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Validated integration of diﬀerential equations with

state-dependent delay

Kevin E. M. Church

Centre de Recherches Math´ematiques, Universit´e de Montr´eal

April 14, 2022

Abstract

We present an implicit method of steps for diﬀerential equations with state-dependent delays and

validated numerics to rigorously enclose solutions of initial-value problems. Our approach uses a

combination of contraction mapping arguments based on a Newton-Kantorovich type theorem and

piecewise polynomial interpolation. Completing multiple steps of integration is challenging, and we

resolve it by smooth interpolation of the previous solution, resulting in an interval-valued polynomial

initial condition for the subsequent step. A set of examples is provided.

1 Introduction

Functional diﬀerential equations (FDE) have been studied for over 200 years, with substantial devel-

opments since the 1950s. Broadly, a FDE of retarded type is a diﬀerential equation

˙x(t) = f(xt),

where fis a functional acting on a function space, and xtdenotes a history function that “windows” the

solution until some time in the past. When this function space consists of the continuous functions on

a compact interval, and fis at least Lipschitz continuous, there is a well-developed theory popularized

by the works of Diekmann, van Gills, Lunel and Walther [9], Hale & Lunel [12] and Krasovskii [19],

among others. The situation is far less clear-cut for systems with so-called state-dependent delay, where

the functional fcan not be understood as Lipschitz continuous on the typical space of continuous

functions. In such cases, there remain several open problems concerning the regularity of the semiﬂow

and so-called solution manifolds and invariant manifolds [20].

Consider the diﬀerential equation with state-dependent delay (DE-SDD), written in the form

˙x(t) = f(x(t), x(t−τ(t, x(t)))).(1)

We assume f:Rd×Rd→Rdand τ:R×Rd→Rare Ckfor some k≥1. More generally, one could

instead allow fand τto only be deﬁned on an open domain Ω ⊆Rd. Equation (1) includes

Time-varying delays: τ(t, x) = r(t) for r:R→R.

Discrete state-dependent delays: τ(t, x) = r(x) for r:Rd→R.

An assumption of causality might be physically reasonable (i.e τ(t, x)≥0), but in many applications

causality must be veriﬁed at the level of a particular solution a posteriori. A classical example

of an equation with time-varying delay is the pantograph equation [22], ˙x(t) = ax(t) + bx(λt) for

λ∈(0,1). As for discrete state-dependent delay problems, these continue to see application in cell

biology [2, 10], electrodynamics [8, 23] and other ﬁelds. In the present work, we will concentrate

on the case of discrete state-dependent delays: that is, formally, τ(t, x) = τ(x) for some function

τ:Rd→R. However, results will be stated with as much generality as possible.

1

In many applications, the delay is an implicit function of the state and is given explicitly. Examples

include the two-body problem in electrodynamics and position control models, among others, and we

refer the reader to the chapter [13] for several detailed examples. In many such situations, however,

it is possible to determine other diﬀerential equations that are satisﬁed by the delay variables. This

generally leads to a state-dependent delay diﬀerential equation of the form (1), perhaps of higher

dimension and with more than one delay. We consider here the case of a single delay, although the

ideas developed in this paper could be extended to the case of multiple delays.

Recently, there has been a surge of interest in computer-assisted proofs in delay diﬀerential equa-

tions. Proiment examples include the proofs of Wright’s conjecture [27], Jones’ conjecture [18]. Sev-

eral works have addressed approaches for validated computation of periodic orbits [24, 26], integration

[21, 24] and parameterization of unstable manifolds [17]. Insofar as validated integration is concerned,

the Taylor methods [24] and Chebyshev spectral methods [21] seem to be the most recent. They are

appropriate for ﬁxed, constant delays. Our objective with the present work is to develop rigorous

numerics for an implicit method of steps, using polynomial interpolation.

1.1 Background on state-dependent delay equations

State-dependent delay equations (SDDE) are very diﬀerent from those with constant delays. Su-

perﬁcially, the biggest change is that they are always nonlinear. More deeply, the semiﬂow of a

state-dependent delay equation is generally only C1on its associated solution manifold. For these

and related notions, the reader may consult [28, 29].

There is a large body of literature on numerical methods for SDDE, and we will make no eﬀort to

describe this here. We will, however, mention some recent work on the development of a posteriori val-

idation approaches to invariant objects of SDDE. There is the work periodic orbits in state-dependent

delayed perturbations of ordinary diﬀerential equations of Yang, Gimeno and de la Llave [31, 30],

and a recent extension of this work [11] has led to computer-assisted proofs of such perturbations of

periodic orbits. Parameterization of quasi-periodic solutions under a small state-dependent delay and

exponential dichotomy assumption was considered by He and de la Llave [14].

1.2 The method of steps

Recall the method of steps for delay diﬀerential equations. Without loss of generality, assume a

constant unit delay and the task of solving the initial-value problem

˙x(t) = f(x(t), x(t−1)), t > 0

x(θ) = φ(θ), θ ∈[−1,0].

The method of steps exploits the fact that for t∈[0,1], we have x(t−1) = φ(t−1). The task of

solving the IVP above is equivalent to solving a sequence of ODE initial-value problems. Denote

φ0=φand consider the sequence of IVPs for n∈N:

˙

φn(t) = f(φn(t), φn−1(t−1)), t ∈[n−1, n]

φn(n−1) = φn−1(n−1).

