Frequency-locked States Research Articles

Glycolytic oscillations under periodic drivings.

In many living organisms displaying circadian rhythms, the intake of energy often occurs in a periodic manner. Glycolysis is a prototypical biochemical reaction that exhibits a self-sustained oscillation under continuous injection of glucose. Here we study the effect of periodic injection of glucose on the glycolytic oscillation from a dynamical systems perspective. In particular, we employ Goldbeter's allosteric model of phosphofructokinase as a model system for glycolytic oscillations, and explore the effect of periodic substrate influx of varying frequencies and amplitudes by building the phase diagrams of Lyapunov exponents and oscillatory periods. When the frequency of driving is tuned around the harmonic and sub/super-harmonic conditions of the natural frequency, the system is entrained to a frequency-locked state, forming an entrainment band that broadens with an increasing amplitude of driving. On the other hand, if the amplitude is substantial, the system may transition, albeit infrequent, to a chaotic state which defies prediction of dynamical behaviour. Our study offers in-depth understandings into the controllability of glycolytic oscillation as well as explaining physical underpinnings that enable the synchronous oscillations among a dense population of cells.

Contrarians Synchronize beyond the Limit of Pairwise Interactions.

We give evidence that a population of pure contrarian globally coupled D-dimensional Kuramoto oscillators reaches a collective synchronous state when the interplay between the units goes beyond the limit of pairwise interactions. Namely, we will show that the presence of higher-order interactions may induce the appearance of a coherent state even when the oscillators are coupled negatively to the mean field. An exact solution for the description of the microscopic dynamics for forward and backward transitions is provided, which entails imperfect symmetry breaking of the population into a frequency-locked state featuring two clusters of different instantaneous phases. Our results contribute to a better understanding of the powerful potential of group interactions entailing multidimensional choices and novel dynamical states in many circumstances, such as in social systems.

Modified Morris–Lecar neuron model: effects of very low frequency electric fields and of magnetic fields on the local and network dynamics of an excitable media

A Morris–Lecar neuron model is considered with electric and magnetic field effects where the electric field is a time-varying sinusoid and magnetic field is simulated using an exponential flux memristor. We have shown that the exposure to electric and magnetic fields have significant effects on the neurons and has exhibited complex oscillations. The neurons exhibit a frequency-locked state for the periodic electric field and different ratios of frequency-locked states with respect to the electric field frequency is also presented. To show the impact of the electric and magnetic fields on network of neurons, we have constructed different types of network and have shown the network wave propagation phenomenon. Interestingly, the nodes exposed to both electric and magnetic fields exhibit more stable spiral waves compared to the nodes exhibited only to the magnetic fields. Also, when the number of layers is increased, the range of electric field frequency for which the layers exhibit spiral waves also increases.

Harmonic Frequency Locking in the Multifrequency Recirculating Planar Magnetron

This paper presents experimental results that demonstrate the first known instance of harmonic frequency locking in a magnetron. The prototype crossed-field high-power microwave source, termed the multifrequency recirculating planar magnetron, consisted of two planar cavity arrays having design frequencies near 1 and 2 GHz, respectively, coupled by smooth-bore electron beam recirculation bends. The magnetron was driven by the MELBA-C modified Marx generator using −300-kV pulses supplying up to 3 kA for approximately 300 ns. Using a 0.17-T axial magnetic field, the harmonic frequency-locked state simultaneously produced microwave pulses of 32± 3 MW at 0.984± 0.001 GHz and 13±2 MW at 1.970±0.002 GHz. In addition, changes in the relative phase difference between the magnetron slow wave structures (SWSs) were correlated with changes in the axial magnetic field magnitude. This correlation, along with the results from experiments that tested each individual planar cavity array in isolation, suggested the locking was facilitated through the second harmonic content of the 1-GHz modulated electron beam recirculating from the 1-to 2-GHz SWS. Analysis of shots conducted near 0.17 T showed a ±17° shot-to-shot variation in the average relative phase difference between the SWSs, and the average variation in this phase difference during a locked shot was ±8°.

