Uncoupled Oscillators Research Articles

Aging and clustering in globally coupled oscillators

A population of coupled nonlinear oscillators may age in such a way that the fraction of non-self-oscillatory elements increases. Following our previous paper [Phys. Rev. Lett. 93, 104101 (2004)], we study the effect of aging in this sense mainly for globally coupled Stuart-Landau oscillators with the emphasis on the structure of the (K,p) phase diagram, where K is the coupling strength and p is the ratio of inactive oscillators. In addition to the aging transition reported previously, such a diagram is shown to be characterized by a hornlike region, which we call a "desynchronization horn," where active oscillators desynchronize to form a number of clusters, provided that uncoupled active oscillators are sufficiently nonisochronous. We also show that desynchronization in such a region can be captured as a type of diffusion-induced inhomogeneity based on a "swing-by mechanism." Our results suggest that the desynchronization horn with some curious properties may be a fairly common feature in aging systems of globally and diffusively coupled periodic oscillators.

Synchronization in oscillator networks with coupling balance

Under the hypothesis of coupling balance, both locally and globally asymptotically synchronization of oscillator networks with nonlinear coupling in general form are studied for the first time. In order to study the globally asymptotically synchronization, we first show that the oscillator network is eventually dissipative if the uncoupled oscillators are eventual dissipative. Then in its absorbing domain, two methods developed recently are combined to study the asymptotically synchronization and two theorems are derived. Numerical simulations confirm the validity of the results.

Noise-Induced Synchronization and Clustering in Ensembles of Uncoupled Limit-Cycle Oscillators

We study synchronization properties of general uncoupled limit-cycle oscillators driven by common and independent Gaussian white noises. Using phase reduction and averaging methods, we analytically derive the stationary distribution of the phase difference between oscillators for weak noise intensity. We demonstrate that in addition to synchronization, clustering, or more generally coherence, always results from arbitrary initial conditions, irrespective of the details of the oscillators.

Global and local responses of coupled dynamic systems

Responses of a master oscillator coupled to a set of satellite oscillators are evaluated. A set is designed by specifying a normalized frequency distribution. When the designed distribution is completely degenerate, the influence of the coupling is global. The coupling generates two global peaks, one on each side of an antiresonance that replaces the peak in the response of the uncoupled master oscillator. The loss factor for these global peaks matches that of the uncoupled master oscillator. As the degeneracy is gradually removed, the evaluated response gradually reveals, within the distribution bandwidth, contributions from individual satellite oscillators. The loss factors and the frequency separation of the two global peaks then become dependent upon the coupling. As the degeneracy is further removed, the local presence of the satellite oscillators assumes the dominant feature in the response whereas the presence of the two global peaks subsides. Since the designed distributions relate loss factors directly to modal overlap parameters, increases in these parameters above unity render this dominance in the response quiescent and saturated. On the other hand, continuous decreases in these parameters below unity increase the fluctuations in that dominant part of the response ad infinitum.

Noether-Symmetry Analysis using Alternative Lagrangian Representations

We have sought to work with an approach to Noether symmetry analysis which uses the properties of infinitesimal point transformations in the space-time (q, t) variable to establish the association between symmetries and conservation laws of a dynamical system. In this approach symmetries are expressed in the form of generators. We have studied the variational or Noether symmetries of two uncoupled Harmonic oscillators and two such oscillators coupled by an interaction. Both these systems can have alternative Lagrangian representations. We have studied in detail how the association between symmetries and conservation laws changes as one alters the analytic or Lagrangian representation. This analysis is carried out with a view to explicitly demonstrate that the correlation between symmetry transformation and corresponding invariant quantity depends crucially on the choice of the analytic representation.

Synchrony of hand-foot coupled movements: is it attained by mutual feedback entrainment or by independent linkage of each limb to a common rhythm generator?

BackgroundSynchrony of coupled oscillations of ipsilateral hand and foot may be achieved by controlling the interlimb phase difference through a crossed kinaesthetic feedback between the two limbs, or by an independent linkage of each limb cycle to a common clock signal. These alternative models may be experimentally challenged by comparing the behaviour of the two limbs when they oscillate following an external time giver, either alone or coupled together.ResultsTen subjects oscillated their right hand and foot both alone and coupled (iso- or antidirectionally), paced by a metronome. Wrist and ankle angular position and Electromyograms (EMG) from the respective flexor and extensor muscles were recorded. Three phase delays were measured: i) the clk-mov delay, between the clock (metronome beat) and the oscillation peak; ii) the neur (neural) delay, between the clock and the motoneurone excitatory input, as inferred from the EMG onset; and iii) the mech (mechanical) delay between the EMG onset and the corresponding point of the limb oscillation. During uncoupled oscillations (0.4 Hz to 3.0 Hz), the mech delay increased from -7° to -111° (hand) and from -4° to -83° (foot). In contrast, the clk-mov delay remained constant and close to zero in either limb since a progressive advance of the motoneurone activation on the pacing beat (neur advance) compensated for the increasing mech delay. Adding an inertial load to either extremity induced a frequency dependent increase of the limb mechanical delay that could not be completely compensated by the increase of the neural phase advance, resulting in a frequency dependent increment of clk-mov delay of the hampered limb. When limb oscillations were iso- or antidirectionally coupled, either in the loaded or unloaded condition, the three delays did not significantly change with respect to values measured when limbs were moved separately.ConclusionThe absence of any significant effect of limb coupling on the measured delays suggests that during hand-foot oscillations, both iso- and antidirectionally coupled, each limb is synchronised to the common rhythm generator by a "private" position control, with no need for a crossed feedback interaction between limbs.

