Synchronization of a large number of continuous one-dimensional stochastic elements with time delayed mean field coupling

We study synchronization as a means of control of collective behavior of an ensemble of coupled stochastic units in which oscillations are induced merely by external noise. We determine the boundary of the synchronization domain of a large number of onedimensional continuous stochastic elements with time delayed non-homogeneous mean-field coupling. Exact location of the synchronization threshold is shown to be a solution of the boundary value problem (BVP) which was derived from the linearized Fokker-Planck equation. Here the synchronization threshold is found by solving this BVP numerically. Approximate analytics is obtained by expanding the solution of the linearized Fokker-Planck equation into a series of eigenfunctions of the stationary Fokker-Planck operator. Bistable systems with a polynomial and piece-wise linear potential are considered as examples. Multistability and hysteresis is observed in the Langevin equations for finite noise intensity. In the limit of small noise intensities the critical coupling strength was shown to remain finite

Dynamical system with plastic self-organized velocity field as an alternative conceptual model of a cognitive system

It is well known that architecturally the brain is a neural network, i.e. a collection of many relatively simple units coupled flexibly. However, it has been unclear how the possession of this architecture enables higher-level cognitive functions, which are unique to the brain. Here, we consider the brain from the viewpoint of dynamical systems theory and hypothesize that the unique feature of the brain, the self-organized plasticity of its architecture, could represent the means of enabling the self-organized plasticity of its velocity vector field. We propose that, conceptually, the principle of cognition could amount to the existence of appropriate rules governing self-organization of the velocity field of a dynamical system with an appropriate account of stimuli. To support this hypothesis, we propose a simple non-neuromorphic mathematical model with a plastic self-organized velocity field, which has no prototype in physical world. This system is shown to be capable of basic cognition, which is illustrated numerically and with musical data. Our conceptual model could provide an additional insight into the working principles of the brain. Moreover, hardware implementations of plastic velocity fields self-organizing according to various rules could pave the way to creating artificial intelligence of a novel type

Delayed feedback control of chaos: Bifurcation analysis

We study the effect of time delayed feedback control in the form proposed by Pyragas on deterministic chaos in the Rossler system. We reveal the general bifurcation diagram in the parameter plane of time delay and feedback strength K which allows one to explain the phenomena that have been discovered in some previous works. We show that the bifurcation diagram has essentially a multi-leaf structure that constitutes multistability: the larger the time delay , the larger the number of attractors that can coexist in the phase space. Feedback induces a large variety of regimes non-existent in the original system, among them tori and chaotic attractors born from them. Finally, we estimate how the parameters of delayed feedback influence the periods of limit cycles in the system

Control of noise-induced oscillations by delayed feedback

We propose a method to control noise-induced motion, based on using delayed feedback in the form of the difference between the delayed and the current states of the system. The method is applied to two different types of systems, namely, a selfoscillator near Andronov-Hopf bifurcation and a threshold system. In both cases we demonstrate that by variation of time delay one can effectively control coherence and timescales of stochastic oscillations. The entrainment of the basic period of oscillations by time delay is discovered. We give explanations of the phenomena observed and provide a theory for the system near bifurcation

Controlling stochastic oscillations close to a Hopf bifurcation by time-delayed feedback

We study the effect of a time-delayed feedback upon a Van der Pol oscillator under the influence of white noise in the regime below the Hopf bifurcation where the deterministic system has a stable fixed point. We show that both the coherence and the frequency of the noise-induced oscillations can be controlled by varying the delay time and the strength of the control force. Approximate analytical expressions for the power spectral density and the coherence properties of the stochastic delay differential equation are developed, and are in good agreement with our numerical simulations. Our analytical results elucidate how the correlation time of the controlled stochastic oscillations can be maximized as a function of delay and feedback strength

Chaos in atmospheric-pressure plasma jets

We report detailed characterization of a low-temperature atmospheric-pressure plasma jet that exhibits regimes of periodic, quasi-periodic and chaotic behaviors. Power spectra, phase portraits, stroboscopic section and bifurcation diagram of the discharge current combine to comprehensively demonstrate the existence of chaos, and this evidence is strengthened with a nonlinear dynamics analysis using two control parameters that maps out periodic, period-multiplication, and chaotic regimes over a wide range of the input voltage and gas flow rate. In addition, optical emission signatures of excited plasma species are used as the second and independent observable to demonstrate the presence of chaos and period-doubling in both the concentrations and composition of plasma species, suggesting a similar array of periodic, quasi-periodic and chaotic regimes in plasma chemistry. The presence of quasi-periodic and chaotic regimes in structurally unbounded low-temperature atmospheric plasmas not only is important as a fundamental scientific topic but also has interesting implications for their numerous applications. Chaos may be undesirable for industrial applications where cycle-to-cycle reproducibility is important, yet for treatment of cell-containing materials including living tissues it may offer a novel route to combat some of the major challenges in medicine such as drug resistance. Chaos in low-temperature atmospheric plasmas and its effective control are likely to open up new vistas for medical technologies

Noise-controlled signal transmission in a multithread semiconductor neuron

We report on stochastic effects in a new class of semiconductor structures that accurately imitate the electrical activity of biological neurons. In these devices, electrons and holes play the role of K+ and Na+ ions that give the action potentials in real neurons. The structure propagates and delays electrical pulses via a web of spatially distributed transmission lines. We study the transmission of a periodic signal through a noisy semiconductor neuron. Using experimental data and a theoretical model we demonstrate that depending on the noise level and the amplitude of the useful signal, transmission is enhanced by a variety of nonlinear phenomena, such as stochastic resonance, coherence resonance, and stochastic synchronization

