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Controlling Mackey--Glass chaos

The Mackey--Glass equation, which was proposed to illustrate nonlinear phenomena in physiological control systems, is a classical example of a simple looking time delay system with very complicated behavior. Here we use a novel approach for chaos control: we prove that with well chosen control parameters, all solutions of the system can be forced into a domain where the feedback is monotone, and by the powerful theory of delay differential equations with monotone feedback we can guarantee that the system is not chaotic any more. We show that this domain decomposition method is applicable with the most common control terms. Furthermore, we propose an other chaos control scheme based on state dependent delays.

The Mackey-Glass equation, which was proposed to illustrate nonlinear phenomena in physiological control systems, is a classical example of a simple looking time delay system with very complicated behavior. Here we use a novel approach for chaos control: we prove that with well chosen control parameters, all solutions of the system can be forced into a domain where the feedback is monotone, and by the powerful theory of delay differential equations with monotone feedback we can guarantee that the system is not chaotic any more.
We show that this domain decomposition method is applicable with the most common control terms. Furthermore, we propose an other chaos control scheme based on state dependent delays.

I. INTRODUCTION
The Mackey-Glass equation µ, p, n, τ > 0, (I. 1) was introduced in 1977 to illustrate some nonlinear phenomena arising in physiological control systems 20 . Here denotes the temporal derivative of a scalar state variable x(t), and the function f (ξ) = pξ 1+ξ n represents a feedback mechanism with time delay τ . The interesting situation is n being large when the function f has a distinctive unimodal shape, and in the paper we consider only this case (at least n > 2). The Mackey-Glass equation provides a benchmark for the application of new techniques for nonlinear delay differential equations as it can generate diverse dynamics, from convergence to oscillations with different characteristics and even chaotic behavior. Despite intensive research over the decades with a number of analytical 4,17,25 , numerical 2,8,22 , and even experimental studies 1,10 , the emergence of such complexity is not fully understood yet.
Recent decades showed a growing interest towards chaos control, and several methods have been proposed and applied 26 . In this paper we use another strategy, which we think is novel in the context of chaos control: instead of controlling a particular unstable periodic orbit, we drive all solutions into a domain where the system is governed by monotone feedback.
The delay differential equation with monotone feedback (where f (x) < 0 for all x, or f (x) > 0 for all x) has been widely studied in the mathematical literature and a comprehensive description is available on its global dynamic behaviors for some classes of monotone nonlinearities 11 , and there have been some further interesting new developments as well recently 13,14 . One important result is a Poincaré-Bendixson type theorem of Mallet-Paret and Sell 21 , which implies that in the case of monotone feedback, bounded solutions converge either to an equilibrium or to a periodic orbit, hence chaotic trajectories are not possible.
The complexity of the Mackey-Glass equation stems from the combination of time delay and the non-monotonicity of the feedback, and in fact chaotic behavior has been proven for a special class of equations with non-monotone delayed feedback 16 . A domain decomposition method has been proposedfor unimodal feedback functions 25 , that provides sufficient conditions such that all solutions eventually enter a domain where f is either increasing or decreasing, and in this case the complicated behavior is excluded. In this paper we take advantage of this idea and propose various schemes that can impose such a situation. After describing the mathematical background in Section 2, in Section 3 we propose additive control terms, and consider the equation with control term u(t). We investigate three typical cases, namely constant perturbation u(t) = k, proportional feedback control u(t) = kx(t), and the delayed feedback controller We shall say that the chaos is controlled if the system shows complicated behavior for k = 0, but all solutions eventually enter and remains in some monotone domain of f for some k = 0, in which case convergence to an equilibrium or to a periodic orbit is guaranteed. In Section 4 we use a different approach: instead of an additive term, we construct a state dependent delay τ = τ (x(t)) in a proper way so that our domain decomposition method is still applicable. It is important to stress that in this case the form of the controlled equations is of (I.1) instead of (I.3), and the delay itself will be the subject to the control. In Section 5 we illustrate our control mechanisms with a set of numerical simulations, and we conclude the paper with a summary and discussion of the interpretation of our results.

