George Seifert

Positively invariant closed sets for systems of delay differential equations

Publisher Summary This chapter describes the positive invariance of closed sets for systems of delay-differential equations. It presents an assumption where {X,〈,〉} is a complete real inner produce space and R is the real line. CB denotes the set of functions continuous and bounded on (−∞, 0] to X, and for φ ∈ CB, the chapter presents a definition of ∥ φ ∥ = sup{φ(s) | s ≤ 0}. Then, {CB, ∥ ∥} is a Banach space over the reals. f(t,φ) denotes a function on R × CB to X, x(t) being a function continuous on (−∞, T) to X where T ≤ ∞. For fixed t t denotes the function x(t + s), s ≤ 0. If t 0 t ). The chapter also discusses the outer normal.

Liapunov-Razumikhin conditions for asymptotic stability in functional differential equations of Volterra type

In [l], B.S. Razumikhin gave conditions for the stability and asymptotic stability of the zero state of systems of ordinary differential equations involving a fixed finite time delay. His conditions make use of Liapunov functions defined on the finite-dimensional state space rather than on the space of functions continuous on the interval of delay; for methods involving functions of the latter type, cf., for example, Yoshizawa 121. In a recent paper [3], the author has used Liapunov functions of Razumikhin type to give conditions sufficient for the stability of the zero state of a system of ordinary differential equations involving an interval of delay which becomes unbounded as t + + co; an example of such a system is an integrodifferential equation of Volterra type:

Liapunov functions and almost periodic solutions for almost periodic systems

Most of the well-known conditions for the existence of almost periodic solutions of systems of ordinary differential equations with almost periodic time dependence involve stability conditions of some sort on a bounded solution; cf. [I], [2], for examples. Although a very general condition developed by L. Amerio [3] does not explicitly involve stability concepts, its use usually involves imposing stability conditions, usually of global type, on bounded solutions. While recently an approach to the existence problem via dynamical systems has yielded conditions in terms of local stability [2], it has also been known that global stability of a conditional type is sufficient for the use of America’s theorem [4], [5]; such conditional stability does not seem to be easily applicable to the dynamical system approach. It is the purpose of this paper to develop conditions, in terms of Liapunov functions, sufficient for the uniqueness of bounded solutions of systems. These conditions essentially impose a type of conditional stability on such bounded solutions, and by use of Amerio’s theorem these again yield sufficient conditions that such bounded solutions be almost periodic. Hence, the Liapunov functions we use are not assumed positive definite near the origin. In fact, our hypotheses in these Liapunov functions are along the lines of those given by LaSalle for certain stability results for autonomous systems [7].

Stability conditions for the existence of almost-periodic solutions of almost-periodic systems☆

It is the purpose of this paper to give sufficient conditions on a system of ordinary differential equations with almost-periodic (a.p. for short) time dependence that it have an a.p. solution. In particular, we assume the existence of a bounded solution with certain stability conditions and use a result due to Amerio [l]. For two-dimensional systems such sufficient conditions have been obtained; cf., for example, [2-41. Certain such conditions have also been given for n-dimensional systems but the stability conditions required seem strong and complicated [5, 61. In recent papers by Hale [7] and Yoshizawa [8], simpler stability conditions have been given to obtain the existence of a.p. solutions of systems of functional-differential equations, which include ordinary differential systems as special cases. The method of these papers involves the use of Liapunov functions, not Amerio’s theorem, and also yields information on the Fourier, exponents of the a.p. solution in terms of those of the function defining the system. For linear systems of differential-difference equations, see also a result due to Bochner [9]. The conditions we give here are somewhat weaker than those given by Hale and Yoshizawa in the above-mentioned papers. In particular, we do not need the Lipschitz conditions on the system required by them. However, our theorems give no information about the Fourier exponents of the a.p. so1ution.l Although we deal explicitly with systems of differential equations, generalizations to functional-differential equations are possible; cf. Section 4 of this paper. ~-.~__ * This research was supported by the National Science Foundation under Grant No. GP-1638. * Since this paper was submitted for publication, it has been found that under the hypotheses of our theorems, one does, in fact, obtain the same information about these Fourier exponents as do Hale and Yoshizawa.

The number of periodic solutions of 2-dimensional periodic systems

On donne des conditions suffisantes sur un systeme a 2 dimensions ω-periodique qui a un nombre fini de solutions ω-periodiques

Almost periodic solutions for delay-differential equations with infinite delays

One of the most widely methods for establishing the existence of almost periodic (a.p. for short) solutions of systems or ordinary differential equations is based on a general result due to Amerio [ 1 J, a generalization of an earlier result for linear a.p. systems due to Favard. In essence, this basic method consists of showing that if a solution is in a compact set in the statespace for all t and has a certain separation property with respect to other such solutions, then if this same property holds for all systems in the socalled hull of the given system, this solution will be a.p. This result in fact relates to a necessary as well as sufficient stability condition for a fairly general dynamical system to have an a.p. solution; cf. [2, Chap. 51. In fact Miller [3] used the idea of embedding the a.p. system in a more general dynamical system and was thus able to obtain a local stability condition for the existence of a.p. solutions without using the separation conditions due to Favard and Amerio. Most stability conditions for the existence of a.p. solutions known prior to Miller’s result were in terms of global stability conditions. For a fairly complete and comprehensive discussion of a.p. solutions for a.p. systems, the books by Fink [4] and Yoshizawa [5] are recommended. For a more general discussion of the idea of embedding a.p. systems in dynamical systems, cf. Sell [6]. For ordinary differential equations with fixed finite time delays, much of the method based on Amerio’s separation result can and has been adapted to obtain similar stability conditions for the existence of a.p. solutions; cf. [7], for example. Miller in [3] shows that local stability conditions can also be used for such delay-differential equations. Results for delay equations with infinite time delays based on Amerio’s separation condition have also been obtained; cf., for example, Hino [B]. However, for such systems which do not involve substantially fading memory, not much seems to have been done; one of the main difficulties in such infinite delay systems arises from the fact that bounded solutions may no longer remain in compact subsets of the state

QUALITATIVE BEHAVIOR OF SOLUTIONS OF A CERTAIN HYBRID SYSTEM

We study the qualitative behavior of solutions of a system of delay-differential equations of the form where α is a positive constant and [t] denotes the greatest integer in t for the specific case f(x)=βx-γx3. We are primarily interested in bounded positive solutions.

Nonsynchronous solutions of a singularly perturbed system with piecewise constant delays

We give conditions under which a system of the form will have constant or periodic solutions (x(t), y( t)) for and sufficiently small, such that for Here [t] denotes the greatest integer function.

QUALITATIVE BEHAVIOR AND STABILITY OF SOLUTIONS OF A HYBRID SYSTEM

In this paper we consider a certain two-dimensional system of delay differential equations with piecewise constant arguments. We find conditions under which this system has constant solutions; i.e. equilibrium points, and their behavior and stability properties. We also find conditions under which certain of these solutions have a type of chaotic behavior. This paper contains results for more general systems than were dealt with in a previous paper [9].

CHAOTIC BEHAVIOR IN CERTAIN SINGULARLY PERTURBED EQUATIONS WITH PIECEWISE CONSTANT DELAYS

Conditions for the existence of chaotic behavior of some solutions of an equation of the form ex′(t)= -x(t)+f(x([t]), λ), where e and λ are positive constants and [t] denotes the greatest integer in t, are given. If e→0, it is shown that solutions tend to piecewise constant functions.