Nonlinear Dynamics in Directly Modulated Semiconductor Lasers with Optical Loop Mirror Feedback

The investigation of the nonlinear dynamics of a semiconductor laser based on nonlinear optical loop mirror (NOLM) feedback using Ge doped optical fiber was carried out experimentally. Animations of compilations of the output power as a function of time series and phase plane with effects of optical feedback level, carrier current and modulation signal strength are demonstrated as a tool to give insight into the laser dynamics. Different dynamic states, including 2×, 4× multiplication and quasi-periodic and periodic frequency-locked pulsing states extended to chaotic behaviour were observed by varying the parameters of modulated frequency and optical feedback strength. The frequency-locked pulsing states were observed to exhibit a harmonic frequency-locking phenomenon and the pulsing frequency is locked to a harmonic nonlinearity in loop instead of the modulated frequency.

延迟光反馈VCSEL的混沌动力学特性

According to the dynamic model of Vertical Cavity Surface Emitting Laser(VCSEL) with delay optical feedback,its chaotic dynamic behaviors were discussed by numerical simulation.And the influence of the external cavity length and optical feedback strength on the dynamic was analyzed.The results show that laser with short external cavity will come back to frequency-locked state after chaos.Laser with long external cavity will come back to period 1 instead of the locked state after chaos.When the external cavity is 3.0 cm,with optical feedback strength increasing,laser turns into chaos by bifurcation,and it can also get period 12 window in chaos.The research results have some theoretical reference value on practical applications.

Chaos suppression and desynchronization phenomena in periodically coupled pendula subjected to localized heterogeneous forces

The suppressory effects of localized heterogeneous periodic pulses on the chaotic behaviour of sinusoidally coupled nonlinear pendula are studied. We assume that when the pendula are driven synchronously, i.e., all driving pulses have the same waveform, the chains display chaotic dynamics. It is shown that decreasing the impulse transmitted by the pulses of a minimal number of pendula results in regularization with the whole array exhibiting frequency synchronization over a wide range of coupling periods. These findings demonstrate that decreasing the impulse transmitted by localized external forces can tame chaos and lead to frequency-locked states in networks of periodically coupled dissipative systems.

Nonlinear Dynamics of Semiconductor Lasers Under Repetitive Optical Pulse Injection

In this paper, nonlinear dynamics of semiconductor lasers under repetitive optical pulse injection are studied numerically. Different dynamical states, including pulsation and oscillation states, are found by varying the intensity and the repetition rate of the injection pulses. The laser is found to enter the chaotic pulsation (CP) states and chaotic oscillation (CO) states through individual period-doubling routes. Mapping and corresponding Lyapunov exponents of these dynamical states are plotted and examined in the parameter space. Moreover, the bandwidths of the chaos states found are investigated, where the bandwidths of the CP states observed at the strong injection regime are two to four times broader than the bandwidths of the CO states found at the weak injection regime. In this paper, frequency-locked states with different winding numbers, the ratio of the oscillation frequency, and the repetition frequency of the injection pulses are also studied. Both the cases for repetition frequency above and below the relaxation oscillation frequency are examined. The winding numbers of the frequency-locked states reveal a Devil's staircase structure, where a Farey tree showing the relations between the neighboring states is constructed.

双频损耗调制掺铒光纤激光器的锁频、准周期与混沌

Dynamic behaviors of the erbium-doped fiber laser (EDFL) with dual-frequency loss modulation are experimentally investigated. Frequency-locked states with their winding numbers which form the devil's staircase are observed in this kind of lasers. In the unlocked regions, the output state changes from quasiperiodicity to chaos under increasing modulation index, which demonstrates a different route to chaos from the conventional loss-modulated EDFLs with a single modulation frequency. The chaos output in the dual-frequency loss-modulated EDFLs shows less harmonic components of the modulation frequency in the corresponding power spectrum, indicating the improvement of the randomness of the chaotic signals.

Optical generation of a precise microwave frequency comb by harmonic frequency locking

A semiconductor laser under negative optoelectronic feedback is applied to the generation of a microwave frequency comb through the nonlinear dynamics. The laser system is operated in a harmonic frequency-locked pulsing state, where its power spectrum is a microwave frequency comb that consists of multiples of a locking frequency. Every frequency component of the comb can be simultaneously stabilized by simply injecting an external microwave modulation at any component of the comb. This phenomenon can be viewed as a kind of microwave injection locking of the laser dynamics.

