Instability control in microwave-frequency microplasma

This paper introduces comprehensive large-signal analyses of modulation dynamics and noise of a chaotic semiconductor laser. The chaos is induced by operating the laser under optical feedback (OFB). Control of the chaotic dynamics and possibility of suppressing the associated noise by sinusoidal modulation are investigated. The studies are based on numerical solutions of a time-delay rate equation model. The deterministic modulation dynamics of the laser are classified into seven regular and irregular dynamic types. Variations of chaotic dynamics and noise with sinusoidal modulation are examined in both time and frequency domains over wide ranges of the modulation depth and frequency. The results showed that chaotic dynamics can be converted into five distinct dynamic types; namely, continuous periodic signal (CPS), continuous periodic signal with relaxation oscillations (CPSRO), periodic pulse (PP), periodic pulse with relaxation oscillations (PPRO) and periodic pulse with period doubling (PPPD). The relative intensity noise (RIN) of these types is characterized when the modulation frequencies are much lower, comparable to, and higher than the resonance frequency. Suppression of RIN to a level 8 dB/Hz higher than the quantum limit was predicted under the CPS type when the modulation frequency is 0.9 times the resonance frequency and the modulation depth is 0.14.     

Control of basins of attraction in a multistable fiber laser

We study how the basins of attraction of coexisting states can be controlled by either harmonic modulation or small noise applied to the pump parameter in a multistable erbium-doped fiber laser. The results of numerical simulations using the three-level laser model display good agreement with previously reported experimental studies on attractor annihilation by periodic modulation. In the laser with stochastic modulation, the attraction basins' volumes have a noise-dependent probabilistic character displaying some resonances for each of the coexisting attractors.     

Stochastic dynamical analysis of a kind of vibro-impact system under multiple harmonic and random excitations

There have been many contributions concerned with non-smooth dynamics. The purpose of this study is focused on the global stochastic dynamics of a kind of vibro-impact oscillator under the multiple harmonic and bounded noisy excitations. The well-known cell-to-cell mapping method is firstly developed to investigate the incursive fractal boundaries between the attracting domains of different random attractors, and a specific Poincaré map is then set up to explore the noise-contaminated dynamical transitions in the system. Lastly, the leading Lyapunov exponents and the surrogate tests are used to identify the noise-contaminated dynamics. It is shown that several random attractors will coexist in the phase space of the randomly driven system by adjusting the parameters’ values, and fractal boundaries may also arise between the attracting domains of different random attractors. Under the joint action of the harmonic excitation and the weak bounded noise excitation, the noisy period-doubling process, similar to a deterministic one, can appear in the Poincaré’s global cross-section by increasing the strength of the bounded noisy excitation. Moreover, the noisy periodic, the noisy chaotic, and the random-dominant dynamics are also distinguished from the noise-contaminated signals.     

Unstable attractors induce perpetual synchronization and desynchronization

Common experience suggests that attracting invariant sets in nonlinear dynamical systems are generally stable. Contrary to this intuition, we present a dynamical system, a network of pulse-coupled oscillators, in which unstable attractors arise naturally. From random initial conditions, groups of synchronized oscillators (clusters) are formed that send pulses alternately, resulting in a periodic dynamics of the network. Under the influence of arbitrarily weak noise, this synchronization is followed by a desynchronization of clusters, a phenomenon induced by attractors that are unstable. Perpetual synchronization and desynchronization lead to a switching among attractors. This is explained by the geometrical fact, that these unstable attractors are surrounded by basins of attraction of other attractors, whereas the full measure of their own basin is located remote from the attractor. Unstable attractors do not only exist in these systems, but moreover dominate the dynamics for large networks and a wide range of parameters.     

Noise-induced extinction in Bazykin-Berezovskaya population model

A nonlinear Bazykin-Berezovskaya prey-predator model under the influence of parametric stochastic forcing is considered. Due to Allee effect, this conceptual population model even in the deterministic case demonstrates both local and global bifurcations with the change of predator mortality. It is shown that random noise can transform system dynamics from the regime of coexistence, in equilibrium or periodic modes, to the extinction of both species. Geometry of attractors and separatrices, dividing basins of attraction, plays an important role in understanding the probabilistic mechanisms of these stochastic phenomena. Parametric analysis of noise-induced extinction is carried out on the base of the direct numerical simulation and new analytical stochastic sensitivity functions technique taking into account the arrangement of attractors and separatrices.     