II. MATHEMATICAL BACKGROUND
Let C = C([−τ, 0], R) denote the Banach space of continuous functions φ : [−τ, 0] → R with the usual sup norm ||φ|| = max −τ ≤s≤0 |φ(s)|. Given its biological interpretation, traditionally only non-negative solutions of (I.1) are studied, hence we restrict our attention the cone and define the corresponding order intervals Every φ ∈ C + determines a unique continuous function also play a crucial role in characterizing the nonlinear dynamics of equation (I.1). A cornerstone of this paper is the following result, which combines Theorem 3.5 (Röst & Wu 25 ) and Theorem 8 (Liz & Röst 17 ), ensuring that the long term dynamics is governed by a monotone part of the feedback function.
Theorem II.1. Let g(x) = µ −1 f (x), and assume g (0) > 1 and K > ξ 0 . Then, if either condition holds, then every solution eventually enters and remains in the domain where f is negative, hence converging to K or to a periodic solution oscillating around K.
The assumption of this theorem means that we are in case (c). Then the interval [α * , β * ] is attractive and invariant 25 , and condition (L) means α > ξ 0 . Results relating attractive invariant intervals of the discrete map f to attractive invariant intervals for (I.1) originate from Ivanov & Sharkovsky 7 , and recently have been successfully used for other problems as well 5,18 . Note that this condition is independent of τ , hence in this situation chaotic behavior can not appear by increasing the delay. The delay dependent condition (T ) is built on earlier works 4,19 .

TERMS
Our aim is to choose our additive control term u(t) from three common classes, in a way that some analogue of Theorem II.1 holds for (I.3).

A. Constant perturbation control
For any k ∈ R, we consider Then the following statements hold: (i) there is a k * < µξ 0 such that for all k ≥ k * , (III.1) has no complicated solution; (ii) there is an explicitly computable k 1 , such that for k < k 1 , (III.1) has no equilibria and solutions become unfeasible; , there are two positive equilibria K 1 and K 2 , and (iv) there exists a k 3 such that for k 2 < k < k 3 , (III.1) has no complicated solutions.
Proof. After using the change of variable y = x − k µ , (III.1) reads as That is with f k (ξ) = f ξ + k µ , thus adding the constant perturbation k has the same effect as shifting the graph of f by k/µ. Note that we are interested only in non-negative solutions x(t) of (III.1), that is y(t) ≥ −k/µ, and we call such solutions feasible.
(ii) We shift the graph of f to the right until equilibria are destroyed. In the critical case . This is a quadratic equation in ξ n µ , and taking its positive root we find When the graph is shifted by k 1 /µ, the tangent line of the shifted graph f k 1 with slope µ is This means that v(t) becomes smaller than −k/µ in finite time, but due to y(t) < v(t), each solution y(t) becomes unfeasible.
(iii) For k 1 < k < 0, there are always two equilibria of (III.2), now we are looking for an other critical value k 2 that separates the cases when the larger equilibrium is on the decreasing part of f k from when both are on the increasing part. The critical case is characterized by one of the equilibria beingξ 0 , that is such that the derivative of the solution is negative at zero, thus unfeasible solutions exist.
To avoid such situations, we restrict our attention to the interval [K 1 * ,ξ 0 * ], where f k is monotone increasing. For solutions with segments from this interval, y(t) =K 1 implies in an interval (k 2 , k 3 ), then an analogue of Theorem II.1 provides the result. Differentiating with respect to k gives shows that the control has no effect on key properties of the nonlinearity in (I.1).
With w = µ − k, Theorem II.1 can be directly applied.
Next we give a delay dependent result.
Theorem III.3. Assume that K > ξ 0 and (L) does not hold forg with some k. Then for sufficiently small delay, (T ) holds forh. Furthermore, the smaller the delay, the larger the range of k that enables chaos control.

