A Volterra integral formulation for determining the periodic solutions of some autonomous, nonlinear, third-order ordinary differential equations

A Volterra integral formulation based on the introduction of a term proportional to the velocity times the square of the (unknown) frequency of oscillation, a new independent variable equal to the original one times the (unknown) frequency of oscillation, the method of variation of parameters and series expansions of both the solution and the frequency of oscillation, is used to determine the periodic solutions to three nonlinear, autonomous, third-order, ordinary differential equations. It is shown that the first term of the series expansion of the frequency of oscillation coincides with that determined from a first-order harmonic balance procedure, whereas the two-term approximation to the frequency of oscillation is shown to be more accurate than that of a parameter perturbation procedure for the second equation considered in this paper. For the third equation, it is shown that the two-term approximation presented in this paper is more accurate than the corresponding one for one of the parameter perturbation methods, and for initial velocities less than one, for the other parameter perturbation approach.     

On elliptic solutions of nonlinear ordinary differential equations

The problem of constructing and classifying exact elliptic solutions of autonomous nonlinear ordinary differential equations is studied. An algorithm for finding elliptic solutions in explicit form is presented.     

Approximate methods based on order reduction for the periodic solutions of nonlinear third-order ordinary differential equations

Four approximate methods based on order reduction, the introduction of a book-keeping parameter and power series expansions for the solution and the frequency of oscillation are used to analyze three autonomous, nonlinear, third-order ordinary differential equations which have analytical periodic solutions. The first method introduces the velocity in both sides of the equation if this (linear) term is not present. The second one is based on the first one but employs a new independent variable, whereas, in the third and fourth techniques, a term equal to the velocity times the square of the unknown frequency of oscillation is introduced in both sides of the equation. The third method uses the original independent variable, whereas the fourth one employs a new independent variable which depends linearly on the unknown frequency of oscillation. It is shown that the second method provides accurate solutions only for initial velocities close to unity, whereas the third one is found to yield very accurate results for the first and second equations considered here and only for large initial velocities for the third one. The fourth technique provides as accurate results as or more accurate results than parameter-perturbation techniques which deal with the third-order equations directly and are based on the expansion of certain constants that appear in the differential equations in terms of a book-keeping parameter.     

Remarks on Barbashin-Ezeilo Problem on third-order nonlinear differential equations

In this paper, we use the frequency domain criteria to initiate a general result on the Barbashin-Ezeilo Problem on third-order nonlinear differential equation. Earlier ideas of Burkin [I.M. Burkin, Orbital stability of second-kind limiting cycles for dynamical systems with cylindrical phase space, Differential Equations 29 (1993) 1262-1264], Leonov [G.A. Leonov, A frequency criterion for the existence of limit cycles of dynamical systems with cylindrical phase spaces, Differential Equations 23 (1987) 1375-1378] on the problem are being improved upon.     

某些非线性三阶常微分方程的几个存在性结论

考察了某些非线性三阶常微分方程的存在性．主要结论的条件涉及非线性项在无穷远处的增长速度．改进和推广了某些现有的结论．     

General nonlinear self-adjoint eigenvalue problem for systems of ordinary differential equations

The general nonlinear self-adjoint eigenvalue problem for systems of ordinary differential equations is considered. A method is proposed for reducing the problem to one for a Hamiltonian system. Results for Hamiltonian systems previously obtained by the authors are extended to this system.     

Some oscillation criteria for second order nonlinear functional ordinary differential equations

The main objective of this article is to study the oscillatory behavior of the solutions of the following nonlinear functional differential equations (a(t)x'(t))' δ1p(t)x'(t) δ2q(t)f(x(g(t))) = 0,for 0 ≤ t0 ≤ t, where δ1 = ±1 and δ2 = ±1. The functions p,q,g : [t0, ∞) → R, f :R → R are continuous, a(t) ＞ 0, p(t) ≥ 0,q(t) ≥ 0 for t ≥ t0, limt→∞ g(t) = ∞, and q is not identically zero on any subinterval of [t0, ∞). Moreover, the functions q(t),g(t), and a(t) are continuously differentiable.     

On certain properties of a nonlinear eigenvalue problem for Hamiltonian systems of ordinary differential equations

Properties of the eigenvalues are examined in a nonlinear self-adjoint eigenvalue problem for linear Hamiltonian systems of ordinary differential equations. In particular, it is proved that, under certain assumptions, every eigenvalue is isolated and there exists an eigenvalue with any prescribed index.     

On the asymptotic behavior of nonoscillatory solutions of second order quasilinear ordinary differential equations

In this paper second order quasilinear ordinary differential equations are considered, and a necessary and sufficient condition for the existence of a slowly growing positive solution is established. Moreover, the precise asymptotic forms as t→∞ of slowly growing positive solutions and slowly decaying positive solutions are obtained.     

Conditional linearizability criteria for a system of third-order ordinary differential equations

We provide linearizability criteria for a class of systems of two third-order ordinary differential equations that is cubically nonlinear in the first derivative, by differentiating a system of second-order quadratically nonlinear ordinary differential equations and using the original system to replace the second derivatives. The procedure developed splits into two cases: those for which the coefficients are constant and those for which they are variables. Both cases are discussed and examples given.     

On the existence of solutions for strongly nonlinear differential equations

The objectives of this paper are twofold. Firstly, to prove the existence of an approximate solution in the mean for some nonlinear differential equations, we also investigate the behavior of the class of solutions which may be associated with the differential equation. Secondly, we aim to implement the homotopy perturbation method (HPM) to find analytic solutions for strongly nonlinear differential equations.     

Periodic solutions of some fourth-order nonlinear differential equations

Using inequality techniques and coincidence degree theory, new results are provided concerning the existence of T-periodic solutions for fourth-order nonlinear differential equations. Two illustrative examples are provided to demonstrate that the results in this paper hold under weaker conditions than the existing ones, and are more effective.     

A new HPM for ordinary differential equations

In this paper, we introduce a new version of the homotopy perturbation method (NHPM) that efficiently solves linear and non‐linear ordinary differential equations. Several examples, including Euler‐Lagrange, Bernoulli and Ricatti differential equations, are given to demonstrate the efficiency of the new method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010