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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]

In this paper, the moderately long-time behaviour of the correlation functions for two charged coupled harmonic oscillators connected to a common heat bath are analysed in the presence of a magnetic field via the quantum Langevin dynamics. Interestingly, it is seen that long-time correlation functions at T → 0 exhibit a power-law decay with coefficients of the power laws being completely different for the two masses, affecting the overall dynamics of the coupled system. The effect of the bath-induced force on mass m 1 mediated by the interaction of m 2 with the common heat bath is studied and the results are highlighted in the presence of an external magnetic field. It is shown that the effect of cyclotron frequency increases the magnitudes of correlation functions at an instant of time by lowering the rate of temporal decay in the presence of a uniform magnetic field. The results in the absence of magnetic field are also presented, which are extremely important for investigating the movements of atoms in a protein molecule at low temperature. [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]

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]

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]

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]

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]

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]

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]

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]

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]

Whereas topological symmetries have been recognized as crucially important to the exploration of synchronization patterns in complex networks of coupled dynamical oscillators, the identification of the symmetries in large-size complex networks remains as a challenge. Additionally, even though the topological symmetries of a complex network are known, it is still not clear how the system dynamics is transited among different synchronization patterns with respect to the coupling strength of the oscillators. We propose here the framework of eigenvector-based analysis to identify the synchronization patterns in the general complex networks and, incorporating the conventional method of eigenvalue-based analysis, investigate the emergence and transition of the cluster synchronization states. We are able to argue and demonstrate that, without a prior knowledge of the network symmetries, the method is able to predict not only all the cluster synchronization states observable in the network, but also the critical couplings where the states become stable and the sequence of these states in the process of synchronization transition. The efficacy and generality of the proposed method are verified by different network models of coupled chaotic oscillators, including artificial networks of perfect symmetries and empirical networks of non-perfect symmetries. The new framework paves a way to the investigation of synchronization patterns in large-size, general complex networks. [ABSTRACT FROM AUTHOR]

We consider the escape of an initially excited dynamical system with two degrees of freedom from a potential well. Three different benchmark well potentials with different topologies are explored. The main challenge is to reveal the basic mechanisms that govern the escape in different regions of the parametric space and to construct appropriate asymptotic approximations for the analytic treatment of these mechanisms. In this study, numerical and analytical tools are used to classify and map the different escape mechanisms for various initial conditions and to offer the analytic criteria predicting the system's behavior for those cases. [ABSTRACT FROM AUTHOR]

The breathing process possesses a complex variability caused in part by the respiratory central pattern generator in the brainstem; however, it also arises from chemical and mechanical feedback control loops, network reorganization and network sharing with nonrespiratory motor acts, as well as inputs from cortical and subcortical systems. The notion that respiratory fluctuations contain hidden information has prompted scientists to decipher respiratory signals to better understand the fundamental mechanisms of respiratory pattern generation, interactions with emotion, influences on the cortical neuronal networks associated with cognition, and changes in variability in healthy and disease-carrying individuals. Respiration can be used to express and control emotion. Furthermore, respiration appears to organize brain-wide network oscillations via cross-frequency coupling, optimizing cognitive performance. With the aid of information theory-based techniques and machine learning, the hidden information can be translated into a form usable in clinical practice for diagnosis, emotion recognition, and mental conditioning. [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]

Non-stationary feature information is common in engineering applications, and its signal features are irregular compared to stationary signals. Because of its strong volatility, the traditional signal analysis method is no longer applicable. In some environments, strong background noise makes it difficult to extract feature information. Adaptive cascade stochastic resonance is an effective method to enhance the stationary signal. The weak characteristic signal is enhanced step by step, but it is difficult to extract non-stationary information under strong noise background. Compared with the classical bistable system, the piecewise linear system can overcome the disadvantage of output saturation and has high output signal-to-noise ratio. Therefore, an adaptive cascaded stochastic resonance method is proposed to extract and enhance the non-stationary weak feature information. Firstly, the simulated non-stationary signal of a faulty bearing is preprocessed. The non-stationary feature information is transformed into the stationary feature information by computed order analysis method. Combined with maximum correlation kurtosis deconvolution filtering, the periodic feature of the characteristic signal is highlighted. Then, the adaptive stochastic resonance in a piecewise linear system and variational mode decomposition are applied to enhance characteristic signal and reduce noise interference. The cascaded mechanism is used to filter the interference signal and enhance the characteristic information step by step. Finally, the effectiveness of the method is verified by experimental signals, which can significantly improve the output characteristic amplitude and signal-to-noise ratio. [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]

