Large-amplitude Relaxation Oscillations Research Articles

Experimental investigation of the period-adding bifurcation route to chaos in plasma.

Since the characteristic timescales of the various transport processes inside the discharge plasma span several orders of magnitude, it can be regarded as a typical fast-slow system. Interestingly, in this work, a special kind of complex oscillatory dynamics composed of a series of large-amplitude relaxation oscillations and small-amplitude near-harmonic oscillations, namely, mixed-mode oscillations (MMOs), was observed. By using the ballast resistance as the control parameter, a period-adding bifurcation sequence of the MMOs, i.e., from L^{s} to L^{s+1}, was obtained in a low-pressure DC glow discharge system. Meanwhile, a series of intermittently chaotic regions caused by inverse saddle-node bifurcation was embedded between the two adjacent periodic windows. The formation mechanism of MMOs was analyzed, and the results indicated that the competition between electron production and electron loss plays an important role. Meanwhile, the nonlinear time series analysis technique was used to study the dynamic behavior quantitatively. The attractor in the reconstructed phase space indicated the existence of the homoclinic orbits of type Γ^{-}. In addition, by calculating the largest Lyapunov exponent (LLE), the chaotic nature of these states was confirmed and quantitatively characterized. With the decrease in the ballast resistance, the return map of the chaotic state gradually changed from the nearly one-dimensional single-peak structure to the multibranch structure, which indicates that the dissipation of the system decreased. By further calculating the correlation dimension, it was shown that the complexity of the strange attractors increased for higher-order chaotic states.

Intermittent Chaos in the CSTR Bray-Liebhafsky Oscillator-Specific Flow Rate Dependence.

Dynamic states with intermittent oscillations consist of a chaotic mixture of large amplitude relaxation oscillations grouped in bursts, and between them, small-amplitude sinusoidal oscillations, or even the quiescent parts, known as gaps. In this study, intermittent dynamic states were generated in Bray–Liebhafsky (BL) oscillatory reaction in an isothermal continuously-fed, well-stirred tank reactor (CSTR) controled by changes of specific flow rate. The intermittent states were found between two regular periodic states and obtained for specific flow rate values from 0.020 to 0.082 min−1. Phenomenological analysis based on the quantitative characteristics of intermittent oscillations, as well as, the largest Lyapunov exponents calculated from experimentally obtained time series, both indicated the same type of behavior. Namely, fully developed chaos arises when approaching to the vertical asymptote which is somewhere between two bifurcations. Hence, this study proposes described route to fully developed chaos in the Bray-Liebhafsky oscillatory reaction as an explanation for experimentally observed intermittent dynamics. This is in correlation with our previously obtained results where the most chaotic intermittent chaos was achieved between the periodic oscillatory dynamic state and stable steady state, generated in BL under CSTR conditions by varying temperature and inflow potassium iodate concentration. Moreover, it was shown that, besides the largest Lyapunov exponent, analysis of chaos in experimentally obtained intermittent states can be achieved by a simpler approach which involves using the quantitative characteristics of the BL reaction evolution, that is, the number and length of gaps and bursts obtained for the various values of specific flow rates.

Three-dimensional doubly diffusive convectons: instability and transition to complex dynamics

Three-dimensional doubly diffusive convection in a closed vertically extended container driven by competing horizontal temperature and concentration gradients is studied by a combination of direct numerical simulation and linear stability analysis. No-slip boundary conditions are imposed on all six container walls. The buoyancy number $N$ is taken to be $-1$ to ensure the presence of a conduction state. The primary instability is subcritical and generates two families of spatially localized steady states known as convectons. The convectons bifurcate directly from the conduction state and are organized in a pair of primary branches that snake within a well-defined range of Rayleigh numbers as the convectons grow in length. Secondary instabilities generating twist result in secondary snaking branches of twisted convectons. These destabilize the primary convectons and are responsible for the absence of stable steady states, localized or otherwise, in the subcritical regime. Thus all initial conditions in this regime collapse to the conduction state. As a result, once the Rayleigh number for the primary instability of the conduction state is exceeded, the system exhibits an abrupt transition to large-amplitude relaxation oscillations resembling bursts with no hysteresis. These numerical results are confirmed here by determining the stability properties of both convecton types as well as the domain-filling states. The number of unstable modes of both primary and secondary convectons of different lengths follows a pattern that allows the prediction of their stability properties based on their length alone. The instability of the convectons also results in a dramatic change in the dynamics of the system outside the snaking region that arises when the twist instability operates on a time scale faster than the time scale on which new rolls are nucleated. The results obtained are expected to be applicable in various pattern-forming systems exhibiting localized structures, including convection and shear flows.

