Solutions Of Ordinary Differential Systems Research Articles

A purely frequency based Floquet-Hill formulation for the efficient stability computation of periodic solutions of ordinary differential systems

Since the founding theory established by G. Floquet more than a hundred years ago, computing the stability of periodic solutions has given rise to various numerical methods, mostly depending on the way the periodic solutions are themselves determined, either in the time domain or in the frequency domain. In this paper, we address the stability analysis of branches of periodic solutions that are computed by combining a pure Harmonic Balance Method (HBM) with an Asymptotic Numerical Method (ANM). HBM is a frequency domain method for determining periodic solutions under the form of Fourier series and ANM is continuation technique that relies on high order Taylor series expansion of the solutions branches with respect to a path parameter. It is well established now that this HBM-ANM combination is efficient and reliable, provided that the system of ODE is first of all recasted with quadratic nonlinearities, allowing an easy manipulation of both the Taylor and the Fourier series. In this context, Hill's method, a frequency domain version of Floquet theory, is revisited so as to become a by-product of the HBM applied to a quadratic system, allowing the stability analysis to be implemented in an elegant way and with good computing performances. The different types of stability changes of periodic solutions are all explored and illustrated through several academic examples, including systems that are autonomous or not, conservative or not, free or forced.

Uniform Boundedness in the Sense of Poisson of Solutions of Systems of Differential Equations and Lyapunov Vector Functions

We introduce several generalizations of the properties of equiboundedness and uniform boundedness of solutions of ordinary differential systems, which are united by the common names of equiboundedness in the sense of Poisson and uniform boundedness in the sense of Poisson. For each of the above-introduced properties, we use the method of Lyapunov vector functions to obtain sufficient criteria for the system to have a certain property. In terms of the upper Dini derivative of the Lyapunov function given by a system, several criteria are established for the solutions of this system to have the relevant type of uniform boundedness in the sense of Poisson.

Multiple solutions of ordinary differential systems with min-max terms and applications to the fuzzy differential equations

In this paper, we investigate the existence of multiple solutions for a class of ordinary differential systems with min-max terms. We present two fundamental results for the existence of solutions. An illustrative example shows that the uniqueness of solution does not hold although the Lipschitz condition is added. Finally, there are some nontrivial applications of the considered theory to fuzzy differential equations with the generalized Hukuhara derivative.

On the asymptotic properties of solutions of ordinary differential systems on a logarithmic scale of growth

1. In this paper, we develop a method for studying the asymptotic properties of solutions of a nonlinear ordinary differential system on a logarithmic scale of growth. The method is based on the theorems proved herein about the existence of limits of characteristic exponents of solutions and a limit of the rate of varia� tion in the components of the solution to the nonlinear system in question. For linear differential systems, the existence of limits of characteristic exponents of solu� tions was fairly thoroughly investigated by O. Perron. In the nonlinear case, many questions remain open despite the theoretical and practical importance of the system under study in the theory of ordinary differen� tial systems. Consider an ordinary differential system

Numerical methods for ordinary differential equations on matrix manifolds

In recent years differential systems whose solutions evolve on manifolds of matrices have acquired a certain relevance in numerical analysis. A classical example of such a differential system is the well-known Toda flow. This paper is a partial survey of numerical methods recently proposed for approximating the solutions of ordinary differential systems evolving on matrix manifolds. In particular, some results recently obtained by the author jointly with his co-workers will be presented. We will discuss numerical techniques for isospectral and isodynamical flows where the eigenvalues of the solutions are preserved during the evolution and numerical methods for ODEs on the orthogonal group or evolving on a more general quadratic group, like the symplectic or Lorentz group. We mention some results for systems evolving on the Stiefel manifold and also review results for the numerical solution of ODEs evolving on the general linear group of matrices.

Fractional step methods for index-1 differential-algebraic equations

In the numerical solution of ordinary differential systems, the method of fractional steps (also known as operator splitting) yields high-order accurate schemes based on separate, computationally convenient treatments of distinct physical effects. Such schemes are equally desirable but much less accurate for semi-explicit index-1 differential-algebraic equations (DAEs). In the first half of this note, it is shown that naı¨ve application to DAEs of standard splitting schemes suffers from order reduction: both first and second-order schemes are only first-order accurate for DAEs. In the second half of this note, a new family of higher-order splitting schemes for semi-explicit index-1 DAEs is developed. The new schemes are based on a deferred correction paradigm in which an error equation is solved numerically, and therefore inherit a simple computationally convenient structure. Higher-order convergence of the new schemes is proved, and numerical results confirm the expected order of accuracy in addition to establishing efficiency.

