Theory Of Ordinary Differential Equations Research Articles

BIFURCATION OF TORI FOR PARABOLIC SYSTEMS OF DIFFERENTIAL EQUATIONS WITH WEAK DIFFUSION

The aim of the present article is to investigate of some properties of quasiperiodic solutions of nonlinear autonomous parabolic systems with the periodic condition. The research is devoted to the investigation of parabolic systems of differential equations with the help of integral manifolds method in the theory of nonlinear oscillations. We prove the existence of quasiperiodic solutions in autonomous parabolic system of differential equations with weak diffusion on the circle. We study existence and stability of an arbitrarily large finite number of tori for a parabolic system with weak diffusion. The quasiperiodic solution of parabolic system is sought in the form of traveling wave. A representation of the integral manifold is obtained. We seek a solution of parabolic system with the periodic condition in the form of a Fourier series in the complex form and introduce the norm in the space of the coefficients in the Fourier expansion. We use the normal forms method in the general parabolic system of differential equations with weak diffusion. We use bifurcation theory for ordinary differential equations and quasilinear parabolic equations. The existence of quasiperiodic solutions in an autonomous parabolic system of differential equations on the circle with small diffusion is proved. The problems of existence and stability of traveling waves in the parabolic system with weak diffusion are investigated.

Equivalence of ill-posed dynamical systems

The problem of topological classification is fundamental in the study of dynamical systems. However, when we consider systems without well-posedness, it is unclear how to generalize the notion of equivalence. For example, when a system has trajectories distinguished only by parametrization, we cannot apply the usual definition of equivalence based on the phase space, which presupposes the uniqueness of trajectories. In this study, we formulate a notion of “topological equivalence” using the axiomatic theory of topological dynamics proposed by Yorke [7], where dynamical systems are considered to be shift-invariant subsets of a space of partial maps. In particular, we study how the type of problems can be regarded as invariants under the morphisms between systems and how the usual definition of topological equivalence can be generalized. This article is intended to also serve as a brief introduction to the axiomatic theory of ordinary differential equations (or topological dynamics) based on the formalism presented in [6].

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO SECOND ORDER DIFFERENTIAL EQUATIONS WITH NONLINEARITIES, THAT ARE COMPOSITIONS OF EXPONENTIAL AND REGULARLY VARYING FUNCTIOS

One of the most actual problems of the modern qualitative theory of ordinary differential equations is the study of nonlinear and, especially, significantly nonlinear non-autonomous differential equations. Among the works in this area related to establishing the asymptotic properties of solutions, the largest part consists of studies of equations with power-law nonlinearities and nonlinearities asymptotically close to power-law nonlinearities, as well as with exponential nonlinearities. The premise of these studies was the study of the Emden–Fowler equation, partial cases of which are used in nuclear physics, gas dynamics, fluid mechanics, relativistic mechanics, and other fields of natural science. The existence conditions and asymptotic representations of a sufficiently wide class of solutions of substantially nonlinear second-order differential equations are found in the paper. This class of solutions was introduced in the works of V. M. Evtukhov for equations of the Emden-Fowler type of the nth order and specified for the equation of the second order. The investigated differential equations contain nonlinearities, which are compositions of exponential and correctly variable when the argument is directed to a special point of the functions. An important difference of this class of equations is the impossibility of even asymptotically representing the nonlinearity in the form of a product of functions, each of which depended either only on the unknown function or only on the derivative of the unknown function. The class of studied solutions contains properly variable solutions of such equations. In the work, asymptotic images are obtained both for the solutions of the studied class and for their first-order derivatives.

Stability analysis of Cohen-Grossberg neural networks with time-varying delay by flexible terminal interpolation method

<abstract><p>In the paper, the existence and uniqueness of the equilibrium point in the Cohen-Grossberg neural network (CGNN) are first studied. Additionally, a switched Cohen-Grossberg neural network (SCGNN) model with time-varying delay is established by introducing a switched system to the CGNN. Based on reducing the conservativeness of the system, a flexible terminal interpolation method is proposed. Using an adjustable parameter to divide the invariant time-delay interval into multiple adjustable terminal interpolation intervals $ (2^{\imath +1}-3) $, more moments when signals are transmitted slowly can be captured. To this end, a new Lyapunov-Krasovskii functional (LKF) is constructed, and the stability of SCGNN can be estimated. Using the LKF method, a quadratic convex inequality, linear matrix inequalities (LMIs) and ordinary differential equation theory, a new form of stability criterion is obtained and specific instances are given to prove the applicability of the new stability criterion.</p></abstract>