C. Pyragas control
A popular control mode is u(t) = k(x(t − τ )) − x(t), and with such term (I.1) becomes that is with F k (ξ) = f (ξ) + kξ. Notice that while the Pyragas control changes the shape of the nonlineariy, it does not change the equilibria of the system.

4n
, all solutions of (III.5) converge to K. When k < 0, then there is aξ such that F k (ξ) < 0, and then solutions with initial functions satisfying φ(0) = 0 and φ(−τ ) =ξ immediately become negative. Since the nonnegative cone is not invariant any more, here we don't discuss Pyragas control with negative k.
When 0 < k < p(n−1) 2 4n then F k (ξ) has a bimodal shape, with local extrema q 1 < q 2 . The numbers q 1 and q 2 can be found as the solutions of f (ξ) = −k, which is quadratic in ξ n , so it is possible to find them explicitly, similarly to case (ii) of Theorem III.1. It is natural to try to apply an analogue of (L) condition in the bimodal case too, and forcing all solutions into the domain (q 1 , q 2 ), where F is monotone decreasing. Nevertheless, the required conditions q 1 < α k and β k < q 2 become analytically intractable, and one can find parameter settings when they fail when k being near either zero or p(n−1) 2

4n
. An other possibility is to forcing solutions to the increasing part of F k thus expecting convergence to K again, so we may require α k > q 2 , that is F k (F k (q 1 )/(µ + k)) > (µ + k)q 2 , but again that seems too involved to find a simply interpretable condition.

IV. STATE DEPENDENT DELAY CONTROL
From Theorem II.1 is is clear that chaos can be controlled by decreasing the delay to a small quantity, since as τ → 0, condition (T ) becomes K > ξ 0 , hence for sufficiently small τ , (T ) is satisfied. However, it may be impossible or very expensive to permanently keep τ small, thus here we explore how can we establish chaos control when we modify the delay only temporarily, depending on the current state. Thus, we consider equation with state dependent delay r(x(t)), where one can interpret r(x(t)) = τ − k(x(t)) with baseline delay τ and delay control k(x(t)). It is reasonable to assume k(x(t)) ≥ 0 and k(x(t)) < τ , then r(x(t)) ∈ (0, τ ]. We say that a solution is slowly oscillatory, if x(t) − K has at most one sign change on each time interval of length τ .
Theorem IV.1. Assume K > ξ 0 and letK < ξ 0 be defined by . Define the following state dependent delay function: Then, solutions of (IV.1) eventually enter the domain where f is negative and slowly oscillatory complicated solutions can not exist.
Remark IV.2. Some recent results of Kennedy 9 , that have not been published yet, suggest that Theorem IV.1. can be extended from slowly oscillatory solutions to all solutions.
While the control scheme in this theorem may seem complicated, what it really means is that when a solution is approaching ξ 0 from above, we decrease the delay in a way that the solution will turn back before reaching ξ 0 , hence forcing it to stay in the domain where f < 0. In particular, k(x) = 0 for x ≥ ξ 0 +ζ, k(x) = τ −τ * for x ≤ ξ 0 and some intermediate control k(x) applied when the solution is in the interval (ξ 0 , ξ 0 + ζ). For such equation with state dependent delay the Poincaré-Bendixson type theorem was proven only to the subset of slowly oscillatory solutions, hence at the current state-of-the-art of the theory we can not say more, but the applicability of this control scheme will be illustrated in the next Section.