Transient Resetting: A Novel Mechanism for Synchrony and Its Biological Examples

The study of synchronization in biological systems is essential for the understanding of the rhythmic phenomena of living organisms at both molecular and cellular levels. In this paper, by using simple dynamical systems theory, we present a novel mechanism, named transient resetting, for the synchronization of uncoupled biological oscillators with stimuli. This mechanism not only can unify and extend many existing results on (deterministic and stochastic) stimulus-induced synchrony, but also may actually play an important role in biological rhythms. We argue that transient resetting is a possible mechanism for the synchronization in many biological organisms, which might also be further used in the medical therapy of rhythmic disorders. Examples of the synchronization of neural and circadian oscillators as well as a chaotic neuron model are presented to verify our hypothesis.

Contrast Enhancement with a Nonlinear Oscillators Network

In this paper, we propose a contrast enhancement realized with a lattice of uncoupled nonlinear oscillators. We show theoretically and numerically that the contrast of a gray scale image is strongly enhanced even if it is initially very weak. An image inversion can also be obtained in real time without needing a reconfiguration of this Cellular Nonlinear Network (CNN). Finally, an electronic implementation of this (CNN) is discussed.

Discretization of frequencies in delay coupled oscillators

We study the dynamics of two mutually coupled chaotic oscillators with a time delayed coupling. Due to the delay, the allowed frequencies of the oscillators are shown to be discretized. The phenomenon is observed in the case when the delay is much larger than the characteristic period of the solitary uncoupled oscillator.

Synchrony of limit-cycle oscillators induced by random external impulses

The mechanism of phase synchronization between uncoupled limit-cycle oscillators induced by common random impulsive forcing is analyzed. By reducing the dynamics of the oscillator to a random phase map, it is shown that phase synchronization generally occurs when the oscillator is driven by weak random impulsive forcing in the limit of large interimpulse intervals. The case where the interimpulse intervals are finite is also analyzed perturbatively for small impulse intensity. For weak Poisson impulses, it is shown that the phase synchronization persists up to the first order approximation.

Image processing with a cellular nonlinear network

A cellular nonlinear network (CNN) based on uncoupled nonlinear oscillators is proposed for image processing purposes. It is shown theoretically and numerically that the contrast of an image loaded at the nodes of the CNN is strongly enhanced, even if this one is initially weak. An image inversion can be also obtained without reconfiguration of the network whereas a gray levels extraction can be performed with an additional threshold filtering. Lastly, an electronic implementation of this CNN is presented.

Synchronization of chaotic systems with parameter driven by a chaotic signal

Abstract Chaos control with driving parameter scheme in uncoupled identical chaotic oscillators is presented. By driving the parameter of chaotic systems using external chaotic signal, synchronization and anti-synchronization can be implemented. Numerical simulations show that either synchronization or anti-synchronization can appear depending significantly on initial condition and on driving strength. The proposed method is particularly suited for a variety of chaotic systems, which cannot couple with each other in engineering.

Estimation of coupling between oscillators from short time series via phase dynamics modeling: Limitations and application to EEG data

We demonstrate in numerical experiments that estimators of strength and directionality of coupling between oscillators based on modeling of their phase dynamics [D. A. Smirnov and B. P. Bezruchko, Phys. Rev. E 68, 046209 (2003)] are widely applicable. Namely, although the expressions for the estimators and their confidence bands are derived for linear uncoupled oscillators under the influence of independent sources of Gaussian white noise, they turn out to allow reliable characterization of coupling from relatively short time series for different properties of noise, significant phase nonlinearity of the oscillators, and nonvanishing coupling between them. We apply the estimators to analyze a two-channel human intracranial epileptic electroencephalogram (EEG) recording with the purpose of epileptic focus localization.

Synchrony of neural Oscillators induced by random telegraphic currents

When a neuron receives a randomly fluctuating input current, its reliability of spike generation improves compared with the case of a constant input current [Mainen and Sejnowski, Science 268, 1503 (1995)]. This phenomenon can be interpreted as phase synchronization between uncoupled nonlinear oscillators subject to a common external input. We analyze this phenomenon using dynamical models of neurons, assuming the input current to be a simple random telegraphic signal that jumps between two values, and the neuron to be always purely self-oscillatory. The internal state of the neuron randomly jumps between two limit cycles corresponding to the input values, which can be described by random phase maps when the switching time of the input current is sufficiently long. Using such a random map description, we discuss the synchrony of neural oscillators subject to fluctuating inputs. Especially when the phase maps are monotonic, we can generally show that the Lyapunov exponent is negative, namely, phase synchronization is stable and reproducibility of spike timing improves.