THE GENERAL KOVALEVSKAYA THEOREM FOR A CAUCHY PROBLEM

The Kovalevskaya theorem is a well-known mathematical result that provides a powerful tool for solving systems of ordinary differential equations (ODEs). However, there has been a longstanding question regarding whether the theorem can be extended to more general systems of partial differential equations (PDEs). In this paper, we address this question by presenting a discussion of the only general Kovalevskaya theorem that can be extended from the theory of ODEs to systems of PDEs. Our discussion begins by reviewing the classical Kovalevskaya theorem for ODEs and its various applications in mathematics and physics. We then introduce the concept of Frobenius manifold, which provides a framework for extending the Kovalevskaya theorem to PDEs. We show how the Frobenius manifold structure can be used to transform a system of PDEs into a system of ODEs, which can then be solved using the classical Kovalevskaya theorem. Overall, our discussion provides a novel perspective on the Kovalevskaya theorem and its potential applications in PDE theory. By extending the theorem to this broader class of equations, we open up new possibilities for solving complex problems in mathematics, physics, and engineering.

The system of ordinary differential questions with rectangular matrices that is partially solved to the derivatives near the pole.

In modern theory of ordinary differential equations and systems of equations with an unknown complex-valued function of a complex variable, a prominent place is occupied by systems of equations that are either unsolved or partially solved with respect to derivatives. The system of ordinary differential equations, which is partially solved with respect to the derivatives, with rectangular matrices around the pole has been studied. The article presents conditions for transforming a system of ordinary differential equations, which is partially solvable with respect to derivatives, to a system of ordinary differential equations with a special form. The theorem with sufficient conditions of the existence at least one solution of the Cauchy problem is proven, some components of the solution are analytic functions in domains with the fixed singularity on the boundary, and the remaining components are functions chosen from a certain class of functions.

Имитационная модель силового трехфазного трансформатора бронестержневой конструкции с витым магнитопроводом

Background. One of the areas of industry digitalization is associated with the concept of digital twins, which allow simulating the operation of real devices in various modes. At present, simulation models of a three-phase rod-type transformer have been developed in the MatLab Simulink SimPowerSystem environment. The models can serve as the basis to design a digital twin of a real device. The problem is to find the most universal method for mathematical description of a transformer of unspecified design to develop its simulation model. The purpose of this study is to develop a simulation model of a three-phase shell-type transformer core with a twisted magnetic circuit and to study this model when the transformer is operating in steady and transient modes. Materials and methods. The authors have applied the methods to model electrical circuits based on the theory of ordinary differential equations, a simulation method using the MatLab Simulink SimPowerSystem software package. Results. The structural futures of the transformers with shell-core design and the main assumptions when developing the simulation model have been assessed. A system of nonlinear equations has been developed to find the flows and MMF generated by the windings of the device. A simulation model of a three-phase shell-core transformer with a twisted magnetic core has been developed. It is based on the obtained equations using the Algebraic Constraint block with the Solver function in MatLab Simulink SimPowerSystem. The results of simulation of dynamic processes in the transformer in various modes are presented. Conclusions. The results of the study can be used to design power transformers in engineering companies and under production conditions. It is also possible to use the developed models in case of power transformer operation to analyze the static and dynamic modes of operation of sections of electrical networks.

The Stability Model of Piano Tone Tuning Based on Ordinary Differential Equations

Abstract Based on the theory of ordinary differential equations, this paper proposes a stable and discriminative method for piano tone tuning. We perform discriminative training on the hidden Markov tone model according to ordinary differential equations’ feature extraction parameters and model parameters. The model can improve the recognition rate of piano tones. This paper trains and tests the MAPS universal dataset for musical transcription of automatic piano tones. The model is uniformly trained on the synthetic part and tested on the real recording part. The experimental study found that the proposed model has high recognition stability and accuracy in piano tone recognition.

