Stable amplitude chimera states and chimera death in repulsively coupled chaotic oscillators

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.     

Effect of amplitude and frequency of limit cycle oscillators on their coupled and forced dynamics

The occurrence of synchronization and amplitude death phenomena due to the coupled interaction of limit cycle oscillators (LCO) has received increased attention over the last few decades in various fields of science and engineering. Studies pertaining to these coupled oscillators are often performed by studying the effect of various coupling parameters on their mutual interaction. However, the effect of system parameters (i.e., the amplitude and frequency) on the coupled interaction of such LCO has not yet received much attention, despite their practical importance. In this paper, we investigate the dynamical behavior of time-delay coupled Stuart–Landau (SL) oscillators exhibiting subcritical Hopf bifurcation for the variation of amplitude and frequency of these oscillators in their uncoupled state. For identical SL oscillators, a gradual increase in the amplitude of LCO shrinks the amplitude death regions observed between the regions of in-phase and anti-phase synchronization leading to its eventual disappearance, resulting in the occurrence of phase-flip bifurcations at higher amplitudes of LCO. We also observe an alternate existence of in-phase and anti-phase synchronization regions for higher values of time delay, whose prevalence of occurrence increases with an increase in the frequency of the oscillator. With the introduction of frequency mismatch, the region of amplitude death. The forced response of SL oscillator shows an asymmetry in the Arnold tongue and the manifestation of asynchronous quenching of LCO. An increase in the amplitude of LCO narrows the Arnold tongue and reduces the region of asynchronous quenching observed in the system. Finally, we compare the coupled and forced response of SL oscillators with the corresponding experimental results obtained from laminar thermoacoustic oscillators and the numerical results from van der Pol (VDP) oscillators. We show that the SL model qualitatively displays many features observed experimentally in coupled and forced thermoacoustic oscillators. In contrast, the VDP model does not capture most of the experimental results due to the limitation in the independent variation of system parameters.

     

Nonlinear dynamics of a non-autonomous network with coupled discrete–continuum oscillators

A network model of a multi-modular floating platform incorporated with a runway structure, viewed as a non-autonomous network with discrete–continuum oscillators, is developed for a general purpose of dynamic analysis. Numerical analysis shows the coupling effect between the two different types of oscillators on various complex dynamics, including sudden leaps, torus motions, beating vibrations, the synergetic effect of phase lock and anti-phase synchronizations. The amplitude death phenomenon, a suppressed weak oscillation state, is studied by using the fundamental solution derived by the averaging method. The parametric domain of the onset of amplitude death is illustrated to show the great significance to the stability design of the floating platform. The effect of the flexural rigidity of the runway on the distribution of amplitude death state is also discussed.     

Two-terminal feedback circuit for suppressing synchrony of the FitzHugh–Nagumo oscillators

An extremely simple analog technique for desynchronization of neuronal FitzHugh–Nagumo-type oscillators is described. Two-terminal feedback circuit has been developed. The feedback circuit, when coupled to a network of oscillators, nullifies the voltage at the coupling node and thus effectively decouples the individual oscillators. Both numerical simulations and hardware experiments have been performed. The results for an array of three mean-field coupled FitzHugh–Nagumo-type oscillators are presented.     

Regular oscillations,chaos, and multistability in a system of two coupled van der Pol oscillators: numerical and experimental studies

In this paper, the dynamics of a system of two coupled van der Pol oscillators is investigated. The coupling between the two oscillators consists of adding to each one’s amplitude a perturbation proportional to the other one. The coupling between two laser oscillators and the coupling between two vacuum tube oscillators are examples of physical/experimental systems related to the model considered in this paper. The stability of fixed points and the symmetries of the model equations are discussed. The bifurcations structures of the system are analyzed with particular attention on the effects of frequency detuning between the two oscillators. It is found that the system exhibits a variety of bifurcations including symmetry breaking, period doubling, and crises when monitoring the frequency detuning parameter in tiny steps. The multistability property of the system for special sets of its parameters is also analyzed. An experimental study of the coupled system is carried out in this work. An appropriate electronic simulator is proposed for the investigations of the dynamic behavior of the system. Correspondences are established between the coefficients of the system model and the components of the electronic circuit. A comparison of experimental and numerical results yields a very good agreement.     

