Chaotic oscillations in a coupled system of bistable oscillators

The influence of the asymmetry of the nonlinear element characteristic on the chaotic oscillations of Chua’s bistable oscillator is studied. It is shown that such asymmetry causes asymmetry of a chaotic attractor that maps the switching of motions between two basins of attraction up to the concentration of oscillations in one basin. Oscillation control in a bistable chaotic self-oscillating system (two coupled Chua’s oscillators) is considered. It is demonstrated that oscillations excited in two basins of attraction may pass to one of them and that oscillations may build up in two basins when they are autonomously excited in different basins. It is also found that chaotic oscillations in a coupled system may be excited at parameter values for which the autonomous chaotic oscillations of partial oscillators are absent. The influence of external noiselike oscillations is investigated.     

Experimental evidence of time-delay-induced death in coupled limit-cycle oscillators

Experimental observations of time-delay-induced amplitude death in two coupled nonlinear electronic circuits that are individually capable of exhibiting limit-cycle oscillations are described. The existence of multiply connected death islands in the parameter space of coupling strength and time delay for coupled identical oscillators is established. The existence of such regions was predicted earlier on theoretical grounds [Phys. Rev. Lett. 80, 5109 (1998); Physica (Amsterdam) 129D, 15 (1999)]. The experiments also reveal the occurrence of multiple frequency states, frequency suppression of oscillations with increased time delay, and the onset of both in-phase and antiphase collective oscillations.     

Amplitude death in nonlinear oscillators with indirect coupling

We investigate the effect of frequency mismatch in two indirectly coupled Rössler oscillators and Hindmarsh–Rose neuron model systems. While identical systems show in-phase or out-of-phase synchronization states when coupled through a dynamic environment, mismatch in the internal frequencies of the systems drives them to a fixed point state, i.e., amplitude death. There is a region in the parameter space of the frequency mismatch and coupling strength where system shows amplitude death. The numerical results of Rössler system are also experimentally verified using piece-wise Rössler circuits.     

Mutual synchronization in a system of two bistable oscillators executing regular and chaotic oscillations

A numerical analysis of a new model describing two coupled modified Chua??s oscillators is conducted. Equations of a partial oscillator differ from classical equations in that the former contain additional delayed feedback in another writing of dimensionless time. Changeover from regular oscillations in the absence of additional feedback to additional-feedback-induced (switchable) chaotic oscillations is studied. It is shown that, when normal regular oscillations, as well as additional-feedback-induced chaotic oscillations, are synchronized, difference oscillations are left. They are absent only when the control parameters of partial oscillators are identical. The application of a harmonic signal allows one to control the oscillations of a chaotic system of coupled modified bistable oscillators.     

Amplitude death in systems of coupled oscillators with distributed-delay coupling

This paper studies the effects of coupling with distributed delay on the suppression of oscillations in a system of coupled Stuart-Landau oscillators. Conditions for amplitude death are obtained in terms of strength and phase of the coupling, as well as the mean time delay and the width of the delay distribution for uniform and gamma distributions. Analytical results are confirmed by numerical computation of the eigenvalues of the corresponding characteristic equations. These results indicate that larger widths of delay distribution increase the regions of amplitude death in the parameter space. In the case of a uniformly distributed delay kernel, for sufficiently large width of the delay distribution it is possible to achieve amplitude death for an arbitrary value of the average time delay, provided that the coupling strength has a value in the appropriate range. For a gamma distribution of delay, amplitude death is also possible for an arbitrary value of the average time delay, provided that it exceeds a certain value as determined by the coupling phase and the power law of the distribution. The coupling phase has a destabilizing effect and reduces the regions of amplitude death.     

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     

Stochastic phase dynamics and noise-induced mixed-mode oscillations in coupled oscillators

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

Dynamics of the entanglement between two oscillators in the same environment

We provide a complete characterization of the evolution of entanglement between two resonant oscillators coupled to a common environment. We identify three phases with different qualitative long time behavior. There is a phase where entanglement undergoes a sudden death. Another phase (sudden death and revival) is characterized by an infinite sequence of events of sudden death and revival of entanglement. In the third phase (no sudden death) there is no sudden death of entanglement, which persists for a long time. The phase diagram is described and analytic expressions for the boundary between phases are obtained. These results are applicable to a large variety of non-Markovian environments. The case of nonresonant oscillators is also numerically investigated.     

