Theory Of Ordinary Differential Equations Research Articles

Nonautonomous bifurcation patterns for one-dimensional differential equations

Although, bifurcation theory of ordinary differential equations with autonomous and periodic time dependence is a major object of research in the study of dynamical systems since decades, the notion of a nonautonomous bifurcation is not yet established. In this article, two different approaches are discussed which are based on special notions of attractivity and repulsivity. Generalizations of the well-known one-dimensional transcritical and pitchfork bifurcation are obtained.

Off-shell Jost solution for a Coulomb-like potential

The theory of ordinary differential equations together with judicious use of boundary conditions and certain properties of higher transcendental functions is exploited to derive a useful analytical expression for the Coulomb-Yamaguchi Jost solution through an $r$-space approach to the problem. Note that the off-shell Jost solution is expressed in its maximal reduced form involving confluent and Gaussian hypergeometric functions. As an application of the Jost solution the off-shell $T$ matrix is also expressed in terms of Gaussian hypergeometric functions.

On boundary value problems for the Cauchy-Riemann and Bitsadze systems of equations

In this paper, we consider boundary value problems for elliptic systems of equations of orders 1 and 2, which are known as the Cauchy‐Riemann and Bitsadze generalized systems of equations. The author’s interest in these systems arose from the theory of partial differential equations; more specifically, it was caused by the following observation. In the theory of ordinary differential equations, the settings of well-posed problems for one second-order equation and a system of secondorder equations virtually coincide. In the theory of partial differential equations, the situation is different. For one second-order elliptic equation, the Dirichlet problem is well-posed if either some constraints are imposed on the coefficients of the equation or the domain in which the problem is considered is small. For second-order systems of equations that are elliptic in the sense of Petrovskii but not strongly elliptic, there exist examples [1] in which the homogeneous Dirichlet problem in a disk of arbitrarily small radius has infinitely many linearly independent solutions. The matrix form of this system of equations is

Self-Contact for Rods on Cylinders

We study self-contact phenomena in elastic rods that are constrained to lie on a cylinder. By choosing a particular set of variables to describe the rod centerline the variational setting is made particularly simple: the strain energy is a second-order functional of a single scalar variable, and the self-contact constraint is written as an integral inequality. Using techniques from ordinary differential equation theory (comparison principles) and variational calculus (cut-and-paste arguments) we fully characterize the structure of constrained minimizers. An important auxiliary result states that the set of self-contact points is continuous, a result that contrasts with known examples from contact problems in free rods.

On the asymptotic integration of nonlinear differential equations

The aim of the present paper is twofold. Firstly, the paper surveys the literature concerning a specific topic in asymptotic integration theory of ordinary differential equations: the class of second order equations with Bihari-like nonlinearity. Secondly, some general existence results are established with regard to a condition that has been found recently to be of significant use in the theory of elliptic partial differential equations.

Topology of an incompressible viscous-fluid flow in a cubic cavity with a moving cover

The mechanisms of formation of three-dimensional jet flows inside large-scale vortex structures in a closed cubic cavity have been considered. The vortex structure of an incompressible viscous-fluid flow in this cavity has been investigated using the qualitative theory of ordinary differential equations. Singular points and their types and positions were determined at different Reynolds numbers on the basis of numerical calculations.

On periodic solutions of ordinary differential equations with discontinuous right-hand side

A new version of the method of translation along trajectories, which does not require the uniqueness of the solution of the Cauchy problem, is applied to the proof of the existence theorem for vector-valued periodic solutions of ordinary differential equations of first and second order. This result is applicable to equations and differential inclusions with discontinuous right-hand side. Several applications of the theorems proved in this paper are considered in cases which are not covered by the classical theory of ordinary differential equations with continuous right-hand side and equations with right-hand side satisfying the Caratheodory conditions.

Remarks on a Lie Theorem on the Integrability of Differential Equations in Closed Form

is integrable by quadratures [1] (for details, see also [2, 3]). More precisely, all of its solutions can be found by “algebraic operations” (including inversion of functions) and “quadratures,” that is, computation of integrals of known functions of one variable. This definition has a local nature. The algebra g generates an n-dimensional solvable Lie group G that acts freely on R. Relations (1) imply that the transformations in G take the trajectories of system (2) to trajectories of the same system. In other words, the solvable group G “permutes” the trajectories of the differential equation (2). Lie treated his theorem as an analog of Galois theory for ordinary differential equations. However, there is a more natural analog of Galois theory, namely, Picard–Vessiot theory, which studies extensions of differential fields by solutions of linear differential equations (e.g., see [4]). Nevertheless, the Lie theorem is one of the key results in the problem of closed-form integrability of nonlinear differential equations. It suffices to note that, in the simplest case, this theorem readily implies the well-known Liouville theorem on the integrability of Hamiltonian differential equations with a complete set of independent first integrals in involution [3]. The following assertion is the main result of the present paper.

Off-shell Jost solution for scattering by a Coulomb field

A relatively uncomplicated mathematical prescription based on the theory of ordinary differential equations together with certain properties of higher transcendental functions is used to obtain a useful analytical expression for the $s$-wave Coulomb off-shell Jost solution.