PERIODIC SOLUTIONS IN ONE-DIMENSIONAL COUPLED MAP LATTICES

IntroductionDynamicsofcoupledmaplattices (CML )haverecentlyattractedconsiderableattention[1- 4].IthasbeenfoundthatCMLsexhibitavarietyofspace_timepatterns:space_timeintermittency ,spatiotemporalchaos,frontsandinterfaces,nonlinearperiodicsolutions.Inthispaper,anone_dimensionalinfinitechainofnonlinearcoupledmapsisconsideredandtheexistenceofperiodicsolutionsisprovedwhenthecouplingstrengthαissmallenough .Ourapproachistheanti_integrablelimitintroducedinRefs.[5-7] .Intheanti_integrablelimit(coup…     

Complexity and emergence of wave dynamics in a chain of sequentially interconnected Chua circuits

This paper considers an ensemble of Chua oscillators bidirectionally coupled in a ring geometry where locally coupled circuits form a closed loop of signal transmission. The spontaneous dynamics of this system is studied numerically for different coupling strength. A transition from periodic to chaotic regimes is observed when the coupling decreases. In the former situation, characterized by high coupling, all the circuits oscillate with pseudo-sinusoidal dynamics on periodic attractors; in the latter they evolve on the same-type of chaotic attractor with a progression of the dynamics from the Chua's spiral to the double scroll as the coupling decreases. The emerging global dynamics is markedly different in the two cases and a phase transition between highly ordered and highly disordered global dynamics is observed. Synchronization and traveling waves moving along the ring are identified in the non-chaotic regime, while spatio-temporal chaos results for very low coupling. Complex patterns formation appears at the “edge of chaos”, for a small couplings interval after the transition between these two regimes.     

Synchronization of four coupled van der Pol oscillators

It is possible that self-excited vibrations in turbomachine blades synchronize due to elastic coupling through the shaft. The synchronization of four coupled van der Pol oscillators is presented here as a simplified model. For quasilinear oscillations, a stability condition is derived from an analysis based on linearizing the original equation around an unperturbed limit cycle and transforming it into Hill’s equation. For the nonlinear case, numerical simulations show the existence of two well-defined regions of phase relationships in parameter space in which a multiplicity of periodic attractors is embedded. The size of these regions strongly depends on the values of the oscillator and coupling constants. For the coupling constant below a critical value, there exists a region in which a diversity of phase-shift attractors is present, whereas for values above the critical value an in-phase attractor is predominant. It is observed that the presence of an anti-phase attractor in the subcritical region is associated with sudden changes in the period of the coupled oscillators. The convergence of the coupled system to a particular periodic attractor is explored using several initial conditions. The study is extended to non-identical oscillators, and it is found that there is synchronization even over a wide range of difference among the oscillator constants.     

Energy localization and white noise-induced enhancement of response in a micro-scale oscillator array

In this work, the authors seek to develop an analytical framework to understand the influence of noise on an array of micro-scale oscillators with special attention to the phenomenon of intrinsic localized modes (ILMs). It was recently shown by one of the authors and co-workers (Dick et al. in Nonlinear Dyn. 54:13, 2008) that ILMs can be realized as nonlinear vibration modes. Building on this work, it is shown here that white noise excitation, by itself, is unable to produce ILMs in an array of coupled nonlinear oscillators. However, in the case of an array subjected to a combined deterministic and random excitation, the obtained numerical results indicate the existence of a threshold noise strength beyond which the ILM at one location in attenuated whilst the localization in strengthened at another location in the array. The numerical results further motivate the formulation of a general analytical framework wherein the Fokker–Planck equation is derived for a typical coupled oscillator cell of the array subjected to a combined white noise and deterministic excitation. With a set of approximations, the moment evolution equations are derived from the Fokker–Planck equation and they are numerically solved. These solutions indicate that once a localization event occurs in the array, a random excitation with noise strength above a threshold value contributes to the sustenance of the event. It is also observed that an excitation with a higher noise strength results in enhanced response amplitudes for oscillators in the center of the array. The efforts presented in this paper, in addition to providing an analytical framework for developing a fundamental understanding of the influence of white noise on the dynamics of coupled oscillator arrays, suggest that noise may be potentially used to manipulate the formation and persistence of ILMs in such arrays. Furthermore, the occurrence of enhanced response amplitudes due to an excitation with a high noise strength indicates that the framework may also be used to investigate stochastic resonance-type phenomena in coupled arrays of nonlinear oscillators including micro-scale oscillator arrays.     