Vallée-Poussin theorem for fractional functional differential equations

An analog of the classical Vallée-Poussin theorem about differential inequality in the theory of ordinary differential equations is developed for fractional functional differential equations. The main results are obtained in a form of a theorem about several equivalent assertions. Among them solvability of two-point boundary value problems with fractional functional differential equation, negativity of Green’s function, and its derivatives and existence of a function v(t) satisfying a corresponding differential inequality. Thus the Vallée-Poussin theorem presents one of the possible “entrances” to assertions on nonoscillating properties and assertions about the negativity of Green’s functions and their derivatives for various two-point problems. Choosing the function v(t) in the condition, we obtain explicit tests of sign-constancy of Green’s functions and their derivatives. It can be stressed that a choice of a corresponding function in the Vallée-Poussin theorem leads to explicit criteria in the form of algebraic inequalities, which, as we demonstrate with examples, cannot be improved. Replacing strict inequalities with non-strict ones will already lead to incorrect statements. In some cases, the well-known results obtained in the form of the Lyapunov inequalities for fractional differential equations can be improved based on ours. Another development, we propose, is a concept to consider fractional functional differential equations. The basis of this concept is to reduce boundary value problems for fractional functional differential equations to operator equations in the space of essentially bounded functions. Fractional functional differential equations can appear in various applications and in the process of construction of equations for a corresponding component of a solution-vector of a system of fractional equations.

Dynamic response of a multi-layered pavement structure with subgrade modulus varying with depth subjected to a moving load

The dynamic response of a three-dimensional multi-layered pavement system subjected to a moving load is calculated by an analytical solution method. In this method, the pavement structure is assumed to be a transverse isotropic viscoelastic multi-layered structure with partially bonded interface, and the subgrade is considered to be a transverse isotropic half-space medium with its modulus varying with depth. The load is considered as a three-dimensional elliptical moving load with a constant speed. This study is divided into three key steps. Firstly, the ordinary differential equations of pavement and subgrade structures are obtained via the double Fourier transform, and the general solutions of dynamic response are obtained according to the theory of ordinary differential equations and the Frobenius method. Secondly, combined with the boundary conditions, the method of wave vector matrix, and the inverse Fourier integral transform, the dynamic responses of pavement structure in time domain are calculated. The accuracy of the analytical solution is verified by comparing against the solution from a finite element method. Lastly, the sensitivity analysis is conducted to assess the influence of four parameters on the dynamic response, namely, the horizontal load, the condition of interlayer contact, the transverse isotropy, and the non-uniform distribution of subgrade moduli. The calculation methods and the corresponding results can further develop a theory of the multi-layered system.

The Existence and Uniqueness of Global Admissible Conservative Weak Solution for the Periodic Single-Cycle Pulse Equation

This paper is devoted to the study of the existence and uniqueness of global admissible conservative weak solutions for the periodic single-cycle pulse equation. We first transform the equation into an equivalent semilinear system by introducing a new set of variables. Using the standard ordinary differential equation theory, we then obtain the global solution to the semilinear system. Next, returning to the original coordinates, we get the global admissible conservative weak solution for the periodic single-cycle pulse equation. Finally, given an admissible conservative weak solution, we find a equation to single out a unique characteristic curve through each initial point and prove the uniqueness of global admissible conservative weak solution without any additional assumptions.

Multi‐step implicit Adams predictor‐corrector network for fire detection

Fire detection methods based on the Convolutional Neural Networks (CNN) have advantages of high accuracy, wide coverage and robustness, receiving significant attention from researchers. Among CNN-based methods, ResNet has achieved better performance than other CNN frameworks in fire detection system, since it uses stacked residual blocks to enlarge the receptive field to overcome the vanishing gradient problem with residual learning. The merits of ResNet can be attributed to the similarity between ResNet and the single-step explicit solver for Ordinary Differential Equations (ODEs), for example, the Euler method. Motivated by the theory of numerical ODE that a multi-step implicit solver has higher accuracy than a single-step explicit solver, the Multi-step Implicit Adams predictor-corrector (MIAPC) network for fire detection is proposed. The MIAPC method is first mapped to a corresponding predictor-corrector Adams block which achieves higher accuracy than a single-step explicit solver. Then, Adaptive Feature Fusion (AFF) and the Spatial Attention Layer (SAL) are utilized to extract hierarchical features from stacked predictor-corrector Adams blocks, forming the corresponding Adams module. Finally, the 4 Adams modules which are made of 4, 6, 8, 10 predictor-corrector Adams blocks and followed by AFF and SAL form the crucial ODE-based approximation part in the proposed network. By adding a simple feature extraction and detection in front of and after the ODE-based approximation part, the MIAPC network is built. Experiments demonstrate that the method achieves 87% accuracy in the challenging test dataset, outperforming existing methods by at least 6%. Besides, the 5.3M model size with inference speed of 4.7 frames/second in CPU and 65.7 frames/second in GPU enables the proposed method to be used in practical applications.

