Vector Fields and Ordinary Differential Equations

This note discusses the definition, the history and the importance of the qualitative theory of ordinary differential equations (ODE). Although such discussions can be found in many textbooks and online lecture notes, some of the information provided are distinct and some are repeated. This article takes the union of the information and fills in the missing details by providing suitable examples from other references. The qualitative theory of ODEs are essential to complement the analytical and numerical methods. It is also required to prove qualitative properties of a dynamical system. Qualitative analysis of ODEs is also needed when exact solution, even if it can be found, is difficult to interpret or visualise. Furthermore, in some applications, only qualitative properties of the system are of interest. The qualitative methods also will provide information on the sensitivity of the system under study to perturbations of change of initial conditions and of various parameters.

The material presented below amounts roughly to the so-called fundamental theorems of Lie and their implications concerning Lie algebras, Lie subgroups and subalgebras. By straightforward and fairly elementary steps, we shall extend the concept of Lie group to include groups admitting coordinate systems in which the functions defining ab, the group product of elements a, b, possess continuous first order derivatives in the coordinates of a and satisfy a Lipschitz condition in those of b. That such groups are equivalent to classical (i.e. analytical) Lie groups was announced in 1936 by van Kampen, although apparently van Kampen published nothing by way of proof beyond a certain decisive uniqueness theorem concerning systems of ordinary differential equations. (This is contained in our theorem (12.1); the proof given below is essentially that of van Kampen [3]). It is conceivable that the condition relative to coordinates of b could be weakened or dispensed with entirely. For, the only part the condition in question plays in our development is to make it certain that the inverse of a certain mapping-the mapping into canonical coordinates-is single-valued. In this connection mention should be made of the paper [1J of Garrett Birkhoff in which it is shown that the existence of continuous first order derivatives of ab with respect to the coordinates of both the a's and the b's is sufficient for defining a Lie group. We shall consider only groups with a finite number of real parameters (whereas the groups considered by Birkhoff are not necessarily finite dimensional). This makes it possible to use the standard existence theorems for systems of ordinary real differential equations. We make no use whatever of the theory of partial differential equations. We maintain throughout a purely local point of view in the sense that we consider only what happens in the neighborhood of the identity. Historical notes. The proof of the uniqueness of square roots (8.5) is due to Claude Chevalley (unpublished). The proof given below of Lie's theorem that a linear system of vector fields which is closed under commutation defines a Lie group is, we believe, due to van der Waerden [4]. We have supplied a number of preliminary lemmas which make the proof rigorously applicable to the case in which the vector fields are only assumed to possess continuous first order derivatives. The theorem in ?17 that every local subgroup of a Lie group is a Lie group is due essentially to E. Cartan [2]. 481

Since the pioneering work of Landau and Nakanishi on general Feynman integrals, it has been known that the singularities of the S-matrix, causal Green's functions and related functions are described by the so called Landau equations. These Landau singularities were physically interpreted as the macroscopic causality by the theoretical physicists working in 5-matrix theory, and the notion of essential support was obtained (Chandler-Stapp [3], lagolnitzer-Stapp [5]). In the branch of mathematics, on the other side, the theory of microfunction has evolved and has been applied powerfully to the general theory of partial differential equations (Sato-Kawai-Kashiwara [11]). It contained the essential support theory as the singularity spectrum of a function (i.e. its support viewed as a microfunction), and was far-reaching because of its close connection with the theory of differential equations. Namely the method of microlocal analysis, based on the theory of holonomic systems, has provided a systematic way of handling functions with natural background and found most effective applications to various problems of mathematics, such as the theory of ^-functions and Fourier transformations (Sato [10], Kashiwara [6]). It was then recognized that the Landau equations give holonomic varieties, which led one to the holonomicity postulate of ^-matrix and related quantities (Sato [10]). In the present paper we shall study the holonomy structure of Landau singularities and Feynman integrals from this standpoint. First we review the notion of Landau varieties. In contrast with

Author investigates the properties of a certain kind of '%'convergence'%' of continuous single-valued fields to multivalued upper semicontinuous fields and shows that this convergence can be used analogously to the usual uniform convergence, to which it reduces when the field is single-valued.

This paper reports study on the mechanism of long-wave breaking in a one-dimensional open channel, without bottom friction, or with linear bottom friction. In general, the wave breaking phenomenon is related to the notion of “blow-up of solution” and can be studied by using the blow-up theory of partial differential equations (PDE). A pair of Riemann invariants are introduced to transform the governing equations into ordinary differential equations (ODE) along the characteristic directions of the governing equations. From the properties of the solutions of the ODE, the time and location of wave breaking are estimated and the criterion of long-wave breaking for the case with linear bottom friction is obtained.