A method for stabilizing the steady states of the globally coupled oscillators is described. The method is based on deactivating a single oscillator. The deactivation of an oscillator is achieved by strong slowering its dynamics. In electrical circuits, deactivating can by implemented by short-circuiting an oscillator with an external large capacitance. Analytical, numerical, and experimental investigations have been carried out using an array of the mean-field coupled FitzHugh–Nagumo type oscillators. [ABSTRACT FROM AUTHOR]

Coupled oscillators are among the simplest composite quantum systems in which the interplay of entanglement and interaction may be explored. We examine the effects of coupling on the quantum fluctuations of the coordinates and momenta of the oscillators in a single-excitation entangled Bell-like state. We discover that coupling acts as a mechanism for noise transfer between one pair of coordinate and momentum and another. Through this noise transfer mechanism, the uncertainty product is lowered, on average, relatively to its non-coupled level for one pair of coordinate and momentum and it is enhanced for the other pair. This novel mechanism for noise transfer may be explored in precision measurements in entanglement-assisted sensing and metrology. [ABSTRACT FROM AUTHOR]

Ecosystems and their embedded ecological communities are almost always by definition collections of oscillating populations. This is apparent given the qualitative reality that oscillations emerge from consumer-resource interactions, which are the elementary building blocks for ecological communities. It is also likely always the case that oscillatory consumer-resource pairs will be connected to one another via trophic cross-feeding with shared resources or via competitive interactions among resources. Thus, one approach to understanding the dynamics of communities conceptualizes them as collections of oscillators coupled in various arrangements. Here we look to the pioneering work of Kuramoto on coupled oscillators and ask to what extent can his insights and approaches be translated to ecological systems. We explore the four ecologically significant coupling arrangements of the simple case of three oscillator systems with both the Kuramoto model and with the classical Lotka-Volterra equations. Our results show that the six-dimensional Lotka-Volterra systems behave strikingly similarly to that of the corresponding Kuramoto systems across all coupling combinations. This qualitative similarity in the results between these two approaches suggests that a vast literature on coupled oscillators may be relevant in furthering our understanding of ecosystem and community organization. [ABSTRACT FROM AUTHOR]

The theory of acoustic gravity waves (AGW) considers free disturbances of the atmosphere within the framework of a single-frequency approach. In this case, the theory implies the existence of two separate types of waves with different natural frequencies: acoustic and gravitational. In the single-frequency approach, wave fluctuations of density, temperature, and velocity are related to each other through the spectral characteristics of the wave, and these relationships are unchanged. However, satellite observations of AGW parameters cannot always be explained within the framework of a single-frequency approach. This paper presents a two-frequency approach to studying AGWs using the model of two coupled oscillators. It is shown that the perturbed movements of the elementary volume of the medium occur simultaneously at two natural frequencies. In this case, the connections between the wave fluctuations of the parameters are determined by the initial conditions, which can be arbitrary. Solutions in real functions for an isothermal atmosphere are obtained. The conditions under which single-frequency AGWs are obtained from the general two-frequency solution are investigated. The AGW waveforms measured from the satellite for velocities and displacements in single-frequency and dual-frequency modes are numerically simulated. The results of simulating two-frequency AGWs agree with the data of satellite measurements. Two-frequency AGWs are not always implemented at two different frequencies. It is shown that, when the frequencies approach each other, the beat effect occurs and two closely related modes become indistinguishable. At the same wavelength, they have one center frequency and one phase velocity. The main feature of the two-frequency approach to the study of AGW is the expansion of the relationships between the wave parameters of the medium. This makes it possible to achieve satisfactory agreement of the model waveforms with the data of satellite measurements. Thus, the use of a two-frequency AGW treatment opens up new possibilities in the interpretation of experimental data. [ABSTRACT FROM AUTHOR]