Exploring the Pullback Attractors of a Low-Order Quasigeostrophic Ocean Model: The Deterministic Case

AbstractA low-order quasigeostrophic double-gyre ocean model is subjected to an aperiodic forcing that mimics time dependence dominated by interdecadal variability. This model is used as a prototype of an unstable and nonlinear dynamical system with time-dependent forcing to explore basic features of climate change in the presence of natural variability. The study relies on the theoretical framework of nonautonomous dynamical systems and of their pullback attractors (PBAs), that is, of the time-dependent invariant sets attracting all trajectories initialized in the remote past. The existence of a global PBA is rigorously demonstrated for this weakly dissipative nonlinear model. Ensemble simulations are carried out and the convergence to PBAs is assessed by computing the probability density function (PDF) of localization of the trajectories. A sensitivity analysis with respect to forcing amplitude shows that the PBAs experience large modifications if the underlying autonomous system is dominated by small-amplitude limit cycles, while less dramatic changes occur in a regime characterized by large-amplitude relaxation oscillations. The dependence of the attracting sets on the choice of the ensemble of initial states is then analyzed. Two types of basins of attraction coexist for certain parameter ranges; they contain chaotic and nonchaotic trajectories, respectively. The statistics of the former does not depend on the initial states whereas the trajectories in the latter converge to small portions of the global PBA. This complex scenario requires separate PDFs for chaotic and nonchaotic trajectories. General implications for climate predictability are finally discussed.

Intermittent chaos in the Bray-Liebhafsky oscillator. Temperature dependence.

Intermittent oscillations as a chaotic mixture of large amplitude relaxation oscillations, grouped in bursts and small-amplitude sinusoidal ones or even quiescent parts between them known as gaps, were found and examined in the Bray-Liebhafsky (BL) reaction performed in CSTR under controlled temperature variations. They were obtained in a narrow temperature range from 61.0 °C to 63.1 °C, where 61.0 °C is the critical temperature for burst emergence from the stable steady state and 63.1 °C is the critical temperature for gap emergence from regular oscillations. Since intermittencies appear gradually from the regular oscillatory state, and no hysteresis was obtained with decreasing/increasing temperature in the vicinity of these two bifurcations, a linear relationship between (τB/τ)(2) and (τG/τ)(2) (where τB, τG and τ denotes duration of bursts, gaps, and whole experiment, respectively), as a function of the temperature as the control parameter, was expected and obtained. Although these intermittent oscillations are chaotic with respect to the lengths of individual gaps as well as bursts, their deterministic behavior related to temperature was additionally established. Thus, the number of bursts or gaps per unit of time (NB/τ and NG/τ) has the form of a normal distribution function over the temperature range in the region where intermittencies are obtained. Temperature dependence of the Lyapunov exponents was also described by a function of the normal distribution form. Hence, we established some regularities in the chaotic behavior of intermittent oscillations that are common in life but difficult for determinations.

Canard cycles in aircraft ground dynamics

The loss of lateral stability of an aircraft turning on the ground is associated with a rapid transition from small-amplitude oscillations to large-amplitude relaxation oscillations over a very small parameter range. This phenomenon is shown to correspond to a canard explosion, which is known to be an important feature of systems that exhibit a separation of time scales. While the industry-tested model of aircraft ground dynamics that we consider does not feature an explicit splitting of time scales, we show that, locally near the transition from stable turning to laterally unstable behaviour, the forward speed of the aircraft acts as a slow variable. The associated family of canard cycles is identified, and the canard explosion is shown to be directly related to the successive saturation of tyre forces at the two main landing gears. We present a canonical two-dimensional slow–fast vector field model that captures the key features of this type of canard explosion; it differs from the canard explosion in the archetypical Van der Pol system in terms of the shape of the associated critical manifold.

