Tensor Products, Positive Linear Operators, and Delay-Differential Equations

Abstract

We develop the theory of compound functional differential equations, which are tensor and exterior products of linear functional differential equations. Of particular interest is the equatioṅwith a single delay, where the delay coefficient is of one sign, say δβ(t) ≥ 0 with δ ∈ {−1, 1}. Positivity properties are studied, with the result that if (−1) k = δ then the k-fold exterior product of the above system generates a linear process which is positive with respect to a certain cone in the phase space. Additionally, if the coefficients α(t) and β(t) are periodic of the same period, and β(t) satisfies a uniform sign condition, then there is an infinite set of Floquet multipliers which are complete with respect to an associated lap number. Finally, the concept of u 0 -positivity of the exterior product is investigated when β(t) satisfies a uniform sign condition.Note that equation (1.1) has a single delay. Typically we assume a signed feedback, that is, β(t) is of constant sign, either positive or negative, for almost every t.Delay-differential equations have been studied for at least 200 years. While some of the early work had its origins in certain types of geometric problems and number theory, much of the impetus for the development of the theory came from studies of viscoelasticity and population dynamics (notably by Volterra; see [43], [44], and [45]), and from control theory (see, for example, Bellman [3]). More recent work has involved models from a wide variety of scientific fields, including nonlinear optics [18], [22], [23], economics [2], [29], biology [12], [30], and as well population dynamics [19], [42]. Classic references for much of the fundamental theory are the books of J.K. Hale and S.M. Verduyn Lunel [15] and of O. Diekmann, S.A. van Gils, S.M. Verduyn Lunel, and H.-O. Walther [10]. For additional material see [1], [13], [14], [16], [17], and [49]. Equations such as (1 .1) can arise as linearizations around solutions of nonlinear equations. Two such very classic yet still challenging nonlinear equations are Wright's Equationẋ (t) = −αx(t − 1)(1 + x(t)), and the Mackey-Glass Equationẋ (t) = −α 1 x(t) + α 2 f (x(t − 1)), where f (x) = x/(1 + x n ). See [48] and [30], as well as the references in [15] for further information on such equations.Abstractly, a linear (evolutionary) process U (t, τ ) : X → X on a Banach space X is a collection of bounded linear operators U (t, τ ), for t ≥ τ , for which U (τ, τ ) = I and U (t, σ)U (σ, τ ) = U (t, τ ) whenever t ≥ σ ≥ τ , with U (t, τ )x varying continuously in (t, τ ) for each fixed x. Linear processes occur as solution maps of a wide variety of nonautonomous linear equations, including of course the finitedimensional caseẋ = A(t)x of an ordinary differential equation. In the case of the delay-differential equation (1.1) the underlying Banach space is X = C([−1, 0]), and we assume that α : R → R and β : R → R are locally integrable functions.Given an abstract linear process U (t, τ ) as above, and given an integer m ≥ 1, one obtains the so-called compoun...

The Unstable Set of a Periodic Orbit for Delayed Positive Feedback

“…It is obviuos that the dimension of C r M < is maximal also in the case r = p, as O p has two Floquet-multipliers outside the unit circle. These observations are in accordance with the recent result [15] of Mallet-Paret and Nussbaum stating that dim C r M < = 3 in more general situations.…”

“…It will easily follow from the results of this paper that dimC r M < = 3 for the periodic solutions p, q, x −1 and x 1 , see Remark 3.7. Recently Mallet-Paret and Nussbaum have shown that the equality dimC r M < = 3 holds in general [15].…”

Periodic orbits of delay equations with monotone feedback and even-odd symmetry

“…The associated period map is then used to characterize the branches of periodic solutions in (1.1) and gives an insight of the mechanisms generating these patterns, in infinite dimensions. Theorem B is a refinement of the spectral theory developed in [MPN13], in the setting of theorem A.…”

“…Section 3 constitutes a summary of the techniques required in proving both theorems A and B and follows the results in [MPS96b,MPS96a,MPN13]. The main tool is the zero number for DDE with monotone feedback.…”

John Mallet-Paret 60

“…The latter are related to their joint work on delay differential equations. In [69] they generalized John's work with George Sell on Floquet multipliers [74], by means of new tensor and exterior products of monodromy operators defined by periodic linear delay differential equations.…”