The relation between the structural properties of the network and its dynamics is a central question in the analysis of dynamical networks. It is especially relevant for complex networks found in real-world applications. This work presents mathematically rigorous analysis of coupled dynamical systems on power law graphs. Specifically, we study large systems of coupled Kuramoto phase oscillators. In the limit as the size of the network tends to infinity, we derive analytically tractable mean field partial differential equation for the probability density function describing the state of the coupled system. The mean field limit is used to establish an explicit formula for the synchronization threshold for coupled phase oscillators with randomly distributed intrinsic frequencies. Furthermore, we study stable spatial patterns generated by the Kuramoto model with repulsive coupling. In particular, we identify a family of stable steady-state solutions having multiple regions with distinct statistical properties. We call these solutions contrast states. Like chimera states, contrast states exhibit coexisting regions of highly localized (coherent) behavior and highly irregular (incoherent) distribution of phases. We provide a detailed mathematical analysis of contrast states in the KM using the Ott–Antonsen ansatz. The analysis of synchronization and contrast states provides new insights into the role of power law connectivity in shaping dynamics of coupled dynamical systems. In particular, we show that despite sparse connectivity, power law networks possess remarkable synchronizability: the synchronization threshold can be made arbitrarily low by varying the parameter of the power law distribution. [ABSTRACT FROM AUTHOR]

We study the effects brought about by adding frustration to the plastic Kuramoto model of neurons; i.e. we simulate for the first time the plastic Kuramoto–Sakaguchi model, in which the nonlocal coupling matrix for the phases is determined from the spike-timing-dependent plasticity rule. We find that frustration leads to the birhythmicity of the synchronous states. The multistability in the dynamics, with frustration as its crucial element, results in hysteresis, even when the learning rule is symmetric. Fine structure in the hysteresis loop reflects the complexity of the underlying energy landscape. A pacemaker is used to probe this landscape. As the neurons spontaneously self-organise, memory is formed by the configuration of synaptic strengths embodied in such synchronous states. We thus have a simple model of synaptic memory. [ABSTRACT FROM AUTHOR]

Many biological and neural systems can be seen as networks of interacting periodic processes. Importantly, their functionality, i.e., whether these networks can perform their function or not, depends on the emerging collective dynamics of the network. Synchrony of oscillations is one of the most prominent examples of such collective behavior and has been associated both with function and dysfunction. Understanding how network structure and interactions, as well as the microscopic properties of individual units, shape the emerging collective dynamics is critical to find factors that lead to malfunction. However, many biological systems such as the brain consist of a large number of dynamical units. Hence, their analysis has either relied on simplified heuristic models on a coarse scale, or the analysis comes at a huge computational cost. Here we review recently introduced approaches, known as the Ott–Antonsen and Watanabe–Strogatz reductions, allowing one to simplify the analysis by bridging small and large scales. Thus, reduced model equations are obtained that exactly describe the collective dynamics for each subpopulation in the oscillator network via few collective variables only. The resulting equations are next-generation models: Rather than being heuristic, they exactly link microscopic and macroscopic descriptions and therefore accurately capture microscopic properties of the underlying system. At the same time, they are sufficiently simple to analyze without great computational effort. In the last decade, these reduction methods have become instrumental in understanding how network structure and interactions shape the collective dynamics and the emergence of synchrony. We review this progress based on concrete examples and outline possible limitations. Finally, we discuss how linking the reduced models with experimental data can guide the way towards the development of new treatment approaches, for example, for neurological disease. [ABSTRACT FROM AUTHOR]

As is known from multiyear theoretical and experimental studies, acoustic−gravity waves (AGW) determine the dynamics and energy balance of the atmospheres of planets and the Sun to a considerable extent. Linear wave perturbations in the atmosphere can be described with a system of second-order equations for the vertical and horizontal components of the perturbed velocity. It follows from this system that small perturbations in the atmosphere may be considered as oscillations of coupled oscillators with two degrees of freedom. This suggests that it is worth studying more thoroughly the linear acoustic−gravitational wave modes in the atmosphere with well-developed methods of the theory of oscillations. To study small perturbations in the Earth's atmosphere, the methods of the theory of coupled oscillatory systems were used. It has been shown that acoustic−gravity waves in an isothermal atmosphere can be considered as a superposition of oscillations that occur simultaneously at two natural frequencies—acoustic and gravitational. The equations for the natural frequencies of oscillations, as well as for the components of the perturbed velocity under specified initial conditions, were derived. From the analysis of temporal changes in the components of the perturbed velocity, new features in their behavior were found. All solutions are presented with the use of real quantities only. This representation is more convenient for comparison with observational data than the complex one commonly used in the theory of acoustic−gravity waves. The conditions under which the usual single-frequency oscillation mode can be realized in the atmosphere are analyzed. The results of the study can be used to explain some features in space-borne observations of wave perturbations in the Earth's atmosphere that do not fit the framework of the known theoretical concepts. [ABSTRACT FROM AUTHOR]

