Periodic Systems Of Differential Equations Research Articles

Periodic Solutions of Periodic Competitive and Cooperative Systems

The periodic solutions, their basins of attraction and invariant manifolds are considered for periodic systems of differential equations which are cooperative or competitive following Hirsch. Competitive and cooperative mappings are introduced which possess the essential features of the Poincare map for such systems. The geometrical properties of these mappings and the discrete dynamical system they generate are the objects of study. The main tools in this study are the Perron–Frobenius theory of positive matrices and invariant manifold theory. A complete description of the “phase portrait” of the discrete dynamical system generated by an orientation preserving planar cooperative map is obtained.

Convergence of the method of trigonometric collocation for nonlinear periodic systems of differential equations with retarded argument

Convergence of the method of trigonometric collocation for nonlinear periodic systems of differential equations with retarded argument

On systems of periodic differential equations

Restrictions on the characteristic exponents of periodic differential equations are derived from certain symmetry properties. In one case these restrictions amount to a stability theorem. In other cases they reduce the number of independent exponents.

A comparison of the method of averaging and Grobner's integral equation†

Abstract The method of averaging is compared to Grobner's method of integral equations as applied to a weakly non-linear periodic system of differential equations. The first and second approximation to the solutions of the differential equations via both techniques are shown to be equivalent.

Stability and existence of a periodic solution

Many authors have discussed the existence of a periodic solution of a periodic system of differential equations under the assumption that the system has a bounded solution which has some kind of stability. LaSalle [Z] has discussed a more general case by considering an asymptotic stability property of motions. Hale [2] and the author [3] have discussed the existence of a periodic solution of functional-differential equations (more generally, the existence of an almost periodic solution) under the assumption that the system has a bounded solution which is uniformly asymptotically stable in the large by applying Liapunov’s second method. Jones [4] also has discussed the existence of a periodic solution by assuming that a set has an asymptotic stability property in some sense. Recently, Sell [5] has shown the existence of a periodic solution of a periodic system of period w under the assumption that the system has a bounded solution which is uniformly asymptotically stable or uniformly asymptotically stable in the large. However, Sell’s definition of uniform asymptotic stability differs from the usual one. (See [a). His type of stability is weaker than the uniform asymptotic stability in the usual sense and it is stronger than the asymptotic stability. Actually, as is seen below, it is also stronger than the equiasymptotic stability. Sell shows that there exists a periodic solution of period KW for some integer K > 1. In this paper, we shall show that Sell’s concept is stronger than the equiasymptotic stability and that for a periodic system Sell’s concept for uniform asymptotic stability of a bounded solution is equivalent to the usual concept, and consequently we have a periodic solution of period w under the assumption that a bounded solution is uniformly asymptotically

Solutions of systems of periodic differential equations

By a system of periodic differential equations referred to in the title we mean a system of the form f(t, U, u’) = 0, where u = u(t), u’ = du/dt, and f(t, u, v) = f(t + r, u, w). Our particular approach will be to break up the vector u into two vector components so that our system becomes f(t, X, y, x’, y’) = 0, where x is an m-dimensional vector, y is n-dimensional, and f is m + n dimensional. Either m or n may be zero, in which case the system is independent of x or y respectively. We shall impose conditions only on x which, however, will yield properties valid for y as well. Throughout the paper, we shall assume that there exists a unique solution through any prescribed point in a region in (t, X, y) space, and that the variables always remain in this region. These requirements will not be explicitly stated. In the first part of the paper we shall discuss systems of arbitrary order. In the second part, these results will be specialized to second order scalar differential equations.