Partially irregular almost periodic solutions of ordinary differential systems

Let $f(t,x)$ be a vector valued function almost periodic in $t$ uniformly for $x$, and let ${\mathrm mod}(f)=L_1\oplus L_2$ be its frequency module. We say that an almost periodic solution $x(t)$ of the system \[ \dot{x} = f (t, x), \quad t\in \mathbb{R}, x\in D \subset \mathbb{R}^n \] is irregular with respect to $L_2$ (or partially irregular) if $({\mathrm mod}(x)+L_1) \cap L_2 = \lbrace 0\rbrace $. Suppose that $ f(t,x) = A(t)x + X(t, x), $ where $A(t)$ is an almost periodic $(n\times n)$-matrix and ${\mathrm mod} (A)\cap {\mathrm mod}(X)= \lbrace 0\rbrace .$ We consider the existence problem for almost periodic irregular with respect to ${\mathrm mod} (A)$ solutions of such system. This problem is reduced to a similar problem for a system of smaller dimension, and sufficient conditions for existence of such solutions are obtained.

On the numerical solution of involutive ordinary differential systems

We propose a method for the numerical solution of ordinary differential systems. The system is considered geometrically as a submanifold in a jet space. The solutions are then certain integral manifolds that can be computed numerically when the system has been transformed into involutive form.

Asymptotic stability for damped strongly non-autonomous systems with applications to holonomic mechanical systems and nonlinear networks

This paper deals with asymptotic stability for the solutions of ordinary differential systems with time dependent restoring potential of the type¶¶\( \nabla_p G(u,u^\prime))^\prime \,\,\, \nabla_u G(u,u^\prime) + q^m (t) f (u) = Q (t,u,u^\prime), \)¶¶ where Q may depends in an essential way on the variable u and can be not tame, a geometrical condition required in previous works. This allows us to extend a stability result concerning nonlinear networks to the non-autonomous case.

Two-step numerical methods for parabolic differential equations

A family of two-stepA-stable methods of maximal order for the numerical solution of ordinary differential systems is developed. If these methods are applied to the stiff, large systems which originate from linear parabolic differential equations they yield a large, sparse set of linear algebraic equations of special form. This set is considerably easier to solve than the algebraic equations which are obtained when using diagonal Obrechkoff methods, which are one-step,A-stable and of maximal order

Bounds for solutions of 2nd order complex differential equations

1. It is well known [1], [2] that upper and lower bounds for the norm of a solution of ordinary differential systems can be given in terms of solutions of related first order scalar equations. However, the independent variable t is taken to be real there. In [3] upper bounds for solutions of a class of 2nd order complex differential equations were obtained. In this paper we derive upper as well as lower bounds for solutions of the complex differential equation

Functional differential systems and extension of Lyapunov's method

Solutions of ordinary differential systems extended to general functional differential equations and considering stability and boundedness

Minimax Solutions of Ordinary Differential Systems

Minimax Solutions of Ordinary Differential Systems

Minimax solutions of ordinary differential systems

The classical existence theorem of Peano states that there is at least one solution1 yi(x), • ■ • , yn(x) of (S) through £ existing in2 £ over some interval Xo — h^x^x0+k, where h, k>0. If for some such interval this solution through £ is not unique, then there are infinitely many solutions through £ and the existence of critical type solutions is a possibility. W. Osgood,3 P. Montel,4 and O. Perron,6 using different methods, considered the case » = 1 and proved the existence of a maximum and a minimum solution. E. Kamke6 gave an example to show that for n > 1 there will not in general be a maximum and a minimum solution through £, but that such solutions do exist provided /?(x, yu • • • , yn), i = l, • • ■ , n, satisfy certain monotone properties with respect to y1( ■ • • , yn. This paper is devoted to establishing the existence of other types of critical solutions and to a consideration of some of their properties.