Electronic Journal of Qualitative Theory of Differential Equations

The limiting version of the Mackey–Glass delay differential equation x′(t) = −ax(t) + b f (x(t − 1)) is considered where a, b are positive reals, and f (ξ) = ξ for ξ ∈ [0, 1), f (1) = 1/2, and f (ξ) = 0 for ξ > 1. For every a > 0 we prove the existence of an ε0 = ε0(a) > 0 so that for all b ∈ (a, a + ε0) there exists a periodic solution p = p(a, b) : R → (0, ∞) with minimal period ω(a, b) such that ω(a, b) → ∞ as b → a+. A consequence is that, for each a > 0, b ∈ (a, a + ε0(a)) and sufficiently large n, the classical Mackey–Glass equation y′(t) = −ay(t) + by(t− 1)/[1 + yn(t− 1)] has an orbitally asymptotically stable periodic orbit, as well, close to the periodic orbit of the limiting equation.


Introduction
The Mackey-Glass equation with positive parameters a, b, τ, n was proposed to model blood production and destruction in the study of oscillation and chaos in physiological control systems by Mackey and Glass [13].This simple-looking differential equation with a single delay attracted the attention of many mathematicians since its hump-shaped nonlinearity causes entirely different dynamics compared to the case where the nonlinearity is monotone.See [16] for a similar equation.There exist several rigorous mathematical results, numerical and experimental studies on the Mackey-Glass equation showing convergence of the solutions, oscillations with different characteristics, and the complexity of the dynamics, see e.g.[1,3,6,7,9,15,[17][18][19]22,23].Despite the intense research, the dynamics is not fully understood yet.
Let R, C and N denote the set of real numbers, complex numbers and positive integers, respectively.Let C be the Banach space C([−1, 0], R) equipped with the norm ϕ = max s∈[−1,0] |ϕ(s)|.For a continuous function u : I → R defined on an interval I, and for so that y 0 = ψ, the restriction y| (0,∞) is differentiable, and equation (E n ) holds for all t > 0. The solutions are easily obtained from the variation-of-constants formula for ordinary differential equations on successive intervals of length one, where Hence it is well known that each ψ ∈ C + uniquely determines a solution y = y n,ψ : [−1, ∞) → R with y n,ψ 0 = ψ, and y n,ψ (t) > 0 for all t ≥ 0. For equation (E ∞ ) with the discontinuous f , we use formula (1.1) It is easy to show that, for any ϕ ∈ C + , there is a unique solution x ϕ of equation (E ∞ ) on [−1, ∞).However, comparing solutions with initial functions ϕ > 1, ϕ ≡ 1, one sees that there is no continuous dependence on initial data in C + .Therefore we restrict our attention to the subset C + r of C + .The choice of C + r as a phase space guarantees not only continuous dependence on initial data, but also allows to compare certain solutions of equations (E ∞ ) and (E n ) for large n.This is not used here, but it is important in [2]. [2]proves that for each ϕ ∈ C + r there is a unique maximal solution r for all t ≥ 0; and if t > 0 and x ϕ (t − 1) = 1, then x ϕ is differentiable at t, and equation (E ∞ ) holds at t.
One of the main results of [2] is as follows.
Theorem 1.1.If the parameters b > a > 0 are given so that (H) equation (E ∞ ) has an ω-periodic solution p : R → R with the following properties: holds then there exists an n * ≥ 4 such that, for all n ≥ n * , equation (E n ) has a periodic solution p n : R → R with period ω n > 0 so that the periodic orbits are hyperbolic, orbitally stable, exponentially attractive with asymptotic phase, moreover, [2] shows that in case b is large comparing to a, namely b > max{ae a , e a − e −a }, then (H) is satisfied.In addition, by using a rigorous computer-assisted technique, [2] gives parameter values a, b such that (H) is valid, and the obtained stable periodic orbits for the Mackey-Glass equation may have complicated structures.
[2] remarks that (H) holds if b > a > 0 and b is sufficiently close to a, and refers to this work for the proof.The aim of this paper is to prove this fact, namely the following result.Theorem 1.2.For every a > 0 there exists an ε 0 = ε 0 (a) > 0 such that for the parameters a, b with b ∈ (a, a + ε 0 ) condition (H) holds.
Theorems 1.1 and 1.2 immediately imply the following result for equation (E n ).
Theorem 1.3.For each a > 0 there exists an ε 0 = ε 0 (a) > 0 such that for every b ∈ (a, a + ε 0 ) there exists an n * = n * (a, b) ≥ 4 so that, for all n ≥ n * , equation (E n ) has a periodic solution p n : R → R with minimal period ω n (a, b) so that the periodic orbits are hyperbolic, orbitally stable, exponentially attractive with asymptotic phase.Moreover, if Note that the papers [8] by Karakostas et al. and [5] by Gopalsamy et al. give conditions for the global attractivity of the unique positive equilibrium of (E n ) for b > a > 0, and n is below a certain constant given in terms of a, b.Theorem 1.3 requires n to be large.
Section 2 contains the proof of Theorem 1.2.The proof requires the study of a special solution of a linear autonomous delay differential equation.
In order to find a periodic solution of (E ∞ ) as stated in Theorem 1.2 we consider the linear autonomous equation . Then, equation (E ∞ ) gives x (t) = −ax(t) for all t > T + 1 as long as x(t − 1) > 1. Hence there exists an ω > T + 1 with x(ω) = 1 and x(t) > 1 for all t ∈ (T, ω).By the fact f (ξ) = 0 for ξ > 1, the solution x does not change on [0, ∞) if ϕ is replaced by x ω , and consequently x(t) = x(t + ω) follows for all t ≥ −1.Therefore the proof of Theorem 1.2 is reduced to the existence of a T > 0 with u(t) We remark that the use of a limiting equation in order to study nonlinear delay differential equations when the nonlinearity is close to its limiting function is not new.We refer to the papers [10-12, 21, 24-26] where the limiting step function reduces the search of periodic solutions to a finite dimensional problem.The limiting Mackey-Glass nonlinearity f is not a step function.The introduction of the limiting Mackey-Glass equation does not reduce the search for periodic solutions to a finite dimensional problem, nevertheless it can simplify it.The paper [14] considered the limiting Mackey-Glass nonlinearity to construct periodic solutions for an equation different from (E n ).The result of [14] is analogous to the case when b is large comparing to a, mentioned above for the Mackey-Glass equation.

The proof of Theorem 1.2
The proof is divided into eight steps.The desired periodic solution of equation (E ∞ ) will be an ω-periodic extension of a function w : [0, ω] → R. We construct w in the remaining part of this section.
Notice that the choice of c depends only on a.
Summarizing the above estimations we obtain that for some continuous function r : with K = (K 1 + K 2 )/(2π).Note that r depends on ε, however K and c are independent of ε.
For the fixed a > 0 set ε 0 = ε * .Observe that c, K, and consequently ξ 0 , ξ 1 , depend only on a. Then relation (2.12) shows that N 0 is also a function of a .Therefore, ε 0 depends only on a.
If b ∈ (a, a + ε 0 ) then the above constructed p(ε) with ε = b − a ∈ (0, ε * ) is clearly an ω(ε)periodic solution of equation (E ∞ ) satisfying (H).Setting ω(a, b) = ω(ε) and σ(a, b) = σ(ε), we see that all statements of Theorem 1.2 are satisfied, and the proof is complete.The typical shape of the periodic solutions obtained in this paper for (E ∞ ) is shown in Figure 2.1 with a = 9, b = 9.7.