V. APPLICATIONS, SIMULATIONS AND DISCUSSION
We investigated a number of possible mechanisms so that with a well chosen control parameter, an otherwise chaotic Mackey-Glass system is forced to show regular behavior.
The Mackey-Glass equation was used to model the rate of change of circulating red blood cells, and most of our results have a meaningful interpretation in this context. For example, u(t) = k with k > 0 may represent medical replacement of blood cells at a constant rate, or u(t) = kx(t) with negative k may represent increased destruction rate of blood cells which can be achieved by administration of antibodies 3 . Our approach is different from typical chaos control methods, since our strategy is to choose a control such that all solutions will be attracted to a domain where the feedback function is monotone, and then some Poincaré-Bendixson type results exclude the possibility of chaotic behavior. By applying this domain decomposition method, which is based on Röst & Wu 25 , instead of stabilizing a particular orbit, we push the full dynamics into a non-chaotic regime.
For u(t) = k, clearly k > 0 helps the cell population, and as Theorem III.1 shows, with sufficiently large k chaos can always be controlled regardless of the delay. A somewhat counterintuitive part of Theorem III.1 is that for some negative k it is possible to force the system to converge to a positive equilibrium, however one has to be careful as the system will collapse if k is below the threshold k 1 . This is illustrated in Fig. 1, where on the left we can see how k > 0 controls a chaotic solution into a periodic one, while in the right we can observe that decreasing k < 0 first regulate the system into periodic behavior, then to convergence, and finally to collapse (i.e. hitting zero in finite time).
The proportional feedback control u(t) = kx(t) again helps the population when k > 0 and when it fully compensates the baseline mortality (κ > µ), the population grows and unbounded (Theorem III.2, (iv)). Yet, controlling chaos is best achieved with k < 0, when the destruction of cells is increased, then with a fine tuning of k the dynamics can be made regular (Theorem III.2, (i) and (ii)), which is shown on the left panel of Fig. 2. If cell destruction is too high, the population goes extinct (Theorem III.2, (iii)). Theorem III.3 gives a delay dependent result, showing that even if the condition (L) fails, chaos control can be achieved by satisfying (T ). We showed that the smaller the delay, the easier the control is, in the sense that we can pick k from a larger range to satisfy (T ). On the right panel of Fig.2., we illustrated this delay dependent feature: when we switched on the control we decreased the delay temporarily to show that with this smaller delay it is a good control, but when we reset the delay at some time later, the delay dependent condition (T ) fails and the solution goes back to irregular mode with the same control.
We also used the popular Pyragas control u(t) = k(x(t − τ ) − x(t)). The conclusion of our Theorem III.4 is that for positive k, the unimodal shape of the nonlinearity turns into a bimodal shape, and when k is large enough (our theorem explicitly tells us how large), then the nonlinearity is transformed into a monotone feedback, as the control term overwhelms the original unimodality. Once we achieved monotonicity, we can use results from Röst and Wu 25 to prove that solutions converge to the positive equilibrium. Figure 3 left shows how such regulation occurs as we increase k. For negative k the non-negative cone is not invariant anymore so we do not consider this possibility. Let us remark that control of Mackey-Glass chaos has been experimentally observed with Pyragas-type control 10 , and here our results give an analytic explanation how and why this happens.
Finally, we considered a very different type of control, taking advantage of some results from the theory of state dependent delays. While it is clear from Theorem (II.1) that chaos can be eliminated when the delay is sufficiently small, in Theorem (IV.1) we constructed a state dependent delay function, that allows us to construct a delay control scheme where the delay is reduced only in a part of the phase space. This is illustrated in the right panel of Figure 3, where we applied delay reduction only in the region x < K, and that was sufficient to drive the irregular solution into periodic behavior.  is plotted. For 0 ≤ t < 50, there is no control (k = 0), and the solution is irregular. At t = 50, we switch on the Pyragas control with k = 0.08 which is too small to cease the irregularity of the solution, which becomes periodic after t = 100 when the control was increased to k = 0.95. Finally, the solution converges K after t = 150 when the control increased further to k = 3.9, when the condition of Theorem III.4 holds. The other parameters were set to µ = 1.08, τ = 3, p = 2, n = 9.65. The initial function was 1 + 0.1e −t . Right: State dependent delay control. A numerical solution to (IV.1) is plotted. We switch on the delay function scheme at t = 31, which drives the solution to a periodic orbit. The horizontal line shows the equilibrium and it is also the boundary for delay reduction, where for the sake of simplicity we used a step function for r(x(t)): the delay is 5 for x > K and 4 for x < K. In the lower part of the graph it is shown when the delay control is on or off. Parameter values are n = 6, p = 2, µ = 1, initial function is 2 * .