We have x(t) = φn(t) whenever t∈[n−1, n], so the solution of the delay IVP can be densely written

as

x=X

n

1[n−1,n)φn.

Here, 1Xis the indicator function on the set Xand by an abuse of notation we deﬁne φn(t) = 0 for

t /∈[n−1, n] so that the above is well-deﬁned.

The Chebyshev spectral approach of [21] makes explicit use of the method of steps by representing

the functions φnat each step as Chebyshev series. The power of this spectral method comes from the

excellent approximation properties of the Chebyshev polynomials and the Banach algebra associated to

the sequence space used in the proofs. Polynomial nonlinearities in ftranslate to cosine convolutions

in the sequence space, and the representation of the derivative operator is diagonally dominant.

2

To compare, the Taylor method of [24] represents solutions as piecewise-Taylor expansions, using a

bootstrapping procedure to get high-order derivatives of the solution needed to step the procedure

forward iteratively, and the step size is not necessarily equal to the delay.

Our initial goal was to develop a fully spectral integrator based on Chebyshev expansion. In

moving from the constant delay case to the non-constant or state-dependent case, there are a some

technical problems that must be either resolved or circumvented. Chieﬂy, self-compositions — for

example, x(t)7→ x(t−r(x(t))) — are highly nonlinear, and are diﬃcult to characterize at the level

of a sequence algebra. The obstructions were so severe that we abandoned the approach. In the next

section, we will overview these obstructions in more detail.

1.3 Interaction of self-compositions with spectral methods

Let x(t) = a0+ 2 P∞

n=1 anTn(t) denote a Chebyshev series, uniformly convergent on [−1,1] and such

that x(t)∈[−1,1]. Let y(t) = b0+ 2 P∞

n=1 bnTn(t) denote another Chebyshev series, and let us

consider the composition y◦x(t) = cn+ 2 Pn≥1cnTn(t). We might ask: how do we compute the

coeﬃcients {cn}given the coeﬃcients {an}and {bn}, and how are the decay of these sets of coeﬃcients

related?

Write the Chebyshev polynomials in the form Tn(t) = Pn

k=0 sn,ktk. Then we have

y(x(t)) = b0+ 2 X

n≥1

bn

n

X

j=0

sn,j

a0+ 2 X

k≥1

akTk(t)

j

=b0+ 2 X

n≥1

bn

n

X

j=0

sn,j

a∗j

0+ 2 X

k≥1

a∗j

kTk(t)

,

where a∗mdenotes the m-fold cosine convolution of awith itself: a∗mdef

= (a∗(m−1) ∗a) and

(u∗v)k=X

n∈Z

u|n|v|k−n|.

It follows that the operation (y, x)7→ y◦xis equivalent at the level of the Chebyshev series coeﬃcients

to the map (b, a)7→ c, where

cn=

∞

X

m=0

bm

m

X

j=0

sm,j a∗j

n.(2)

The previous derivation has a fairly direct consequence to numerical computations. If the coeﬃ-

cients {an}are ﬁnitely-supported with an= 0 for n≥N+ 1, then a self-composition b=a◦awill

have support for n∈ {0,...,N2}. Hence, rigously evaluating a numerical defect can be expensive

even for a low-order approximation, and the amount of information that is lost by taking a ﬁnite-mode

projection gets quadratically worse for higher-order approximation.

Far more problematic from the point of view of rigorous Chebyshev (or Fourier) spectral methods:

if one prescribes geometric decay of the coeﬃcients — for example, ||a||ν

def

=|a0|+ 2 P∞

n=1 νn|an|<∞

for some ν > 1 — it is generally false that ||a◦2||ν<∞. To see this, consider x(t) = t(1 + t2)−1.

The poles of xare at ±iand it is therefore analytic on the Bernstein ellipse Bρfor any ρ < 1 + √2,

with the associated Chebyshev series expansion being uniformly convergent there. Let {an}denote

the Chebyshev coeﬃcients of x(t) = t(1 + t2)−1. Then ||a||ν<∞for ν < √2. However, consider the

self-composition x◦xand the associated Chebyshev coeﬃcients a◦2.x◦xhas six poles, all imaginary,

but for the present discussion it suﬃces to list only the two smallest ones: t∗=±i(2−1(3 −√5))1/2.

Let ρ∗∈(1,1 + √2) be such that t∗is inside the Berenstein ellipse Bρ∗. Then ||a||ν∗<∞for

ν∗=ρ∗−1, but x◦xhas a pole on the in the interior or Bρ∗. In particular, ||a◦2||ν∗=∞. See Figure

1.