The dynamics of memristor-based RLC circuit is explored on the effects of magnetic field coupling through a transformer in the circuit. With the increasing strength of magnetic field coupling, the two coupled RLC circuits firstly break the reflectional symmetry by transiting from double-scroll attractors to single-scroll attractors and then break the translational symmetry with two attractors in different sizes and shapes. The two single-scroll attractors are locked in antiphase synchronous states for positive magnetic field coupling while transit to a new kind of dynamic regime, named as central symmetrical antiphase synchronization (CSAS) for negative magnetic field coupling. Moreover, the single-scroll attractors in inphase synchronization or in CSAS may coexist with asynchronous states in some interval of coupling strength. The dynamics of the anti-synchronization and the CSAS of the coupled memristor-based circuits have potential applications in nonvolatile memories, neuromorphic devices to simulate learning, adaptive and spontaneous behaviors. [ABSTRACT FROM AUTHOR]

We investigate how the structure of interactions between coupled oscillators influences the formation of asynchronous patterns in a multilayer network by formulating a simple, general multilayer oscillator model. We demonstrate the analysis of this model in three-oscillator systems, illustrating the role of interactions among oscillators in sustaining differences in both the phase and amplitude of oscillations leading to the formation of asynchronous patterns. Finally, we demonstrate the generalizability of our model's predictions through comparison with a more realistic multilayer model. Overall, our model provides a useful approach for predicting the types of asynchronous patterns that multilayer networks of coupled oscillators which cannot be achieved by the existing methods which focus on characterizing the synchronous state. [ABSTRACT FROM AUTHOR]

Motivated by the dynamics of microscale oscillators with thermo-optical feedback, a simplified third-order model capturing the key features of these oscillators is developed, where each oscillator consists of a displacement variable coupled to a temperature variable. Further, the dynamics of a pair of such oscillators coupled via a linear spring is analyzed. The analytical procedures used are the variational equation method and the two-variable expansion method. It is shown that the analytical results are in agreement with the results of numerical integration. The bifurcation structure of the system is revealed through a bifurcation diagram. [ABSTRACT FROM AUTHOR]

Coupled oscillators can exhibit complex spatiotemporal dynamics. Here, we study the propagation of nonlinear waves into an unstable state in dissipative coupled oscillators. To this, we consider the dissipative Frenkel–Kontorova model, which accounts for a chain of coupled pendulums or Josephson junctions and coupling superconducting quantum interference devices. As a function of the dissipation parameter, the front that links the stable and unstable state is characterized by having a transition from monotonous to non-monotonous profile. In the conservative limit, these traveling nonlinear waves are unstable as a consequence of the energy released in the propagation. Traveling waves into unstable states are peculiar of dissipative coupling systems. When the coupling and the dissipation parameter are increased, the average front speed decreases. Based on an averaging method, we analytically determine the front speed. Numerical simulations show a quite fair agreement with the theoretical predictions. To show that our results are generic, we analyze a chain of coupled logistic equations. This model presents the predicted dynamics, opening the door to investigate a wider class of systems. [ABSTRACT FROM AUTHOR]

We explore the dynamical transitions facilitated by the symmetry breaking dynamical states in a directly coupled limit-cycle oscillators, when the oscillators are interacting through attractive and repulsive couplings. Initially, we show that the transition from out-of-phase synchronized (OPS) state to in-phase synchronized (IPS) state is mediated by the onset of symmetry breaking oscillatory (SBO) state using two coupled systems. Similarly, we find the emergence of symmetry breaking oscillation death (AOD) state when the system transits from oscillation death (OD) state to nontrivial amplitude death (NAD) state. Further, we deduce the analytical stability conditions for all the observed dynamical states, namely the IPS, OPS, NAD, OD, including SBO and AOD states. Additionally, the investigation on symmetry breaking dynamics in a ring of unidirectionally coupled oscillators reveals that the breaking of symmetry leads to the traveling chimera (TC) and symmetry breaking incoherent oscillation death states. We also report that the transient nature of TC state obeys a power-law relation near the stable TC region as a function of the coupling strength. From the results, it is evident that the trade-off between the attractive and the repulsive couplings induces the symmetry breaking dynamics in the entire range of the system frequency. [ABSTRACT FROM AUTHOR]