Singular unlocking transition in the Winfree model of coupled oscillators

The Winfree model consists of a population of globally coupled phase oscillators with randomly distributed natural frequencies. As the coupling strength and the spread of natural frequencies are varied, the various stable states of the model can undergo bifurcations, nearly all of which have been characterized previously. The one exception is the unlocking transition, in which the frequency-locked state disappears abruptly as the spread of natural frequencies exceeds a critical width. Viewed as a function of the coupling strength, this critical width defines a bifurcation curve in parameter space. For the special case where the frequency distribution is uniform, earlier work had uncovered a puzzling singularity in this bifurcation curve. Here we seek to understand what causes the singularity. Using the Poincaré-Lindstedt method of perturbation theory, we analyze the locked state and its associated unlocking transition, first for an arbitrary distribution of natural frequencies, and then for discrete systems of N oscillators. We confirm that the bifurcation curve becomes singular for a continuum uniform distribution, yet find that it remains well behaved for any finite N , suggesting that the continuum limit is responsible for the singularity.

Controlling chaos with localized heterogeneous forces in oscillator chains

The effects of decreasing the impulse transmitted by localized periodic pulses on the chaotic behavior of homogeneous chains of coupled nonlinear oscillators are studied. It is assumed that when the oscillators are driven synchronously, i.e., all driving pulses transmit the same impulse, the chains display chaotic dynamics. It is shown that decreasing the impulse transmitted by the pulses of the two free end oscillators results in regularization with the whole array exhibiting frequency synchronization, irrespective of the chain size. A maximum level of amplitude desynchrony as the pulses of the two end oscillators narrow is typically found, which is explained as the result of two competing universal mechanisms: desynchronization induced by localized heterogeneous pulses and oscillation death of the complete chain induced by drastic decreasing of the impulse transmitted by such localized pulses. These findings demonstrate that decreasing the impulse transmitted by localized external forces can suppress chaos and lead to frequency-locked states in networks of dissipative systems.

Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback

Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback are studied both numerically and experimentally. Mappings of the dynamic states and bifurcation diagrams are compared between a delayed negative optoelectronic feedback system and a delayed positive optoelectronic feedback system. Both systems follow a quasiperiodic route to chaos, where regular pulsing, quasiperiodic pulsing, and chaotic pulsing states are observed. Frequency-locked pulsing states are also found in a delayed negative optoelectronic feedback system, but not in a delayed positive optoelectronic feedback system. These frequency-locked pulsing states are experimentally observed to exhibit a harmonic frequency-locking phenomenon, where the pulsing frequency is locked to a harmonic of the delay loop frequency instead of the delay loop frequency itself. The rotation numbers of these frequency-locked pulsing states show a Devil's staircase structure.

Transition from oscillatory to excitable regime in a system forced at three times its natural frequency.

The effect of a temporal modulation at three times the critical frequency on a Hopf bifurcation is studied in the framework of amplitude equations. We consider a complex Ginzburg-Landau equation with an extra quadratic term, resulting from the strong coupling between the external field and the unstable modes. We show that, by increasing the intensity of the forcing, one passes from an oscillatory regime to an excitable one with three equivalent frequency-locked states. In the oscillatory regime, topological defects are one-armed phase spirals, while in the excitable regime they correspond to three-armed excitable amplitude spirals. Analytical results show that the transition between these two regimes occurs at a critical value of the forcing intensity. The transition between phase and amplitude spirals is confirmed by numerical analysis and it might be observed in periodically forced reaction-diffusion systems.

Flow transitions of a low-Prandtl-number fluid in an inclined 3D cavity

We present a numerical and theoretical investigation on the natural convection of a low Prandtl number fluid ( Pr=0.025) in 2D and 3D side-heated enclosures tilted α=80° with respect to the vertical position. The choice of this inclination angle comes from a previous linear stability analysis of the basic (plane-parallel) flow that predicts the same critical Ra for longitudinal oscillatory and stationary transversal modes. In both the 2D and 3D enclosures the first transition gradually leads to a transversal stationary centered shear roll. In the 2D geometry the flow becomes time-dependent and multicellular (3 rolls) at the onset of a Hopf bifurcation, followed by subsequent period-doubling. On the other hand, in the 3D enclosure, the onset of oscillations is due to a fully three-dimensional standing wave composed of three counter-rotating longitudinal rolls. The further evolution of the 3D flow qualitatively agrees with previous experiments (J. Crystal Growth, 102 (1990) pp. 54–68): a quasiperiodic flow followed by a frequency locked state. The main contribution of this work is the analysis of the flow structure underlying the secondary frequency: a transversal wave composed of two shear rolls that coexist with the three longitudinal cells. This is the first numerical work that explicitly illustrates this scenario which was suggested at the onset of the biperiodic regime in many of the previous experiments.