For the purposes of solving an initial-value problem associated to a DE-SDD like (1), the self-

composition barrier is not initially present. Indeed, the implicit method of steps introduced in Section

2.1 transforms an initial-value problem for a DD-SDD into a boundary-value problem for an ordinary

diﬀerential equation. This ordinary diﬀerential equation depends on the initial condition. Its solution

3

-2 -1 0 1 2

-1.5

-1

-0.5

0

0.5

1

1.5

Figure 1: The Berenstein ellipses Bρfor

ρ= 1 + √2 (solid line) and ρ=ρ∗def

=

2.1039 (dashed line) are plotted together

with the poles of x(t) (circles) and x◦x(t)

(stars). Since the interior of Bρ∗contains

no poles of x, its sequence of Chebyshev

coeﬃcients has ﬁnite || · ||ν∗norm, with

ν∗=ρ∗−1=1.1039. However, x◦x

has a pole in the interior of this ellipse, so

its sequence of Chebyshev coeﬃcients has

inﬁnite || · ||ν∗norm.

will then solve the initial-value problem on an interval of ﬁnite length. However, to solve the initial-

value problem on a longer interval, the computed solution must be fed forward as the initial condition

for a new initial-value problem. Viewing this process more globally, the feedforward structure is

“lower triangular” but non-trivially coupled. Solving the initial-value problem

˙x(t) = f(x(t), x(t−τ(t, x(t)))), x(θ) = φ(θ), θ ≤0

on a suitably long time interval (such that two implicit steps occur; see later Section 2.1) is equivalent,

after suitable transformations, to a ﬁxed-point problem of the form

x1(t) = φ(0) + Zt

0

F1(x1, φ)(s)ds

x2(t) = x1(1) + Zt

0

F2(x2, x1)(s)ds,

where Fj:C0([0,1],Rd)×C0([0,1],Rd)→C0([0,1],Rd) for j= 1,2 is a function that is related to

the vector ﬁeld fbut, importantly, features a compositional term. Speciﬁcally, Fj(y, z)(s) contains a

term of the form y(sδ −τ(sδ, z(s))) for some constant δ. At the level of a ﬁxed-point problem, the

pair of integral equations above deﬁne (almost; see Section 2.2) a map on C0([0,1],Rd)2that contains

a self-composition, due to the presence of F2. Therefore, moving to a purely spectral approach would

still require understanding a composition formula such as (2). At present, this seems out of reach.

For these reasons, we will avoid the use of purely spectral methods.

1.4 Structure of the paper

In the present work, we develop an implicit method of steps that is applicable to diﬀerential equations

with state-dependent delay. This is done in Section 2. Speciﬁcally, the ﬁrst Section 2.1 is more

computational, while Section 2.2 is more theoretically-minded, exploring how to translate the implicit

method of steps to a zero-ﬁnding problem. In Section 3 we review some necessary background on

interpolation and introduce the rigorous numerics framework for the zero-ﬁnding problem of the

previous section. Section 4 contains the theoretical background for rigorous multiple integration

steps, as well as extensive details concerning practical implementation. Several examples concerning

a scalar state-dependent delay equation appear in 5.

1.5 Notation

Given a function F:X1×X2×·· · ×Xn→Yfor Banach spaces X1,...,Xnand Y, we denote DjF

the partial Fr´echet derivative of Fwith respect to variables in Xj. For an interval I, we denote by

4

I◦its interior. If Iis closed we denote I◦

+= [inf I,sup I). IR denotes the set of real intervals, and

we deﬁne IRddef

= (IR)dfor any d≥1.

2 Solving DE-SDD

In this section we present the implicit method of steps and an associated zero-ﬁnding problem.

2.1 The implicit method of steps

Our modiﬁcation to the method of steps will be based on the idea that, initially, the time lag t−τ(t, x)

should be in the domain of the initial condition. In many situations the monotonicity of the time

lag is assumed, so once the time lag leaves the domain of the initial condition, it can never return.

To keep the method general, and allow for potential non-monotonicity of the time lag, we need to

introduce a piece of machinery that glues solutions together.

Deﬁnition 1. Let g1: [a, b]→Rdand g2: [b, c]→Rdsatisfy g1(a) = g2(b). The annealing of g1and

g2is the unique function g1∪g2: [a, c]→Rdsuch that g1∪g2|[a,b]=g1and g1∪g2|[b,c]=g2.

Consider the initial-value problem

˙x(t) = f(x(t), x(h(t, x(t)))),(3)

x(θ) = φ(θ), θ ∈ J (4)

for φ0:J → Rd, with Jbeing a closed, non-degenerate interval. Here, the interpretation is that

h(t, x(t)) represents the lagged time. Speciﬁcally, h(t, x) = t−τ(t, x), for the delay τ(t, x). We deﬁne

the implicit method of steps to be the following algorithm.

0. Initialize solution: x

7→

φ,I

7→

J,T

7→

∅. Deﬁne t0= sup Jand verify h(t0, φ(t0)) ∈ J◦

+.

1. Solve the following boundary-value problem for ψ: [t0, t1]→Rd,t1> t0:

˙

ψ(t) = f(ψ(t), φ(h(t, ψ(t)))), t ∈[t0, t1] (5)

ψ(t0) = φ(t0) (6)

h(t1, ψ(t1)) = t0.(7)

2. Update the solution: x

7→

x∪ψ,I

7→

J ∪ [t0, t1], T

7→

T ∪ {t0, t1}.

3. Update the initial data: φ

7→

x,J

7→

I. Deﬁne t0= sup J.

4. Return to step 1.

Proposition 1. Suppose f,τand φare continuous. Every time step 4 of the implicit method of steps

is reached, the function x:I → Rdis a solution of the initial-value problem (3)–(4). In particular, if

f,τand φare Lipschitz continuous, it is the unique solution deﬁned on that interval.