The conventional practice of establishing the structure of coupling between the elements of complex systems from experimental observations of their oscillations (time series) treated in terms of the Wiener–Granger causality revealed some problems of measurement noise hindering the obtaining of reliable results. In particular, the presence of observational noise can produce the effect of spurious coupling (SC) that leads to the erroneous conclusion that there is mutual (bidirectional) coupling between two elements that are in fact coupled unidirectionally. A quantitative analysis of this phenomenon has been carried out, and recommendations are given how to reduce the probability of error. It is shown that the SC effect is usually manifested only in the presence of strong noise comparable with the level of observed oscillations, although rare situations are possible in which the SC effect takes place at much lower noises. [ABSTRACT FROM AUTHOR]

In the context of the Kuramoto model of coupled oscillators with distributed natural frequencies interacting through a time-delayed mean-field, we derive as a function of the delay exact results for the stability boundary between the incoherent and the synchronized state and the nature in which the latter bifurcates from the former at the critical point. Our results are based on an unstable manifold expansion in the vicinity of the bifurcation, which we apply to both the kinetic equation for the single-oscillator distribution function in the case of a generic frequency distribution and the corresponding Ott–Antonsen (OA)-reduced dynamics in the special case of a Lorentzian distribution. Besides elucidating the effects of delay on the nature of bifurcation, we show that the approach due to Ott and Antonsen, although an ansatz, gives an amplitude dynamics of the unstable modes close to the bifurcation that remarkably coincides with the one derived from the kinetic equation. Further more, quite interestingly and remarkably, we show that close to the bifurcation, the unstable manifold derived from the kinetic equation has the same form as the OA manifold, implying thereby that the OA-ansatz form follows also as a result of the unstable manifold expansion. We illustrate our results by showing how delay can affect dramatically the bifurcation of a bimodal distribution. [ABSTRACT FROM AUTHOR]

Rotating waves are a fascinating feature of a wide array of complex systems, particularly those arising in the study of many chemical and biological processes. With many rigorous mathematical investigations of rotating waves relying on the model exhibiting a continuous Euclidean symmetry, this work is aimed at understanding these nonlinear waves in the absence of such symmetries. Here we will consider a spatially discrete lattice dynamical system of Ginzburg–Landau type and prove the existence of rotating waves in the anti-continuum limit. This result is achieved by providing a link between the work on phase systems stemming from the study of identically coupled oscillators on finite lattices to carefully track the solutions as the size of the lattice grows. It is shown that in the infinite square lattice limit of these phase systems that a rotating wave solution exists, which can be extended to the Ginzburg–Landau system of study here. The results of this work provide a necessary first step in the investigation of rotating waves as solutions to lattice dynamical systems in an effort to understand the dynamics of such solutions outside of the idealized situation where the underlying symmetry of a differential equation can be exploited. [ABSTRACT FROM AUTHOR]

Herein, we characterize the dynamics of traveling pulses developed in tunnel diode (TD) oscillators that form a closed ring via neighbor-to-neighbor coupling by lossy inductors for multiphase oscillation. It is found that the development of rotary traveling pulses having no sink–source pairs is guaranteed by sufficiently slow application of TD bias voltage. The properties of such rotary pulses are investigated, including the bias dependence of their velocity and their bifurcational structure. Based on these properties, we propose a scheme for multiphase oscillation with a significant number of channels. In addition, this paper describes several experimental observations that validate the numerically obtained properties of rotary traveling pulses. [ABSTRACT FROM AUTHOR]

We demonstrate that the Douady–Earle extension can be computed by solving the specific set of ODE's. This system of ODE's has several interpretations in Mathematical Physics, such as Kuramoto model of coupled oscillators or Josephson junction arrays. This method emphasizes the key role of conformal barycenter in some theories of Mathematical Physics. On the other hand, it indicates that variations of Kuramoto model might be used in different problems of computational quasiconformal geometry. The idea can be extended to compute the D–E extension in higher dimensions as well. [ABSTRACT FROM AUTHOR]

