Research of Chaotic Dynamics of 3D Autonomous Quadratic Systems by Their Reduction to Special 2D Quadratic Systems

New results about the existence of chaotic dynamics in the quadratic 3D systems are derived. These results are based on the method allowing studying dynamics of 3D system of autonomous quadratic differential equations with the help of reduction of this system to the special 2D quadratic system of differential equations

Stability of Neural Ordinary Differential Equations with Power Nonlinearities

The article presents a study of solutions of ODEs system with a specialnonlinear part, which is a continuous analogue of an arbitrary recurrent neural network(neural ODEs). As a nonlinear part of the mentioned system of differential equations, weused sums of piecewise continuous functions, where each term is a power function. (Theseare activation functions.) The use of power activation functions (PAF) in neural networksis a generalization of well-known the rectified linear units (ReLU). In the present timeReLU are commonly used to increase the depth of trained of a neural network. Therefore,the introduction of PAF into neural networks significantly expands the possibilities ofReLU. Note that the purpose of introducing power activation functions is that theyallow one to obtain verifiable Lyapunov stability conditions for solutions of the systemdifferential equations simulating the corresponding dynamic processes. In turn, Lyapunovstability is one of the guarantees of the adequacy of the neural network model for theprocess under study. In addition, from the global stability (or at least the boundedness)of continuous analog solutions it follows that learning process of the corresponding neuralnetwork will not diverge for any training sample.The article presents a study of solutions of ODEs system with a special nonlinear part, which is a continuous analogue of an arbitrary recurrent neural network (neural ODEs). As a nonlinear part of the mentioned system of differential equations, we used sums of piecewise continuous functions, where each term is a power function. (These are activation functions.) The use of power activation functions (PAF) in neural networks is a generalization of well-known the rectified linear units (ReLU). In the present time ReLU are commonly used to increase the depth of trained of a neural network. Therefore, the introduction of PAF into neural networks significantly expands the possibilities ofReLU. Note that the purpose of introducing power activation functions is that they allow one to obtain verifiable Lyapunov stability conditions for solutions of the system differential equations simulating the corresponding dynamic processes. In turn, Lyapunov stability is one of the guarantees of the adequacy of the neural network model for the process under study. In addition, from the global stability (or at least the boundedness) of continuous analog solutions it follows that learning process of the corresponding neural network will not diverge for any training sample

Univariate Time Series Analysis with Hyper Neural ODE

Neural ordinary differential equations (NODE) are ordinary differential equations whose right-hand side is determined by a neural network. Hyper NODE (hNODE) is a special type of neural network architecture, which is aimed at creating such NODE system that regulates its own parameters based on known input data. The article uses a new approach to the study of one-dimensional time series, the basis of which is the hNODE system. This system takes into account the relationship between the input data and its latent representation in the network and uses an explicit parametrization when controlling the latent flow. The proposed model is tested on artificial time series of data. The influence of some activation functions (besides sigmoid and hyperbolic tangent) on the quality of the forecast is also considered

Systems of Singular Differential Equations as the Basis for Neural Network Modeling of Chaotic Processes

Currently, systems of neural ordinary differential equations (ODEs) have become widespread for modeling various dynamic processes. However, in forecasting tasks, priority remains with the classical neural network approach to building a model. This is due to the fact that by choosing the neural network architecture, a more accurate approximation of the trajectories of a dynamic system can be achieved. It is known that the accuracy of the mentioned approximation significantly depends on the settings of the neural network parameters and their initial values. In this regard, the main idea of the article is that the initial values of the neural network parameters are taken to be equal to the parameters of the neural ODE system obtained by modeling the same process, which will then be simulated using a neural network. Subsequently, the singular ODE system was used to adjust the parameters of the LSTM (Long Short Term Memory) neural network. The results obtained were used to model the process of epilepsy

Modeling of Chaotic Processes by Means of Antisymmetric Neural ODEs

The main goal of this work is to construct an algorithm for modeling chaotic processes using special neural ODEs with antisymmetric matrices (antisymmetric neural ODEs) and power activation functions (PAFs). The central part of this algorithm is to design a neural ODEs architecture that would guarantee the generation of a stable limit cycle for a known time series. Then, one neuron is added to each equation of the created system until the  approximating properties of this system satisfy the well-known Kolmogorov theorem on the approximation of a continuous function of many variables. Inaddition, as a result of such an addition of neurons, the cascade of bifurcations that allows generating a chaotic attractor from stable limit cycles is launched. We also consider the possibility of generating a homoclinic orbit whose bifurcations lead to the appearance of a chaotic attractor of another type. In conclusion, the conditions under which the found attractor adequately simulates the chaotic process are discussed. Examples are given

