Algorithm for Rigorous Integration of Delay Differential Equations and the ComputerAssisted Proof of Periodic Orbits in the Mackey–Glass Equation
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Abstract
We present an algorithm for the rigorous integration of delay differential equations (DDEs) of the form \(x'(t)=f(x(t\tau ),x(t))\). As an application, we give a computerassisted proof of the existence of two attracting periodic orbits (before and after the first perioddoubling bifurcation) in the Mackey–Glass equation.
Keywords
Computerassisted proofs Delay differential equations Periodic orbit Topological methods Interval arithmeticMathematics Subject Classification
34K13 65G30 65Q201 Introduction
There are many important works that establish the existence and the shape of a (global) attractor under various assumptions on f in Eq. (1). Much is known about systems of the form \(\dot{x} = \mu x(t) + f\left( x(t1)\right) \) when f is strictly monotonic, either positive or negative [9]. Let us mention here a few developments in this direction. MalletParet and Sell used discrete Lyapunov functionals to prove a Poincaré–Bendixsontype theorem for special kind of monotone systems [19]. Krisztin et al. have conducted a thorough study on systems having a monotone positive feedback, including studies on the conditions needed to obtain the shape of a global attractor; see [11] and references therein. In the case of a monotonic positive feedback f and under some assumptions on the stationary solutions, Krisztin and Vas proved that there exist large amplitude slowly oscillatory periodic solutions (LSOPs) which revolve around more than one stationary solution. Together with their unstable manifolds, connecting them with the classical spindlelike structure, they constitute the full global attractor for the system [10]. In a recent work, Vas showed that f may be chosen such that the structure of the global attractor may be arbitrarily complicated (containing an arbitrary number of unstable LSOPs) [30].
LaniWayda and Walther were able to construct systems of the form \(\dot{x} = f\left( x(t1)\right) \) for which they proved the existence of transversal homoclinic trajectory, and a hyperbolic set on which the dynamics are chaotic [13].
Srzednicki and LaniWayda proved, by the use of the generalized Lefshetz fixed point theorem, the existence of multiple periodic orbits and the existence of chaos for some periodic, toothshaped (piecewise linear) f [12].
The results from [10, 12, 13, 30], while impressive, are established for functions which are close to piecewise affine ones. The authors of these works construct equations where an interesting behavior appears; however, it is not clear how to apply their techniques for some wellknown equations.
In recent years, there appeared many computerassisted proofs of various dynamical properties for ordinary differential equations and (dissipative) partial differential equations by an application of arguments from the geometric theory of dynamical systems plus the rigorous integration; see, for example, [2, 7, 20, 29, 32, 36] and references therein. By the computerassisted proof, we understand a computer program which rigorously checks assumptions of abstract theorems. This paper is an attempt to extend this approach to the case of DDEs by creating a rigorous forwardintime integration scheme for Eq. (1). By the rigorous integration we understand a computer procedure which produces rigorous bounds for the true solution. In the case of DDEs, the integrator should reflect the fact that, after the integration time longer than the delay \(\tau \), the solution becomes smoother, which gives the compactness of the evolution operator. Having an integrator, one should be able to directly apply standard tools from dynamics such as Poincaré maps, various fixed point theorem. In this paper, as an application, we present computerassisted proofs of the existence of two stable periodic orbits for Mackey–Glass equation; however, we do not prove that these orbits are attracting.
There are several papers that deal with computerassisted proofs of periodic solutions to DDEs [8, 14, 33], but the approach used there is very different from our method. These works transform the question of the existence of periodic orbits into a boundary value problem (BVP), which is then solved by using the Newton–Kantorovich theorem [8, 14] or the local Brouwer degree [33]. It is clear that the rigorous integration may be used to obtain more diverse spectrum of results. There are also several interesting results that apply rigorous numerical computations to solve problems for DDEs [3, 4], but they do not rely on the rigorous, forwardintime integration of DDEs.
The rest of the paper is organized as follows. Section 2 describes the theory and algorithms for the integration of Eq. (1). Section 3 defines the notion of the Poincaré map and discusses computation of the Poincaré map using the rigorous integrator. Section 4 presents an application of the method to prove the existence of two stable periodic orbits in the Mackey–Glass equation (Eq. (2)). Here, we investigate case for \(n = 6\) (before the first perioddoubling bifurcation) and for \(n = 8\) (after the first perioddoubling bifurcation). To the best of our knowledge, these are the first rigorous proofs of the existence of these orbits. The presented method has been also successfully used by the first author to prove the existence of multiple periodic orbits in some other nonlinear DDEs [25].
1.1 Notation
We use the following notation. For a function \(f : \mathbb {R} \rightarrow \mathbb {R}\), by \(f^{(k)}\), we denote the kth derivative of f. By \(f^{[k]}\), we denote the term \(\frac{1}{k!}\cdot f^{(k)}\). In the context of piecewise smooth maps by \(f^{(k)}(t^)\) and \(f^{(k)}(t^+)\), we denote the onesided derivatives f w.r.t. t.
