A. B. Potapov

Robust chaos in neural networks

Abstract We consider the problem of creating a robust chaotic neural network. Robustness means that chaos cannot be destroyed by arbitrary small change of parameters [Phys. Rev. Lett. 80 (1998) 3049]. We present such networks of neurons with the activation function f ( x )=|tanh s ( x − c )|. We show that in a certain range of s and c the dynamical system x k +1 = f ( x k ) cannot have stable periodic solutions, which proves the robustness. We also prove that chaos remains robust in a network of weakly connected such neurons. In the end, we discuss ways to enhance the statistical properties of data generated by such a map or network.

On the concept of stationary Lyapunov basis

We propose the concept of stationary Lyapunov basis ‐ the basis of tangent vectors e .i/ .x/ defined at every point x of the attractor of the dynamical system, and show that one can reformulate some algorithms for calculation of Lyapunov exponents i so that eachi can be treated as the average of a function Si.x/. This enables one to use measure averaging in theoretical arguments thus proposing the rigorous basis for a number of ideas for calculation of Lyapunov exponents from time series. We also study how the Lyapunov vectors in Benettin’s algorithm converge to the stationary basis and show that this convergence rate determines continuity of the field of stationary Lyapunov vectors.

Allee Effect and Control of Lake System Invasion

We consider the model of invasion prevention in a system of lakes that are connected via traffic of recreational boats. It is shown that in presence of an Allee effect, the general optimal control problem can be reduced to a significantly simpler stationary optimization problem of optimal invasion stopping. We consider possible values of model parameters for zebra mussels. The general N-lake control problem has to be solved numerically, and we show a number of typical features of solutions: distribution of control efforts in space and optimal stopping configurations related with the clusters in lake connection structure.

Distortions of reconstruction for chaotic attractors

Abstract We consider the problem of quality evaluation for delay reconstruction of chaotic attractors from scalar time series. The proposed approach is based upon local linear analysis of the reconstruction mapping. It is shown that some linear distortions in principle could be corrected, but usually nonlinear distortions make it dificult to efficiently apply this technique. The new algorithms of estimating the scale of nonlinear distortion and “irrelevance time” (upper limit for the reconstruction window) from a time series are proposed. Both of them do not require pricise knowledge of minimal embedding dimension for the attractor studied.

Hybrid system increases efficiency of ballast water treatment

Summary 1. Ballast water has been a principal pathway of non-indigenous species introduction to global ports for much of the 20th century. In an effort to reduce the scale of this pathway, and recognizing forthcoming global regulations that will supplant ballast water exchange (BWE) with ballast water treatment (BWT), we explore whether a combined hybrid treatment of BWE and chlorination (Cl) exceeds individual effects of either BWE or chlorination alone in reducing densities of bacteria, microplankton and macroplankton. 2. Five full-scale trials were conducted on an operational bulk carrier travelling between Canada and Brazil. 3. The hybrid treatment generally had the lowest final densities among all treatments for putative enterococci, Escherichia coli and coliform bacteria, as well as microplankton and macroplankton, with the former two being synergistically lower than individual treatments alone. Microplankton abundance in the hybrid treatment was significantly but antagonistically reduced relative to individual treatments alone. Macroplankton final density was lowest in the hybrid treatment, though the interaction between treatments was not significant. 4. Synthesis and applications. In most cases, the combined hybrid treatment of ballast water exchange (BWE) and chlorination reduced population densities of indicator organisms in ballast water below those proposed by the International Maritime Organization’s D-2 performance standards. BWE alone was often ineffective at reducing bacterial and macroplankton densities. Even when performance standards are implemented globally, continued use of BWE could further reduce risk of invasions to freshwater ecosystems that receive ballast water from foreign sources by accentuating the decline in propagule pressure and enhancing demographic constraints for putative invaders.

