Dynamics of coexisting rotating waves in unidirectional rings of bistable Duffing oscillators.

We study the dynamics of multistable coexisting rotating waves that propagate along a unidirectional ring consisting of coupled double-well Duffing oscillators with different numbers of oscillators. By employing time series analysis, phase portraits, bifurcation diagrams, and basins of attraction, we provide evidence of multistability on the route from coexisting stable equilibria to hyperchaos via a sequence of bifurcations, including the Hopf bifurcation, torus bifurcations, and crisis bifurcations, as the coupling strength is increased. The specific bifurcation route depends on whether the ring comprises an even or odd number of oscillators. In the case of an even number of oscillators, we observe the existence of up to 32 coexisting stable fixed points at relatively weak coupling strengths, while a ring with an odd number of oscillators exhibits 20 coexisting stable equilibria. As the coupling strength increases, a hidden amplitude death attractor is born in an inverse supercritical pitchfork bifurcation in the ring with an even number of oscillators, coexisting with various homoclinic and heteroclinic orbits. Additionally, for stronger coupling, amplitude death coexists with chaos. Notably, the rotating wave speed of all coexisting limit cycles remains approximately constant and undergoes an exponential decrease as the coupling strength is increased. At the same time, the wave frequency varies among different coexisting orbits, exhibiting an almost linear growth with the coupling strength. It is worth mentioning that orbits originating from stronger coupling strengths possess higher frequencies.

Coloured-noise-induced transport in a model of the thermochemical reactor.

In this paper, effects of coloured noise on the stochastic excitement in a model of the thermochemical flow reactor are studied. Transport phenomena associated with noise-induced generation of large-amplitude oscillations are investigated depending on the correlation time of coloured noise. We study how probability of the noise-induced excitement is related to the stochastic sensitivity of the system to coloured noise with certain correlation characteristics. Parameter zones of the high stochastic sensitivity are found and discussed in connection with occurrence of resonance. This article is part of the theme issue 'Transport phenomena in complex systems (part 2)'.

Stochastic phenomena in pattern formation for distributed nonlinear systems.

We study a stochastic spatially extended population model with diffusion, where we find the coexistence of multiple non-homogeneous spatial structures in the areas of Turing instability. Transient processes of pattern generation are studied in detail. We also investigate the influence of random perturbations on the pattern formation. Scenarios of noise-induced pattern generation and stochastic transformations are studied using numerical simulations and modality analysis. This article is part of the theme issue 'Patterns in soft and biological matters'.

Asymmetry in electrical coupling between neurons alters multistable firing behavior.

The role of asymmetry in electrical synaptic connection between two neuronal oscillators is studied in the Hindmarsh-Rose model. We demonstrate that the asymmetry induces multistability in spiking dynamics of the coupled neuronal oscillators. The coexistence of at least three attractors, one chaotic and two periodic orbits, for certain coupling strengths is demonstrated with time series, phase portraits, bifurcation diagrams, basins of attraction of the coexisting states, Lyapunov exponents, and standard deviations of peak amplitudes and interspike intervals. The experimental results with analog electronic circuits are in good agreement with the results of numerical simulations.

Coherence enhanced intermittency in an optically injected semiconductor laser.

We report on the experimental observation of coherence enhancement of noise-induced intermittency in a semiconductor laser subject to optical injection from another laser at the boundary of the frequency-locking regime. The intermittent switches between locked and unlocked states occur more regularly at a certain value of the injecting laser pump current. A shape of probability distribution of the experimental inter-spike-interval fluctuations is used to quantitatively characterize the intermittent behavior.

Memristive Neural Networks for Predicting Seizure Activity.