Nonlinear dynamics of two harmonic oscillators coupled by Rayleigh type self-exciting force

The present paper reports some interesting phenomena observed in the nonlinear dynamics of two self-excitedly coupled harmonic oscillators. The system under consideration consists of two mechanical oscillators coupled by the Rayleigh type self-exciting force. Both autonomous and nonautonomous cases for weakly coupled systems are analyzed. When the natural frequencies of the two oscillators are close to each other, only one mode of oscillation exists. As two modes of oscillations get locked to a single mode, the system is said to be in a mode-locked condition. Under a mode-locked condition, the oscillators can oscillate with only a single frequency. However, when two oscillators are sufficiently detuned, the mode-locking condition does not persist and two distinct modes of oscillations emerge. Under these circumstances, particularly when detuning is large, one of the oscillators, depending on the initial conditions, oscillates with much larger amplitude as compared to the other oscillator, and hence mode localization is observed. When one of the oscillators is subject to a harmonic excitation, at two different frequencies, termed here as the decoupling frequencies, the coupling between the oscillators is almost lost, resulting in almost zero response of the unexcited oscillator. Analytical and numerical results are presented to analyze the above mentioned phenomena. Some potential applications of the aforesaid phenomena are also discussed.     

Stability of the synchronization manifold in nearest neighbor nonidentical van der Pol-like oscillators

We investigate the stability of the synchronization manifold in a ring and in an open-ended chain of nearest neighbor coupled self-sustained systems, each self-sustained system consisting of multi-limit cycle van der Pol oscillators. Such a model represents, for instance, coherent oscillations in biological systems through the case of an enzymatic-substrate reaction with ferroelectric behavior in a brain waves model. The ring and open-ended chain of identical and nonidentical oscillators are considered separately. By using the Master Stability Function approach (for the identical case) and the complex Kuramoto order parameter (for the nonidentical case), we derive the stability boundaries of the synchronized manifold. We have found that synchronization occurs in a system of many coupled modified van der Pol oscillators, and it is stable even in the presence of a spread of parameters.     

耦合阻尼对非保守耦合振子能量分布与功率流的影响

作为非保守耦合系统统计能量分析的基础,本文研究了耦合阻尼对非保守耦合系统能量分布与功率流的影响.在给出各项有关的损耗因子和耦合损耗因子的定义后,本文从理论上推导了非保守耦合振子间各项功率流与振子平均振动能量之间关系的理论表达式,以及功率平衡方程式和振子能量比的表达式.理论分析和数值计算的结果表明,非保守耦合振子之间的原始功率流和附加功率流以及总功率流不仅取决于两振子的平均振动能量之差,而且取决于振子的平均振动能量之和,总功率流的方向即与两振子能量相对大小有关,也与耦合性质有关;小耦合阻尼是非保守耦合的特例,由此特例不足以得到非保守耦合情况的一般特点。     

Periodic and quasiperiodic motion for complex non-linear systems

A damped complex non-linear system corresponding to two coupled non-linear oscillators with a periodic damping force is investigated by an asymptotic perturbation method based on Fourier expansion and time rescaling. Four coupled equations for the amplitude and the phase of solutions are derived. Phase-locked solutions with period equal to the damping force period are possible only if the oscillators amplitudes are equal. On the contrary, if the oscillators amplitudes are different, periodic solutions exist only with a period different from the damping force period. These solutions are stable only for perturbations that conserve the phase difference and the square amplitude sum of the oscillators. Energy considerations are used in order to study existence and characteristics of quasiperiodic motion. We demonstrate that modulated motion can be also obtained for appropriate values of the detuning parameter and in this case an approximate analytic solution is easily constructed. If the detuning parameter decreases the modulation period increases and then diverges, an infinite-period bifurcation occurs and the resulting motion becomes unbounded. Analytic approximate solutions are checked by numerical integration.     

