On the Global Attractor of Delay Differential Equations with Unimodal Feedback

We give bounds for the global attractor of the delay differential equation $x'(t) =-\mu x(t)+f(x(t-\tau))$, where $f$ is unimodal and has negative Schwarzian derivative. If $f$ and $\mu$ satisfy certain condition, then, regardless of the delay, all solutions enter the domain where f is monotone decreasing and the powerful results for delayed monotone feedback can be applied to describe the asymptotic behaviour of solutions. In this situation we determine the sharpest interval that contains the global attractor for any delay. In the absence of that condition, improving earlier results, we show that if the d5A5Aelay is sufficiently small, then all solution enter the domain where $f'$ is negative. Our theorems then are illustrated by numerical examples using Nicholson's blowflies equation and the Mackey-Glass equation.

In particular, they consider the case when f is a unimodal function, which is the situation for the famous Nicholson's blowflies equation and the Mackey-Glass model.
In that reference, the authors have proved several results on the global dynamics of Eq. (1.1), and they also formulated some open problems. It is our purpose to prove new results in the direction initiated in [11], and also to answer some of the open questions. We show the applicability of our results for different cases of the Nicholson's blowflies equation where µ, p, γ are positive parameters (see, e.g., [7] for a biological interpretation); and the Mackey-Glass equation [8] x ′ (t) = −µx(t) + px(t − τ ) 1 + x(t − τ ) n . (1.3) Following [11], we assume that f is unimodal. More precisely, the following hypothesis will be required: (U) f (x) ≥ 0 for all x ≥ 0, f (0) = 0, and there is a unique Much is known about the global picture of the dynamics of Eq. (1.1), when f is a monotone function. However, unimodal feedback may lead to very complicated and still not completely understood dynamics. See [4,5,6] and references thereof.
We shall use the function g(x) = µ −1 f (x). Notice that the equilibria of (1.1) are the fixed points of g, and that, under condition (U), function g has at most two fixed points x = 0 and x = K > 0.
Thus, we shall assume that g ′ (0) > 1 and K > x 0 . Ivanov and Sharkovsky [3, Theorem 2.3] proved that an invariant and attracting interval [α, β] for g is also invariant and attracting for (1.1) for all values of the delay τ , that is, for any nonzero solution x of (1.1). Similar results were proven using a slightly different approach in [2,11].
It is clear that we can choose β = g(x 0 ), α = g(β) = g 2 (x 0 ) to get an attracting invariant interval [α, β] for the map g (here, and in the following, g 2 denotes the composition g•g). Thus, this interval contains the global attractor associated to Eq (1.1) for all values of τ . Using this fact, Röst and Wu obtain sufficient conditions to ensure that every solution of (1.1) enters the domain where f ′ is negative. In this case the asymptotic behaviour of the solutions is governed by monotone delayed feedback and the comprehensive theory of monotone dynamics is applicable, as it was demonstrated in [11]. In particular, since a Poincaré-Bendixson type theorem is available for (1.1) when f ′ is negative, this kind of conditions exclude the possibility of solutions with complicated asymptotic behaviour (the ω-limit set can only be the positive equilibrium K or a periodic orbit). We include here the main results of [11] in this direction Theorem 1. Every solution of (1.1) enters the domain where f ′ is negative if any of the following conditions holds: Notice that the first condition in Theorem 1 is independent of the delay, while condition (L τ ) shows that even if f is unimodal, the solutions of (1.1) have the same asymptotic behaviour as in the case of monotone decreasing feedback for all sufficiently small delay τ .
An open problem suggested in [11] is the following: under condition (L), find the sharpest invariant and attracting interval containing the global attractor of (1.1) for all τ . (Numerical experiments performed in [11] show that J = [α, β] seems to be a very sharp bound). To avoid confusion we remark that the global attractor A is a subset of the function space C([−τ, 0], R), and saying that an interval [a, b] contains the global attractor we mean that for each φ ∈ A, we have a ≤ φ(s) ≤ b for any s ∈ [−τ, 0].
Our main results in this note are the following: 1. We completely solve this problem in the case of Nicholson's blowflies and Mackey-Glass equations, obtaining an interesting dichotomy result for (1.2) and (1.3) when condition (L) holds (Theorem 6).
2. We give a weaker delay-dependent condition different from (L τ ) under which the statement of Theorem 1 remains valid. In other words, we can determine a τ * that is larger than τ * in Theorem 1, such that all solutions enter the domain where f ′ is negative, if τ < τ * . Moreover, we provide some examples showing that the new condition significantly improves (L τ ) in certain situations.