Clustering of noise-induced oscillations.

The subject of our study is clustering in a population of excitable systems driven by Gaussian white noise and with randomly distributed coupling strength. The cluster state is frequency-locked state in which all functional units run at the same noise-induced frequency. Cooperative dynamics of this regime is described in terms of effective synchronization and noise-induced coherence.

Transition to Chaos in the Rayleigh-Bénard Convection

Direct numerical simulations of the Rayleigh-Benard convection in a rectangular box are run using the MAC method at the assigned values of the external parameters such that the Prandtl number P r = 5 and the box aspect ratios Γ x =3.5 and Γ y =2. The grid consists of 84 ×48 ×24 uniformly divided cubic cells. The results show that with increase of the Rayleigh number the oscillatory motion undergoes successive transitions in such a way as mono-periodic → bi-periodic → frequency-locked state → chaotic, which are in favorable agreement with the available experimental results due to Gollub and Benson obtained under the same settup of the external parameters P r , Γ x , and Γ y .

Phase locking of Josephson junction arrays achieved by a non-traditional bias scheme

We propose to modify the usual biasing scheme of a two dimensional Josephson junction array by connecting the wires in a special cross-type fashion. Numerical simulations show that a simple two dimensional circuit of Josephson junctions with the new biasing scheme, to the contrary of the traditional scheme, one can achieve a stable frequency locked state without the use of an external load. We also show that four of these circuits can be series connected and that the output voltage is further increased. The new scheme may dramatically improve the performance of Josephson junction microwave oscillators.

Simplification of the quasiperiodic route to chaos in agonist-induced vasomotion by iterative circle maps.

We have shown that the patterns of vasomotion induced by histamine in isolated rabbit ear resistance arteries can be described in terms of iterative circle maps that model the dynamics of coupled nonlinear oscillators. Cyclopiazonic acid (CPA), an inhibitor of the sarcoplasmic reticulum Ca(2+)-adenosinetriphosphatase pump, consistently transformed chaotic behavior into characteristic periodic oscillations known as mixed-mode responses, which consist of mixtures of large- and small-amplitude excursions and represent frequency-locked states. Quasiperiodicity, which reflects the interaction of oscillators with incommensurate frequencies, was also observed, although in a smaller number of experiments. The patterns of mixed-mode complexes found at different CPA concentrations allowed the derivation of firing numbers, i.e., number of large oscillations/sum of number of small and large oscillations, and the sequences in which they emerged conformed to Farey arithmetic. Two-dimensional return maps derived by Poincaré section of phase space representations of the dynamics were used to compute the mean number of rotations per iteration on the circle, i.e., the winding number. Plots of winding number against firing number revealed a devil's staircase-type structure. Experiments with verapamil, a voltage-operated L-type Ca(2+)-channel antagonist, confirmed that influx of extracellular Ca2+ was essential to sustain chaos, quasiperiodicity, and mixed-mode responses. Nonlinear coupling between cytosolic and membrane events in rabbit ear arteries thus results in a self-organized dynamics that collapses to that predicted by the theory of simple circle maps.

A model for phase transitions in human hand movements during multifrequency tapping

In bimanual tapping, abrupt transitions between frequency ratios were observed when movement frequency was gradually increased. The transition routes showed individual tendencies, not necessarily in agreement with predictions derived from the sine circle map. Therefore, a more detailed theoretical model of coupled oscillators was developed. In the model the interaction function is a polynomial of coupling terms which allow for specific frequency locks. The magnitudes of these coupling terms are related to the amplitude of oscillation and the order of the frequency lock. Because increase in movement frequency is associated with a drop in amplitude, it results in differential loss of stability of the allowed frequency ratios. New frequency-locked states may be attained by detuning the stiffness parameters of the component oscillators. The model accounts for both free-running solutions and individual tendencies in transition routes. The relative weights of the coupling terms are influenced by practice and intention.