Optimization with delay-induced bifurcations

Optimization is finding the best solution, which mathematically amounts to locating the global minimum of some cost function. Optimization is traditionally automated with digital or quantum computers, each having their limitations and none guaranteeing an optimal solution. Here, we conceive a principle behind optimization based on delay-induced bifurcations, which is potentially implementable in non-quantum analog devices. Often, optimization techniques are interpreted via a particle moving in multi-well energy landscape and to prevent confinement to a non-global minima they should incorporate mechanisms to overcome barriers between the minima. Particularly, simulated annealing digitally emulates pushing a fictitious particle over a barrier by random noise, whereas quantum computers utilize tunneling through barriers. In our principle, the barriers are effectively destroyed by delay-induced bifurcations. Although bifurcation scenarios in nonlinear delay-differential equations can be very complex and are notoriously difficult to predict, we hypothesize, verify, and utilize the finding that they become considerably more predictable in dynamical systems, where the right-hand side depends only on the delayed variable and represents a gradient of some potential energy function. By tuning the delay introduced into the gradient descent setting, thanks to global bifurcations destroying local attractors, one could force the system to spontaneously wander around all minima. This would be similar to noise-induced behavior in simulated annealing but achieved deterministically. Ideally, a slow increase and then decrease of the delay should automatically push the system toward the global minimum. We explore the possibility of this scenario and formulate some prerequisites.Can optimization problem be solved without either relatively slow digital computers or fast, but costly and difficult to make quantum computers? Instead of employing complex algorithms or expensive technologies, could we rely on a fast analog device operating in a semi-automatic manner? We conceive and prove a principle behind optimization, which uses bifurcations caused by time delay in nonlinear dynamical systems and could potentially be implemented in analog circuits. We hypothesize and verify that, in a special class of delay equations involving the gradient of the cost function, an increase of delay induces a chain of global bifurcations, which effectively remove the barriers between the minima and enable the system to “explore” the vicinities of all minima. Thus, we propose and demonstrate an alternative way for a candidate optimizer to overcome the barriers. The delay plays the role of random noise in standard algorithm-based optimization settings, such as simulated annealing, or of the tunneling effect in quantum computers, and could be promising for obtaining a global minimum. We consider various configurations of the cost function with five minima with slightly different forms of the barrier-breaking mechanism and demonstrate that in many cases, the system converges to the global minimum as desired. Testing with real-life problems and hardware implementation of the proposed principle are the tasks for the future. Given that none of the existing optimization approaches can fully guarantee the optimal solution, that digital computers can be too slow, and quantum computers require expensive technologies, our approach could represent an attractive alternative deserving further exploration.</div

Mathematical consistency and long-term behaviour of a dynamical system with a self-organising vector field

A dynamical system with a plastic self-organising velocity vector field was introduced in [Janson & Marsden 2017] as a mathematical prototype of new explainable intelligent systems. Although inspired by the brain plasticity, it does not model or explain any specific brain mechanisms or processes, but instead expresses a hypothesised principle possibly implemented by the brain. The hypothesis states that, by means of its plastic architecture, the brain creates a plastic self-organising velocity vector field, which embodies self-organising rules governing neural activity and through that the behaviour of the whole body. The model is represented by a two-tier dynamical system, in which the observable behaviour obeys a velocity field, which is itself controlled by another dynamical system. Contrary to standard brain models, in the new model the sensory input affects the velocity field directly, rather than indirectly via neural activity. However, this model was postulated without sufficient explication or theoretical proof of its mathematical consistency. Here we provide a more rigorous mathematical formulation of this problem, make several simplifying assumptions about the form of the model and of the applied stimulus, and perform its mathematical analysis. Namely, we explore the existence, uniqueness, continuity and smoothness of both the plastic velocity vector field controlling the observable behaviour of the system, and the of the behaviour itself. We also analyse the existence of pullback attractors and of forward limit sets in such a non-autonomous system of a special form. Our results verify the consistency of the problem and pave the way to constructing more models with specific pre-defined cognitive functions

Supplementary Information Files for Delay-induced homoclinic bifurcations in modified gradient bistable systems and their relevance to optimization

Supplementary Information Files for Delay-induced homoclinic bifurcations in modified gradient bistable systems and their relevance to optimizationNonlinear dynamical systems with time delay are abundant in applications but are notoriously difficult to analyze and predict because delay-induced effects strongly depend on the form of the nonlinearities involved and on the exact way the delay enters the system. We consider a special class of nonlinear systems with delay obtained by taking a gradient dynamical system with a two-well “potential” function and replacing the argument of the right-hand side function with its delayed version. This choice of the system is motivated by the relative ease of its graphical interpretation and by its relevance to a recent approach to use delay in finding the global minimum of a multi-well function. Here, the simplest type of such systems is explored for which we hypothesize and verify the possibility to qualitatively predict the delay-induced effects, such as a chain of homoclinic bifurcations one by one eliminating local attractors and enabling the phase trajectory to spontaneously visit vicinities of all local minima. The key phenomenon here is delay-induced reorganization of manifolds, which cease to serve as barriers between the local minima after homoclinic bifurcations. Despite the general scenario being quite universal in two-well potentials, the homoclinic bifurcation comes in various versions depending on the fine features of the potential. Our results are a pre-requisite for understanding general highly nonlinear multistable systems with delay. They also reveal the mechanisms behind the possible role of delay in optimization.<br