For \(F : \mathbb {R}^m \rightarrow \mathbb {R}^n\) by DF(z), we denote the matrix \(\left( \frac{\partial F_i}{\partial x_j}(z) \right) _{i \in \{1, \ldots , n\}, j \in \{1, \ldots , m\}}\).
For a given set A, by \(\mathrm{cl}\,(A)\) and \(\mathrm{int}\,(A)\), we denote the closure and interior of A, respectively (in a given topology, e.g., defined by the norm in the considered Banach space).
Let \(A = \varPi _{i=1}^n [a_i, b_i]\) for \(a_i \le b_i\), \(a_i, b_i \in \mathbb {R}\). Then, we call A an interval set (a product of closed intervals in \(\mathbb {R}^n\)). For any \(A \subset \mathbb {R}^n\), we denote by hull(A) a minimal interval set, such that \(A \subset hull(A)\). If \(A \subset \mathbb {R}\) is bounded then \(hull(A) = [\inf (A), \sup (A)]\). For sets \(A \subset \mathbb {R}\), \(B \subset \mathbb {R}\), \(a \in \mathbb {R}\) and for some binary operation \(\diamond : \mathbb {R} \times \mathbb {R} \rightarrow \mathbb {R}\) we define \(A \diamond B = \left\{ a \diamond b: a \in A, b \in B\right\} \) and \(a \diamond A = A \diamond a = \{ a \} \diamond A\). Analogously, for \(g : \mathbb {R} \rightarrow \mathbb {R}\) and a set \(A \in \mathbb {R}\) we define \(g(A) = \{g(a) \  \ a \in A\}\).
For \(v \in \mathbb {R}^n\) by \(\pi _i v\) for \(i \in \{1, 2, \ldots , n \}\), we denote the projection of v onto the ith coordinate. For vectors \(u, v \in \mathbb {R}^n\) by \(u \cdot v\), we denote the standard scalar product: \(u \cdot v = \sum _{i=1}^{n} \pi _i v \cdot \pi _i u\)
We denote by \(C^r\left( D, \mathbb {R}\right) \) the space of all functions of class \(C^r\) over a compact set \(D \subset \mathbb {R}\), equipped with the supremum \(C^r\) norm: \(\Vert g \Vert = \sum _{i=0}^{r} \sup _{x \in D}  g^{(i)}(x) \). In case \(D = [\tau , 0]\), when \(\tau \) is known from the context, we will write \(C^k\) instead of \(C^k\left( [\tau ,0], \mathbb {R}\right) \).
For a given function \(x : [1, a) \rightarrow \mathbb {R}\), \(a \in \mathbb {R}_+ \cup \{ \infty \}\) for any \(t \in [0, a)\) we denote by \(x_t\) a function such that \(x_t(s) = x(t+s)\) for all \(s \in [1, 0]\).
We will often use a symbol in square brackets, e.g., [r], to denote a set in \(\mathbb {R}^m\). Usually it will happen in formulas used in algorithms, when we would like to stress the fact that a given variable represents a set. If both variables r and [r] are used simultaneously, then usually r represents a value in [r]; however, this is not implied by default and it will be always stated explicitly. Please note that the notation [r] does not impose that the set [r] is of any particular shape, e.g., an interval box. We will always explicitly state if the set is an interval box.
For any set X by mid(X), we denote the midpoint of hull(X) and by diam(X) the diameter of hull(X).
1.2 Basic Properties of Solutions to DDEs
For the convenience of the reader, we recall (without proofs) several classical results for DDEs [5].
Lemma 1
[Continuous (local) semiflow] If f is (locally) Lipshitz, then \(\varphi \) is a (local) continuous semiflow on \(C^0([\tau , 0], \mathbb {R})\).
Lemma 2
[Smoothing property] Assume f is of class \(C^m\), \(m > 0\). Let \(n \in \mathbb {N}\) be given and let \(t \ge n \cdot \tau \). If \(x_0 \in C^0\) then \(x_t = \varphi (t, x_0)\) is of class at least \(C^{\min (m+1, n)}\).
The smoothing of solutions gives rise to some interesting objects in DDEs [31]. Assume for a while that \(f \in C^\infty \). Then for any \(n \ge 0\), there exists a set (in fact a manifold) \(M^n \subset C^n\), such that \(M^n\) is forward invariant under \(\varphi \).
Notice that \(M^n \subset M^k\) for \(k \le n\) and \(\varphi (k \tau , \cdot ) : M^n \rightarrow M^{n+k}\).
2 Rigorous Integration of DDEs
This section is a reorganized excerpt from the Ph.D. dissertation of the first author (Robert Szczelina). A detailed analysis of results from numerical experiments with the proposed methods, more elaborate description of the algorithms, and detailed pseudocodes of the routines can be found in the original dissertation [24].