Impact of stochasticity in immigration and reintroduction on colonizing and extirpating populations

A thorough quantitative understanding of populations at the edge of extinction is needed to manage both invasive and extirpating populations. Immigration can govern the population dynamics when the population levels are low. It increases the probability of a population establishing (or reestablishing) before going extinct (EBE). However, the rate of immigration can be highly fluctuating. Here, we investigate how the stochasticity in immigration impacts the EBE probability for small populations in variable environments. We use a population model with an Allee effect described by a stochastic differential equation (SDE) and employ the Fokker-Planck diffusion approximation to quantify the EBE probability. We find that, the effect of the stochasticity in immigration on the EBE probability depends on both the intrinsic growth rate (r) and the mean rate of immigration (p). In general, if r is large and positive (e.g. invasive species introduced to favorable habitats), or if p is greater than the rate of population decline due to the demographic Allee effect (e.g., effective stocking of declining populations), then the stochasticity in immigration decreases the EBE probability. If r is large and negative (e.g. endangered populations in unfavorable habitats), or if the rate of decline due to the demographic Allee effect is much greater than p (e.g., weak stocking of declining populations), then the stochasticity in immigration increases the EBE probability. However, the mean time for EBE decreases with the increasing stochasticity in immigration with both positive and negative large r. Thus, results suggest that ecological management of populations involves a tradeoff as to whether to increase or decrease the stochasticity in immigration in order to optimize the desired outcome. Moreover, the control of invasive species spread through stochastic means, for example, by stochastic monitoring and treatment of vectors such as ship-ballast water, may be suitable strategies given the environmental and demographic uncertainties at introductions. Similarly, the recovery of declining and extirpated populations through stochastic stocking, translocation, and reintroduction, may also be suitable strategies.

LEARNING, EXPLORATION AND CHAOTIC POLICIES

We consider different versions of exploration in reinforcement learning. For the test problem, we use navigation in a shortcut maze. It is shown that chaotic ∊-greedy policy may be as efficient as a random one. The best results were obtained with a model chaotic neuron. Therefore, exploration strategy can be implemented in a deterministic learning system such as a neural network.

Correlation integral as a tool for distinguishing between dynamics and statistics in time series data

Abstract We propose a new test for time series data for proper choice of processing technique: dynamical or statistical. It is based upon the normalized slope of the correlation integral ϕ(∈, m) = m −1 d(ln C(∈)) d ln ∈ , where m is the embedding dimension. It is shown that when ϕ does not tend to 0 on the resolved range of scales as m grows, then there will be serious limitations for dynamical methods even if the data are dynamical by nature. In the latter case it means that the length of time series does not allow to resolve small scales, and on large scales the delay reconstruction for any m mixes true and false neighbours of points and therefore restricts the application of dynamical techniques, such as estimating Lyapunov exponents or predicting time series.

On mean field fluctuations in globally coupled logistic-type maps

Abstract We consider mean field fluctuations in globally coupled map x n+1 (i) = f(x n (i)) + ϵ( 1 N ) ∑ j=1 N x n (j) when the local map f has a flat power-law top, e.g. f(x) = 1 − a|x|β with β > 1. It is shown that in the thermodynamic limit N → ∞ the amplitude of mean field fluctuations is O (ϵ 1 (β−1) ) for ϵ ⪡ 1 which is in good agreement with numerical calculations. We also demonstrate the relation between this problem and behaviour of averages as functions of the parameter a in 1D maps. For the latter, we give both theoretical grounds and experimental evidence that the “mean deviation” of an average value behaves as a power of the deviation of the parameter, e.g. ∫|〈x〉(a + δa) − 〈x〉(a)| d a ∼ |δa| 1 β .

Complex ordering and stochastic oscillations in a class of reaction-diffusion systems with small diffusion

SummaryWe consider four models of partial differential equations obtained by applying a generalization of the method of normal forms to two-component reaction-diffusion systems with small diffusionut=εDuxx+(A+εA1)u+F(u),u ∈ ℝ2. These equations (quasinormal forms) describe the behaviour of solutions of the original equation forε → 0.One of the quasinormal forms is the well-known complex Ginzburg-Landau equation. The properties of attractors of the other three equations are considered. Two of these equations have an interesting feature that may be called asensitivity to small parameters: they contain a new parameterϑ(ε)=−(aε−1/2 mod 1) that influences the behaviour of solutions, but changes infinitely many times whenε → 0. This does not create problems in numerical analysis of quasinormal forms, but makes numerical study of the original problem involvingε almost impossible.