Chaotic Relaxation Oscillations Research Articles

Nonlinear dynamics of a self-mixing thin-slice solid-state laser subjected to Doppler-shifted optical feedback.

Chaotic oscillations of a linearly polarized single longitudinal-mode thin-slice Nd:GdVO_{4} laser placed in a self-mixing laser Doppler velocity scheme were dynamically characterized in terms of the intensity probability distribution, joint time-frequency analysis, and short-term Fourier transformation of temporal evolutions, and the degree of disorder in the amplitude and phase of the long-term temporal evolutions. The transition from chaotic relaxation oscillations (ROs) to chaotic spiking oscillations (SOs) was explored via the chaotic itinerancy (CI) regime by increasing the feedback ratio toward the laser from a rotating scattering object. The intensity probability distribution was found to change from an exponential decay in the RO regime to an inverse power law in the SO regime, which manifests itself in self-organized critical behavior, while stochastic subharmonic frequency locking among the two periodicities of RO and SO takes place in the CI regime featuring quantum-noise (spontaneous-emission)-induced order in the amplitude and phase of the spiking oscillations. All of the experimental results were reproduced by numerical simulations of a model equation of a single-mode self-mixing solid-state laser subjected to Doppler-shifted optical feedback from a rotating scattering object.

Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem

<p style='text-indent:20px;'>We consider an ecological model consisting of two species of predators competing for their common prey with explicit interference competition. With a proper rescaling, the model is portrayed as a singularly perturbed system with one fast (prey dynamics) and two slow variables (dynamics of the predators). The model exhibits a variety of rich and interesting dynamics, including, but not limited to mixed-mode oscillations (MMOs), featuring concatenation of small and large amplitude oscillations, relaxation oscillations and bistability between a semi-trivial equilibrium state and a coexistent oscillatory state. More interestingly, in a neighborhood of <i>singular Hopf</i> bifurcation, long lasting transient dynamics in the form of chaotic MMOs or relaxation oscillations are observed as the system approaches the periodic attractor born out of supercritical Hopf bifurcation or a semi-trivial equilibrium state respectively. The transient dynamics could persist for hundreds or thousands of generations before the ecosystem experiences a regime shift. The time series of population cycles with different types of irregular oscillations arising in this model stem from a biological realistic feature, namely, by the variation in the intraspecific competition amongst the predators. To explain these oscillations, we use bifurcation analysis and methods from <i>geometric singular perturbation theory</i>. The numerical continuation study reveals the rich bifurcation structure in the system, including the existence of codimension-two bifurcations such as fold-Hopf and generalized Hopf bifurcations.

Quasi-periodic and chaotic relaxation oscillations in a laser model with variable delayed optoelectronic feedback

We derive asymptotically discrete mappings which determine the dynamics of relaxation spikes in a model of laser with optoelectronic feedback in the pump circuit. The time delay in the feedback path varies periodically. Bifurcations of mapping’s attractors correspond to the appearance of quasi-periodic and chaotic spiking with special properties.

Spontaneous Slow Oscillations and Sequential Patterns Due to Short-Term Plasticity in a Model of the Cortex

We study a realistic model of a cortical column taking into account short-term plasticity between pyramidal cells and interneurons. The simulation of leaky integrate-and-fire neurons shows that low-frequency oscillations emerge spontaneously as a result of intrinsic network properties. These oscillations are composed of prolonged phases of high and low activity reminiscent of cortical up and down states, respectively. We simplify the description of the network activity by using a mean field approximation and reduce the system to two slow variables exhibiting some relaxation oscillations. We identify two types of slow oscillations. When the combination of dynamic synapses between pyramidal cells and those between interneurons accounts for the generation of these slow oscillations, the end of the up phase is characterized by asynchronous fluctuations of the membrane potentials. When the slow oscillations are mainly driven by the dynamic synapses between interneurons, the network exhibits fluctuations of membrane potentials, which are more synchronous at the end than at the beginning of the up phase. Additionally, finite size effect and slow synaptic currents can modify the irregularity and frequency, respectively, of these oscillations. Finally, we consider possible roles of a slow oscillatory input modeling long-range interactions in the brain. Spontaneous slow oscillations of local networks are modulated by the oscillatory input, which induces, notably, synchronization, subharmonic synchronization, and chaotic relaxation oscillations in the mean field approximation. In the case of forced oscillations, the slow population-averaged activity of leaky integrate-and-fire neurons can have both deterministic and stochastic temporal features. We discuss the possibility that long-range connectivity controls the emergence of slow sequential patterns in local populations due to the tendency of a cortical column to oscillate at low frequency.