2.1 Finite Representation of “Sufficiently Smooth” Functions
Here, we would like to present the basic blocks used in the algorithm for the rigorous integration of Eq. (1). The idea is to implement the Taylor method for Eq. (1) based on the piecewise polynomial representation of the solutions plus a remainder term. We will work on the equally spaced grid and we will fix the step size of the Taylor method to match the selected grid.
Remark 1
In this section, for the sake of simplicity of presentation, we assume that \(\tau = 1\). All computations can be easily redone for any delay \(\tau \).
We also assume that r.h.s. f of Eq. (1) is “sufficiently smooth” for various expressions to make sense. The class of f in (1) restricts the possible order of the Taylor method that can be used in our algorithms, that is, if f is of class \(C^n\), then we can use Taylor method of order at most n. Therefore, thorough the paper it can be assumed that \(f \in C^\infty \). This is a reasonable assumption in the case of applications of computerassisted proofs where r.h.s. of equations are usually presented as a composition of elementary functions. The Mackey–Glass equation (2) is a good example (away from \(x = 1\)).
We fix two integers \(n \ge 0\) and \(p > 0\) and we set \(h = \frac{1}{p}\).
Definition 1

g is \((n+1)\)times differentiable on \((i \cdot h, i \cdot h + h)\),

\(g^{(k)}\left( i \cdot h^+\right) \) exists for all \(k \in \{0,\ldots ,n+1\}\) and \(\lim \limits _{\xi \rightarrow 0^+} g^{(k)}(i \cdot h+\xi ) = g^{(k)}(i \cdot h^+)\),

\(g^{(n+1)}\) is continuous and bounded on \((i \cdot h, i \cdot h + h)\).
In our approach, we store the piecewise Taylor expansion as a finite collection of coefficients \(g^{[k]}( i \cdot h)\) and interval bounds on \(g^{[n+1]}(\cdot )\) over the whole interval \([i \cdot h, i \cdot h + h]\) for \(i \in \{1,\ldots ,p\}\). Our algorithm for the rigorous integration of (1) will then produce rigorous bounds on the solutions to (1) for initial functions defined by such piecewise Taylor expansion.
Please note that we are using here a word functions instead of a single function, as, because of the bounds on \(g^{[n+1]}\) over intervals \([i\cdot h, i \cdot h + h]\), the finite piecewise Taylor expansion describes an infinite set of functions in general. This motivates the following definitions.
Definition 2
Let \(g \in C^n_p\) and let \(\mathbb {I} : \{ 1, \ldots , p\} \times \{ 0, \ldots , n\} \rightarrow \{1, \ldots , p \cdot (n+1) \}\) be any bijection.

\({\bar{g}}_{}^{0,[0]} := \pi _{1} a\),

\( {\bar{g}}_{}^{i,[k]} := \pi _{1+\mathbb {I}(i,k)} a\),

\({\bar{g}}_{}^{i,[n+1]} := \pi _i B\)
Definition 3
Please note that the minimal (p, n)representation \(\bar{g}\) of g defines (p, n)fset \(G \subset C^n_p\), which, in general, contains more than the sole function g. Also, in general, for any (p, n)fset G there are functions \(g \in G\) which are discontinuous at grid points \(i \cdot h\) (see (5)). Sometimes, we will need to assume higher regularity, therefore we define:
Definition 4
Please note that, in general, \(C^n_p \supset Supp(G) \ne Supp^{(0)}(G) \subset C^0\). It may also happen that \(Supp^{(k)}(G)=\emptyset \) for nonempty G even for \(k=0\).
Now we present three simple facts about the convexity of the support sets. These properties will be important in the context of the computerassisted proofs and in an application of Theorem 11 to (p, n)fsets in Sect. 4.
Lemma 3

If \([\bar{g}] \subset [\bar{f}]\), then \(Supp([\bar{g}]) \subset Supp([\bar{f}])\).

If \([\bar{g}]\) is a convex set in \(\mathbb {R}^m\), then \(Supp([\bar{g}])\) is a convex set in \(C^n_p\).

If \([\bar{g}]\) is a convex set in \(\mathbb {R}^m\), then \(Supp([\bar{g}]) \cap C^k\) is a convex set for any \(k \ge 0\).
We omit the easy proof.
To extract information on \(g^{[k]}\left( {i \over p} + \varepsilon \right) \) for any i and k having only information stored in a (p, n)representation, we introduce the following definition.