Proof. Let x: [inf J, t0+)→Rddenote any solution of (3)–(4). If h(t0, φ(t0)) ∈ J ◦

+, then

h(t0, ψ(t0)) < t0. Since xis continuous, t∗

1= inf{t > t0:h(t, x(t)) = t0}exists (or t∗

1=∞)

and t∗

1> t0. Without loss of generality, assume t∗

1< t0+. Then h(t, x(t)) ∈ J for t∈[t0, t∗

1], so

x(h(t, x(t)) = φ(h(t, x(t)). It follows that xis a solution of the boundary-value problem (5)–(7), with

t1=t∗

1. The result then proceeds by induction. Uniqueness if a consequence of the Picard-Lindel¨of

theorem for ordinary diﬀerential equations.

We refer to an implicit step as one cycle of 1–4 in the above algorithm. The set Tshould be understood

as a being an ordered set. It is a construct that keeps track of the terminal time in the boundary

condition (6)–(7). Aside from a bookkeeping device, we will use it to deﬁne a special class of solution.

Deﬁnition 2. Write T={t0, t1, . . . }for ordered times tj< tj+1. We say that the solution x:

I → Rdsupplied by the method of steps is m-steps disjoint if the graph of t7→ h(t, x(t)) intersects

{t0,...,tm−1} × Rdexactly mtimes, for t∈(t0, tm]. In general, a solution of the IVP (3)–(4) is

m-steps disjoint if it is equal on the interval [t0, tm]to a solution supplied by the implicit method of

steps, and that solution is m-steps disjoint.

5

024

-2

0

2

4

024

-0.4

-0.2

0

0.2

0.4

0.6

Figure 2: For a disjoint-step solution x(t) of the initial-value problem (plotted right), the times tjfor

j= 0,...,5 correspond to unique zeroes of the map t7→ h(t, x(t)) −tj. For visualization, the time

lag function t7→ h(t, x(t)) is plotted on the left, where stars indicate intersections with the level sets

h(t, x(t)) = tj. On the right plot, the pieces of curve between individual stars correspond to the segments

ϕ1, . . . , ϕ5. The horizontal length of each segment is uneven, which is typical of the implicit method of

steps in the scope of DE-SDD. Plots are for illustrative purposes only.

The idea behind such m-steps disjoint solutions is that they can be computed by a simpler version

of the implicit method of steps; see Figure 2. We will call it the implicit method of disjoint steps.

0. Initialize data: φ0

7→

φ,T

7→

∅,j

7→

1. Deﬁne t0= sup Jand verify h(t0, φ0(t0)) ∈ J◦

+.

1. Solve the following boundary-value problem for φj: [tj−1, tj]→Rd:

˙

φj(t) = f(φj(t), φj−1(h(t, φj(t)))), t ∈[tj−1, tj] (8)

φj(tj−1) = φj−1(tj−1) (9)

h(tj, φj(tj)) = tj−1.(10)

2. If j=m, proceed to step 3. Otherwise, set j

7→

j+ 1 and return to step 1.

3. Build the solution: x

7→

φ0∪φ1∪ · ·· ∪ φm.

In this version of the implicit method of steps, the solution xis only computed at the end. Segment

φj+1 can be computed using only the previous segment, φj, rather than the entire previously-computed

history. One can then prove an analogue of Proposition 1. The proof is straightforward and is omitted.

Proposition 2. The implicit method of disjoint steps produces a solution x: [t0, tm]→Rdof the

initial-value problem (3)–(4), and this solution is m-steps disjoint.

Remark 1. If φis not on the solution manifold — that is, ˙

φ(t0)̸=f(φ(t0), φ(h(t0, φ(t0)))) — then

the solution of the initial-value problem will have a discontinuous derivative at time t0. Therefore,

regardless the smoothness of φ, the solution loses regularity in a neighbourhood of t0. This means that

in the “standard” implicit method of steps, the vector ﬁeld (5) might only be (Lipschitz) continuous,

even though the initial condition is smooth. This is related to the so-called splicing condition of

integration of DE-SDD [3], and is one of the reasons solving such equations is numerically challenging.

To compare, in the disjoint steps version of the method, the vector ﬁeld maintains the regularity of

the initial condition φas long as fand hare also suﬃciently smooth.

6

2.2 Out-of-bounds evaluation and a zero-ﬁnding problem

The next step is to transform (5)–(7) into a suitable zero-ﬁnding problem. Assume without loss of

generality that dom(φ) = J= [−1,0]. Denote δ=t1and perform a re-scaling of time by deﬁning

˜

ψ(t) = ψ(tδ), substituting this into (5)–(7) and dropping the tildes. We get

˙

ψ(t) = δf (ψ(t), φ(h(tδ, ψ(t)))), t ∈[0,1] (11)

ψ(0) = φ(0) (12)

0 = h(δ, ψ(1)).(13)

We want to transform this modiﬁed boundary-value problem into a zero-ﬁnding problem on a

suitable function space. A standard approach is to integrate the diﬀerential equation and use the

boundary condition to obtain an integral equation. However, since the domain of φis only the

interval [−1,0], it is possible (in fact, likely) that for ψin a general vector space of functions, the

composition t7→ h(tδ, ψ(t)) will include points outside of the domain [−1,0] of φ. We refer to this

apparent issue as the out-of-bounds evaluation problem.