In this paper, amplitude death (AD) is investigated theoretically and numerically in N globally delay-coupled fractional-order oscillators. Due to the presence of fractional-order derivative and coupling delay, Laplace transform method has been utilized to obtain the characteristic equations. Then, based on Lyapunov stability, we theoretically get the boundaries and number of death islands. It has been found that with the introduction of the fractional-order derivative, many more death islands emerge, and the oscillation quenching dynamics are facilitated. We find AD only occurs between two critical fractional-order derivatives α c - (lower-bounded value) and α c + (upper-bounded value) which are affected by natural frequency and system size. With the increment of system size, the oscillation quenching dynamics are weakened. The number of death islands is closely geared to the fractional-order derivative and the system size. Furthermore, the results from numerical simulations best confirm the theoretical analyses. [ABSTRACT FROM AUTHOR]

Collective behaviors of coupled oscillators are an important issue in the field of nonlinear dynamics and complex networks, which have attracted much attention in recent years. For some systems satisfying certain symmetries, the dynamics of high-dimensional space can be reduced to that of low-dimensional subspace in terms of the Watanabe-Strogatz theory. In this paper, the dynamics of the transition to synchronization on the coupled star networks are studied, and we identify a two-step phase transition in the system both from the macroscopic and from the microscopic viewpoints. Theoretically, the low-dimensional order-parameter dynamical equation is developed to get analytical insights, and the roles of the coupled hubs in the synchronization process are uncovered. Physically, the two-phase transition steps correspond to different bifurcations in phase space. Theoretical analysis is examined by performing the numerical simulations, and the results and analysis proposed in this paper are helpful to understand the phase transition in general heterogenous networks. [ABSTRACT FROM AUTHOR]

Dynamical patterns in complex networks of coupled oscillators are of both theoretical and practical interest, yet to fully reveal and understand the interplay between pattern emergence and network structure remains to be an open problem. Among the many outstanding issues, a fundamental one is how the network structure affects the stability of dynamical patterns. To address this issue, we focus on the spiral wave patterns and investigate the effects of systematically added random links on their stability and dynamical evolutions. We find that, as the network structure deviates more from the regular topology and thus becomes increasingly more complex, an originally stable spiral wave pattern can disappear but different types of patterns can emerge. In addition, short-distance links added to a small region containing the spiral tip can have a more significant effect on the wave pattern than long-distance connections. As more random links are introduced into the network, distinct pattern transitions can occur, such as the transition of the spiral wave pattern to a global synchronization state, to a chimera-like state, or to a pinned spiral wave. About the transition points, the network dynamics are highly sensitive to small structural perturbations in that the addition of even a single link can change the pattern from one type to another. These findings provide additional insights into the pattern dynamics in complex networks, a problem that is relevant to many physical, chemical, and biological systems. [ABSTRACT FROM AUTHOR]

Amplitude chimera states, representing a spontaneous symmetry breaking of a population of coupled identical oscillators into two distinct clusters with one oscillating in spatial coherent amplitude, while the other displaying oscillations in a spatially incoherent manner, have been observed as a kind of transient dynamics in the process of transition to the in-phase synchronization in coupled limit-cycle oscillators. Here, we obtain a kind of stable amplitude chimera state in the chaotic regime of a system of repulsively coupled Lorenz oscillators. With the increment of the coupling strength, the coupled oscillators transit from spatiotemporal chaos to amplitude chimera states then to coherent oscillation death or chimera death states. Moreover, the number of clusters in amplitude chimera patterns has a power-law dependence on the number of coupled neighbors. The amplitude chimera and the chimera death states coexist at certain coupling strength. Moreover, the amplitude chimera and the amplitude death patterns are related to the initial condition for given coupling strength. Our findings of amplitude chimera states and chimera death states in coupled chaotic system may enrich the knowledge of the symmetry-breaking-induced pattern formation. [ABSTRACT FROM AUTHOR]