Definition 5
We will omit subscript \(\bar{g}\) in \(c^{i,[k]}_{\bar{g}}(\varepsilon )\) if it is clear from the context. The following lemma follows immediately from the Taylor formula, so we skip the proof:
Lemma 4
Remark 2
The task of obtaining family \(\mathbb {F}_f\) by directly and analytically applying the chain rule may seem quite tedious, especially, if one will be required to supply this family as implementations of computer procedures. It turns out, that this is not the case for a wide class of functions. In fact, only the r.h.s. of Eq. (1) needs to be implemented and the derivatives may be obtained by the means of the automatic differentiation (AD) [21, 26]. We use Taylor coefficients \(x^{[k]}\) to follow the notation and implementation of AD in the CAPD library [1] which provide a set of rigorous interval arithmetic routines used in our programs.
2.2 One Step of the Integration with FixedSize Step \(h = \frac{1}{p}\)
 1.
computing coefficients \({\bar{x}}_{h}^{1,[k]}\) for \(k \in \{1, \ldots , n\}\),
 2.
computing the remainder \({\bar{x}}_{h}^{1,[n+1]}\),
 3.
computing the estimate for \(x_h(0)\) (stored in \({\bar{x}}_{h}^{0,[0]}\)).
Forward Part: Subroutine 2
Assume for a moment that we have some a priori estimates for x([0, h]), i.e., a set \(Z \subset \mathbb {R}\) such that \(x(\left[ 0, h]\right) \subset Z\). We call this set the rough enclosure of x on the interval [0, h]. Having rough enclosure Z, we could apply Eq. (7) (as in the case of Subroutine 1) to obtain the estimates on \(x^{[k]}\left( [0, h]\right) \) for \(k > 0\). So the question is: how to find a candidate Z and prove that \(x(\left[ 0, h]\right) \subset Z\)? The following lemma gives a procedure to test the later.
Lemma 5
Proof
Using Lemma 5, a heuristic iterative algorithm may be designed such that it starts by guessing an initial Y, and then it applies Eq. (8) to obtain Z. In a case of failure of the inclusion, i.e., \(Z \not \subset Y\), a bigger Y is taken in the next iteration. Please note that this iteration may never stop or produce unacceptably big Y, especially when the step size h is large. Finding a rough enclosure is the only place in the algorithm of the integrator that can in fact fail to produce any estimates. In such a case, we are not able to proceed with the integration, and we signalize an error.
Remark 3
Please note that the term \(a^*\) is computed the same way as other coefficients in Subroutine 1 and the rough enclosure do not influence this term. In fact this is the \(n+1\)th derivative of the flow w.r.t. time. It is possible to keep track of those coefficients during the integration and after p steps (full delay) those coefficients may be used to build a \((p,n+1)\)representation of the solutions—this is a direct reflection of consequences of Lemma 2.
This fact is also important for the compactness of the evolution operator—an essential property that allows for an application of the topological fixed point theorems in infinitedimensional spaces.
Forward Part: Subroutine 3
The Integrator: Altogether
2.3 Reducing the Wrapping Effect
The wrapping effect arises when one intends to represent a result of some evaluation on sets as a simple interval set. Figure 2 illustrates this when we consider the rotation of the square.
One of the mostly used and efficient methods for reducing the impact of the wrapping effect and the dependency problem was proposed by Lohner [15]. In the context of the iteration of maps and the integration of ODEs, he proposed to represent sets by parallelograms, i.e., interval sets in other coordinate systems. In the sequel, we follow [35], and we sketch the Lohner methods briefly.
Method 0 (Interval Set): Representation of \([r_k]\) by an interval box and the direct evaluation of (16) is equivalent to directly computing \(I_h([x_{k+1}])\). This method is called an interval set and is the least effective.
Method 2 (Cuboid): this is a modification of Method 1. In this method, we choose \(U \in [A_k][B_{k+1}]\), and we do the floating point approximate QR decomposition of \(U = Q \cdot R\), where Q is close to an orthogonal matrix. Next we obtain matrix [Q] by applying the interval (rigorous) Gram–Schmidt method to Q, so there exist orthogonal matrix \(\tilde{Q} \in [Q]\) and \(Q^{1} = Q^{T} \in [Q]^T\). We set \(B_{k+1} = [Q]\), \(B_{k+1}^{1} = [Q]^{T}\).
2.4 Optimization Exploiting the Block Structure of \(D\varPhi (x)\)
If we have an arbitrary matrix C, then the cost of computing \([A_k] \cdot C\) by a standard algorithm for the matrix multiplication is of order \(O(n^3 \cdot p^3)\) in both the scalar addition and multiplication operations (we remind, that p, n are the parameters of (p, n)representation). In the case of the optimized algorithm, the block structure and sparseness of \([A_k]\) reduce the computational cost to \(O(n^2 \cdot p^2)\) in scalar additions and \(O(n^3)\) in scalar multiplications.