The aim of the study is to assess the possibilities of predicting epileptiform activity using the neuronal activity data recorded from the hippocampus and medial entorhinal cortex of mice with chronic epileptiform activity. To reach this goal, a deep artificial neural network (ANN) has been developed and its implementation based on memristive devices has been demonstrated. Materials and Methods: The biological part of the investigation. Young healthy outbred CD1 mice were used in our study. They were divided into two groups: control (n=6) and the group with induced chronic epileptiform activity (n=6). Local field potentials (LFP) were recorded from the hippocampus and medial entorhinal cortex of the mice of both groups to register neuronal activity. The LFP recordings were used for deep ANN training. Epileptiform activity in mice was modeled by intraperitoneal injection of pilocarpine (280 mg/kg). LFP were recorded in the awake mice a month after the induction of epileptiform activity.Mathematical part of the investigation. A deep long short-term memory (LSTM) ANN capable of predicting biological signals of neuronal activity in mice has been developed. The ANN implementation is based on memristive devices, which are described by the equations of the redox processes running in the memristive thin metal-oxide-metal films, e.g., Au/ZrO2(Y)/TiN/Ti and Au/SiO2(Y)/TiN/Ti. In order to train the developed ANN to predict epileptiform activity, a supervised learning algorithm was used, which allowed us to adjust the network parameters and train LSTM on the described recordings of neuronal activity. Results: After training on the LFP recordings from the hippocampus and medial entorhinal cortex of the mice with chronic epileptiform activity, the proposed deep ANN has demonstrated high values of evaluation metric (root-mean-square error, RMSE) and successfully predicted epileptiform activity shortly before its occurrence (40 ms). The results of the numerical experiments have shown that the RMSE value of 0.019 was reached, which indicates the efficacy of proposed approach. The accuracy of epileptiform activity prediction 40 ms before its occurrence is a significant result and shows the potential of the developed neural network architecture. Conclusion: The proposed deep ANN can be used to predict pathological neuronal activity including epileptic seizure (focal) activity in mice before its actual occurrence. Besides, it can be applied for building a long-term prognosis of the disease course based on the LFP data. Thus, the proposed ANN based on memristive devices represents a novel approach to the prediction and analysis of pathological neuronal activity possessing a potential for improving the diagnosis and prognostication of epileptic seizures and other diseases associated with neuronal activity.

Generation of cnoidal waves by a laser system with a controllable saturable absorber.

We demonstrate the cnoidal wave formation in a two-laser system with a saturable absorber in the cavity of one of the lasers. Another laser is used to activate the saturable absorber in order to control the pulse shape, width, intensity and frequency. Using the three-level laser model based on the Statz--De Mars equations, we show that for any value of the saturable absorber parameter there exists a certain modulation frequency for which the pulse shape is very close to a soliton shape with less than 5% error at the pulse base. Such a device may be prominent for optical communication and laser engineering applications.

Synchronization of semiconductor lasers with coexisting attractors.

We study synchronization of unidirectionally coupled optical bistable systems. In particular, we consider two semiconductor lasers with an external cavity, which exhibit, when isolated, coexistence of two different attractors: fixed point and chaos, fixed point and one periodic orbit, and two periodic orbits with different periods. The analysis is performed with a cross-correlation function between the master and slave laser oscillations calculated with model equations based on the Lang-Kobayashi approach. Depending on both the laser operating point and the coupling strength, different bifurcations (Hopf, period doubling, saddle node, torus, and crisis) and diverse dynamical regimes (steady state, periodicity, quasiperiodicity, bistability, and chaos) occur in the route from asynchronous motion to complete synchronization. We show some similarities and differences between synchronization of monostable and bistable lasers.

Experimental approach to the study of complex network synchronization using a single oscillator.

We propose an experimental setup based on a single oscillator for studying large networks formed by identical unidirectionally coupled systems. A chaotic wave form generated by the oscillator is stored in a computer to adjust the signal according to the desired network configuration to feed it again into the same oscillator. No previous theoretical knowledge about the oscillator dynamics is needed. To visualize network synchronization we introduce a network synchronization bifurcation diagram that should prove to be an effective tool for analysis, design, and optimization of complex networks.

Prebifurcation noise amplification in a fiber laser.

We report on the experimental evidence of noise amplification in an erbium-doped fiber laser in the vicinity of saddle-node, period-doubling, and crisis bifurcations. We demonstrate this interesting phenomenon by analyzing the laser bifurcation diagrams and power spectra. Numerical simulations on the base of an advanced laser model display good agreement with the experimental results.

Chimera state in complex networks of bistable Hodgkin-Huxley neurons.

In this paper we study a chimera state in complex networks of bistable Hodgkin-Huxley neurons with excitatory coupling, which manifests as a termination of spiking activity of a part of interacting neurons. We provide a detailed investigation of this phenomenon in scale-free, small-world, and random networks and show that the chimera state is robust to the network topology. Nevertheless, network topological properties determine the stability of spatiotemporal states and therefore affect the excitability of the chimera state in the whole network. In particular, the scale-free network whose higher degree nodes are more stable to small perturbations is least exposed to chimera formation and exhibits an abrupt transition from a spiking to a silent regime. On the other hand, small-world and random networks are more likely to provide transitions to the chimera state.

Phase-locking phenomenon in a semiconductor laser with external cavities.

Phase-locked solutions are found numerically in a semiconductor laser with one and two external cavities. Different periodic, quasiperiodic, chaotic, and steady-state regimes form Arnold's tongues in bi-dimensional parameter spaces of the length and feedback strengths of the external cavities and the pump parameter. This rich structure gives additional possibility for controlling complex dynamics and chaos in a semiconductor laser with external cavities by properly adjusting their lengths and feedback strengths.

Experimental evidence of deterministic coherence resonance in coupled chaotic systems with frequency mismatch.

