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.     

On the self-adjoint nonlinear eigenvalue problem for Hamiltonian systems of ordinary differential equations with singularities

Certain properties of the nonlinear self-adjoint eigenvalue problem for Hamiltonian systems of ordinary differential equations with singularities are examined. Under certain assumptions on the way in which the matrix of the system and the matrix specifying the boundary condition at a regular point depend on the spectral parameter, a numerical method is proposed for determining the number of eigenvalues lying on a prescribed interval of the spectral parameter.     

Nonlinear eigenvalue problem for second-order Hamiltonian systems

The nonlinear self-adjoint eigenvalue problem for a Hamiltonian system of two ordinary differential equations is examined under the assumption that the matrix of the system is a monotone function of the spectral parameter. Certain properties of eigenvalues that were previously established by the authors for Hamitonian systems of arbitrary order are now worked out in detail and made more precise for the above system. In particular, a single second-order ordinary differential equation is analyzed.     

On the index of the boundary value problem for a homogeneous Hamiltonian system of differential equations

The index of the homogeneous self-adjoint boundary value problem for the Hamiltonian systems of ordinary differential equations is introduced. It is assumed that the system has a nontrivial solution. The relationship between the index of an eigenvalue of the nonlinear eigenvalue problem and the index of the corresponding homogeneous problem is established. Properties of the index of the problem and those of the eigenvalue are examined.     

On two classes of autonomous third-order nonlinear ordinary differential equations

Two autonomous, nonlinear, third-order ordinary differential equations whose dynamics can be represented by second-order nonlinear ordinary differential equations for the first-order derivative of the solution are studied analytically and numerically. The analytical study includes both the obtention of closed-form solutions and the use of an artificial parameter method that provides approximations to both the solution and the frequency of oscillations. It is shown that both the analytical solution and the accuracy of the artificial parameter method depend greatly on the sign of the nonlinearities and the initial value of the first-order derivative.     

Positive solutions for systems of nonlinear eigenvalue problems for functional differential equations

Values of?λ?are determined for which there exist positive solutions of the system of functional differential equations, u″?+?λa(t)f(v _t )?=?0,v″?+?λb(t)g(u _t )?=?0, for 0?t?u(s)?=?v(s)?=?φ(s), ?r?≤?s?≤?0, and the boundary conditions u(0)?=?v(0)?=?φ(0)?=?u(1)?=?v(1)?=?0. A Guo–Krasnosel'skii fixed point theorem is applied.     

On stability of equilibrium points in nonlinear fractional differential equations and fractional Hamiltonian systems

In this article, a brief stability analysis of equilibrium points in nonlinear fractional order dynamical systems is given. Then, based on the first integral concept, a definition of planar Hamiltonian systems with fractional order introduced. Some interesting properties of these fractional Hamiltonian systems are also presented. Finally, we illustrate two examples to see the differences between fractional Hamiltonian systems with their classical order counterparts.© 2014 Wiley Periodicals, Inc. Complexity 21: 93–99, 2015     

非线性常微分系统的稳定性

用矩阵的Lozinskii测度的方法,得到了非线性常微分系统的一些稳定性准则,导出了关于非线性系统,x′=A（t）x（t）＋R（t,x）稳定性的充要条件．     

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.     

Smoothness of generalized solutions of boundary value problems for certain degenerate nonlinear ordinary differential equations

The variational method is applied to the study of a boundary value problem of the first kind for a class of nonlinear ordinary differential equations of order 2r with strong degeneracy at the endpoints of the interval (a, b). An inequality is obtained in which the norm of the solutionU of the problem under study in the sense ofW _p, ^r (a, b) is estimated from above by the norms of the given functions (x) andF(x).Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 882–890, June, 1998.     

On uniqueness criteria for systems of ordinary differential equations

We present some new uniqueness criteria for the Cauchy problem

     

Global bifurcation results for some fourth-order nonlinear eigenvalue problem with a spectral parameter in the boundary condition

This paper is devoted to the study of global bifurcation from infinity of nontrivial solutions of a nonlinear eigenvalue problem for ordinary differential equations of fourth order with a spectral parameter in the boundary condition. We prove the existence of two families of unbounded continua of nontrivial solutions to this problem, which emanate from bifurcation points in $ℝ\times \{\infty \}$ and possess oscillatory properties of eigenfunctions (and their derivatives) of the corresponding linear problem in some neighborhoods of these bifurcation points.     

Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations

In this paper, we study the three-point boundary value problems for systems of nonlinear second order ordinary differential equations of the form

Under some conditions, we show the existence and multiplicity of positive solutions of the above problem by applying the fixed point index theory in cones.     

A method for solving nonlinear spectral problems for a class of systems of differential algebraic equations

A method for calculating eigenvalues of a nonlinear spectral problem for one class of linear differential algebraic equations is proposed under the assumption of an analytical dependence on spectral parameter of the matrices appearing in the system of equations and the matrices determining boundary conditions.     

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     

Integral dispersion equation method to solve a nonlinear boundary eigenvalue problem

In this work a nonlinear eigenvalue problem for a nonlinear autonomous ordinary differential equation of the second order is considered. This problem describes the process of propagation of transverse-electric electromagnetic waves along a plane dielectric waveguide with nonlinear permittivity. We demonstrate, as far as we know, a new method that allows one to derive an equation w.r.t. spectral parameter (the dispersion equation) which contains all necessary information about the eigenvalues. The method is based on a simple idea that the distance between zeros of a periodic solution to the differential equation is the same for the adjacent zeros. This method has no connections with the perturbation theory or the notion of a bifurcation point. Theorem of equivalence between the eigenvalue problem and the dispersion equation is proved. Periodicity of the eigenfunctions is proved, a formula for the period is found, and zeros of the eigenfunctions are determined. The formula for the distance between adjacent zeros of any eigenfunction is given. Also theorems of existence and localization of the eigenvalues are proved.     

Numerical methods based on multipoint Hermite interpolating polynomials for solving the Cauchy problem for stiff systems of ordinary differential equations

Families of A-, L-, and L(δ)-stable methods are constructed for solving the Cauchy problem for a system of ordinary differential equations (ODEs). The L(δ)-stability of a method with a parameter δ ∈ (0, 1) is defined. The methods are based on the representation of the right-hand sides of an ODE system at the step h in terms of two-or three-point Hermite interpolating polynomials. Comparative results are reported for some test problems. The multipoint Hermite interpolating polynomials are used to derive formulas for evaluating definite integrals. Error estimates are given.