Multistable remote synchronization in a star-like network of non-identical oscillators

In the context of networks of coupled oscillators, remote synchronization happens when phase difference between non-adjacent units become constant, even though there is no global phase-locking. We study such regime considering a star-like network of Stuart-Landau oscillators. As previous works, our setup comprises peripheral nodes with different but close natural frequencies and the central node frequency detuned from them. The main contribution here is to numerically report multistability under intermediate coupling values: some initial condition yield remote synchronization, with quasi-periodic motion; while others do not converge to synchronized states. By using a Gaussian distribution to select the initial phases of the oscillators, we found that relatively small value of the standard deviation and absolute value of the mean of this distribution far from a specific range of values seem to favor remote synchronization in the multistability region. This phenomenon is extensively analyzed for both cases, considering a fixed coupling value.     

Synchronization of Colpitts oscillators with different orders

In this paper, we consider chaos synchronization between chaotic Colpitts oscillators with different orders, consisting of standard and improved version of Colpitts oscillators. Firstly, the normalized state equation of the improved version of the Colpitts oscillator designed to operate in the ultrahigh frequency range are presented. It is found that this version is described by fourth-order nonlinear differential equations. The equations of motion are solved numerically using the Runge–Kutta algorithm and simulations demonstrate chaos in the microwave frequencies range. Secondly, the problem of synchronization dynamics of third and fourth orders systems in the chaotic states is also investigated, and a controller is proposed based on stability theory by constructing the Lyapunov function, to ensure synchronization between both oscillators. Computer experiments demonstrate the effectiveness and feasibility of the proposed technique for these oscillators.     

Dynamical properties and chaos synchronization of improved Colpitts oscillators

In this paper, the dynamics and synchronization of improved Colpitts oscillators designed to operate in ultrahigh frequency range are considered. The model is described by a continuous time four-dimensional autonomous system with an exponential nonlinearity. The system is integrated numerically and various bifurcation diagrams and corresponding graphs of largest 1D Lyapunov exponent are plotted to summarize different scenarios leading to chaos. It is found that the oscillator moves from the state of fixed point motion to chaos via the usual paths of period-doubling, intermittency and interior crisis routes when monitoring the bias (i.e. power supply) in tiny ranges. In order to promote chaos-based synchronization designs of this type of oscillators, a synchronization strategy based upon the design of a nonlinear state observer is successfully adapted. The suggested approach enables synchronization to be achieved via a scalar transmitted signal which represents a suitable feature for communication applications. Numerical simulations are performed to demonstrate the effectiveness and feasibility of the proposed technique.     

Parametric generation of robust chaos with time-delayed feedback and modulated pump source

We consider a chaos generator composed of two parametrically coupled oscillators whose natural frequencies differ by factor of two. The system is driven by modulated pump source on the third harmonic of the basic frequency, and on each next period of pumping the excitation of the oscillator of doubled frequency is stimulated by the signal from the oscillator of the basic frequency undergoing quadratic nonlinear transformation and time delay. Using qualitative analysis and numerical results, we argue that chaotic dynamics in the system corresponds to hyperbolic strange attractor. It is a kind of Smale–Williams solenoid embedded in the infinite-dimensional state space of the stroboscopic map of the time-delayed system.     

Bifurcation analysis of a simple 3D oscillator and chaos synchronization of its coupled systems

Tama?evi?ius et al. proposed a simple 3D chaotic oscillator for educational purpose. In fact the oscillator can be implemented very easily and it shows typical bifurcation scenario so that it is a suitable training object for introductory education for students. However, as far as we know, no concrete studies on bifurcations or applications on this oscillator have been investigated. In this paper, we make a thorough investigation on local bifurcations of periodic solutions in this oscillator by using a shooting method. Based on results of the analysis, we study chaos synchronization phenomena in diffusively coupled oscillators. Both bifurcation sets of periodic solutions and parameter regions of in-phase synchronized solutions are revealed. An experimental laboratory of chaos synchronization is also demonstrated.     

Phase-locking for dynamical systems on the torus and perturbation theory for mathieu-type problems

Summary When several oscillators are coupled together and the parameters of their coupling are varied, the oscillators pass through so-called phase-locked regimes. In physical terms this means that the oscillators tend to synchronize their motion. To describe this phenomenon, we frame the concepts ofpartial phase andphase-locking. A partial phase of a toral flow puts emphasis on how orbits of the flow drift around the torus in some fixed direction. The partial phase is locked if it grows in time along some orbit slower than any linear function. When a toral flow is given by a trigonometric polynomial, its phase-locked regions are quite narrow. With the coupling amplitude increasing, each region grows in width as some power of the amplitude. That power can be calculated in terms of both the partial phase and degree of the trigonometric polynomial.     