We present the experimental evidence of deterministic coherence resonance in unidirectionally coupled two and three Rössler electronic oscillators with mismatch between their natural frequencies. The regularity in both the amplitude and the phase of chaotic fluctuations is experimentally proven by the analyses of normalized standard deviations of the peak amplitude and interpeak interval and Lyapunov exponents. The resonant chaos suppression appears when the coupling strength is increased and the oscillators are in phase synchronization. In two coupled oscillators, the coherence enhancement is associated with negative third and fourth Lyapunov exponents, while the largest first and second exponents remain positive. Distinctly, in three oscillators coupled in a ring, all exponents become negative, giving rise to periodicity. Numerical simulations are in good agreement with the experiments.

Deterministic coherence resonance in coupled chaotic oscillators with frequency mismatch.

A small mismatch between natural frequencies of unidirectionally coupled chaotic oscillators can induce coherence resonance in the slave oscillator for a certain coupling strength. This surprising phenomenon resembles "stabilization of chaos by chaos," i.e., the chaotic driving applied to the chaotic system makes its dynamics more regular when the natural frequency of the slave oscillator is a little different than the natural frequency of the master oscillator. The coherence is characterized with the dominant component in the power spectrum of the slave oscillator, normalized standard deviations of both the peak amplitude and the interpeak interval, and Lyapunov exponents. The enhanced coherence is associated with increasing negative both the third and the fourth Lyapunov exponents, while the first and second exponents are always positive and zero, respectively.

Synchronization of intermittent behavior in ensembles of multistable dynamical systems.

We propose a methodology to analyze synchronization in an ensemble of diffusively coupled multistable systems. First, we study how two bidirectionally coupled multistable oscillators synchronize and demonstrate the high complexity of the basins of attraction of coexisting synchronous states. Then, we propose the use of the master stability function (MSF) for multistable systems to describe synchronizability, even during intermittent behavior, of a network of multistable oscillators, regardless of both the number of coupled oscillators and the interaction structure. In particular, we show that a network of multistable elements is synchronizable for a given range of topology spectra and coupling strengths, irrespective of specific attractor dynamics to which different oscillators are locked, and even in the presence of intermittency. Finally, we experimentally demonstrate the feasibility and robustness of the MSF approach with a network of multistable electronic circuits.

Controlling the multistability of nonlinear systems with coexisting attractors.

A method for controlling generalized multistability is suggested. The method implies the application of a small harmonic modulation with properly chosen frequency and amplitude to a system parameter. The possibility of control is demonstrated with the example of the Hénon map with coexisting period-1, period-3, and period-9 attractors. It is shown that one or more coexisting attractors can lose their stability and the attractor can undergo crisis when the control frequency is close to the relaxation oscillation frequency or its subharmonics of the corresponding attractor.

Synchronization effects in a dual-wavelength class-B laser with modulated losses.

Different types of synchronization: in-phase, antiphase, phase, and lag synchronization, as well as amplitude death have been found theoretically in a dual-wavelength class-B laser with modulated losses in one of the channels. Depending on the laser parameters, oscillations in master and slave channels can be either completely or partially synchronized. The conditions for the dual-wavelength regime have been established. The analysis has been performed on the basis of transfer functions of the master and slave channels.

Experimental observation of two-state on-off intermittency.

We observe a two-state on-off intermittency in a diode laser with an external cavity experimentally. The control parameter is the length of the external cavity that is periodically modulated. We demonstrate that the system exhibits an intermittent behavior between a two-mode chaotic regime and a single-mode state when the control parameter passes through the bifurcation point. Power-law scaling of the average laminar time with a critical exponent of -1 is found as a function of both the amplitude and frequency of the external modulation.

Experimental demonstration of attractor annihilation in a multistable fiber laser.

We report on the experimental open-loop control of generalized multistability in a system with coexisting attractors. The experimental system is an erbium-doped fiber laser with pump modulation of the diode laser. We demonstrate that additional weak harmonic modulation of the diode current annihilates one or two stable limit cycles in the laser. The ability of the method to select a desired state is illustrated through a codimension-two bifurcation diagram in the parameter space of the frequency and amplitude of the control modulation. We identify main resonances on the bifurcation lines (annihilation curves) and evaluate conditions for attractor annihilation.

Multistate intermittency and extreme pulses in a fiber laser.

In our recent Letter [Phys. Rev. Lett. 107, 274101 (2011)], we demonstrated that slow random perturbations of a system parameter were responsible for the emergence of rogue waves in a fiber laser with coexisting attractors. In this paper we investigate how the probability of a particular state to appear in multistate intermittency can be controlled by low-pass noise filtering. We show that the probability of some states depends nonmonotonously on the noise amplitude and cutoff frequency. The conditions for the emergence of extreme pulses in a erbium-doped fiber laser are analyzed numerically and experimentally.