To explore the conditions ensuring amplitude death is always a topic of research for understanding the nature of real systems. However, the previous studies mainly focus on the coupling, that is, continuous or time-invariant. Here we would bring the discontinuous (on-off) coupling to the coupled nonidentical Stuart-Landau oscillators and Rössler oscillators, respectively. We show that the domains of amplitude death can be effectively enlarged along two directions of coupling strength and parameter mismatch. Specifically, the range of coupling strength is extended infinitely for the appropriate on-off rate and on-off period although it usually is considered to be bounded to induce the amplitude death for continuous coupling. Moreover, our findings are of great importance and have many potential applications for the research of neuroscience and biology where the interaction between the neurons or cells is usually intermittent or discontinuous. [ABSTRACT FROM AUTHOR]

A mathematical model of an unstable system in the form of inverted coupled pendulums is developed and simulated. Dynamics of such a system is investigated, and the stability zones are identified in the explicit form. The algorithm of stabilization of the pendulums near the vertical position is constructed and verified. [ABSTRACT FROM AUTHOR]

We consider a system of two masses connected by linear springs and in contact with a belt moving at a constant velocity. One of the masses can collide with a fixed rigid obstacle. The contact forces between the masses and the belt are given by Coulomb’s laws. Moreover, when the colliding mass is in contact with the obstacle, we assume that a perfect elastic impact occurs. Several periodic orbits including contact against the fixed obstacle followed by slip and stick phases are obtained in analytical form. [ABSTRACT FROM AUTHOR]

We present cointegration analysis as a method to infer the network structure of a linearly phase coupled oscillating system. By defining a class of oscillating systems with interacting phases, we derive a data generating process where we can specify the coupling structure of a network that resembles biological processes. In particular we study a network of Winfree oscillators, for which we present a statistical analysis of various simulated networks, where we conclude on the coupling structure: the direction of feedback in the phase processes and proportional coupling strength between individual components of the system. We show that we can correctly classify the network structure for such a system by cointegration analysis, for various types of coupling, including uni-/bi-directional and all-to-all coupling. Finally, we analyze a set of EEG recordings and discuss the current applicability of cointegration analysis in the field of neuroscience. [ABSTRACT FROM AUTHOR]

Dynamical synchronous modes in a network of four nearly identical chemical oscillators unidirectionally coupled via inhibitory pulse coupling with time delay τ (when a spike in one oscillator inhibits the next oscillator in the circle after time delay τ), are obtained experimentally. The Belousov-Zhabotinsky reaction is used as a chemical oscillator. The existence of four main modes is confirmed experimentally: in-phase (IP) oscillations; an anti-phase (AP) mode, in which any two neighboring oscillators have a phase shift equal to half of global period T; a walk mode (W), in which oscillators produce consecutive spikes in the direction of the connection with a phase shift between neighboring oscillators equal to T/4; and a walk-reverse mode (WR), when the oscillators produce consecutive spikes (with phase shift T/4), but in the direction opposite the connections (the mode opposite to the W mode). In addition to the main modes, OS modes in which at least one of the four oscillators is suppressed, and '2+1+1' modes in which two neighboring oscillators produce spikes simultaneously and the phases of the third and the fourth oscillators are shifted by T/3 and 2 T/3, respectively, are found. It is shown that the modes found experimentally correspond to those found in simulations. [ABSTRACT FROM AUTHOR]

The responses of electrically coupled neuronal network to external stimulus injected on a single neuron are investigated. Stimulating the largest-degree neuron in the network, it is found that as the intensity of the stimulus increases, the network will be transiting from the resting to firing states and then restoring to the resting state, thereby showing a bounded firing region in the parameter space. Furthermore, it is found that as the coupling strength among the neurons decreases, the firing region is gradually expanded and, at the weak couplings, it could be separated into several disconnected subregions. By a simplified network model, we conduct a detailed analysis on the bifurcation diagram of the network dynamics in the two-dimensional parameter space spanned by stimulating intensity and coupling strength, and, by introducing a new coefficient named effective stimulus, explore the underlying mechanisms for the modified firing region. It is revealed that the coupling strength and stimulating intensity are equally important in evoking the network, but with different mechanisms. Specifically, the effective stimuli are shifted up globally by increasing the stimulating intensity, while are drawn closer by increasing the coupling strength. The dynamical responses of small-world and random complex networks to external stimulus are also investigated, which confirm the generality of the observed phenomena. The findings shed new lights on the collective behaviors of complex neuronal networks and might help our understandings on the recent experimental results. [ABSTRACT FROM AUTHOR]