The space–time variety of the hyperradiance from phase-locked soliton oscillators

The hyperradiance from phase-locked soliton oscillators is investigated by using the numerical simulation method for the perturbed sine-Gordon equation. Space–time variety for the emitted power from phase-locked soliton oscillators have been diffusely exhibited for the two magnetically coupled long Josephson junctions, operated in singlefluxon modes and involving the family of solutions. We derive some simulation results of space–time character, having the extensive physics meaning, for the theory for superradiance from phase-locked oscillators.     

Arnol’d tongues for a resonant injection-locked frequency divider: Analytical and numerical results

In this paper we consider a resonant injection-locked frequency divider which is of interest in electronics, and we investigate the frequency locking phenomenon when varying the amplitude and frequency of the injected signal. We study both analytically and numerically the structure of the Arnol’d tongues in the frequency–amplitude plane. In particular, we provide exact analytical formulae for the widths of the tongues, which correspond to the plateaux of the devil’s staircase picture. The results account for numerical and experimental findings presented in the literature for special driving terms and, additionally, extend the analysis to a more general setting.     

Frequency–energy plots of steady-state solutions for forced and damped systems,and vibration isolation by nonlinear mode localization

We study the structure of the periodic steady-state solutions of forced and damped strongly nonlinear coupled oscillators in the frequency–energy domain by constructing forced and damped frequency – energy plots (FEPs). Specifically, we analyze the steady periodic responses of a two degree-of-freedom system consisting of a grounded forced linear damped oscillator weakly coupled to a strongly nonlinear attachment under condition of 1:1 resonance. By performing complexification/averaging analysis we develop analytical approximations for strongly nonlinear steady-state responses. As an application, we examine vibration isolation of a harmonically forced linear oscillator by transferring and confining the steady-state vibration energy to the weakly coupled strongly nonlinear attachment, thereby drastically reducing its steady-state response. By comparing the nonlinear steady-state response of the linear oscillator to its corresponding frequency response function in the absence of a nonlinear attachment we demonstrate the efficacy of drastic vibration reduction through steady-state nonlinear targeted energy transfer. Hence, our study has practical implications for the effective passive vibration isolation of forced oscillators.     

Chaotic synchronization of two complex nonlinear oscillators

Synchronization is an important phenomenon commonly observed in nature. It is also often artificially induced because it is desirable for a variety of applications in physics, applied sciences and engineering. In a recent paper [Mahmoud GM, Mohamed AA, Aly SA. Strange attractors and chaos control in periodically forced complex Duffing’s oscillators. Physica A 2001;292:193–206], a system of periodically forced complex Duffing’s oscillators was introduced and shown to display chaotic behavior and possess strange attractors. Such complex oscillators appear in many problems of physics and engineering, as, for example, nonlinear optics, deep-water wave theory, plasma physics and bimolecular dynamics. Their connection to solutions of the nonlinear Schrödinger equation has also been pointed out.In this paper, we study the remarkable phenomenon of chaotic synchronization on these oscillator systems, using active control and global synchronization techniques. We derive analytical expressions for control functions and show that the dynamics of error evolution is globally stable, by constructing appropriate Lyapunov functions. This means that, for a relatively large set initial conditions, the differences between the drive and response systems vanish exponentially and synchronization is achieved. Numerical results are obtained to test the validity of the analytical expressions and illustrate the efficiency of these techniques for inducing chaos synchronization in our nonlinear oscillators.     

Origin of arrhythmias in a heart model

An investigation of the nonlinear dynamics of a heart model is presented. The model compartmentalizes the heart into one part that beats autonomously (the x oscillator), representing the pacemaker or SA node, and a second part that beats only if excited by a signal originating outside itself (the y oscillator), representing typical cardiac tissue. Both oscillators are modeled by piecewise linear differential equations representing relaxation oscillators in which the fast time portion of the cycle is modeled by a jump. The model assumes that the x oscillator drives the y oscillator with coupling constant $\alpha$. As $\alpha$ decreases, the regular behavior of y oscillator deteriorates, and is found to go through a series of bifurcations. The irregular behavior is characterized as involving a large amplitude cycle followed by a number n of small amplitude cycles. We compute critical bifurcation values of the coupling constant, ${\alpha }_n$, using both numerical methods as well as perturbations.     

Convergence of analytic gradient-type systems with periodicity and its applications in Kuramoto models

We consider the convergence of gradient-type systems with periodic and analytic potentials. The main tool is the celebrated Łojasiewicz inequality which is valid for any analytic function. Our results show that the convergence of such systems with periodic and analytic potentials is unconditional to the initial data; in other words, any trajectory converges to some equilibrium. As direct applications, we can show that any trajectory converges to phase-locked state for the first- and second-order Kuramoto models on a symmetric network with attractive–repulsive forces and identical natural frequencies. In particular, the inertial Kuramoto model with identical oscillators converges to phase-locked state for any initial configuration.     