Malonic acid concentration as a control parameter in the kinetic analysis of the Belousov–Zhabotinsky reaction under batch conditions

The influence of the initial malonic acid concentration [MA]0 (8.00 x 10(-3) < or = [MA]0 < or = 4.30 x 10(-2) mol dm(-3)) in the presence of bromate (6.20 x 10(-2) mol dm(-3)), bromide (1.50 x 10(-5) mol dm(-3)), sulfuric acid (1.00 mol dm(-3)) and cerium sulfate (2.50 x 10(-3) mol dm(-3)) on the dynamics and the kinetics of the Belousov-Zhabotinsky (BZ) reactions was examined under batch conditions at 30.0 degrees C. The kinetics of the BZ reaction was analyzed by the earlier proposed method convenient for the examinations of the oscillatory reactions. In the defined region of parameters where oscillograms with only large-amplitude relaxation oscillations appeared, the pseudo-first order of the overall malonic acid decomposition with a corresponding rate constant of 2.14 x 10(-2) min(-1) was established. The numerical results on the dynamics and kinetics of the BZ reaction, carried out by the known skeleton model including the Br2O species, were in good agreement with the experimental ones. The already found saddle node infinite period (SNIPER) bifurcation point in transition from a stable quasi-steady state to periodic orbits and vice versa is confirmed by both experimental and numerical investigations of the system under consideration. Namely, the large-amplitude relaxation oscillations with increasing periods between oscillations in approaching the bifurcation points at the beginning and the end of the oscillatory domain, together with excitability of the stable quasi-steady states in their vicinity are obtained.

Wavelength switching dynamics in injected fiber ring lasers

We experimentally and theoretically investigate the wavelength switching dynamics of an injection-seeded Er doped fiber ring laser. The fiber ring laser is seeded by an external cavity laser diode that is amplitude modulated with a long period 50% duty cycle wave form so that the return to steady state can be investigated. It is found that the laser suffers from severe large-amplitude relaxation oscillations and slow decay times that limit the available wavelength switching speed.

BIFURCATION AND CHAOTIC TRANSITION TO RELAXATION OSCILLATIONS IN AN INDUCTIVELY DRIVEN ELECTRONEGATIVE PLASMA

Regular and chaotic relaxation oscillations in charged particle densities, light and floating potential are seen in low-pressure inductive discharges, in the transition between lower power capacitive operation and higher power inductive operation, if the plasma is electronegative, i.e. contains negative ions. As pressure or power is varied to cross a threshold, either from stable capacitive or stable inductive operation, the instability goes through a series of oscillatory, sometimes chaotic, states to large scale relaxation oscillations between higher and lower densities. A volume-averaged model, to describe the instability, contains time varying densities of charged species and temperature, indicating that the separation of the time scales of electron and negative ion motion is the driver of the complicated dynamics. The particle and energy balance equations are integrated to produce the dynamical behavior. A phase plane description of negative ions versus electrons is used to gain understanding of the instability. The theory shows that, depending on specific choices of parameters, as power is increased, the oscillations, born at Hopf bifurcation, can grow to a large amplitude relaxation oscillations, sometimes traversing chaotically between basins of attraction. The model qualitatively agrees with experimental observations which show the same range of behaviors, but with quantitative differences to be explained.

Multimodal regimes in a compartmental model of the dopamine neuron

We study chains of strongly electrically coupled relaxation oscillators modeling dopamine neurons. When individual oscillators are in the regime close to an Andronov–Hopf bifurcation (AHB), the coupled system exhibits a variety of oscillatory behavior. We show that the proximity of individual oscillators to the AHB has a significant impact on the system dynamics in a wide range of parameters. It manifests itself through a family of stable multimodal periodic solutions that are composed out of large-amplitude relaxation oscillations and small-amplitude oscillations. This family of solutions has a rich bifurcation structure. The waveform and the period vary greatly across the family. The structure and bifurcations of the stable periodic solutions of the coupled system are investigated using numerical and analytic techniques.

