobiology experiments often reveal intriguing non-linear phenomena, which require mathematical models and computer simulations for their interpretation. One example is shown here, where the two circadian oscillators located in the eyes of the mollusk Bulla gouldiana were isolated and measured in vitro. By maintaining one eye under control conditions and manipulating the period of the second eye, Page and Nalovic (1992) obtained a diversity of results, including synchronized and desynchronized eyes, associated to weak coupling and period differences. A subset of eye pairs, however, showed increasing phase angle followed by phase jumps. These occur and have been satisfactorily modeled in more complex systems where two zeitgebers play clear entraining roles. However, simulations of a simple model of free-running, two mutually coupled limit-cycle oscillators with unilateral change in oscillator period failed completely to reproduce these phase jumps. Here we explain how phase jumps arise in two-zeitgeber systems and then show the closest but unsatisfying, intermediate model that was fit to the Bulla system.

This work considers the response of an array of oscillators, each with cubic nonlinear stiffness, in the presence of global reactive and dissipative coupling. The interplay between the excitation, global coupling, and nonlinearity gives rise to steady-state solutions for the population in which the response of each individual element depends on that of every other component. The present analysis continues the work presented in Borra et al. (J Sound Vib 393:232–239, 2017. https://doi.org/10.1016/j.jsv.2016.12.021), in which a continuum formulation was introduced to study the steady-state response as the number of oscillators increases. However, that work considered only parameter values and excitation levels for which the equilibrium distribution was unique. In the present work, the individual resonators are excited to sufficient amplitude to allow for multiple coexisting equilibrium population distributions. The method of multiple scales is then applied to the system to describe evolution equations for the amplitude and phase of each resonator. Because of the global nature of the coupling, this leads to an integro-differential equation for the stationary populations. Moreover, the characteristic equation used to determine the stability of these states is also an integral equation and admits both a discrete and continuous spectrum for its eigenvalues. The equilibrium structure of the system is studied as the reactive and dissipative coupling parameters are varied. For specific families of the equilibrium distributions, two-parameter bifurcation sheets can be constructed. These sheets are connected as individual resonators transition between different branches for the corresponding individual resonators. The resulting one-parameter bifurcation curves are then understood in terms of the collections of these identified bifurcation sheets. The analysis is demonstrated for a system of N = 10 coupled resonators with mass detuning and extended results with N = 100 coupled resonators are illustrated. [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]