Numerical Analysis of Ordinary and Delay Differential Equations

Periodic solutions of delay differential equations (DDEs) splitting into stable and unstable branches are examined in an infinite-dimensional space for fixed and multiple time delays. The center manifold theorem and the classical Hopf bifurcation theorem for the study of periodic solutions of ordinary differential equations (ODEs) are employed to reduce the infinite-dimensional character of the DDEs to finite-dimensional ODEs. Using integral averaging method, the vector field of the ODEs is converted and averaged into amplitude a and phase φ relations. From these relations bifurcation equations of the form ℑ(a, φ) = 0 for the solution branches are derived.

The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found

Periodic Solutions in the Golden Sixties: The Birth of a Continuation Theorem

Introduction Suggestions for the readerSuggestions for the reader Fredholm mappings of index zero and linear boundary value problems Degree theory for some classes of mappings Duality theorems for several fixed point operators associated to periodic problems for ordinary differential equations Existence theorems for equations in normed spaces Boundary value problems for second order nonlinear vector differential equations Periodic solutions of ordinary differential equations with one-sided growth restrictions Bound sets for functional differential equations The index of isolated zeros of some mappings Bifurcation theory Periodic solutions of autonomous ordinary differential equations around an equilibrium References.

Ordinary differential equations on singular spaces, Z. Bartosiewicz stability in delayed neural networks, J. Belair a condition on multi-existence of periodic solutions for a differential delay equation, Y. Cao control of global economic growth - will the centre hold?, E.N. Chukwu asymptotic behaviour of the Titchmarsh-Weyl coefficient for a coupled second order system, S.L. Clark stability problems for systems of nuclear reactors, C. Cordoneanu comparison theorems for disconjugate linear differential equations, M. Gaudenzi oscillation results for higher order nonlinear neutral delay equations with periodic coefficients, J.R. Graef, M.K. Grammatikopoulos and P.W. Spikes an implicit differential equation related to epidemic models, K.P. Hadeler and R. Shonkwiler the relationship between stability under disturbances and uniform stability in a periodic integrodifferential equation, Y. Hamaya on the asymptotic stability of the equilibrium of the damped oscillator, L. Hatvani on higher order nonlinear differential-difference equations, U. An Der Heiden vector field approximations flow homogencity, H. Hermes hopf bifurcation for a differential-difference equation from climate modeling, G. Hetzer shock layer behaviour for vector boundary value problems, S.J. Kirschvink properties of solutions of nth order equations, W.A.J. Kosmala small solutions to BVP's at resonance with nonhomogeneous nonlinearity, L. Lefton vibrational control of time delay systems, B. Lehman bifurcation set and compound eyes in a perturbed cubic Hamiltonian system, J. Li and Z. Lu gevrey character of formal solutions of nonlinear differential equations, X. Liu finite-difference schemes having the correct linear stability properties for all finite step-sizes, R.E. Mickens evolution of surface functionals and differential equations, Y. Li and J.S. Muldowney some remarks on stability properties in functional differential equations with infinite delay, S. Murakami and T. Yoshizawa periodic orbits of the Froeschle's map, A. Olvera and C. Vargas rotated vector fields, global families of limit cycles and Hilbert's 16th problem, L.M. Perko green's matrices and disconjugacy of a vector difference equation, A. Peterson the poincare manifold for the general case of a planar flow, W. Rivera hopf bifurcation in a class of ODE systems related to climate modeling, P.G. Schmidt on second order two point boundary value problems at resonance, M. Hihnala and S. Seikkala separatrix connections of quadratic gradient vector fields, D.S. Shafer. Part contents.

First, we prove a necessary and sufficient condition for global in time existence of all solutions of an ordinary differential equation (ODE). It is a condition of one‐sided estimate type that is formulated in terms of so‐called proper functions on extended phase space. A generalization of this idea to stochastic differential equations (SDE) and parabolic equations (PE) allows us to prove similar necessary and sufficient conditions for global in time existence of solutions of special sorts: L1‐complete solutions of SDE (this means that they belong to a certain functional space of L1 type) and the so‐called complete Feller evolution families giving solutions of PE. The general case of equations on noncompact smooth manifolds is under consideration.

Methods used for ordinary differential equations cannot generally be used to solve delay differential equation, which is reflected in the choice of suitable software. The paper shows possibilities of software solution of delay differential equations. The aim of the paper is to present the possibilities of current software packages and programme systems (e.g. Matlab, Maple, R). The paper further shows how the aforementioned equations can be used in solutions of dynamical models. The conclusion of the papers demonstrates a solution of a specific dynamical model -- the Phillips curve applied to the Czech Republic. Setting up the model required the use of analytic and synthetic methods, dynamical modelling and solving the system of two delay differential equations. In the conclusion the authors claim that although the quality of the available software, suitable for solving delay differential equations, is not as good as that of software used for solving ordinary differential equations, such software that meets basic requirements can be found. Therefore such software can be used as a supportive tool for exact modelling methods in practice.

