Periodic solutions for first-order cubic non-autonomous differential equation with bifurcation analysis

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

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.

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.

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.

The behavior of a considerable number of physical phenomena is described by periodic bifurcating solutions of quadratic Ordinary Differential Equations. The purpose of this paper is to present an algorithm consisting of a combination of Power series and Fourier Series (PES) which reduces the problem of solving the differential equations to a recursive sequence of linear algebraic ones. This algorithm is based on two theorems that, with easily verifiable algebraic conditions, ensure the existence of such solutions.

Numerical solution for high order differential equations using a hybrid neural network—Optimization method

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

When estimating the error of numerical solutions of ordinary differential equations, it is proposed to use the regularization of error estimation algorithms for approximate solutions. Regularization is considered as the use of additional a priori information about the numerical solution. A priori information is estimated by the method of the inverse error analysis, which implements the transformation of the statement of the problem of evaluating the accuracy of the numerical solution. In this case, the problem is selected so that the approximate solution is the exact solution of the transformed problem, close to the original problem. It is also possible to add some additional restrictions to the conditions of the problem. This allows us to solve the incorrectly posed problems of estimating errors in numerical solutions, which in most cases are unstable. It is known that earlier in the numerical solution of linear algebra problems the inverse error analysis was used by Wilkinson J. and also Voevodin V.V., application of the inverse error analysis in other areas of numerical analysis can be noted. Using direct and inverse error analysis Voevodin V.V. effectively calculated majorants of rounding errors in the most important methods of linear algebra and significantly developed the method of inverse error analysis.. However, the regularization process associated with the reverse analysis of errors has not been previously studied. The article discusses the characteristics of regularization in the inverse analysis of errors in the numerical solution of ordinary differential equations with initial data. The regularization of error estimation algorithms has advantages in the numerical solution of ordinary differential equations of real models of physics and technology.

THIS investigation is primarily concerned with the solution of boundary-value problems by means of automatic computing equipment. Specifically, it is concerned with the solution of ordinary differential equations in which certain conditions (boundary values) are specified at two or more values of the independent variable. Such problems are generally called boundary-value problems; however, it must be emphasized that this report is restricted to the solution of ordinary differential equations, and not to the solution of partial differential equations.

Non-linear oscillations of a third-order differential equation

A novel genetic algorithm has been developed and applied to the solution of ordinary differential equations. The algorithm solves the equations by a process of breeding better candidate solutions from a family of estimates, and learns to retain the best features as it progresses. This self-learning system is intrinsically parallel, and capable of handling linear and nonlinear equations, both stiff and non-stiff. Genetic algorithms are a key element in artificial intelligence and machine learning, playing a significant role in optimization and robotics. In this document, the application of genetic algorithms to the solution of ordinary differential equations is presented as a radically different way of approaching numerical simulations, and is of importance to many disciplines.

This paper presents solutions of second-order ordinary differential equations on bond problems to estimate crack widths of reinforced concrete (RC) members subjected to repeated cyclic loads. Up to now, bond behaviors under repeated cyclic loads have usually been calculated by a difference scheme. In contrast, this research adopts analytical solutions of the differential equations to calculate bond behaviors in finite element models where a number of concrete cracks occur. The conventional differential equation for the bond problem has been available only under monotonic loading conditions. This paper extends the equation to cases where the reinforcement becomes plastic and the bond is subject to unloading/reloading conditions. The bond stress-slip relationship is modeled by a multilinear model that is composed of an envelope made of five lines, unloading/reloading paths, and a negative friction path. The solution of the differential equation is classified into 25 cases when the bond stress-slip relationship is on the envelope and seven cases when on the unloading/reloading paths or the negative friction path. The solutions are implemented into the smeared crack-based finite-element method (FEM) to evaluate the crack widths of a RC beam subject to repeated cyclic flexural/shear loads. The FEM program is incorporated with the discrete-like crack procedure developed by the authors and calculates stress redistributions due to cracking using the solutions of the differential equations. Typical distributions of reinforcement stress, bond stress, and slip along the reinforcement are presented.

Damped and driven oscillators are generally modeled with a nonautonomous second-order nonlinear ordinary differential equation including a sinusoidal driving forcing term, such as the forced Duffing equation and the forced Holmes–Rand equation. These equations have been extensively studied during the last century and the last two decades. In the early 1990s, Abarbanel, Rabinovich and Sushchik proposed replacing the sinusoidal forcing term with a “force controlled by the movements of the oscillator itself”, i.e. by the product of two variables: the first being the solution of the oscillator itself, while the second is the solution of a first-order nonlinear ordinary differential equation. They referred to the resulting autonomous dynamical system of two coupled nonlinear ordinary differential equations as a “controlled nonlinear oscillator”. To that end, they introduced a change of variables and parameters to transform the “controlled nonlinear oscillator” that corresponds to a particular case of the forced Duffing equation into the Lorenz system. The aim of this work is to show that their idea can be further generalized and applied to many other dynamical systems, including the forced Holmes–Rand equation, Chua’s cubic circuit, Chen’s system and the forced Helmholtz oscillator. It is proved that a certain class of three-dimensional dynamical systems can be rewritten into the form of “generalized controlled nonlinear oscillators”, which can then be transformed into various Lorenz-like systems. Such a transformation could be very useful for the study of intermittent chaos.

In this paper, sufficient criteria for the existence of solutions to uniform asymptotic stability and boundedness problems associated with certain second order nonlinear non autonomous ordinary differential equation are established with the aid of Lyapunov's direct method. Furthermore, the appropriate complete Lyapunov function is given explicitly. Our results complement some well known results on the second order differential equations in the literature.

The problem of constructing and classifying elliptic solutions of nonlinear differential equations is studied. An effective method enabling one to find any elliptic solution of an autonomous nonlinear ordinary differential equation is described. The method does not require integrating additional differential equations. Much attention is being paid to the case of elliptic solutions with several poles in a parallelogram of periods. With the help of the method we find elliptic solutions up to the fourth order inclusively of an ordinary differential equation with a number of physical applications. The method admits a natural generalization and can be used to find elliptic solutions satisfying systems of ordinary differential equations.