Lemma 2. Assume that (L) holds. Then,J = [ᾱ,β] is an attracting invariant interval for the map g.
Proof. We have already proved thatJ is invariant. Next, we prove that it is attracting.
Since h = g 2 is monotone increasing inJ, andᾱ andβ are respectively the minimal and the maximal fixed points of h in [α, β], it follows that  [11] x the intervalJ is given byJ It is easy to check that this interval does not attract every orbit associated to the map g, since there is a period four orbit given by Numerical experiments from [11] suggest thatJ also does not attract every orbit of (2.2) for large values of τ .
We recall here that the nonlinearity f in some important examples of Eq. (1.1) (including the Mackey-Glass and Nicholson's blowflies models) fulfills the following additional assumption: (S) f is three times differentiable, and (Sf )(x) < 0 whenever f ′ (x) = 0, where Sf denotes the Schwarzian derivative of f , defined by The following proposition is a consequence of Singer's results [12].
The second statement can be easily proved using the same arguments.

On the other hand, if g does not have a unique globally attracting 2-periodic solution, under condition (L), intervalJ is still the smallest globally attracting interval for the difference equation
x n+1 = g(x n ), n = 0, 1, . . . We conjecture that the conclusion of Proposition 5 remains valid without assumption (S).
As a consequence of Corollary 3, Remark 1 and Propositions 4 and 5, we get the following dichotomy for Equation (  According to Theorem 1, when condition (L) does not hold, it is still possible to find a delaydependent condition (L τ ) under which every solution of (1.1) enters the domain where f ′ is negative and the theory of monotone delayed feedback can be applied to describe the asymptotic behaviour of our equation with unimodal feedback [11]. Next we give a different condition in the same direction. The proof is based on the following lemma proved in [2] (see also [7, Lemma 5.1]). Theorem 8. Assume that the following condition holds: Then, every solution of (1.1) enters the domain where f ′ is negative. Notice that g 1 (x) lies between g(x) and K. In particular, g 1 (x 0 ) > x 0 . On the other hand, since f satisfies (U), it is clear that g 1 also meets the same condition, except that g 1 (0) > 0. Moreover, g 1 has the only fixed point K.  Since and therefore x(t) > x 0 for all sufficiently large t. The proof is complete.
As a byproduct of the proof of Theorem 8, we get the following result, which is of independent interest to obtain sharper bounds for the global attractor of Eq. (1.1).
Proposition 10. Assume that (L) does not hold. Then (L τ ) implies (L ′ τ ). In other words, (L ′ τ ) gives a better estimate than (L τ ) for the possible delays that still guarantee that every solution enters the domain where f ′ is negative.
Remark 5. Notice that (L τ ) always fails if θ ≥ 1 (or equivalently τ ≥ 1/µ), but we have not used this fact in the proof.

Examples
In this section, we use the Nicholson's blowflies equation and the Mackey-Glass equation with different parameters in order to illustrate our results in Section 2.
After a change of variables, one can always write (1.2) in the form  This interval may be strengthened, since the smallest invariant and attracting interval for (3.1) independent of the delay τ is given by the unique 2-cycle of g, that is, According to Proposition 5, there exists a slowly oscillating periodic solution of (3.1) whose minimum and maximum values get closer and closer toᾱ andβ as τ tends to infinity. See Figure 2.A, where two distinct solutions are presented and the horizontal lines indicateᾱ andβ.
On the other hand, one can check that condition (L' τ ) in Theorem 8 holds for τ < τ * = 1.46534, showing how (L' τ ) gives an estimate significantly sharper than (L τ ).
It is interesting to notice that from Theorem 2.1 in [7] it follows that the positive equilibrium K is globally attracting for (3.1) if τ < 1.1935. Hence, the information provided by Theorem 1 is not very useful here, whereas Theorem 8 can be applied for τ between 1.1935 and 1.46534.