Finite-dimensional approximations are developed for retarded delay differential equations (DDEs). The DDE system is equivalently posed as an initial-boundary value problem consisting of hyperbolic partial differential equations (PDEs). By exploiting the equivalence of partial derivatives in space and time, we develop a new PDE representation for the DDEs that is devoid of boundary conditions. The resulting boundary condition-free PDEs are discretized using the Galerkin method with Legendre polynomials as the basis functions, whereupon we obtain a system of ordinary differential equations (ODEs) that is a finite-dimensional approximation of the original DDE system. We present several numerical examples comparing the solution obtained using the approximate ODEs to the direct numerical simulation of the original non-linear DDEs. Stability charts developed using our method are compared to existing results for linear DDEs. The presented results clearly demonstrate that the equivalent boundary condition-free PDE formulation accurately captures the dynamic behaviour of the original DDE system and facilitates the application of control theory developed for systems governed by ODEs.

Ordinary differential equations (ODEs) are a fundamental tool for modeling dynamical systems in various scientific fields. However, solving ODEs analytically can be challenging, and numerical methods can be computationally expensive or inaccurate. In recent years, deep neural networks (DNNs) have emerged as a promising alternative for solving ODEs. This paper focuses on the utilization of DNNs to solve first and second-order ODEs. Existing works in this field are reviewed, with an emphasis on identifying their limitations. Subsequently, an approach is proposed to overcome these limitations. The proposed method utilizes a combination of a deep neural network and various activation functions to achieve accurate and efficient solutions for ordinary differential equations (ODEs). Deep neural networks (DNNs) have shown significant potential in solving ordinary differential equations (ODEs) due to their ability to approximate complex functions and their capacity to learn from data. Traditional numerical methods for solving ODEs have some limitations such as requiring large amounts of computation, limited accuracy, and difficulty in handling high-dimensional systems. The aim is to overcome these limitations by leveraging the capabilities of DNNs, that have the potential to learn and generalize from data, provide more accurate solutions, requiring fewer computational resources, and being able to handle high-dimensional systems. This paper aims to explore the use of DNNs in solving 1st and 2nd order differential equations, and proposes a novel approach for improving the accuracy and efficiency of the solution process.

The Painlevé approach for finding solitary wave solutions of nonlinear nonintegrable differential equations

Genetic oscillatory networks can be mathematically modeled with delay differential equations (DDEs). Interpreting genetic networks with DDEs gives a more intuitive understanding from a biological standpoint. However, it presents a problem mathematically, for DDEs are by construction infinitely-dimensional and thus cannot be analyzed using methods common for systems of ordinary differential equations (ODEs). In our study, we address this problem by developing a method for reducing infinitely-dimensional DDEs to two- and three-dimensional systems of ODEs. We find that the three-dimensional reductions provide qualitative improvements over the two-dimensional reductions. We find that the reducibility of a DDE corresponds to its robustness. For non-robust DDEs that exhibit high-dimensional dynamics, we calculate analytic dimension lines to predict the dependence of the DDEs’ correlation dimension on parameters. From these lines, we deduce that the correlation dimension of non-robust DDEs grows linearly with the delay. On the other hand, for robust DDEs, we find that the period of oscillation grows linearly with delay. We find that DDEs with exclusively negative feedback are robust, whereas DDEs with feedback that changes its sign are not robust. We find that non-saturable degradation damps oscillations and narrows the range of parameter values for which oscillations exist. Finally, we deduce that natural genetic oscillators with highly-regular periods likely have solely negative feedback.

This work focuses on the existence of quasi-periodic solutions for ordinary and delay differential equations (ODEs and DDEs for short) with an elliptic-type degenerate equilibrium point under quasi-periodic perturbations. We prove that under appropriate hypotheses there exist quasi-periodic solutions for perturbed ODEs and DDEs near the equilibrium point for most parameter values, then apply these results to the delayed van der Pol’s oscillator with zero-Hopf singularity.

This textbook provides a broad introduction to continuous and discrete dynamical systems. With its hands-on approach, the text leads the reader from basic theory to recently published research material in nonlinear ordinary differential equations, nonlinear optics, multifractals, neural networks, and binary oscillator computing. Dynamical Systems with Applications Using Python takes advantage of Python’s extensive visualization, simulation, and algorithmic tools to study those topics in nonlinear dynamical systems through numerical algorithms and generated diagrams. After a tutorial introduction to Python, the first part of the book deals with continuous systems using differential equations, including both ordinary and delay differential equations. The second part of the book deals with discrete dynamical systems and progresses to the study of both continuous and discrete systems in contexts like chaos control and synchronization, neural networks, and binary oscillator computing. These later sections are useful reference material for undergraduate student projects. The book is rounded off with example coursework to challenge students’ programming abilities and Python-based exam questions. This book will appeal to advanced undergraduate and graduate students, applied mathematicians, engineers, and researchers in a range of disciplines, such as biology, chemistry, computing, economics, and physics. Since it provides a survey of dynamical systems, a familiarity with linear algebra, real and complex analysis, calculus, and ordinary differential equations is necessary, and knowledge of a programming language like C or Java is beneficial but not essential.

Dedicated to Professor Yuzan He on the Occasion of his 80th BirthdayIn this paper, we employ the complex method to obtain all meromorphic solutions of an auxiliary ordinary differential equation at first and then find out all meromorphic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations. Our result shows that all rational and simply periodic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations are solitary wave solutions, the method is more simple than other methods, and there exist some rational solutions wr,2(z) and simply periodic solutions ws,2(z) that are not only new but also not degenerated successively by the elliptic function solutions. Copyright © 2016 John Wiley & Sons, Ltd.

Solution of nonlinear partial differential equations from base equations