One way to resolve the out-of-bounds evaluation problem would be to restrict the range of any

ψ, and pose the integral equation on function space determined by such a restriction. However, this

would result in a zero-ﬁnding problem on a manifold, which we would rather avoid. To circumvent

this problem, let Ekφdenote a speciﬁc Ckextension of φto the real line, for k≤k. That is,

Ekφ[−1,0] =φ, (14)

and Ekφis Ck. Such an extension can always be constructed. For example, we can deﬁne

Eky(x) =

y(x), x ∈[α, β]

Pk

r=0

(x−α)r

r!y(r)(α), x < α

Pk

r=0

(x−β)r

r!y(r)(β), x > β

In this case, Ekyis simply a degree kTaylor extension of y. In what follows we will drop the subscript

on E, but the reader should keep in mind that k≤kwill always need to be chosen large enough so

that any subsequent derivatives can be given meaning.

With the extension Eφdeﬁned, we replace every nontrivial instance of φin (5) with Eφ. This gives

us

˙

ψ(t) = δf (ψ(t),Eφ(h(tδ, ψ(t)))), t ∈[0,1] (15)

ψ(0) = φ(0) (16)

0 = h(δ, ψ(1)).(17)

We then have the following lemma, whose proof is omitted.

Lemma 3. Let ψbe a solution of (15)–(17), with φ:J → Rd. If h(tδ, ψ(t)) ∈ J for t∈[0,1], then

ψis a solution of (11)–(13).

As we will see in Section 3.6, verifying the inclusion condition of this lemma is simple once we have

rigorously computed a solution of (15)–(17). We can now integrate (15) and use the initial condition

(16) to derive a zero-ﬁnding problem. The following lemma can be proven in a straightforward manner.

Lemma 4. ψ: [0,1] →Rdis a solution of (15)–(17) if and only if F(ψ, δ ) = 0 and t7→ h(tδ, ψ(t))

has range in [−1,0], where F:C([0,1],Rd)×R→C([0,1],Rd)×Ris deﬁned by

F(ψ, δ) = −ψ(t) + Rt

0δf (ψ(s),Eφ(h(sδ, ψ(s))))ds

h(δ, ψ(1)) (18)

With respect to the supremum norm on C([0,1],Rd),Fis ktimes continuously diﬀerentiable provided

the same is true of f,hand φ. In that case ψis k+ 1 times continuously diﬀerentiable.

7

3 Interpolation and rigorous numerics for a single im-

plicit step

With Lemma 4 in mind, our focus shifts to proving the existence of a zero of the map Fin (18). A

secondary goal is to rigorously check the inclusion t7→ h(tδ, ψ(t)) ∈ J = dom(φ). To accomplish

this, we will represent approximate solutions ψusing piecewise polynomial interpolants of a given

order and design a Newton-like operator for (18) that can be expected to contract in some closed ball

around a candidate interpolant ψand approximate terminal integration time δ∈R.

3.1 Piecewise reformulation of F

We will use piecewise Chebyshev interpolants for our data. Since these interpolants will not be

continuous unless additional constraints are speciﬁed, we reformulate the operator Fon a space of

piecewise-continuous functions. Given m∈N, let ∆m={s0, s1,...,sm}be a mesh of [0,1], with

0 = s0< s1<··· < sm= 1. For k∈N, let Ck

∆m([0,1],Rd) denote the space of functions that are k

times continuously diﬀerentiable on [0,1] \∆m, with the derivatives up to order khaving only jump

discontinuities on the mesh, and whose derivatives up to order kare continuous from the left. The

following lemma is simple to prove.

Lemma 5. Introduce a nonlinear map g:C0

∆m([0,1],Rd)×R→C1

∆m([0,1],Rd),

g(u, δ)(t) = u(s−

j) + Zt

sj

δf (u(s),Eφ(h(sδ, u(s)))ds, t ∈(sj, sj+1] (19)

and g(u, δ)(0) = φ(0), while in the case j= 0, we replace u(s−

j)with φ(0).genjoys the following

properties.

1. gis well-deﬁned.

2. If ψ=g(ψ, δ )and h(δ, ψ (1)) = 0, then ψ∈C0([0,1],Rd)and F(ψ, δ) = 0, and vice-versa.

3. Deﬁne µ:C1

∆m([0,1],Rd)→C0

∆m([0,1],Rd)to be the embedding µ(u) = u. Then µ◦gis k

times continuously diﬀerentiable, provided the same is true of f,hand φ.

As remarked in [4], the piecewise reformulation of the bootstrapped Picard operator has a few

advantages over a purely continuous one. We refer the reader to the discussion therein. Thanks to

Lemma 5, we can deﬁne a modiﬁcation of the map F.

Lemma 6. Deﬁne a map G:C0

∆m([0,1],Rd)×R→C0

∆m([0,1],Rd)×R,

G(ψ, δ) = −u+g(u, δ )

h(δ, ψ(1)) .(20)

If G(ψ, δ) = 0, then ψ∈C([0,1],Rd)and F(ψ , δ) = 0. The regularity assertions for Gand ψin this

case match those of Lemma 4.

