The aim of the paper was to analyze the given nonlinear problem by different methods of computation of the Lyapunov exponents (Wolf method, Rosenstein method, Kantz method, the method based on the modification of a neural network, and the synchronization method) for the classical problems governed by difference and differential equations (Hénon map, hyperchaotic Hénon map, logistic map, Rössler attractor, Lorenz attractor) and with the use of both Fourier spectra and Gauss wavelets. It has been shown that a modification of the neural network method makes it possible to compute a spectrum of Lyapunov exponents, and then to detect a transition of the system regular dynamics into chaos, hyperchaos, and others. The aim of the comparison was to evaluate the considered algorithms, study their convergence, and also identify the most suitable algorithms for specific system types and objectives. Moreover, an algorithm of calculation of the spectrum of Lyapunov exponents based on a trained neural network has been proposed. It has been proven that the developed method yields good results for different types of systems and does not require a priori knowledge of the system equations.

Chronic venous disease (CVD) occurs in a substantial proportion of the world’s population. If the onset of CVD looks like a cosmetic defect, over time, it might be transformed into serious problems that will require surgical intervention. The aim of this work is to use deep learning (DL) methods for automatic classification of the stage of CVD for self-diagnosis of a patient by using the image of the patient’s legs. The images of legs with CVD required for DL algorithms were collected from open Internet resources using the developed algorithms. For image preprocessing, the binary classification problem “legs–no legs” was solved based on Resnet50 with accuracy of 0.998. The application of this filter made it possible to collect a dataset of 11,118 good-quality leg images with various stages of CVD. For classification of various stages of CVD according to the CEAP classification, the multi-classification problem was set and resolved by using four neural networks with completely different architectures: Resnet50 and transformers such as data-efficient image transformers (DeiT) and a custom vision transformer (vit-base-patch16-224 and vit-base-patch16-384). The model based on DeiT without any tuning showed better results than the model based on Resnet50 did (precision = 0.770 (DeiT) and 0.615 (Resnet50)). vit-base-patch16-384 showed the best results (precision = 0.79). To demonstrate the results of the work, a Telegram bot was developed, in which fully functioning DL algorithms were implemented. This bot allowed evaluating the condition of the patient’s legs with fairly good accuracy of CVD classification.

In this part of the paper, the theory of nonlinear dynamics of flexible Euler–Bernoulli beams (the kinematic model of the first-order approximation) under transverse harmonic load and colored noise has been proposed. It has been shown that the introduced concept of phase transition allows for further generalization of the problem. The concept has been extended to a so-called noise-induced transition, which is a novel transition type exhibited by nonequilibrium systems embedded in a stochastic fluctuated medium, the properties of which depend on time and are influenced by external noise. Colored noise excitation of a structural system treated as a system with an infinite number of degrees of freedom has been studied.

One of the critical problems in multiclass classification tasks is the imbalance of the dataset. This is especially true when using contemporary pre-trained neural networks, where the last layers of the neural network are retrained. Therefore, large datasets with highly unbalanced classes are not good for models’ training since the use of such a dataset leads to overfitting and, accordingly, poor metrics on test and validation datasets. In this paper, the sensitivity to a dataset imbalance of Xception, ViT-384, ViT-224, VGG19, ResNet34, ResNet50, ResNet101, Inception_v3, DenseNet201, DenseNet161, DeIT was studied using a highly imbalanced dataset of 20,971 images sorted into 7 classes. It is shown that the best metrics were obtained when using a cropped dataset with augmentation of missing images in classes up to 15% of the initial number. So, the metrics can be increased by 2–6% compared to the metrics of the models on the initial unbalanced data set. Moreover, the metrics of the rare classes’ classification also improved significantly–the True Positive value can be increased by 0.3 or more. As a result, the best approach to train considered networks on an initially unbalanced dataset was formulated.

A mathematical model for studying the nonlinear response of the endwall of a narrow channel filled with a viscous fluid to the vibration of the channel’s upper wall was formulated. The channel, formed by two parallel, rigid walls, was investigated. The right end-channel wall was supported by a nonlinear spring. At the end of the left channel, the fluid flowed into a cavity with constant pressure. The upper channel wall oscillated according to a given law. As a result of the interaction between the endwall and the upper wall via a viscous fluid, the forced, nonlinear oscillations of the channel endwall arose. The fluid motion was considered in terms of the hydrodynamic lubrication theory. The endwall was studied as a spring-mass system with a nonlinear cubic restoring force. The coupled hydroelasticity problem was formulated, and it was shown that the problem under consideration was reduced to a single equation in the form of the Duffing equation. The nonlinear hydroelastic response of the end wall was determined by means of the harmonic balance method. The results of numerical experiments on nonlinear hydroelastic response behavior and a comparison with the case when the support spring is linear were presented. The obtained results are of a fundamental nature and can be used in modeling various devices and systems that have narrow channels filled with viscous fluid and are subjected to vibrations on one side of the channel. For example, coolant pipes are subjected to vibrations from the engine. Of particular interest is the application of the presented solution to the mathematical modeling of nano- and micro-spacecraft systems with fluids since the proposed decision allows for the consideration of some boundary effects, which is important for nano- and micro-spacecraft due to their small size.

Monitoring moving bio-objects is currently of great interest for both fundamental and practical research. The advent of deep-learning algorithms has made it possible to automate the qualitative and quantitative analysis of the behavior of bio-objects recorded in video format. When processing such data, it is necessary to consider additional factors, such as background noise in the frame, the speed of the bio-object, and the need to reflect information about the previous (past) and subsequent (future) pose of the bio-object in one video frame. The preprocessed dataset must be suitable for verification by experts. This article proposes a method for preprocessing data to identify the behavior of a bio-object, a clear example of which is experiments on laboratory animals with the collection of video data. The method is based on combining information about a behavioral event presented in a sequence of frames with the addition of a native image and subsequent boundary detection using the Sobel filter. The resulting representation of a behavioral event is easily perceived by both human experts and neural networks of various architectures. The article presents the results of training several neural networks on the obtained dataset and proposes an effective neural network architecture (F1-score = 0.95) for identifying discrete events of biological objects’ behavior.