Large Amplitude pH Oscillations in the Hydrogen Peroxide−Dithionite Reaction in a Flow Reactor

The kinetics of the reaction between S2O4 2- and H2O2 is found to be very complex both in a closed reactor and in a continuous-flow stirred tank reactor (CSTR) in an unbuffered aqueous solution. The main products of the oxidation are SO4 2- and S2O6 2- in excess H2O2. The reaction is accompanied by acidification of the initially alkaline mixture. The measured pH-time traces show multiple inflection points in a closed reactor, suggesting that the reaction takes place in several distinct steps. Transient formation of SO 3 2- /HSO3 - buffer system is observed in the early stage of the reaction. Next, the sulfite ions are oxidized further to sulfate ions in an autocatalytic reaction. Simultaneously a small amount of dithionate ions is also formed. The pH exhibits very large amplitude relaxation oscillations in a CSTR in a narrow range of input concentrations, flow rate, and temperature. A simple empirical rate law model consisting of three redox reactions and three protonation equilibria is proposed. Numerical simulations based on this model agree well with the experimental results. The key to the pH-regulated oscillation is the H + -autocatalysis that results from the more rapid oxidation of the protonated than of the unprotonated sulfite intermediate.

On the nature of the spontaneous oscillations observed for the Koper-Sluyters electrocatalytic reaction

Abstract The study of an electrocatalytic mechanism is presented. Closed form solutions are given for the steady-state behaviour, facilitating the linear stability analysis. It is shown that two types of oscillations may be exhibited for this self-oscillating electrocatalytic reaction: small amplitude harmonic oscillations and large amplitude relaxation oscillations. The existence of these oscillations and their form depend on the reaction conditions. The Hopf bifurcation is supercritical and small amplitude harmonic oscillations exist only over a very narrow range of electrode potentials. The electrocatalytic reaction behaviour is that of a so-called ‘canard’.

Megahertz-repetition-rate Nd:glass fiber laser

Stable trains of 100-200-ns-long pulses are obtained at 0.46-1.06-MHz repetition rates in a Nd:glass fiber laser by driving large-amplitude relaxation oscillations at or below their resonance. Numerical simulations agree qualitatively with temporal profiles of the output power. We show that the maximum pulse-repetition rate of a relaxation-oscillation laser increases as the square root of pump power at low P and that its theoretical upper limit is close to the pulse rate of a mode-locked laser.

Diode-pumped Nd:YLF all-in-one laser

A high-power longitudinally end-diode-pumped Nd:YLF all-in-one laser has been constructed and tested. The compact single-cavity laser generates pulses with durations of 28 ps. Pulse energies of ~120 or -500 microJ are measured at a 1-kHz repetition rate when the all-in-one laser is pumped with a 3- or 15-W diode laser, respectively. Controlled prelasing is utilized to prevent the buildup of large-amplitude relaxation oscillations and to minimize the prelasing leakage. Pulse shaping in the all-in-one laser is analyzed in terms of the Kuizenga-Siegman theory.

Variability of intensity of interstellar maser lines due to induced Compton scattering

Induced (nonlinear) Compton interaction between spectrally narrow and incoherent radiation, corresponding to interstellar maser lines (OH and H/sub 2/O) with a high brightness temperature (T/sub b/approx. =10/sup 15/ K), and the surrounding rarefied plasma is considered in order to interpret the strong variability of intensity for unsaturated maser lines. The narrow amplified line of width deltav exhibits large-amplitude relaxation oscillations, generating at each oscillation a broad satellite at the Compton-Doppler distance >>deltav, on the longward side of the line, which behaves as a photon soliton moving downward on the frequency axis at constant speed and constant amplitude.