In the sections that follow, it will be beneﬁcial to have a compact representation of the integrand

in (19). With φﬁxed, deﬁne ˜

f: [0,1] ×Rd×R→Rdby

˜

f(t, u, δ) = δf (u, Eφ(h(tδ, u))).(21)

3.2 Piecewise polynomial interpolation

Here we review some necessary material on piecewise polynomial interpolation, speciﬁcally with the

Chebyshev points of the second kind. Fix some k∈N. Given a mesh ∆m, let ∆m,k0denote a reﬁned

mesh such that the k−1 Chebyshev points of the second kind are inserted between sjand sj+1.

Formally,

sj,ℓ =sj+xk

ℓ+ 1

2(sj+1 −sj), = 1,...,k−1,

xk

ℓ= cos k−

kπ= 0,...,k,

8

with the symbolic extension sj,0=sjand sj,k =sj+1.

Deﬁne Sd

m,k ⊂C0

∆m([0,1],Rd), the space of piecewise polynomial functions on ∆m. Precisely, u∈

Sd

m,k if u∈C0

∆m([0,1],Rd) and u|(sj,sj+1)is a (d-vector) polynomial of degree kfor j= 0,...,m−1.

We then have a projection operator Πd

m,k :C0

∆m([0,1],Rd)→Sd

m,k deﬁned by the following property:

u= Πm,kuis the unique function in Sd

m,k such that u|(sj,sj+1)=pj|(sj,sj+1)for a degree kpolynomial

pj, where

pj(sj,ℓ) =

u(sj,ℓ), ̸= 0, k

u(s+

j,ℓ), = 0

u(s−

j,ℓ), =k.

The following interpolation error bounds are proven in [4].

Proposition 7. For all u∈Ck+1

∆m([0,1],R),

||u−Π1

m,ku||∞≤Ckmax

j=0,...,m−1 (sj+1 −sj)k+1 sup

t∈(sj,sj+1)

dk+1u

dtk+1 (t)!(22)

where Ck=1

(k+1)!22k.

Proposition 8. Fix ∈Nsuch that 1≤≤k. For all u∈Cℓ

∆m([0,1],R),

||u−Π1

m,ku||∞≤˜

Ck,ℓ max

j=0,...,m−1 (sj+1 −sj)ℓsup

t∈(sj,sj+1)

dℓu

dtℓ(t)!,(23)

˜

Ck,ℓ = min

(1 + Λk)π

4ℓ(k+ 1 −)!

(k+ 1)! ,1

!2ℓ

⌊ℓ−1

2⌋

X

q=0

1

4q −1

2q! 2q

q!

,(24)

with Λkbeing the Lebesgue constant. Speciﬁcally,

Λk= sup

x∈[−1,1]

k

X

i=0 |Lk

i(x)|, Lk

i(x) = Y

j̸=i

x−xk

j

xk

i−xk

j

,

which satisﬁes Λk≤1 + 2

πln(k+ 1), and

Λk=1

k

k−1

X

j=0

cot 2j+ 1

kπfor kodd.

Sd

m,k is isomorphic to Rmd(k+1). There are several ways this isomprphism can be realized, depend-

ing on how one would like to store the data associated to an element of Sd

m,k. We will take the most

direct approach and encode u∈Sd

m,k using its values at the points sj,ℓ, which we denote uj,ℓ. Then,

we have the expansion

u(t) =

k

X

ℓ=0

uj,ℓLk

ℓt−sj

sj+1 −sj−sj+1 −t

sj+1 −sj, t ∈(sj, sj+1), j = 0,...,m−1,(25)

for Lk

n: [−1,1] →Rthe Lagrange function

Lk

n(x) = Y

j̸=n

x−xk

j

xk

n−xk

j

.

When necessary, we will index the components of uj,ℓ using a superscript, u(i)

j,ℓ. Similarly, we

will index components of u∈C0

∆m([0,1],Rd) using superscripts, u(i)for i= 1,...,d. The induced

isomorphism between the function t7→ uand the coeﬃcients {uj,ℓ}will not be explicitly written.

9

3.3 Newton-like operator

From this point on, we will assume that f,τand φare ktimes continuously diﬀerentiable. We will

ﬁx mand k(generally distinct from k) so that we can drop the subscript indices on the projection

operator Πd

m,k and interpolant space Sd

m,k, and write them simply as Πdand Sd. Deﬁne a projection

operator

Π : C0

∆m([0,1],Rd)×R→Sd×R,(u, δ)7→ Π(u, δ) = (Πdu, δ ).

Let Π∞=I−Π be the complementary projector, and decompose X=C0

∆m([0,1],Rd)×Ras an

internal direct sum

X= Π(X)⊕Π∞(X)def

=Xd

m,k ⊕Xd

∞.

Then Xd

∞= Πd(C0

∆m([0,1],Rd)) × {0}, so we will identify elements of Xd

∞with their associated

elements of C0

∆m([0,1],Rd).