Time delay induced partial death patterns with conjugate coupling in relay oscillators

We investigate the dynamics of delay-coupled relay oscillators with conjugate (or dissimilar) coupling and find the partial death with the phase-flip transition. This phenomenon is quite general and occurs for the limit cycle as well as chaotic relay oscillators. In the regime of partial death, parts of the system oscillate with large amplitude, while other element stays at rest. Using the Stuart-Landau and Rössler oscillators, we demonstrate that partial amplitude death is a robust dynamical state in coupled oscillators. We also studied the mismatch delay and find different types of dynamical pattern with partial death.     

反馈控制棘轮的定向输运效率研究

研究了反馈耦合布朗棘轮中粒子处于负载力、时变外力及噪声作用下的定向输运问题.详细讨论了外力作用时间的不对称性、外势空间的不对称性及外力周期等对反馈耦合棘轮中粒子输运效率的影响.研究发现,外力的时间不对称度能促进反馈棘轮中粒子的定向输运,随时间不对称度的增大,反馈棘轮中粒子能获得较大的效率.然而,外势空间的不对称度能有效抑制耦合棘轮中粒子的扩散,达到增强耦合粒子定向输运的效果.同时还发现,存在最优的噪声强度能使耦合粒子的输运效率达到最大.     

Amplitude and phase effects on the synchronization of delay-coupled oscillators

We consider the behavior of Stuart-Landau oscillators as generic limit-cycle oscillators when they are interacting with delay. We investigate the role of amplitude and phase instabilities in producing symmetry-breaking/restoring transitions. Using analytical and numerical methods we compare the dynamics of one oscillator with delayed feedback, two oscillators mutually coupled with delay, and two delay-coupled elements with self-feedback. Taking only the phase dynamics into account, no chaotic dynamics is observed, and the stability of the identical synchronization solution is the same in each of the three studied networks of delay-coupled elements. When allowing for a variable oscillation amplitude, the delay can induce amplitude instabilities. We provide analytical proof that, in case of two mutually coupled elements, the onset of an amplitude instability always results in antiphase oscillations, leading to a leader-laggard behavior in the chaotic regime. Adding self-feedback with the same strength and delay as the coupling stabilizes the system in the transverse direction and, thus, promotes the onset of identically synchronized behavior.     

A novel mixed-synchronization phenomenon in coupled Chua’s circuits via non-fragile linear control

Dynamical variables of coupled nonlinear oscillators can exhibit different synchronization patterns depending on the designed coupling scheme.In this paper,a non-fragile linear feedback control strategy with multiplicative controller gain uncertainties is proposed for realizing the mixed-synchronization of Chua’s circuits connected in a drive-response configuration.In particular,in the mixed-synchronization regime,different state variables of the response system can evolve into complete synchronization,anti-synchronization and even amplitude death simultaneously with the drive variables for an appropriate choice of scaling matrix.Using Lyapunov stability theory,we derive some sufficient criteria for achieving global mixed-synchronization.It is shown that the desired non-fragile state feedback controller can be constructed by solving a set of linear matrix inequalities (LMIs).Numerical simulations are also provided to demonstrate the effectiveness of the proposed control approach.     

The dynamics of two coupled Van der Pol oscillators with attractive and repulsive coupling

We consider a variant of two coupled Van der Pol oscillators with both attractive and repulsive mean-field interactions. In the presence of attractive coupling, the system is in the complete synchrony, while repulsive coupling shows anti-synchronization state leading to suppression of oscillations with increasing interaction strength. The coupled system with both attractive and repulsive interactions shows competitive tendencies of being complete synchronization and anti-synchronization resulting in the stabilization of the fixed point. We have also studied the effect of the damping coefficient of the VdP oscillator on the nature of the transition from oscillatory to a steady-state. These oscillators stabilize to unstable equilibrium point or coupling dependent inhomogeneous steady state via second or first-order transitions respectively depending upon the damping coefficient and coupling strength. These transitions are analyzed in the parameter plane by analytical and numerical studies of the two coupled Van der Pol oscillators.     