3 Poincaré Map for Delay Differential Equations
3.1 Definition of a Poincaré Map
Definition 6
Remark 4
Please note that the requirement \(x \in W \cap C^{n+1}\) in (18) is essential to guarantee that \(\dot{x}\) and thus \(l(\dot{x})\) in (18) are well defined, as, for \(x_0 \in C^{n+1}\) and \(t > 0\), it might happen that \(\varphi (t, x_0)\) is of class \(C^k\), \(k < n+1\) (the loss of regularity), but Lemma 2 states that we only need to “long enough” integrate the initial functions to get rid of this problem completely. These two phenomena are illustrated in the following example.
This shows for any \(x_0 \in C^n\), if \(\omega > n \cdot \tau \), then we have “only” \(\varphi (\omega , x_0) \in C^n\) in a general case. On the other hand, “long enough” integration time \(\omega \) can be used to guarantee that every initial function \(x \in C^0\) has a well defined image in \(C^n\) under mapping \(\varphi (\omega , \cdot )\). This is essential in the following construction of a Poincaré map for DDEs (Theorem 5 and Definition 7).
Theorem 5
Assume \( n \in \mathbb {N}\), \(V \subset C^0\). Let S be a local transversal \(C^n\)section for (1) and let W be as in Definition 6. Let \(\omega = (n+1) \cdot \tau \).
Proof
Definition 7
Finally, we state the last and the most important theorem that will allow us to apply topological fixed point theorems to \(P_{\ge \omega }\).
Theorem 6
Consider Poincaré map (after \(\omega \)) \(P_{\ge \omega }: S \supset V \rightarrow S\) for some section S under the same assumptions as in Theorem 5, especially assume \((n+1) \cdot \tau \le \omega \le t_S(V) \in [t_1, t_2]\).
Assume additionally that \(\varphi ([\omega ,t_2],V)\) is bounded in \(C^n\).
Then the map \(P_{\ge \omega }\) is continuous and compact in \(C^n\), i.e., if \(K \subset V\) is bounded, then \(\mathrm{cl}\,(P_{\ge \omega }(K))\) is compact in \(C^n\).
Proof
By Theorem 5, \(P_{\ge \omega }\) is well defined for any \(x_0 \in V\) since \(t_1 \ge \omega \ge (n+1) \cdot \tau \) and \(P_{\ge \omega }(x_0) \in C^{n+1}\).
The continuity follows immediately from the continuity of \(t_{S}\) (Theorem 5) and \(\varphi \) (Lemma 1).
Let \(D = P_{\ge \omega }(V)\). From our assumptions, it follows that D is bounded in \(C^n\). A known consequence of the Arzela–Ascoli Theorem is that, if \(D \subset C^{n}\) is closed and bounded, \(x \in D\), \(x^{(n+1)}\) exists, and there is M such that \(\sup _{t} \left x^{(n+1)}(t)\right \le M\) for all \(x \in D\), then D is compact (in \(C^{n}\)norm). Therefore, to finish the proof, it is enough to show that there is a uniform bound on \(P_{\ge \omega }(x)^{(n+1)}\). For this, it is sufficient to have a uniform bound on \((\varphi (t,x))^{(n+1)}\) for \(t \in [\omega ,\sup _{x \in V} t_S]\). The existence of this bound follows from boundedness of derivatives up to order n and formula (7). \(\square \)
The restriction on the transition time may seem a bit unnatural since each solution becomes \(C^\infty \) eventually, as discussed in Remark 4. In fact, it should be possible to work directly with the solutions on the \(C^n\) solution manifold \(M^n\) (i.e., \(M^n \subset C^n\) and \(\varphi (t, M^n) \subset M^n\) for all \(t \ge 0\)). When we restrict the flow to the solutions manifold \(M^n\), then we do not need to demand that the transition time to the section is bigger than \(\omega = (n+1) \cdot \tau \). Instead, to obtain the compactness, we need to shift the set forward only by one full delay. Therefore, we obtain the following theoretical result:
Theorem 7
Consider Poincaré map (after \(\tau \)) \(P_{\ge \tau }: S \supset V \rightarrow S\) for some section S, where \(V \subset M^n\). Let \(t_1\) and \(t_2\) be like in Theorem 5.
Assume that \(\varphi ([0,t_2],V)\) is bounded in \(C^n\).
Then the map \(P_{\ge \tau }: V \rightarrow S \cap M^n\) is continuous and compact in \(C^n\), i.e., if \(K \subset V\) is bounded, then \(\mathrm{cl}\,(P_{\ge \omega }(K))\) is compact in \(C^n\).
At the present stage of the development of our algorithm, we do not have the constructive parametrization of the manifold \(M^n\), therefore we need to use the “long enough” integration time \(\omega \) in the rigorous numerical computations.