Let r0, r∞>0 be two positive parameters. Introduce norms on the factors of X,

||(u, δ)||Xd

m,k = max max

(i,j,ℓ)|u(i)

j,ℓ|,1

r0|δ|,||v||Xd

∞= max

i=1,...,d ||v(i)||∞(26)

and also deﬁne a norm on X:

||(u, δ)|| = max ||Π(u, δ)||Xd

m,k ,1

r∞||Π∞(u, δ)||Xd

∞.(27)

For all r0, r∞>0, (X, || · ||) is a Banach space.

Let p∈ {1,...,k+ 1}be ﬁxed, and consider the map G:X→Xdeﬁned in (20), and ˜

G:Xd

m,k →

Xd

m,k,

˜

G(u, δ) = ΠG(u, δ).

By Lemma 6, each of Gand ˜

Gis ktimes continuously diﬀerentiable. By Lemma 5 and preceding

results, zeroes of Gare (almost) in one-to-one correspondence to solutions of the boundary-value

problem (15)–(17).

Let w= (u, δ)∈Xd

m,k be a candidate for a numerical zero of the function ˜

G. Let Am,k :Xd

m,k →

Xd

m,k be an injective linear operator, and introduce a Newton-like operator T:X→X,

T w = (Πw−Am,kΠG(w)) + Π∞(G(w) + w).(28)

In practice, we will verify directly that the matrix representation of A†

m,k on Rmd(k+1)+1 (recall,

Sk

m,d ∼Rmd(k+1)) is invertible, and Am,k will be chosen to be its inverse (up to conjugation by

coordinate maps).

Remark 2. Tcan alternatively be written T w =w−AG(w), where A:X→Xis the bounded linear

operator deﬁned by AΠw=Am,kΠwand AΠ∞w=−Π∞w. In this context, Ais interpreted as an

approximate inverse of DG(w).

Introduce the ball

Br(w) = {z∈X:||w−z|| < r}

centered at z∈Xwith radius r > 0. Let Br(w) denote its closure. The following theorem provides

suﬃcient conditions under which the operator Tis contractive on the closure of (and maps into the

interior of) a ball centered at the numerical zero wof G. It is similar to a theorem from [7] and is

proven in the same way.

Theorem 9. Let T:X→Xbe continuously diﬀerentiable, Am,k injective, and w∈Xd

m,k. Suppose

there exist bounds Y,Zsuch that

||Π(T w −w)||Xd

m,k ≤Y(29)

||Π∞(T w −w)||Xd

∞≤Y∞(30)

sup

ξ1∈Br(0) ||Π(DT (w+ξ1)ξ2)||Xd

m,k ≤Z(r) (31)

sup

ξ1∈Br(0) ||Π∞(DT (w+ξ1)ξ2)||Xd

∞≤Z∞(r) (32)

10

for all ξ2∈B1(0). If there exists r > 0such that the radii polynomials satisfy

pdef

=Y+rZ(r)−r < 0 (33)

p∞def

=Y∞+rZ∞(r)−rr∞<0,(34)

then Thas a unique ﬁxed point in Br(w). Hence, Ghas a unique zero in Br(u).

Being based on piecewise polynomial interpolation, the Yand Zbounds can be computed using

much the same machinery as in [4]. The diﬀerences are that our vector ﬁelds are non-autonomous

(because of the delays) and we have a boundary condition. In the following two sections we derive

these bounds in general, but the presentation will be rather terse since many of the computations are

similar to those of the cited reference.

3.4 Technical bounds for computer-assisted proofs

In this section we provide explicit formulas for the bounds Yand Zfrom Theorem 9. We also discuss

some details concerning the implementation of the operator A†

m,k =D˜

G(w) and the approximate

inverse, Am,k. Before deriving the bounds, we introduce a few pieces of notation. Given w= (u, δ)∈

Xd

m,k, we write u=π1wand δ=π2w. Partition Am,k as a block operator

Am,k =A11 A12

A21 A22

such that

Am,k(π1w+π2w) = (A11 π1w+A12 π2w, A21 π1w+A22π2w).(35)

We also use the same projection (i.e. using π1, π2) notation for an element of X. By way of (25), the

isomorphism of Sd

m,k with Rmd(k+1) induces a matrix representation of A11 as a map on Rmd(k+1).

We will use the same symbol, A11 , for this matrix operator. Similarly, A12 can be identiﬁed with

an element of Rmd(k+1),A21 can be identiﬁed with an element of the dual (Rmd(k+1))∗, hence a row

vector, and A22 is a real number. Finally, deﬁne the polynomial function mj: [−1,1] →[sj, sj+1],

mj(t) = sj+1 −sj

2t+sj+1 +sj

2.

By construction, mj(xk

ℓ) = sj,ℓ and

u(mj(t)) =

k

X

ℓ=0

uj,ℓLk

ℓ(t), t ∈(−1,1).