On global asymptotic controllability of planar affine nonlinear systems

In this paper, we present a necessary and sufficient condition for globally asymptotic controllability of the general planar affine nonlinear systems with single-input. This result is obtained by introducing a new method in the analysis, which is based on the use of some basic results in planar topology and in the geometric theory of ordinary differential equations.

Critical points and positive solutions of singular elliptic boundary value problems

Usually we do not think there is variational structure for singular elliptic boundary value problems, so it cannot be considered by using critical points theory. In this paper, we use critical theory on certain convex closed sets to solve positive solutions for singular elliptic boundary value problems, especially use the ordinary differential equation theory of Banach spaces to obtain new results on the existence of multiple positive solutions. The method is useful for other singular problems.

Stability of Solutions of a Quasilinear Index-2 Tractable Differential Algebraic Equation by the Lyapunov Second Method

The Lyapunov second method is an important tool in the qualitative theory of ordinary differential equations. In this paper, we consider the behavior of solutions of quasilinear index-2 tractable differential algebraic equations. Using the Lyapunov second method, we prove sufficient conditions for the stability of zero solution of such equations.

Revisiting fuzzy differential equations

We revisit the original formulation of fuzzy differential equations (FDEs) and provide examples to show that the theory of FDEs is richer than the corresponding theory of ordinary differential equations (ODEs) and therefore need not be rejected. The alternative formulation in terms of differential inclusions does not always have advantage over ODEs, as claimed. We also discuss a stability result via Lyapunov functions in this setup.

Anti-Lyapunov method for systems of discrete equations

In the present paper problems concerning the qualitative behaviour of solutions of discrete equations are considered. A general principle which gives a guarantee that the graph of at least one solution stays in a prescribed domain is formulated. As a consequence, the auxiliary tool—namely, a method of regular polyfacial sets—is developed and an application of the general principle is given. This application is based, with exceptions, on the fact that points of the boundary of a domain considered are strict egress points. Such a situation is often (in the theory of ordinary differential equations) termed the Anti-Lyapunov case. Illustrative applications and examples are given and open problems are formulated.

From Bloch model to the rate equations

We consider Bloch equations which govern the evolution of the density matrix of an atom (or: a quantum system) with a discrete set of energy levels. The system is forced by a time dependent electric potential which varies on a fast scale and we address the long time evolution of the system. We show that the diagonal part of the density matrix is asymptotically solution to a linear Boltzmann equation, in which transition rates are appropriate time averages of the potential. This study provides a mathematical justification of the approximation of Bloch equations by rate equations, as described in <em>e.g.</em> [Lou91]. The techniques used stem from manipulations on the density matrix and the averaging theory for ordinary differential equations. Diophantine estimates play a key role in the analysis.

On the solutions of quasilinear elliptic equations with a polynomial-type reaction term

We study existence and boundedness of solutions for the quasilinear elliptic equation $-\Delta_{m} u=\lambda(1+u)^p$ in a bounded domain $\Omega$ with homogeneous Dirichlet boundary conditions. The assumptions on both the parameters $\lambda$ and $p$ are fundamental. Strange critical exponents appear when boundedness of solutions is concerned. In our proofs we use techniques from calculus of variations, from critical-point theory, and from the theory of ordinary differential equations.

Sign properties of Green's functions for disconjugate dynamic equations on time scales

This paper continues the development of disconjugacy of higher order dynamic equations on time scales. Two-point conjugate type boundary value problems for general disconjugate dynamic equations on time scales are studied and the sign properties of associated Green's functions are established. As expected, the results unify known results from the theories of ordinary differential equations and finite difference equations.

A Reality Proof in PT-Symmetric Quantum Mechanics

We review the proof of a conjecture concerning the reality of the spectra of certain PT-symmetric quantum mechanical systems, obtained via a connection between the theories of ordinary differential equations and integrable models. Spectral equivalences inspired by the correspondence are also discussed.

The method of variation of constants and multiple time scales in orbital mechanics.

The method of variation of constants is an important tool used to solve systems of ordinary differential equations, and was invented by Euler and Lagrange to solve a problem in orbital mechanics. This methodology assumes that certain "constants" associated with a homogeneous problem will vary in time in response to an external force. It also introduces one or more constraint equations. We show that these constraints can be generalized in analogy to gauge theories in physics, and that different constraints can offer conceptual advances and methodological benefits to the solution of the underlying problem. Examples are given from linear ordinary differential equation theory and from orbital mechanics. However, a slow driving force in the presence of multiple time scales contained in the underlying (homogeneous) problem nevertheless requires special care, and this has strong implications to the analytic and numerical solutions of problems ranging from celestial mechanics to molecular dynamics. (c) 2003 American Institute of Physics.

Integral averages and oscillation criteria for half-linear partial differential equation

The technique of weighted integral averages, known in the oscillation theory of ordinary differential equations, is extended to the half-linear partial differential equation (E) div(∥ ∇u∥ p−2 ∇u)+c(x)|u| p−2u=0, p>1. This technique is used to obtain new oscillation criteria for (E) on the unbounded domains.