Artificial Neural Network as a Universal Model of Nonlinear Dynamical Systems
We suggest a universal map capable of recovering the behavior of a wide range of dynamical systems given by ODEs. The map is built as an artificial neural network whose weights encode a modeled system. We assume that ODEs are known and prepare training datasets using the equations directly without computing numerical time series. Parameter variations are taken into account in the course of training so that the network model captures bifurcation scenarios of the modeled system. The theoretical benefit from this approach is that the universal model admits applying common mathematical methods without needing to develop a unique theory for each particular dynamical equations. From the practical point of view the developed method can be considered as an alternative numerical method for solving dynamical ODEs suitable for running on contemporary neural network specific hardware. We consider the Lorenz system, the Rцssler system and also the Hindmarch – Rose model. For these three examples the network model is created and its dynamics is compared with ordinary numerical solutions. A high similarity is observed for visual images of attractors, power spectra, bifurcation diagrams and Lyapunov exponents.
We study chaotic plane sections of some particular family of triply periodic surfaces. The question about possible behavior of such sections was posed by S. P. Novikov. We prove some estimations on the diffusion rate of these sections using the connection between Novikov’s problem and systems of isometries—some natural generalization of interval exchange transformations. Using thermodynamical formalism, we construct an invariant measure for systems of isometries of a special class called the Rauzy gasket, and investigate the main properties of the Lyapunov spectrum of the corresponding suspension flow.
We present a functional integration method for the averaging of continuous productsPt ofN×N random matrices. As an application, we compute exactly the statistics of the Lyapunov spectrum ofPt. This problem is relevant to the study of the statistical properties of various disordered physical systems, and specifically to the computation of the multipoint correlators of a passive scalar advected by a random velocity field. Apart from these applications, our method provides a general setting for computing statistical properties of linear evolutionary systems subjected to a white-noise force field.
In this lecture, the applications of the Pyragas time-delay feedback control technique and Leonov analytical approach for estimation of Lyapunov dimension and topological entropy in the framework of studying the Eden’s conjecture are discussed. The problem of reliable numerical computation of the mentioned dimension-like characteristics along the trajectories over large time intervals is demonstrated.
Contributions in this volume focus on computationally efficient algorithms and rigorous mathematical theories for analyzing large-scale networks. Researchers and students in mathematics, economics, statistics, computer science and engineering will find this collection a valuable resource filled with the latest research in network analysis. Computational aspects and applications of large-scale networks in market models, neural networks, social networks, power transmission grids, maximum clique problem, telecommunication networks, and complexity graphs are included with new tools for efficient network analysis of large-scale networks.
This proceeding is a result of the 7th International Conference in Network Analysis, held at the Higher School of Economics, Nizhny Novgorod in June 2017. The conference brought together scientists, engineers, and researchers from academia, industry, and government.
There is a diverse variety of demographic data that can be analyzed with modern methods of data mining to achieve better results. On the one hand, the main chosen task is to compare different methods for the next event prediction and gender prediction, on the other hand, we pay special attention to interpretable patterns describing demographic behavior in the studied problems. There were considered interpretable methods as decision trees and their ensembles and semi- or non-interpretable methods, such as the SVM method with different customized kernels tailored for demographers' needs and neural networks, respectively. The best accuracy results were obtained with two-channel Convolutional Neural Networks.
Building of adequate dynamical models of microblogging social networks is a topical task that is of interest from both theoretical and practical aspects. Experimental and theoretical results of studies related to choice of the adequate model are presented. The choice was made between two models: a nonlinear dynamical system and a nonlinear random dynamical system. By results of the fractal analysis of observable network time series and defining their probability density function it was established that the nonlinear random dynamical system was more adequate than the nonlinear dynamical system. The character of the observable time series was also explored. The possibility that microblogging social networks can be analyzed by means of Tsallis entropy and self-organized criticality is examined.
An approach to the detection of hidden information (stegocontainers) in the audio data of MP3 files based on neural network modeling is considered. A multilayer perceptron is used as the instrumental model of the neural network. The structural components of the MP3 file are analyzed: fields containing related information (song titles, album, information about the author, lyrics, etc.), and frames, and fragmented sets of encoded audio data. Useful data are highlighted. A procedure is proposed for presenting audio data of any MP3 file as a uniform set of features of a relatively small size. The dimension of the feature set (data set) can be selected from the range [100-520], in accordance with the minimum and maximum frame size, depending on the compression quality of a single audio file when encoded in MP3 format. Modern software packages for encrypting and decrypting stegocontainers into MP3 files are being investigated. Based on selected software implementations, a database of examples (data sets) is formed from pre-processed MP3 files both containing the stegocontainer and without the stegocontainer. The structure of the neural network for steganalysis of MP3 files is determined experimentally, it is trained and tested. The test results of the neural network system allow us to state its high efficiency
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.