Entropy control in discrete-and continuous-time dynamic systems

The dynamics of a modified logistic mapping are considered for a system with the order parameter modulated by an external signal. It is shown that the Kolmogorov-Sinay entropy changes with changing modulation depth, while the harmonic signals and white noise can be used as a modulating signal. The conditions for the excitation of regular and strange nonchaotic attractors in the phase space are established.     

A memristive map with coexisting chaos and hyperchaos

By introducing a discrete memristor and periodic sinusoidal functions, a two-dimensional map with coexisting chaos and hyperchaos is constructed. Various coexisting chaotic and hyperchaotic attractors under different Lyapunov exponents are firstly found in this discrete map, along with which other regimes of coexistence such as coexisting chaos, quasi-periodic oscillation, and discrete periodic points are also captured. The hyperchaotic attractors can be flexibly controlled to be unipolar or bipolar by newly embedded constants meanwhile the amplitude can also be controlled in combination with those coexisting attractors. Based on the nonlinear auto-regressive model with exogenous inputs (NARX) for neural network, the dynamics of the memristive map is well predicted, which provides a potential passage in artificial intelligence-based applications.     

Spatial symmetry breaking and coexistence of attractors in a nonlinear ring cavity

We consider a passive optical system consisting of a ring cavity and a homogeneously broadened two-level medium. We find that a spatial modulation of the input beam imposes processes of competition between the external modulation frequency and the internal space frequencies which emerge from modulation instabilities. This leads to symmetry breaking phenomena when both the amplitude and the wavelength of the modulation are increased and results in the coexistence of periodic attractors. We numerically analyze the corresponding bifurcations and find that they can be explained by a generalization of the concept of cooperative frequency locking although the bifurcating attractors are periodic and are not related to linear resonator modes.     

Photonic integrated device for chaos applications in communications

A novel photonic monolithic integrated device consisting of a distributed feedback laser, a passive resonator, and active elements that control the optical feedback properties has been designed, fabricated, and evaluated as a compact potential chaotic emitter in optical communications. Under diverse operating parameters, the device behaves in different modes providing stable solutions, periodic states, and broadband chaotic dynamics. Chaos data analysis is performed in order to quantify the complexity and chaoticity of the experimental reconstructed attractors by applying nonlinear noise filtering.     

Prevalence of unstable attractors in networks of pulse-coupled oscillators

We present and analyze the first example of a dynamical system that naturally exhibits attracting periodic orbits that are unstable. These unstable attractors occur in networks of pulse-coupled oscillators, and become prevalent with increasing network size for a wide range of parameters. They are enclosed by basins of attraction of other attractors but are remote from their own basin volume such that arbitrarily small noise leads to a switching among attractors.     

Effect of noise on the relaxation to an invariant probability measure of nonhyperbolic chaotic attractors

We study the influence of external noise on the relaxation to an invariant probability measure for two types of nonhyperbolic chaotic attractors, a spiral (or coherent) and a noncoherent one. We find that for the coherent attractor the rate of mixing changes under the influence of noise, although the largest Lyapunov exponent remains almost unchanged. A mechanism of the noise influence on mixing is presented which is associated with the dynamics of the instantaneous phase of chaotic trajectories. This also explains why the noncoherent regime is robust against the presence of external noise.     

Dynamics of a three DOF mechanical system with dry friction

The dynamics of a three-block mechanical system is investigated: each block is pulled by a belt and is subjected to linear elastic and nonlinear frictional forces which induce oscillations in the system. The study of the full dynamics of the system is partially reduced to the study of a two-dimensional map; its attractors, their basins of attraction and their Lyapunov exponents provide a powerful tool to understand the dynamic behaviour of the full mechanical system which possesses rich dynamics characterised by periodic, quasi-periodic, chaotic and hyper-chaotic attractors.     

Fast convergence of spike sequences to periodic patterns in recurrent networks

The dynamical attractors are thought to underlie many biological functions of recurrent neural networks. Here we show that stable periodic spike sequences with precise timings are the attractors of the spiking dynamics of recurrent neural networks with global inhibition. Almost all spike sequences converge within a finite number of transient spikes to these attractors. The convergence is fast, especially when the global inhibition is strong. These results support the possibility that precise spatiotemporal sequences of spikes are useful for information encoding and processing in biological neural networks.     