Strong coupling of nonlinear electronic and biological oscillators: reaching the "amplitude death" regime

Interaction between an electronic and a biological circuit has been investigated for a pair of electrically connected nonlinear oscillators, with a spontaneously oscillating olivary neuron as the single-cell biological element. By varying the coupling strength between the oscillators, we observe a range of behaviors predicted by model calculations, including a reversible low-energy dissipation "amplitude death" where the oscillations in the coupled system cease entirely.     

Bifurcation, amplitude death and oscillation patterns in a system of three coupled van der Pol oscillators with diffusively delayed velocity coupling

In this paper, we study a system of three coupled van der Pol oscillators that are coupled through the damping terms. Hopf bifurcations and amplitude death induced by the coupling time delay are first investigated by analyzing the related characteristic equation. Then the oscillation patterns of these bifurcating periodic oscillations are determined and we find that there are two kinds of critical values of the coupling time delay: one is related to the synchronous periodic oscillations, the other is related to eight branches of asynchronous periodic solutions bifurcating simultaneously from the zero solution. The stability of these bifurcating periodic solutions are also explicitly determined by calculating the normal forms on center manifolds, and the stable synchronous and stable phase-locked periodic solutions are found. Finally, some numerical simulations are employed to illustrate and extend our obtained theoretical results and numerical studies also describe the switches of stable synchronous and phase-locked periodic oscillations.     

First order transition to oscillation death through an environment

The emergence of dynamical abrupt transitions for the first time in an ensemble of identical limit-cycle and chaotic oscillators coupled via a common environment is reported. The transition from the oscillatory state to the death state and vice versa, in these networks of oscillators are found not only discontinuous as well as irreversible in the parameter space. This first order phase transition in these systems is termed as Explosive Death. The occurrence of such transition is studied in details by using an appropriate order parameter for both limit-cycle and chaotic oscillators, in particular, Stuart–Landau and Rössler oscillators. The backward transition point for this phenomenon is obtained analytically using linear stability analysis and is found to be consistent with the numerical results.     

Synchronization of time-continuous chaotic oscillators

Considering a system of two coupled identical chaotic oscillators, the paper first establishes the conditions of transverse stability for the fully synchronized chaotic state. Periodic orbit threshold theory is applied to determine the bifurcations through which low-periodic orbits embedded in the fully synchronized state lose their transverse stability, and the appearance of globally and locally riddled basins of attraction is discussed, respectively, in terms of the subcritical, supercritical nature of the riddling bifurcations. We show how the introduction of a small parameter mismatch between the interacting chaotic oscillators causes a shift of the synchronization manifold. The presence of a coupling asymmetry is found to lead to further modifications of the destabilization process. Finally, the paper considers the problem of partial synchronization in a system of four coupled R?ssler oscillators.     

Distributed delays facilitate amplitude death of coupled oscillators

Coupled oscillators are shown to experience amplitude death for a much larger set of parameter values when they are connected with time delays distributed over an interval rather than concentrated at a point. Distributed delays enlarge and merge death islands in the parameter space. Furthermore, when the variance of the distribution is larger than a threshold, the death region becomes unbounded and amplitude death can occur for any average value of delay. These phenomena are observed even with a small spread of delays, for different distribution functions, and an arbitrary number of oscillators.     

Autonomous coupled oscillators with hyperbolic strange attractors

We propose several examples of smooth low-order autonomous dynamical systems which have apparently uniformly hyperbolic attractors. The general idea is based on the use of coupled self-sustained oscillators where, due to certain amplitude nonlinearities, successive epochs of damped and excited oscillations alternate. Because of additional, phase sensitive coupling terms in the equations, the transfer of excitation from one oscillator to another is accompanied by a phase transformation corresponding to some chaotic map (in particular, an expanding circle map or Anosov map of a torus). The first example we construct is a minimal model possessing an attractor of the Smale-Williams type. It is a four-dimensional system composed of two oscillators. The underlying amplitude equations are similar to those of the predator-pray model. The other three examples are systems of three coupled oscillators with a heteroclinic cycle. This scheme presents more variability for the phase manipulations: in the six-dimensional system not only the Smale-Williams attractor, but also an attractor with Arnold cat map dynamics near a two-dimensional toral surface, and a hyperchaotic attractor with two positive Lyapunov exponents, are realized.