A Nonlinear Theory of the Kuroshio Extension Bimodality

Abstract The Kuroshio Extension (KE) flow in the North Pacific Ocean displays a very distinctive decadal variability of bimodal character involving two completely different states (a large-meander “elongated” state and a small-meander “contracted” state) connected by very asymmetric temporal transitions. Although such a flow has been widely studied by means of a suite of mathematical models and by using several observational platforms, a satisfactory theoretical framework answering quite elementary questions is still lacking, the main question being whether such variability is induced by a time-varying wind forcing or, rather, by intrinsic oceanic mechanisms. In this context, the chaotic relaxation oscillation produced by a process-oriented model of the KE low-frequency variability, with steady climatological wind forcing, was recently recognized to be in substantial agreement with altimeter data. Here those model results are further compared with a comprehensive altimeter dataset. The positive result of such a comparison allows the conclusion that a minimal model for the KE bimodality has been identified and that, consequently, nonlinear intrinsic oceanic mechanisms are likely to be the main cause of the observed variability. By applying the methods of nonlinear dynamical systems theory, relevant dynamical features of the modeled flow are then explained, such as the origin of the relaxation oscillation as a consequence of a homoclinic bifurcation, the spatiotemporal character of the bimodal behavior, and the degree of predictability of the flow in the different stages of the oscillation (evaluated through a field of finite-time Lyapunov exponents and the corresponding Lagrangian time series).

On the Crucial Role of Basin Geometry in Double-Gyre Models of the Kuroshio Extension

Abstract The decadal chaotic relaxation oscillation obtained in a recent double-gyre model study of the Kuroshio Extension intrinsic low-frequency variability was found to compare surprisingly well with the real variability of the jet as revealed by altimeter data, despite the high degree of idealization of the model. In this note it is shown that elements of realism in the basin geometry, present in that study and absent in previous double-gyre models applied to the Kuroshio Extension, play a crucial role in shaping the low-frequency variability of the jet, and can explain the good performance of the model. A series of numerical experiments with different basin geometries of increasing degrees of simplicity are analyzed. If the schematic western boundary representing the coastline south of Japan is removed, the strong decadal variability completely disappears and only a very weak periodic oscillation about an elongated state of the jet is found. If the large zonal width of the basin (representing correctly the extension of the North Pacific Ocean) is reduced by a half, then the total meridional Sverdrup transport is reduced by the same factor, and so is the intensity of the Kuroshio and Oyashio western boundary currents: as a result, the modeled Kuroshio Extension is totally unrealistic in shape and is steady. If both simplifications are introduced the resulting jet is, again, totally unrealistic, yielding a weak periodic bimodal cycle. On the basis of these results, two main conclusions are drawn: (i) the introduction of appropriate geometrical elements of realism in double-gyre model studies of the Kuroshio Extension is essential, and (ii) the Kuroshio Extension intrinsic low-frequency variability would be dramatically different if the southwestern coastline of Japan were more meridionally oriented.

Nonlinear dynamics in thin-slice Nd:YAG ceramic lasers: Coupled local-mode laser model

The authors propose a coupled spatially distributed local-mode laser model for understanding self-induced high-speed modulations in Nd:YAG ceramic lasers with laser-diode end pumping, which critically depend on pump positions. In addition to complicated high-speed modulations, quasiperiodic and chaotic relaxation oscillations have been demonstrated experimentally and reproduced by numerical simulation of the model of coupled local-mode lasers. Q-switching-like periodic spiking pulsations have been also observed and reproduced numerically by assuming saturable absorber type of inclusions in grain boundaries.