This chapter treats the theory of ordinary differential equations, both linear and nonlinear.Sections 1–4 establish existence and uniqueness theorems for ordinary differential equations. The first section gives some examples of first-order equations, mostly nonlinear, to illustrate certain kinds of behavior of solutions. The second section shows, in the presence of continuity for a vectorvalued F satisfying a “Lipschitz condition,” that the first-order system y′ = F(t, y) has a unique local solution satisfying an initial condition y (t0) = y0. Since higher-order equations can always be reduced to first-order systems, these results address existence and uniqueness for nth-order equations as a special case. Section 3 shows that the solutions to a system depend well on the initial condition and on any parameters that are present in F. Section 4 applies these results to existence of integral curves for a vector field and to construction of coordinate systems from families of integral curves.Sections 5–8 concern linear systems. Section 5 shows that local solutions of linear systems may be extended to global solutions and that in the homogeneous case the vector space of global solutions has dimension equal to the size of the system. The method of variation of parameters reduces the solution of any linear system to the solution of a homogeneous linear system. Sections 6–7 identify explicit solutions to nth-order linear equations and first-order linear systems. The “Jordan canonical form” of a square matrix plays a role in the case of a system. Section 8 discusses power-series solutions to second-order homogeneous linear equations whose coefficients are given by convergent power series, as well as solutions that arise in the case of regular singular points. Two kinds of special functions are mentioned that result from this study-Legendre polynomials and Bessel functions.KeywordsOrdinary Differential EquationUniqueness TheoremDiagonal EntryLipschitz ConditionIntegral CurfThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, a single-species population model with distributed delay and Michaelis-Menten type harvesting is established. Through an appropriate transformation, the mathematical model is converted into a two-dimensional system. Applying qualitative theory of ordinary differential equations, we obtain sufficient conditions for the stability of the equilibria of this system under three cases. The equilibrium A1 of system is globally asymptotically stable when br−c>0 and η<0. Using Poincare-Bendixson theorem, we determine the existence and stability of limit cycle when br−c>0 and η>0. By computing Lyapunov number, we obtain that a supercritical Hopf bifurcation occurs when η passes through 0. High order singularity of the system, such as saddle node, degenerate critical point, unstable node, saddle point, etc, is studied by the theory of ordinary differential equations. Numerical simulations are provided to verify our main results in this paper.

Based on the superposition theory of partial differential equation (PDE), a new method is introduced here to solve complicated problems of flow field. In this method, a problem of flow field with complex boundary conditions and source items is divided into several problems of flow fields with simple boundary conditions and source items, and then the solution of a complicated problem of flow field can be determined by the solutions’ superposition of several simple problems of flow fields. For an instance, by using a software in computational fluid dynamics (CFD), a linear seeping problem in porous media is solved by numerical simulation as well as solutions’ superposition of two simpler problems deriving from the proto-problem, and thus the application feasibility of superposition theory in numerical simulation of flow field is verified. More importantly, the application of the new method in CFD can decrease the number of necessary numerical examination greatly.

A nonstandard approach to averaging theory for ordinary differential equations and functional differential equations is developed. We define a notion of perturbation and we obtain averaging results under weaker conditions than the results in the literature. The classical averaging theorems approximate the solutions of the system by the solutions of the averaged system, for Lipschitz continuous vector fields, and when the solutions exist on the same interval as the solutions of the averaged system. We extend these results to perturbations of vector fields which are uniformly continuous in the spatial variable with respect to the time variable and without any restriction on the interval of existence of the solution.

This book draws together information previously available only in hard-to-find journals to introduce the most recent research in six key areas of numerical analysis: guaranteed error bounds for ordinary differential equations; computational methods for differential equations; numerical solution of differential-algebraic equations; boundary element methods; perturbation theory for infinite dimensional dynamical systems; and delay differential equations. Excellent up-to-date bibliographies are included, but the essential material is expertly presented to avoid lengthy searches. The research assumes a relatively low level of prerequisite knowledge, making it an important tool for graduate students and researchers in computational mathematics and in applications areas in physics and engineering.

Invariance and stability of fuzzy systems

In this chapter we introduce the basic notions of the theory of ordinary differential equations. Besides establishing the existence and uniqueness of solutions, we study the class of linear differential equations with constant coefficients, as well as their perturbations. In particular, we show how to solve linear differential equations by computing exponentials of matrices, and we establish the Variation of parameters formula for the perturbations of these equations.

The study of partial differential equations is an enormous field. If we recall that the theory of ordinary differential equations is subsumed within it, we can get a vague idea of its size. It goes almost without saying, therefore, that any short introduction, such as this chapter can only be an introduction in a very limited sense. For that reason we can include here only one not too satisfactory existence theorem before turning our attention to some properaties of first order and second order equations. We emphasize again that what follows is not in any sense complete.

In this book we study the degree theory and some of its applications in analysis. It focuses on the recent developments of this theory for Sobolev functions, which distinguishes this book from the currently available literature. We begin with a thorough study of topological degree for continuous functions. The contents of the book include: degree theory for continuous functions, the multiplication theorem, Hopf`s theorem, Brower`s fixed point theorem, odd mappings, Jordan`s separation theorem. Following a brief review of measure theory and Sobolev functions and study local invertibility of Sobolev functions. These results are put to use in the study variational principles in nonlinear elasticity. The Leray-Schauder degree in infinite dimensional spaces is exploited to obtain fixed point theorems. We end the book by illustrating several applications of the degree in the theories of ordinary differential equations and partial differential equations.

Nowadays, the measurements within a duct have to be conducted using in-duct microphone array, which is unable to provide information of complete acoustic solutions across the test section. In this work, an imaging method of acoustic spinning modes propagating within a circular duct simply with surface pressure informa- tion is introduced. The proposed method is developed in a theoretical way and is demonstrated by a numerical simulation case. The fundamental idea behind the testing method was originally developed in control theory for ordinary differential equations. Spinning mode propagation, however, is formulated in partial differential equations. A finite difference technique is used to reduce the associated partial differential equations to a clas- sical form in control theory. The observer method can thereafter be applied straightforwardly. The algorithm is recursive and thus could be operated in real-time. The observer error of the method is analyzed then. A numerical simulation for a straight circular duct is conducted. The acoustic solutions on the test section can be reconstructed with a good agreement to analytical solutions. The results suggest the potential and applications of the proposed method.