Self-synchronization of nonlinear oscillations in the presence of fluctuations

A statistical mechanical theory is presented for the self-organization of a macroscopic oscillation with the presence of external fluctuations in a system of Van der Pol oscillators coupled through dissipative interactions. Starting from Langevin equations for the Van der Pol oscillators, the static and dynamic characteristics are studied. The threshold condition is given by the relative size between the fluctuation and the interaction. The transitions between synchronous and asynchronous phases are well discussed by a Landau-type equation. The steady state value of the order parameter and the onset time are compared between the theory and the computer experiments and a good agreement is obtained.     

Analysis on Patterns of Globally Coupled Phase Oscillators with Attractive and Repulsive Interactions

The Hong-Strogatz (HS) model of globally coupled phase oscillators with attractive and repulsive interactions reflects the fact that each individual (oscillator) has its own attitude (attractive or repulsive) to the same environment (mean field). Previous studies on HS model focused mainly on the stable states on Ott-Antonsen (OA) manifold. In this paper, the eigenvalues of the Jacobi matrix of each fixed point in HS model are explicitly derived, with the aim to understand the local dynamics around each fixed point. Phase transitions are described according to relative population and coupling strength. Besides, the dynamics off OA manifold is studied.     

Analysis on Patterns of Globally Coupled Phase Oscillators with Attractive and Repulsive Interactions

The Hong-Strogatz(HS) model of globally coupled phase oscillators with attractive and repulsive interactions reflects the fact that each individual(oscillator) has its own attitude(attractive or repulsive) to the same environment(mean Seld).Previous studies on HS model focused mainly on the stable states on Ott-Antonsen(OA)manifold.In this paper,the eigenvalues of the Jacobi matrix of each fixed point in HS model are explicitly derived,with the aim to understand the local dynamics around each fixed point.Phase transitions are described according to reJative population and coupling strength.Besides,the dynamics off OA manifold is studied.     

耦合分数阶双稳态振子的同步、反同步与振幅死亡

研究了耦合分数阶振子的同步、反同步和振幅死亡等问题. 基于P-R振子在特定参数下的双稳态特性, 利用最大条件Lyapunov指数、最大Lyapunov指数和分岔图等数值方法分析发现, 通过选取初始条件和耦合强度, 可以控制耦合振子呈现混沌同步、混沌反同步、全部振幅死亡同步、全部振幅死亡反同步和部 分振幅死亡等丰富的动力学现象. 基于蒙特卡罗方法的原理, 在初始条件相空间中随机选取耦合振子的初始位置, 计算不同耦合强度下耦合振子的全部振幅死亡态、部分振幅死亡态和非振幅死亡态的比例, 从统计学角度表征了耦合分数阶双稳态振子的动力学特征. 几种有代表性的双稳态振子的吸引域进一步证明了统计方法的计算结果. 关键词： 振幅死亡 吸引域 双稳态     

Oscillation death in a coupled van der Pol–Mathieu system

We report an investigation of the oscillation death (OD) of a parametrically excited coupled van der Pol–Mathieu (vdPM) system. The system can be considered as a pair of harmonically forced van der Pol oscillators under a double-well potential. The two oscillators are coupled with a cubic nonlinearity. We have shown that the system arrives at an OD regime when coupling strength crosses a threshold value at which the system undergoes saddle-node bifurcation and two limit cycles coalesce onto a fixed point of the system. We have further shown that this nonautonomous system possesses a centre manifold corresponding to the OD regime.     

Synchronization and anti-synchronization of chaos in an extended Bonhöffer–van der Pol oscillator using active control

In this paper, we investigate the complete synchronization and anti-synchronization (AS) of double-scroll chaotic attractor exhibited by an extended Bonhöffer–van der Pol (BVP) oscillator, using active control technique. In both synchronization schemes, the oscillators show good transient performance; while the AS state is further shown to correspond with complete inverse synchronization. Numerical simulations are also presented to verify the theoretical results.     

Power delivered by an array of Van der Pol oscillators coupled to a resonant cavity

An array of Van der Pol oscillators coupled to an RLC load is considered both theoretically and experimentally. It is found that the oscillators are active when the capacitance of the capacitor coupling the array to the load is below a critical value increasing with the number of oscillators. The power delivered to the load by the array of active oscillators increases with the number of oscillators till a limiting value increasing with the quality factor of the load. Good agreement is obtained between the theoretical and experimental results.     