Dynamical Order in Systems of Coupled Noisy Oscillators

We investigate a dynamical order induced by coupling and/or noise in systems of coupled oscillators. The dynamical order is referred to a one-dimensional topological structure of the global attractor of the system in the context of random skew-product flows. We show that if the coupling is sufficiently strong, then the system exhibits one dimensional dynamics regardless of the strength of noise. If the coupling is weak, then it is shown numerically that the system also exhibits one dimensional dynamics provided the noise is sufficiently strong. We also show that for any coupling and any noise, the system has a unique rotation number and hence all the oscillators tend to oscillate with the same frequency eventually (frequency locking). Dedicated to Professor Pavol Brunovsky on the occasion of his 70th birthday.     

The largest transversal Lyapunov exponent and master stability function from the perturbation vector and its derivative dot product (TLEVDP)

The behavior of systems of coupled nonlinear oscillators and, connected with it, the synchronization phenomena are of significant interest in many areas of science. One of the most important problems in this field is the stability of the synchronous state. The most often applied tool which allows one to quantify this stability is the largest Transversal Lyapunov Exponent (TLE) and, connected with it, Master Stability Function (MSF) theory (Pecora and Carroll in Phys. Rev. Lett. 80:2109, 1998). Thus there is still need to elaborate fast and simple methods of TLE calculation. The new method of the TLE estimation is presented in this paper. It applies the perturbation vector and its derivative Dot Product to calculate the largest Lyapunov exponent in the direction transversal to the synchronization manifold. The method (TLEVDP) of the TLE calculation is very simple in its background and numerical application. It does not require calculations of the Jacobi matrix eigenvalues and orthonormalization in each integration step. Thus it is fast and simple. Used methodology applies the method of Master Stability Function proposed by Pecora and Carroll (in Phys. Rev. Lett. 80:2109, 1998) which is undependable on the system type. Moreover on the base of the one MSF calculation one can predict complete synchronization of the sets of oscillators coupled in any coupling scheme. The theoretical improvement of the method is introduced. Results of the numerical simulations are shown and compared with other publications. Investigations of the system of Duffing oscillators coupled in ring scheme as well as the coupled Van der Pol oscillators made with use of the (TLEVDP) method are presented. Fast stabilization of the TLE value was shown.     

Effect of parameter mismatch and time delay interaction on density-induced amplitude death in coupled nonlinear oscillators

Oscillations in the coupled systems can be suppressed by varying density of mean field. The presence of parameter mismatch or time delay interaction in such systems enhances the amplitude death (AD) region in parameter space. This behavior of stability of steady state is analyzed by analytical as well as numerical studies of specific cases of limit-cycle and Rössler oscillators. Experimental evidence of AD is also shown, using an electronic version of the Chua’s oscillator.     

Nonlinear normal modes of a self-excited system driven by parametric and external excitations

Vibrations of nonlinear coupled parametrically and self-excited oscillators driven by an external harmonic force are presented in the paper. It is shown that if the force excites the system inside the principal parametric resonance then for a small excitation amplitude a resonance curve includes an internal loop. To find the analytical solutions, the problem is reduced to one degree of freedom oscillators by applications of Nonlinear Normal Modes (NNMs). The NNMs are formulated on the basis of free vibrations of a nonlinear conservative system as functions of amplitude. The analytical results are validated by numerical simulations and an essential difference between linear and nonlinear modes is pointed out.     

Effect of configuration symmetry on synchronization in a Van der Pol ring with nonlocal interactions

Nonlinear Dynamics - This paper discusses the influence of configuration symmetry on synchronization of coupled Van der Pol oscillators in a ring, where the inherent symmetry in an even number ring...     

Heteroclinic motion and energy transfer in coupled oscillator with nonlinear magnetic coupling

In this paper, we consider two coupled oscillators exhibiting both transient chaos and energy transfer from mechanical to electrical oscillators. Melnikov method is applied to these oscillators with linear damping and strongly nonlinear coupling terms in order to study the possibility of existence of chaos and transversal heteroclinic orbits and their control in a dynamical system. The energy transfer is studied using a qualitative measure of the system which can be obtained by computing the energy dissipated in it. At last, the numerical simulation is carried out for this system.