Revealing hidden dynamics from time-series data by ODENet

To derive the hidden dynamics from observed data is one of the fundamental but also challenging problems in many different fields. In this study, we propose a new type of interpretable network called the ordinary differential equation network (ODENet), in which the numerical integration of explicit ordinary differential equations (ODEs) is embedded into the machine learning scheme to build a general framework for revealing the hidden dynamics buried in massive time-series data efficiently and reliably. ODENet takes full advantage of both machine learning algorithms and ODE modeling. On one hand, the embedding of ODEs makes the framework more interpretable benefiting from the mature theories of ODEs. On the other hand, the schemes of machine learning enable data handling, paralleling, and optimization to be easily and efficiently implemented. From classical Lotka-Volterra equations to chaotic Lorenz equations, the ODENet exhibits its remarkable capability in handling time-series data even in the presence of large noise. We further apply the ODENet to real actin aggregation data, which shows an impressive performance as well. These results demonstrate the superiority of ODENet in dealing with noisy data, data with either non-equal spacing or large sampling time steps over other traditional machine learning algorithms.

Global conservative solution for a dissipative Camassa-Holm type equation with cubic and quartic nonlinearities

This paper is devoted to the global conservative solutions of a dissipative Camassa-Holm type equation with cubic and quartic nonlinearities. We first transform the equation into an equivalent semilinear system by introducing a new set of variables. Using the standard ordinary differential equation theory, we then obtain the global solutions of the semilinear system. Returning to the original variables, we get the global conservative solution of the equation. Finally, we show that the peakon solutions of the equation still conserve in .

On the Existence and Stability of Variable Order Caputo Type Fractional Differential Equations

In the theory of differential equations, the study of existence and the uniqueness of the solutions are important. In the last few decades, many researchers have had a keen interest in finding the existence–uniqueness solution of constant fractional differential equations, but literature focusing on variable order is limited. In this article, we consider a Caputo type variable order fractional differential equation. First, we present the existence–uniqueness of a solution of the considered problem. Secondly, By borrowing the idea from the theory of ordinary differential equations, we extend the continuation theorem for the variable order fractional differential equation. Further, we prove the global existence results. Finally, we present different types of Ulam–Hyers stability results, which have never been studied before for the Caputo type variable order fractional differential equation.

Analysis of an Eco-Epidemiological Model Considering Effect of Harvesting and Prey Refuge

In this work, we analyze an eco-epidemiological model with the disease in the prey, considering a constant proportion of harvesting of either species or a prey refuge. The positive invariant set, the conditions of existence, and locally asymptotically stability of the equilibria are studied using the stability theory of ordinary differential equation. The global stability of border equilibria by constructing Lyapunov functions and permanence of the system by comparison theoremare proved. The numerical simulation further proved the correctness of the theoretical analysis. The result indicates that overfishing would lead to population extinction and a reasonable fishing strategy should keep the coexistence of populations.

ODE-RU: a dynamical system view on recurrent neural networks

<abstract><p>The core of the demonstration of this paper is to interpret the forward propagation process of machine learning as a parameter estimation problem of nonlinear dynamical systems. This process is to establish a connection between the Recurrent Neural Network and the discrete differential equation, so as to construct a new network structure: ODE-RU. At the same time, under the inspiration of the theory of ordinary differential equations, we propose a new forward propagation mode. In a large number of simulations and experiments, the forward propagation not only shows the trainability of the new architecture, but also achieves a low training error on the basis of main-taining the stability of the network. For the problem requiring long-term memory, we specifically study the obstacle shape reconstruction problem using the backscattering far-field features data set, and demonstrate the effectiveness of the proposed architecture using the data set. The results show that the network can effectively reduce the sensitivity to small changes in the input feature. And the error generated by the ordinary differential equation cyclic unit network in inverting the shape and position of obstacles is less than $ 10^{-2} $.</p></abstract>