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.

Burst synchronization in two thin-slice solid-state lasers incoherently coupled face to face

We studied the dynamic behavior of laser-diode-pumped nonidentical thin-slice solid-state lasers, coupled face to face, with orthogonally polarized emissions. When such an incoherent mutual optical coupling was introduced, the coupled lasers exhibited slow fluctuations of transverse-mode patterns, and isolated lasers exhibited stable transverse-mode patterns. When one laser exhibited nonorthogonal multi-transverse-mode operations without coupling, simultaneous random bursts of chaotic relaxation oscillations took place in both lasers over time with coupling. A plausible physical interpretation is proposed in terms of the simultaneous excitation of chaotic relaxation oscillations in both lasers through resonances that stem from interference-induced modulation of one laser at a swept beat frequency of the fluctuating nonorthogonal mode pair. Observed instabilities were well reproduced by numerical simulation of the model equation.

Hard transition to chaotic dynamics in Alfvén wave fronts

The derivative nonlinear Schrödinger (DNLS) equation, describing propagation of circularly polarized Alfvén waves of finite amplitude in a cold plasma, is truncated to explore the coherent, weakly nonlinear, cubic coupling of three waves near resonance, one wave being linearly unstable and the other waves damped. In a reduced three-wave model (equal dampings of daughter waves, three-dimensional flow for two wave amplitudes and one relative phase), no matter how small the growth rate of the unstable wave there exists a parametric domain with the flow exhibiting chaotic relaxation oscillations that are absent for zero growth rate. This hard transition in phase-space behavior occurs for left-hand (LH) polarized waves, paralleling the known fact that only LH time-harmonic solutions of the DNLS equation are modulationally unstable, with damping less than about (unstable wave frequency)2/4×ion cyclotron frequency. The structural stability of the transition was explored by going into a fully 3-wave model (different dampings of daughter waves, four-dimensional flow); both models differ in significant phase-space features but keep common features essential for the transition.

Observation of laser transverse modes analogous to a SU(2) wave packet of a quantum harmonic oscillator

We report the generation of a new type of laser transverse mode that is analogous to a SU(2) elliptical wave packet of a quantum harmonic oscillator. Experimental results show that using a doughnut pump profile to excite an isotropic microchip laser in a spherical cavity can generate the elliptical transverse modes. The formation of elliptical transverse modes is found to be a spontaneous locking process of Hermite-Gaussian modes within the same family. The chaotic relaxation oscillation caused by the interaction of two nearly degenerate elliptical modes is also observed.

Convective flows in rapidly rotating spheres and their dynamo action

The present understanding of convection in rotating spherical fluid shells is reviewed and some recent results on coherent structures and magnetic field generation are considered in detail. The constraints exerted by the Coriolis force leads to unusual properties not found in nonrotating systems, such as vacillations, localized convection, and chaotic relaxation oscillations. The central role played by the differential rotation generated by the Reynolds stress of convection in the case of Prandtl numbers of the order unity or less is emphasized. Magnetic fields generated through the dynamo process offer new degrees of freedom. Through the braking of the differential rotation the Lorentz force contributes to an enhanced heat transport.

Formation of optical vortex lattices in solid-state microchip lasers: Spontaneous transverse mode locking

We experimentally investigate pattern formation in a solid-state microchip laser with a large Fresnel number. Controlling the reflectivity of the output coupler can generate the stable square pattern of optical vortex lattices. The formation of square vortex lattices is found to be a spontaneous process of transverse mode locking within almost-degenerated mode families. The frequency of self-induced oscillation in square vortex lattices agrees well with the numerical calculation. The chaotic relaxation oscillation is found in square vortex lattices due to the multi-longitudinal-mode operation.