3.2 Rigorous Computation of Poincaré Maps
The restriction of the integration procedure \(I_h\) (Sect. 2) to the fixedsize step \(h = \frac{1}{p}\) is a serious obstacle when we consider computation of Poincaré map \(P_{\ge \omega } : V \rightarrow S\). Obviously, if we assume for simplicity that \(\omega = q \cdot {\tau \over p}\), \(q \in \mathbb {N}\) and \(t_S(V) \subset \omega + [\varepsilon _1, \varepsilon _2]\) with \(0< \varepsilon _1< \varepsilon _2< {\tau \over p}\), then we have to find a method to compute image of the set after small time \(\varepsilon \in [\varepsilon _1, \varepsilon _2]\). The definition of the (p, n)representation together with Eq. (5) give a hint how to compute the value of the function (and the derivatives up to the order n) for some intermediate time \(0< \varepsilon < h\). But again, we face yet another obstacle, as computing the (p, n)fset representing \(\varphi (\varepsilon , x_0)\) for all initial functions \(x_0\) in some given (p, n)fset turns out to be impossible. It can be seen from the very same example as in Remark 4. In the example, \(x_\varepsilon \) would be only \(C^0\) at \(t = \varepsilon \). So if \(\varepsilon \) is not a multiple of h, then, for any \(n > 0\), there is no (p, n)representation of \(x_\varepsilon \), unless we restrict the computations to the set \(C^n_p \cap C^{n+1}\) (or to the solutions manifold \(M^n\)). This is again a reason for an appearance of the “long enough” integration time in Definition 7.
This discussion motivates the following definition and lemma.
Definition 8
Remark 8
\(I_\varepsilon (\bar{x}_0)\) is constructed in such a way that it contains all solutions to (4) for initial functions \(x_0 \in Supp(\bar{x}_0) \cap M^{n+1}\) after time \(t = \varepsilon \).
Theorem 9
Proof
Since \(q \ge n \cdot p\), then \(q \cdot h \ge n \cdot \tau \) and the proof follows from Lemmas 2, 4 and Definition 2. \(\square \)
Now, the application of \(I_h\) and \(I_\varepsilon \) to compute \(P_{\ge \omega }\) is straightforward.
 1.
a section S;
 2.
a (p, n)fset \(\bar{x}_0 \subset S\);
 3.
\(\omega = (n+1) \cdot \tau \);
 1.
\(q \in \mathbb {N}\), \(0< \varepsilon _1< \varepsilon _2 < {\tau \over p}\) such that \(t_S(\bar{x}_0) \subset q \cdot {\tau \over p} + [\varepsilon _1, \varepsilon _2]\) for \(\omega \le q \cdot {\tau \over p}\);
 2.
(p, n)fset \(\bar{x}_{\ge \omega }\) such that \(P_{\ge \omega }(x_0) \in \bar{x}_{\ge \omega }\) for all \(x_0 \in \bar{x}_0\);
 1.
do at least \((n+1) \cdot p\) iterations of \(I_h\) to guarantee the \(C^{n+1}\) regularity of solutions for all initial functions (so the map \(P_{\ge (n+1) \cdot \tau }\) is well defined and compact).
 2.find \(q \ge (n+1) \cdot p\) and \(\varepsilon _1\), \(\varepsilon _2 < h\) (for example by the binary search algorithm) such that for the assumptions of Theorem 5 are guaranteed for section S, \(t_1 = q \cdot h + \varepsilon _1\), \(t_2 = q \cdot h + \varepsilon _2\) and set W defined by$$\begin{aligned} W := C^n \cap \left( I_{[\varepsilon _1, \varepsilon _2]} \circ I_h^q (\bar{x}_0) \right) . \end{aligned}$$
 3.
By assumptions and by Theorems 5 and 9, we know that we have \(P_{t_1}(x_0) \in W \cap S\) for each \(x_0 \in V\). Moreover, by Theorem 6, the map \(P_{t_1}\) is compact (in \(C^n\)).
Remark 10
We can use the decomposition of \(I_\varepsilon \) into \(\varPhi _\varepsilon \) and \(R_\varepsilon \) such that the Lohner algorithm can again be used in the last step of the integration as described in Sect. 2.3. We skip the details and refer to the source code documentation of the library available at [24].
Now, the question arises: How to represent the section S in a manner suitable for computation of the program \(P_\ge \omega \)?
3.3 (p, n)Sections
Since we are using the (p, n)representations to describe functions in \(C^n\), it is advisable to define sections in such a way that it would be easy to rigorously check whether \(x \in S\) for all functions represented by a given (p, n)fset. The straightforward way is to require l in the definition of S to depend only on representation coefficients \({\bar{x}}_{}^{i,[k]}\).
Definition 9
3.4 Choosing an Optimal Section
We will discuss the problem of choosing optimal section in the ODEs case. Later, in Sect. 4, we will apply a heuristic procedure based on this discussion to obtain a good candidate for an optimal section in the DDEs setting.
Under the above assumptions, we have the following lemma.