On synchronized solutions of coupled Van‐der‐Pol equations

Two linearly coupled Van‐der‐Pol oscillators are considered in the case of a small frequency detuning of the oscillators as well as weak coupling. The occurring resonant Hopf bifurcation leading to synchronized motions is examined and analytical approximations of the oscillation amplitudes and the synchronous frequency are derived. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A simple adaptive control for full and reduced-order synchronization of uncertain time-varying chaotic systems

In this paper, a simple adaptive feedback control is proposed for full and reduced-order synchronization of time-varying and strictly uncertain chaotic systems. Our method uses only one feedback gain with parameter adaptation law and converges very fast even in the presence of noise. For full synchronization, a drive-response system consisting of two second-order identical parametrically excited oscillators achieve global synchronization; while for reduced-order synchronization, the dynamical evolution of a second-order parametrically driven oscillator is synchronized with the projection of a third-order time-varying chaotic system. The effectiveness of our approach is demonstrated using numerical simulations.     

Experimental reconsideration of spatio‐temporal dynamics observed in fluid‐elastic oscillator arrays from complex system viewpoint: From vibrating pipes in heat exchangers to waving plants in agricultural fields

Transition from local complexity to global spatio‐temporal dynamics in a two‐dimensional array of fluid‐elastic oscillators is examined experimentally with an apparatus comprising 90‐1000 cantilevered rods in a wind tunnel as the Reynolds number (based on rod diameter) is increased from 200 to 900. A cluster‐pattern entropy measure is introduced as a quantitative measure of local complexity. As the intensity of interaction among neighboring elements (in this case, frequency of collisions among rods) increases, a set of the elements (in this case, a rod‐array) achieves globally better‐organized behavior. On the basis of accelerometer data, the rod impact rate versus flow velocity shows a power‐law scaling relation. Video images reveal that, initially, each rod moves individually; then clusters consisting of several rods emerge. Finally, global wave‐like motion occurs at higher flow velocities. Each wave‐like motion has its specific frequency and spatial wavelength, which vary according to wind velocity. © 2007 Wiley Periodicals, Inc. Complexity 12: 36–47, 2007     

Vacuum and Thermal Energies for Two Oscillators Interacting Through A Field

We consider a simple (1+1)-dimensional model for the Casimir–Polder interaction consisting of two oscillators coupled to a scalar field. We include dissipation in a first-principles approach by allowing the oscillators to interact with heat baths. For this system, we derive an expression for the free energy in terms of real frequencies. From this representation, we derive the Matsubara representation for the case with dissipation. We consider the case of vanishing intrinsic frequencies of the oscillators and show that the contribution from the zeroth Matsubara frequency is modified in this case and no problem with the laws of thermodynamics appears.     

Network of phase-locking oscillators and a possible model for neural synchronization

In order to model the synchronization of brain signals, a three-node fully-connected network is presented. The nodes are considered to be voltage control oscillator neurons (VCON) allowing to conjecture about how the whole process depends on synaptic gains, free-running frequencies and delays. The VCON, represented by phase-locked loops (PLL), are fully-connected and, as a consequence, an asymptotically stable synchronous state appears. Here, an expression for the synchronous state frequency is derived and the parameter dependence of its stability is discussed. Numerical simulations are performed providing conditions for the use of the derived formulae. Model differential equations are hard to be analytically treated, but some simplifying assumptions combined with simulations provide an alternative formulation for the long-term behavior of the fully-connected VCON network. Regarding this kind of network as models for brain frequency signal processing, with each PLL representing a neuron (VCON), conditions for their synchronization are proposed, considering the different bands of brain activity signals and relating them to synaptic gains, delays and free-running frequencies. For the delta waves, the synchronous state depends strongly on the delays. However, for alpha, beta and theta waves, the free-running individual frequencies determine the synchronous state.     

Bifurcation structure of a discrete neuronal equation

We present the bifurcations diagram of a threshold automation with memory. This automation has a unique attractor which is periodic if the memory is bounded, periodic or Cantorian if it is unbounded. We show that the associated rotation number is an increasing piecewise constant function of the threshold parameter. If the memory is unbounded, this function is a devil staircase.     

Frequency dependent iteration method for forced nonlinear oscillators

A new iteration method for nonlinear vibrations has been developed by decomposing the periodic solution in two parts corresponding to low and high harmonics. For a nonlinear forced oscillator, the iteration schema is proposed with different formulations for these two parts. Then, the schema is deduced by using the harmonic balance technique. This method has proven to converge to the periodic solutions provided that a convergence condition is satisfied. The convergence is also demonstrated analytically for linear oscillators. Moreover, the new method has been applied to Duffing oscillators as an example. The numerical results show that each iteration schema converges in a domain of the excitation frequency and it can converge to different solutions of the nonlinear oscillator.