Robustness of the Noise-Induced Phase Synchronization in a General Class of Limit Cycle Oscillators

We show that a wide class of uncoupled limit-cycle oscillators can be in-phase synchronized by common weak additive noise. An expression of the Lyapunov exponent is analytically derived to study the stability of the noise-driven synchronizing state. The result shows that such a synchronization can be achieved in a broad class of oscillators with little constraint on their intrinsic property. On the other hand, the leaky integrate-and-fire neuron oscillators do not belong to this class, generating intermittent phase slips according to a power law distribution of their intervals.

Entanglement in the Interaction Between Two Quantum Oscillators Systems

The fundamental quantum dynamics of two interacting oscillator systems are studied in two different scenarios. In one case, both oscillators are assumed to be linear, whereas in the second case, one oscillator is linear and the other is a non-linear, angular-momentum oscillator; the second case is, of course, more complex in terms of energy transfer and dynamics. These two scenarios have been the subject of much interest over the years, especially in developing an understanding of modern concepts in quantum optics and quantum electronics. In this work, however, these two scenarios are utilized to consider and discuss the salient features of quantum behaviors resulting from the interactive nature of the two oscillators, i.e., coherence, entanglement, spontaneous emission, etc., and to apply a measure of entanglement in analyzing the nature of the interacting systems. ... For the coupled linear and angular-momentum oscillator system in the fully quantum-mechanical description, we consider special examples of two, three, four-level angular momentum systems, demonstrating the explicit appearances of entanglement. We also show that this entanglement persists even as the coupled angular momentum oscillator is taken to the limit of a large number of levels, a limit which would go over to the classical picture for an uncoupled angular momentum oscillator.

Uncoupled oscillator model for interpreting second harmonic generation measurements of oriented chiral systems

Molecular modeling calculations of the electronically resonant nonlinear optical susceptibility of the archetypal chiral molecule binaphthol are shown to be qualitatively reproduced by considering two uncoupled naphthol molecules oriented in a manner similar to that in binaphthol. This remarkably simple uncoupled oscillator model reliably predicts the signs and relative magnitudes of the key elements within the molecular susceptibility tensor of binaphthol. These results suggest that surprisingly simple and intuitive physical models can often be constructed for qualitatively interpreting chiral-specific phenomena in nonlinear optical investigations of oriented chiral chromophores in thin films and at surfaces.

Coupling schemes for cluster synchronization in coupled Josephson equations

We propose an approach to constructing different coupling schemes to stabilize selected cluster synchronization patterns for coupled Josephson equations. In particular, we select a coupling scheme to create synchronization with frequency ratio m 1 : m 2 : ⋯ : m n , which can be arbitrarily chosen and is independent of the frequencies of the uncoupled oscillators. We also discuss coupled discrete systems.

A NONLINEAR OSCILLATORS NETWORK DEVOTED TO IMAGE PROCESSING

A contrast enhancement and image inverting tool using a lattice of uncoupled nonlinear oscillators is proposed. We show theoretically and numerically that the gray scale picture contrast is strongly enhanced even if this one is initially very small. An image inversion can be also obtained in real time with the same Cellular Nonlinear Network (CNN) without reconfiguration of the network. A possible electronic implementation of this CNN is finally discussed.

Resonance and chaotic trajectories in magnetic field reversed configuration

The nonlinear dynamics of a single ion in a field reversed configuration (FRC) were investigated. FRC is a toroidal fusion device which uses a specific type of magnetic field to confine ions. As a result of angular invariance, the full three-dimensional Hamiltonian system can be expressed as two coupled, highly nonlinear oscillators. Due to the high nonlinearity in the equations of motion, the behavior of the system is extremely complex, showing different regimes, depending on the values of the conserved canonical angular momentum and the geometry of the fusion vessel. Perturbation theory and averaging were used to derive an integrable Hamiltonian and frequencies of the two degrees of freedom. The derived equations were then used to find resonances and compare to Poincare surface-of-section plots. A regime was found where the nonlinear resonances were clearly separated by KAM curves. The structure of the observed island chains was explained. The condition for the destruction of KAM curves and the onset of strong chaos was derived, using Chirikov island overlap criterion, and shown qualitatively to depend both on the canonical angular momentum and geometry of the device. After a brief discussion of the adiabatic regime, the paper goes on to explore the degenerate regime that sets in at higher values of angular momenta. In this regime, the unperturbed Hamiltonian can be approximated as two uncoupled linear oscillators. In this case, the system is near-integrable, except in cases of a universal resonance, which results in large island structures, due to the smallness of nonlinear terms, which bound the resonance. The linear force constants, dominant in this regime, were derived and the geometry for a large one-to-one resonance identified. The above analysis showed good agreement with numerical simulations and was able to explain characteristic features of the dynamics.