Computing Arnol′d tongue scenarios

A famous phenomenon in circle-maps and synchronisation problems leads to a two-parameter bifurcation diagram commonly referred to as the Arnol′d tongue scenario. One considers a perturbation of a rigid rotation of a circle, or a system of coupled oscillators. In both cases we have two natural parameters, the coupling strength and a detuning parameter that controls the rotation number/frequency ratio. The typical parameter plane of such systems has Arnol′d tongues with their tips on the decoupling line, opening up into the region where coupling is enabled, and in between these Arnol′d tongues, quasi-periodic arcs. In this paper, we present unified algorithms for computing both Arnol′d tongues and quasi-periodic arcs for both maps and ODEs. The algorithms generalise and improve on the standard methods for computing these objects. We illustrate our methods by numerically investigating the Arnol′d tongue scenario for representative examples, including the well-known Arnol′d circle map family, a periodically forced oscillator caricature, and a system of coupled Van der Pol oscillators.     

Phase compactons in chains of dispersively coupled oscillators

We study the phase dynamics of a chain of autonomous oscillators with a dispersive coupling. In the quasicontinuum limit the basic discrete model reduces to a Korteveg-de Vries-like equation, but with a nonlinear dispersion. The system supports compactons: solitary waves with a compact support and kovatons which are compact formations of glued together kink-antikink pairs that may assume an arbitrary width. These robust objects seem to collide elastically and, together with wave trains, are the building blocks of the dynamics for typical initial conditions. Numerical studies of the complex Ginzburg-Landau and Van der Pol lattices show that the presence of a nondispersive coupling does not affect kovatons, but causes a damping and deceleration or growth and acceleration of compactons.     

Obtaining certificates for complete synchronisation of coupled oscillators

In this paper, we provide a novel reformulation of sufficient conditions that guarantee global complete synchronisation of coupled identical oscillators to make them computationally implementable. To this end, we use semidefinite programming techniques. For the first time, we can efficiently search for and obtain certificates for synchronisability and, additionally, also optimise associated cost functions. In this paper, a Lyapunov-like function (certificate) is used to certify that all trajectories of a networked system consisting of coupled dynamical systems will eventually converge towards a common one, which implies synchronisation. Moreover, we establish new conditions for complete synchronisation, which are based on the so called Bendixson’s Criterion for higher dimensional systems. This leads to major improvements on the lower bound of the coupling constant that guarantees global complete synchronisation. Importantly, the certificates are obtained by analysing the connection network and the model representing an individual system only. In order to illustrate the strength of our method we apply it to a system of coupled identical Lorenz oscillators and to coupled van der Pol oscillators.     

Sequential activity and multistability in an ensemble of coupled Van der Pol oscillators

In this paper collective dynamics of an ensemble of inhibitory coupled Van der Pol oscillators are studied. It was found that a stable heteroclinic contour and a stable heteroclinic channel between saddle cycles exist. These heteroclinic structures are responsible for the sequential activity of different oscillations. The corresponding bifurcations leading to the appearance of heteroclinic trajectories are analyzed.     

Phase-flip transition in nonlinear oscillators coupled by dynamic environment

We study the dynamics of nonlinear oscillators indirectly coupled through a dynamical environment or a common medium. We observed that this form of indirect coupling leads to synchronization and phase-flip transition in periodic as well as chaotic regime of oscillators. The phase-flip transition from in- to anti-phase synchronization or vise-versa is analyzed in the parameter plane with examples of Landau-Stuart and Ro?ssler oscillators. The dynamical transitions are characterized using various indices such as average phase difference, frequency, and Lyapunov exponents. Experimental evidence of the phase-flip transition is shown using an electronic version of the van der Pol oscillators.     

Mixed synchronization in chaotic oscillators using scalar coupling

We report experimental evidence of mixed synchronization in two unidirectionally coupled chaotic oscillators using a scalar coupling. In this synchronization regime, some of the state variables may be in complete synchronization while others may be in anti-synchronization state. We extended the theory by using an adaptive controller with an updating law based on Lyapunov function stability to include parameter fluctuation. Using the scheme, we implemented a cryptographic encoding for digital signal through parameter modulation.     