Discrete Processes and Chaos in Systems of Ordinary Differential Equations

A method for constructing a one-dimensional discrete mapping describing a certain periodic process in a general system of ordinary autonomous differential equations is proposed. The resulting discrete mapping is then used to prove the existence of chaos in the original system of differential equations

A new approach to the problem of diagnostics of cerebral cortex diseases using chaotic dynamics methods

An modeling attempt of behavior process of brain electric impulses for some patient by solutions of 3Dsystem of autonomous quadratic differential equations is undertaken. This system of differential equations was got from a multivariatetimes series with the help of polynomial averages and least squares method. Further, with the help of the got system a question about existence of chaotic attractor in this system is studied. In this case, the presence of chaotic attractor makes it possible to interpretas the absence of disease and vice versa

Singular Differential Equations and their Applications for Modeling Strongly Oscillating Processes

The normal system of ordinary differential equations, whose right-hand sides are the ratios of linear and nonlinear positive functions, is considered. A feature of these ratios is that some of their denominators can take on arbitrarily small nonzero values. (Thus, the modules of the corresponding derivatives can take arbitrarily large value.) In the sequel, the constructed system of differential equations is used to model strongly oscillating processes (for example, processes determined by the rhythms of electroencephalograms measured at certain points in the cerebral cortex). The obtained results can be used to diagnose human brain diseases

Exhibitions of Japanese children’s drawings in the USSR: Depicting Japan, showing the world

The article is devoted to the history of exhibitions of Japanese children’s drawings in the Soviet Union in 1920s – 1980s, as well as to the critical interpretation and perception by the Soviet audience of the artistic works of Japanese children. The importance of such events can be seen not only in the artistic value of the exposition material, but also in the influence of the expositions on the image of Japan in mass consciousness. The material is devoted to key exhibition projects related to the presentation of Japanese children’s art, in particular, the “Exhibition of Children’s Books and Children’s Art of Japan” in 1928, as well as a series of international exhibitions “I See the World,” held in the USSR since the late 1960s. The greatest attention is paid to the peculiarities of Soviet art criticism towards Japanese children’s drawing in the pre-war and post-war period, as well as the influence of Soviet ideology on the interpretation of children’s art from Japan. The author comes to the conclusion that the approach to the exhibitions was characterized by ideological indoctrination, as well as certain stereotypes about Japan, which created a request for exoticization of the creative products of the Japanese children. During the initial period of the Russian-Japanese cultural ties, despite the controversial nature of the Soviet art criticism of Japanese children’s drawings, the exhibition had substantial importance for the cultural ties of the two countries. In the post-war period, not only mono- national exhibitions, but also large projects involving multiple countries drew attention to various creative works of Japanese children. Since the early 1990s, the past importance of such exhibitions as an important element of cultural exchange receded, which is also true for the present times, despite the episodic exhibition projects of this sort in various regions of Russia. The “propaganda” component of children’s drawings faded. It is, however, regrettable that such exhibitions stopped attracting public attention due to the lack of interest of the media to these initiatives, as well as of systematic study of the works of Japanese children from the point of view of art studies and psychology. The article is based on documents, many of which are being introduced into scientific circulation for the first time, from the following archives: the State Archive of the Russian Federation (GARF), the Russian State Archive of Literature and Art (RGALI), the Central State Archive of St. Petersburg

ON EQUIVALENCE OF LINEAR CONTROL SYSTEMS AND ITS USAGE TO VERIFICATION OF THE ADEQUACY OF DIFFERENT MODELS FOR A REAL DYNAMIC PROCESS

A problem of description of algebraic invariants for a linear control system thatdetermine its structure is considered. With the help of these invariants, the equivalence problem of two linear time-invariant control systems with respect to actions of some linear groups on the spaces of inputs, outputs, and states of these systems is solved. The invariants are used to establish the necessary equivalence conditions for two nonlinear systems of differential equations generalizing the well-known Hopfield neural network model. Finally, these conditions are applied to establish the adequacy of two neural network models designed to describe the behavior of a real dynamic process given by two different sets of time series