Lemma 6
Proof
The other direction of the second assertion is obvious. \(\square \)
In simple words, Lemma 6 states that choosing the left eigenvector of the monodromy matrix \(\frac{\partial \varphi }{\partial x}(T,x_0)\) gives a section such that the return time to this section is constant in the first order approximation.
4 The Existence of Periodic Orbits in Mackey–Glass Equation
The Mackey–Glass system (2) is one of the best known delay differential equations. The original work of Mackey and Glass [17] spawned wide attention, being cited by many papers with a broad spectrum of topics: from theoretical mathematical works to neural networks and electrical engineering. Numerical experiments show that, as either parameter \(\tau \) [17] or n [16] is increased, the system undergoes a series of perioddoubling bifurcations, and they lead to the creation of an apparent chaotic attractor.
In this section, we present computerassisted proofs of the existence of attracting periodic orbits in Mackey–Glass system (2). We use the classical values of parameters: \(\tau = 2\), \(\beta = 2\) and \(\gamma = 1\), and we investigate the existence of periodic orbits with \(n = 6\) (before the first period doubling) and \(n = 8\) (after the first period doubling) [16]. We would like to stress, that we are not proving that these orbits are attracting. This would require some \(C^1\)estimates for the Poincaré map defined by (2).
4.1 Outline of the Method for Proving Periodic Orbits
 1.
find a good, finite representation of bounded sets in the phase space \(C^k\) (or in other suitable function space),
 2.
choose suitable section S and some a priori initial set V on the section,
 3.
compute image of V by Poincaré map \(P_{\ge \omega }\) on section S,
 4.
prove that the map \(P_{\ge \omega }\), the set V, and the set \(W := P_{\ge \omega }(V)\) all satisfy assumptions of some fixed point theorem so that it implies the existence of a fixed point for \(P_{\ge \omega }\) in V. This gives rise to the periodic orbit in Eq. (1).
Theorem 11
Let X be a Banach space, let \(V \subset X\) be nonempty, convex, bounded set and let \(P : V \rightarrow X\) be continuous mapping such that \(P(V) \subset K \subset V\) and K is compact. Then the map P has a fixed point in V.
Theorem 11 is suitable for proving the existence of periodic orbits for which there is a numerical evidence that they are attracting. The unstable periodic orbits can be treated by adopting the covering relations approach from [6], which may be applied in the context of infinitedimensional phase space (for such an adaptation in the context of dissipative PDEs; see [37]).
In Sect. 3.4, we have presented some theoretical background on the selection of a suitable section that is the foundation of Step 2. Now, we would like to put more emphasis on technical details, as the procedure in Step 2 introduces some difficulties due to large size of the data defining (p, n)representations. In the proofs we use (32, 4) and (128, 4)fsets with representation sizes \(m = 193\) and \(m = 769\), respectively. Thus we are not able to simply “guess” good coordinates or refine them “by hand”—we need an automated way to do that.
The following discussion is a bit technical and involves some heuristics, and thus it is probably relevant only for people interested in implementing their own version of the software. Those interested only in the actual proofs of the existence of periodic orbits should move to Sect. 4.3.
4.2 Finding Suitable Section and Good Initial Set for a ComputerAssisted Proof
 1.
find a good numerical approximation \(x_0\) of a periodic solution to Eq. (1),
 2.
find a good section S—by this we mean the difference between transition times \(t_1\) and \(t_2\) (as defined in Theorem 5) is as small as possible (in the vicinity of \(x_0\)),
 3.
choose a good coordinate frame in S for the initial (p, n)representation, and then choose the (p, n)fset \(V \subset S\), such that \(x_0 \in V\) and \(P_{\ge \omega }(V) \subset V\).
Step 1. Since we are looking for an attracting orbit, we start by nonrigorously integrating forward in time an initial function \(\hat{x}_0 \equiv 1.1\) for some arbitrary, long time \(T_{iter}\), until we see that \(\hat{x}_{T_{iter}}\) approach the apparently stable periodic orbit. Then, we refine \(\hat{x}_{T_{iter}}\) by the Newton algorithm applied to \(x \mapsto \hat{P}_{\ge \omega }(x)  x\), where the map \(\hat{P}_{\ge \omega }\) is a nonrigorous version of \(P_{\ge \omega }\) defined as a first return map for semiflow \(\hat{\varphi }\) to a simple section \(S = \{x \  \ x(0) = \hat{x}_{T_{iter}}(0)\}\). The output of this step is a numerical candidate for the periodic solution \(x_0\), given by its (p, n)representation \(\bar{x}_0\) such that \(\bar{x}_0\) and \(P_{\ge \omega }(\bar{x}_0)\) are close.