Synchronization of coupled oscillators on small-world networks

We investigate the synchronous dynamics of Kuramoto oscillators and van der Pol oscillators on Watts-Strogatz type small-world networks. The order parameters to characterize macroscopic synchronization are calculated by numerical integration. We focus on the difference between frequency synchronization and phase synchronization. In both oscillator systems, the critical coupling strength of the phase order is larger than that of the frequency order for the small-world networks. The critical coupling strength for the phase and frequency synchronization diverges as the network structure approaches the regular one. For the Kuramoto oscillators, the behavior can be described by a power-law function and the exponents are obtained for the two synchronizations. The separation of the critical point between the phase and frequency synchronizations is found only for small-world networks in the theoretical models studied.     

Amplitude death induced by mixed attractive and repulsive coupling in the relay system

The amplitude death (AD) phenomenon is found in the relay system in the presence of the mixed couplings composed of attractive coupling and repulsive coupling. The generation mechanism of AD is revealed and shows that the middle oscillator achieving AD is a prerequisite to further suppress oscillation of the outermost oscillators for the paradigmatic Stuart-Landau and Rössler models. Moreover, regarding the Stuart-Landau relay system as a small motif of star network, we also observe that the mixed couplings can facilitate AD state of the whole network system. Particularly, the threshold of coupling strength is invariable with the change of network size. Our findings may shed a new insight to explore the effects of hybrid coupling on complex systems, also provide a new strategy to control dynamic behaviors in engineering science and neuroscience fields.     

Repulsive synchronization in an array of phase oscillators

We study the dynamics of a repulsively coupled array of phase oscillators. For an array of globally coupled identical oscillators, repulsive coupling results in a family of synchronized regimes characterized by zero mean field. If the number of oscillators is sufficiently large, phase locking among oscillators is destroyed, independently of the coupling strength, when the oscillators' natural frequencies are not the same. In locally coupled networks, however, phase locking occurs even for nonidentical oscillators when the coupling strength is sufficiently strong.     

Adaptation to synchronization in phase-oscillator networks

We introduce an adaptation algorithm by which an ensemble of coupled oscillators with attractive and repulsive interactions is induced to adopt a prescribed synchronized state. While the performance of adaptation is controlled by measuring a macroscopic quantity, which characterizes the achieved degree of synchronization, adaptive changes are introduced at the microscopic level of the interaction network, by modifying the configuration of repulsive interactions. This scheme emulates the distinct levels of selection and mutation in biological evolution and learning.     

Resonance tongues in a system of globally coupled FitzHugh-Nagumo oscillators with time-periodic coupling strength

We investigate the dynamics of a population of globally coupled FitzHugh-Nagumo oscillators with a time-periodic coupling strength. While for synchronizing global coupling, the in-phase state is always stable, the oscillators split into several cluster states for desynchronizing global coupling, most commonly in two, irrespective of the coupling strength. This confines the ability of the system to form n:m locked states considerably. The prevalence of two and four cluster states leads to large 2:1 and 4:1 subharmonic resonance regions, while at low coupling strength for a harmonic 1:1 or a superharmonic 1:m time-periodic coupling coefficient, any resonances are absent and the system exhibits nonresonant phase drifting cluster states. Furthermore, in the unforced, globally coupled system the frequency of the oscillators in a cluster state is in general lower than that of the uncoupled oscillator and strongly depends on the coupling strength. Periodic variation of the coupling strength at twice the natural frequency causes each oscillator to keep oscillating with its autonomous oscillation period.     

Robust synchronization and anti-synchronization of identical Φ6 oscillators via adaptive sliding mode control

The main goal of this paper is to propose the single input robust adaptive sliding mode controllers to accomplish synchronization and anti-synchronization between two identical Φ⁶ Duffing or Van der Pol oscillators with unmodel dynamics and external disturbances. Unlike directly eliminating the nonlinear dynamics by active control and sliding mode control in the literature, the proposed sliding mode controllers include the equivalent control part, which is only proportional to the synchronized error states, and the switching control part, where the discontinuous control functions have adaptive feedback gains. Sufficient conditions are provided based on the Lyapunov stability theorem and numerical simulations are performed to verify the effectiveness of presented schemes.