Multivalued stochastic resonance in a model of an excitable neuron

We study the effect of the periodic small signals and noise in the Hindmarsh–Rose model of neuron. We found that the stochastic and the chaotic relaxation oscillations can transform into qualitatively different periodic oscillations under the action of the same small signal. The type of the steady oscillations depends on the moment the signal is turned on. The obtained results might be used to qualitatively explain ambiguity of the image sensing by the human eye.

Instabilities in a laser-diode-pumped microchip solid-state laser with feedback

Random chaotic relaxation oscillation burst generations have been observed in laser-diode-pumped microchip multimode lasers coupled to a single-mode optical ber. Dynamical characterizations have been carried out by the singular value decomposition method and the joint time-frequency analysis, and the observed instability has been identi ed to be a random switching between stable and chaotic operation. @S1050-2947~99!50511-8#

Observation of Fibre Laser Instabilities Through Modulation of the Cavity Optical Length

Abstract A Nd-doped, single-mode, double-clad fibre laser (FL) is coiled around a resonant piezoelectric ceramic ring; this induces modulation of the cavity optical length and two distinct classes of laser intensity instabilities are observed as the ring drive voltage V P-P is increased. The first class involves a slowly varying intensity, with characteristic time scale in the microsecond range. For fixed drive-voltage frequency f m ≈ 20 kHz, we consistently observe subharmonics f m/2 and f m/4 in the FL output intensity spectrum; noise in the system precludes the observation of higher subharmonic bifurcations. Small-amplitude random relaxation oscillations of the unmodulated FL are strongly amplified as V P-P is increased and transition to a state of large-amplitude chaotic relaxation oscillations occurs for V P-P ≈ 4 V; a value of 2·6 has been measured for the correlation dimension of the chaotic attractor. The second class of laser instability occurs simultaneously for V P-P ≳ 10 V, and the FL output c...

Stochastic and chaotic relaxation oscillations

For relaxation oscillators stochastic and chaotic dynamics are investigated. The effect of random perturbations upon the period is computed. For an extended system with additional state variables chaotic behavior can be expected. As an example, the Van der Pol oscillator is changed into a third-order system admitting period doubling and chaos in a certain parameter range. The distinction between chaotic oscillation and oscillation with noise is explored. Return maps, power spectra, and Lyapunov exponents are analyzed for that purpose.

Instabilities in Semiconductors Including Chaotic Phenomena

A dynamic systems approach to nonlinear charge transport and electrical instabilities in semiconductors is presented, and illustrated by specific examples of physical transport mechanisms which yield oscillatory instabilities: Impact ionization of shallow impurities at liquid helium temperature as a mechanism for selfgenerated chaotic dielectric relaxation oscillations; Poole-Frenkel emission from deep levels at room-temperature as a mechanism for low-frequency oscillations in semiinsulating GaAs; Impact ionization and optical phonon emission as a mechanism for global bifurcations and hysteresis of oscillations in p-Ge; Breathing current filaments.

Nonequilibrium phase transitions and chaos in semiconductors

Generation-recombination-induced nonequilibrium phase transitions in semiconductors were first noted by Landsberg and Pimpale (1976) [ J. Phys. C 9, 1243]. During the past decade the view of a semiconductor as a nonlinear dynamic system governed by generation-recombination processes has stimulated much theoretical and, more recently, experimental work. We briefly review these developments, which include first- and second-order phase transitions (breakdown and threshold switching), the spontaneous emergence of spatial structures (current filamentation), and chaos, and discuss three recent examples, viz. Auger recombination induced tristability, optically controlled threshold switching and current filamentation, and chaotic dielectric relaxation oscillations.

Origin of Chaotic Relaxation Oscillations in an Optically Pumped Molecular Laser

We show that pump-induced modifications to the gain and dispersion profiles can induce a weak side-band modulation which acts as a driving term to sustain chaotic relaxation oscillations in a single-mode, resonant, optically pumped molecular laser. Our study suggests that a careful two- or more-parameter unfolding of higher-codimension bifurcations should lead to a wealth of nonlinear dynamical behavior within easy access of current experiments.