Step 2. This is an essential step, as numerical experiments with the rigorous integrator have shown that the choice of a good section is the key factor to obtain tight bounds on the image of the Poincaré map. We use an observation from Sect. 3.4, and we find the left eigenvector \(\hat{l}\) of the matrix \(\frac{\partial \hat{\varphi }}{\partial x}(T,x_0)\) corresponding to eigenvalue 1, where T is an apparent period of the approximate periodic orbit for the nonrigorous semiflow \(\hat{\varphi }\).
Observe that we are not very careful in the choice of coordinates on the section—we simply choose some basis orthonormal to the normal vector \(\hat{l}\) of the section hyperplane. Definitely better choice would be to use approximate eigenvectors of the Poincaré map, but in the case of strongly attracting periodic orbits, it is enough to choose a good section. Observe also, that the orthonormal matrix is easy to invert rigorously, which is an important step in the comparison of the initial set and its image by the Poincaré map.
4.3 Attracting Periodic Orbits in Mackey–Glass Equation for \(n = 6\) and \(n = 8\)
In this section we present two theorems about the existence of periodic orbits in Mackey–Glass equation. As they depend heavily on the estimates obtained from the rigorous numerical computations, we would like to discuss first the textual presentation of numbers used in this section and how they are related to the input/output values used in rigorous computations.
In the proofs, we refer to computer programs mg_stable_n6 and mg_stable_n8. Their source codes, together with instructions on the compilation process, can be downloaded from [23]. The codes were tested on a laptop with Intel^{®} Core^{™} I72860QM CPU (2.50 GHz), 16 GB RAM under 64bit Linux operating system (Ubuntu 12.04 LTS) and C/C++ compiler gcc version 4.6.3.
4.3.1 Case \(n=6\)
Our first result is for the periodic orbit for the parameter value before the first perioddoubling bifurcation.
With \(n=6\), numerical experiments clearly show that the minimal period of the periodic orbit is around 5.58. In our proof, however, due to the problem with the loss of the regularity at the grid points, thus the need to use the “long enough” transition time, we consider the second return to the section.
\(Re \lambda \)  − 0.0437  − 0.0437  0.0030  0.0030  − 0.0028  0.0019  − 0.0003  − 0.0003  0.0005 
\(Im \lambda \)  0.0793  − 0.0793  0.0097  − 0.0097  0.0018  − 0.0018  
\( \lambda \)  0.0905  0.0905  0.0102  0.0102  0.0028  0.0019  0.0019  0.0019  0.0005 
Therefore, the contraction appears to be quite strong, so the choice of good coordinates on the section appears to be not important.
We obtained the following theorem.
Theorem 12
Proof
The execution of the program realizing this proof took around 12 seconds on 2.50 GHz machine.
The diameter of the estimation for period T (also for the last step \([\varepsilon _1, \varepsilon _2]\)) obtained from the computerassisted proof is close to \(1.15 \cdot 10^{4}\).
4.3.2 Case \(n=8\)
For \(n=8\) we consider the periodic orbit after the first period doubling. This time the period of the orbit is long enough to overcome the initial loss of regularity, so we consider the first return Poincaré map.
\(Re \lambda \)  0.3090  − 0.1359  − 0.0067  \(\,7.58 \cdot 10^{4}\)  \(6.58 \cdot 10^{4}\)  \(1.23 \cdot 10^{4}\)  \( 2.184 \cdot 10^{5}\) 
\(Im \lambda \)  \( 6.265 \cdot 10^{6}\)  
\( \lambda \)  0.3090  0.1359  0.0067  \(7.58 \cdot 10^{4}\)  \(6.58 \cdot 10^{4}\)  \(1.23 \cdot 10^{4}\)  \( 2.272 \cdot 10^{5}\) 
Theorem 13
Proof
The diameter of the estimation for period T (also for the last step \([\varepsilon _1, \varepsilon _2]\)) obtained from the computerassisted proof is close to \(3.899 \cdot 10^{6}\). A graphical representation of the estimates obtained in the proof is found in Fig. 5.
The execution time was around 12 min. This increase when compared to \(n=6\) is due to much larger representation size in this case which affects the complexity of matrix and automatic differentiation algorithms which we are using.
5 Outlook and Future Directions

An extension of the integration algorithm to the systems of delay equations in \(\mathbb {R}^k\) for \(k>1\). This is rather straightforward and it does not require any new ideas;

A different representation of function sets. Currently, we use the piecewise Taylor expansions, but other approaches, like the Chebyshev polynomials, might be better as they may produce better approximations on longer intervals;

avoiding the loss of the regularity at the beginning of the integration, which imposes the requirement for the transition time to section \(t_S\) to be “long enough.” The complete solution would be to confine the initial condition to the invariant set \(M^n \subset C^n\). We are currently working on this matter;
The ultimate goal is to establish tools to prove chaotic dynamics in general DDEs, such as Mackey–Glass equation.
Notes
Acknowledgements
Research has been supported by Polish National Science Centre Grants 2011/03B/ST1/04780 and 2016/22/A/ST1/00077.
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