Excitability of periodic and chaotic attractors in semiconductor lasers with optoelectronic feedback

The dynamics of a semiconductor laser with AC-coupled nonlinear optoelectronic feedback has been experimentally studied. A period doubling sequence of small periodic and chaotic attractors is observed, each of them displaying excitable features. This scenario is found also in a simplified physical model of the system, thus extending the concept of excitability, usually associated to fixed points, also to the case of higher-dimensional attractors.     

Vulnerability to paroxysmal oscillations in delayed neural networks: a basis for nocturnal frontal lobe epilepsy?

Resonance can occur in bistable dynamical systems due to the interplay between noise and delay (τ) in the absence of a periodic input. We investigate resonance in a two-neuron model with mutual time-delayed inhibitory feedback. For appropriate choices of the parameters and inputs three fixed-point attractors co-exist: two are stable and one is unstable. In the absence of noise, delay-induced transient oscillations (referred to herein as DITOs) arise whenever the initial function is tuned sufficiently close to the unstable fixed-point. In the presence of noisy perturbations, DITOs arise spontaneously. Since the correlation time for the stationary dynamics is ～τ, we approximated a higher order Markov process by a three-state Markov chain model by rescaling time as t?→?2sτ, identifying the states based on whether the sub-intervals were completely confined to one basin of attraction (the two stable attractors) or straddled the separatrix, and then determining the transition probability matrix empirically. The resultant Markov chain model captured the switching behaviors including the statistical properties of the DITOs. Our observations indicate that time-delayed and noisy bistable dynamical systems are prone to generate DITOs as switches between the two attractors occur. Bistable systems arise transiently in situations when one attractor is gradually replaced by another. This may explain, for example, why seizures in certain epileptic syndromes tend to occur as sleep stages change.     

Stabilization of chaotic oscillations in systems with a hyperbolic-type attractor

It has been shown that the chaotic dynamics of systems with nearly hyperbolic-type attractors can be stabilized by periodic parametric perturbations.     

Complex transient dynamics of hidden attractors in a simple 4D system

A simple four-dimensional system with only one control parameter is proposed in this paper. The novel system has a line or no equilibrium for the global control parameter and exhibits complex transient transition behaviors of hyperchaotic attractors, periodic orbits, and unstable sinks. Especially, for the nonzero-valued control parameter, there exists no equilibrium in the proposed system, leading to the formation of various hidden attractors with complex transient dynamics. The research results indicate that the dynamics of the system shows weak chaotic robustness and depends greatly on the initial states.     

Mode-competition noise associated with microwave modulation of multimode semiconductor lasers

We analyze the mode-competition (MC) phenomenon and the associated noise in multimode semiconductor lasers at microwave modulation. The study is based on the multimode rate-equation model, which takes into account the mechanisms of modal gain suppression. The MC is evaluated by the correlation coefficients between oscillating modes in the laser cavity. We show that an increase in the modulation depth changes the mode coupling from anticorrelation to positive correlation and then to complete coupling, which corresponds to emission of periodic pulses. The frequency spectra of relative intensity noise (RIN) exhibit sharp peaks at the modulation frequency and higher harmonics. The increase in the modulation depth is associated with suppression of the total and modal RIN under high-frequency modulation and with noise enhancement under low-frequency modulation.     

Coherence resonance and stochastic resonance in directionally coupled rings

In coupled systems, symmetry plays an important role for the collective dynamics. We investigate the dynamical response to noise with and without weak periodic modulation for two classes of ring systems. Each ring system consists of unidirectionally coupled bistable elements but in one class, the number of elements is even while in the other class the number is odd. Consequently, the rings without forcing show at a certain coupling strength, either ordering (similar to anti-ferromagnetic chains) or auto-oscillations. Analysing the bifurcations and fixed points of the two ring classes enables us to explain the dynamical response measured to noise and weak modulation. Moreover, by analysing a simplified model, we demonstrate that the response is universal for systems having a directional component in their stochastic dynamics in phase space around the origin.     

Dynamics of unidirectionally coupled bistable Hénon maps

We study dynamics of two bistable Hénon maps coupled in a master-slave configuration. In the case of coexistence of two periodic orbits, the slave map evolves into the master map state after transients, which duration determines synchronization time and obeys a −1/2 power law with respect to the coupling strength. This scaling law is almost independent of the map parameter. In the case of coexistence of chaotic and periodic attractors, very complex dynamics is observed, including the emergence of new attractors as the coupling strength is increased. The attractor of the master map always exists in the slave map independently of the coupling strength. For a high coupling strength, complete synchronization can be achieved only for the attractor similar to that of the master map.