APPROXIMATION OF BOUNDARY VALUE PROBLEMS FOR INTEGRO-DIFFERENTIAL EQUATIONS WITH DELAY

In mathematical modeling of physical and technical processes, the evolution of which depends on prehistory, we arrive at differential equations with a delay. With the help of such equations it was possible to identify and describe new effects and phenomena in physics, biology, technology. An important task for differential-functional equations is to construct and substantiate finding approximate solutions, since there are currently no universal methods for finding their precise solutions. Of particular interest are studies that allow the use of methods of the theory of ordinary differential equations for the analysis of delay differential equations. Schemes for approximating differential-difference equations by special schemes of ordinary differential equations are proposed in the works N. N. Krasovsky, A. Halanay, I. M. Cherevko, L. A. Piddubna, O. V. Matwiy in various functional spaces. The purpose of this paper is to apply approximation schemes of differential-difference equations to approximation of solutions of boundary-value problems for integro-differential equations with a delay. The paper presents sufficient conditions for the existence of a solution of the boundary value problem for integro-differential equations with many delays. The scheme of its approximation by a sequence of boundary value problems for ordinary integro-differential equations is proposed and the conditions of its convergence are investigated. A model example is considered to demonstrate the given approximation scheme.

A novel multivariate grey system model with conformable fractional derivative and its applications

To further improve the effectiveness and precision of multivariate time series forecasting, a conformable fractional derivative multivariate grey system model is proposed in this work. Firstly, the general solution of the model is deduced by the grey theory, the definition of the conformable fractional derivative, the theory of ordinary differential equations and the two-point trapezoidal approximation formula. Then the ordinary least squares estimation method is utilized to derive linear system parameters, and the ant lion optimizer algorithm is used to search the optimal fractional order. Finally the newly constructed model is applied to forecast the urban consumption per capita of China. The total mean absolute percentage error of the new model is only 0.6434%, in comparison with ones obtained from GM(1,1), DGM(1,1), CFGM(1,1), CFNGM(1,1,k,c), CFGM(1,1,D), MLR, GMCT(1,N), RDGM(1,N), FGMC(1,N) and GMC(r,N), which are 4.0334%, 4.0837%, 3.4751%, 3.6629%, 2.2129%, 1.0487%, 784.1107%, 1.0168%, 0.7655% and 0.7954% respectively. The results show the new model has a better performance compared with other models.

Modeling thermal processes using electrical equivalent circuits in digital twins of technical devices

One of the promising areas of digitalization of the economy today is associated with the concept of digital twins of technical systems. Digital twins based on simulation models of technical devices, which are calibrated according to the results of experimental studies on a real-time device are of great interest. Such models allow us to conduct preventive analysis of the operation consequences of these devices in various modes. Thus, a problem occurs to develop parametrically coupled models of physical processes of various natures that underlie the principles of operation of these devices. Currently, for these purposes, simulation software packages are used. MatLab Simulink software is the most popular one. However, not all such software packages provide tools to work with chain models of all physical processes that are of the user interest. Thus, the purpose of this article is to develop methods to construct digital twins of technical devices using models of arbitrary physical processes (in particular, thermal) based on electrical equivalent circuits. It unifies the task of modeling processes based on circuit theory. To develop the method to construct digital twins, the authors have applied the phenomenon of isomorphism of equations of physical processes based on the theory of circuits that is based on the theory of ordinary differential equations. The simulation has been carried out in the MatLab Simulink environment using the SimScape library for modeling physical processes. Assumptions typical for circuit theory are made during simulation. A method has been developed to construct simulation models based on the use of electrical equivalent circuits of physical processes of an arbitrary nature. In contrast to existing approaches, where the analogy method is used to simulate one of the processes of the researcher interest, it is proposed to develop a single integrated model of all physical processes underlying the operation of this class of devices. It will reduce the level of requirements for simulation systems by limiting the requested functionality of these systems to electrical circuits only. The proposed method can be used as the basis for the development of digital twins of technical devices that allow simulating their operation in arbitrary modes, considering a variety of related factors of different physical nature.