3.4.1 Ybound

We write Y= max{Y1, r −1

0Y(2)}, where each of Y(1) and Y(2) will be deﬁned such that

max

i,j,ℓ |[π1(Am,k ˜

G(w))](i)

j,ℓ| ≤ Y(1) (36)

|π2Am,k ˜

G(w)| ≤ Y(2) (37)

If Y(1) and Y(2) satisfy the above, then (29) holds. Implementing the left-hand side of (37) is

straightforward, so we will not discuss the Y(2) bound. For the Y(1) bound, we have

π1(Am,kG(w)) = A11 (g(u, δ)−u) + A22 h(δ, u(1)) (38)

A22h(δ , u(1)) can be computed directly, so we focus on computing g(u, δ). Recall

g(u, δ)(t) = u(s−

j) + Zt

sj

δf (u(s),Eφ(h(sδ, u(s)))ds, t ∈(sj, sj+1].(39)

11

Computing [g(u, δ)]j,ℓ requires evaluating the integrand above at sj,ℓ . If all functions involved are

polynomials, this can be done directly and exactly. If not, one must generally resort to numerical

quadrature. The implementation of this integral is, arguably, the hardest part about completing

proofs with multiple steps, the latter of which is discussed in Section 4.

Once the integral (39) is implemented (in the sense that we can rigorously enclose it), we can

obtain a suitable bound Y(1) in (36). Combined with the previous discussion concerning Y(2), we can

compute a Ybound satisfying (29).

3.4.2 Y∞bound

A suitable Y∞bound as in (30) is straightforward to compute using Proposition 7. Namely, we can

choose

Y∞≥Ckmax

j=0,...,m−1 (sj+1 −sj)k+1 sup

t∈(sj,sj+1)

dk

dtk˜

f(t, u(t), δ)!(40)

where the norms on the right-hand side are the ∞norm on Rd, and ˜

fis deﬁned in (21). To implement

this bound in practice, we subdivide [sj, sj+1 ] with an interval mesh and compute coarse enclosures of

the derivative using interval arithmetic. In the case of multiple steps, we incorporate the error from

previous steps into the derivatives; see Section 4 for details.

Remark 3. The Y∞bound is often the limiting bound in obtaining a small radius rin a computer-

assisted proof. Increasing kis generally expensive if one is interested in doing proofs with multiple

integration steps, while increasing mrequires ever more memory. See Section 3.7 for further discus-

sion.

3.4.3 Zbound

To begin, we write the components of ΠDT (w+ξ1)ξ2in the following form:

ΠDT (w+ξ1)ξ2= (Π −Am,kΠDG(w+ξ1)) ξ2

= (I−Am,kA†

m,k)Πξ2+Am,k (A†

m,k −DΠG(w))ξ2+Am,k (DΠG(w)−DΠG(w+ξ1))) ξ2,(41)

taking into account that the projection operator Π, being bounded, commutes with the Fr´echet

deriavtive. We will identify Z0,Z1and Z2such that

Z0≥ ||(I−Am,kA†

m,k)Πξ2||Xd

m,k ,(42)

Z1≥ ||Am,k(A†

m,k −DΠG(w))ξ2||Xd

m,k ,(43)

Z2≥ ||Am,k(DΠG(w+ξ1)−DΠG(w))ξ2||Xd

m,k (44)

for all ξ1∈Br(0) and ξ2∈B1(0). Then, we can take Zdef

=Z0+Z1+Z2, and the bound (31) will be

satisﬁed.

Z0bound

The Z0bound is trivial to compute. Indeed, since Πξ2∈Xd

m,k ∼Rmd(k+1)+1, we can take Z0to

be the operator norm of I−Am,k A†

m,k on Xd

m,k. Due to the isomorphism, this reduces to a ﬁnite-

dimensional matrix norm computation, appropriately weighted so it is consistent with the norm on

Xd

m,k. We omit the details.

12

Z1bound

Denote Π∞ξ2=ξ∞

2. Before we begin, we remark that ξ∞

2(sj) = 0 for j= 0,...,m due to the

interpolation. Similarly, ξ∞

2(s−

j) = 0. This will be used a few times.

Observe that DΠG(w)ξ2=A†

m,kΠξ2+DΠG(w)Π∞ξ2. Therefore, we need only construct Z1such

that Z1≥ ||Am,kDΠG(w)Π∞ξ2||.Denote ξ2= (u2, δ2) and Π∞ξ2=ξ∞

2. Then

|π2DΠG(w)ξ∞

2|=|D1h(δ, u(1))0 + D2h(δ, u(1))ξ∞

2(1)|= 0 (45)

since ξ∞

2(1) = 0, and the other zero being due to ξ∞

2having trivial component in R. As for the π1

component, since ξ∞

2(s−

j) = 0 for j= 0,...,m,

[π1DΠG(w)ξ∞

2]j,ℓ =Zsj,ℓ

sj

D2˜

f(s, u(s), δ)ξ∞

2ds.

Since |ξ∞

2(s)| ≤ r∞because ||ξ2||X≤1, we can majorize as follows:

|[π1DΠG(w)ξ∞

2](i)

j,ℓ| ≤ r∞|sj,ℓ −sj|sup

t∈[sj,sj,ℓ]|D2˜

f(t, u(t), δ)(i)|.

Combining the above with (35) and (45), it follows that if we deﬁne ρ∈Sd

m,k by

ρ(i)

j,ℓ =|sj,ℓ −sj|sup

t∈[sj,sj,ℓ]|D2˜

f(t, u(t), δ)(i)|,(46)

then we can take any Z1such that

Z1≥r∞max

i,j,ℓ ||(abs(A11)ρ)(i)

j,ℓ||,(47)

for abs(A11) being the matrix whose entries are the absolute values of the entries of A11. To implement

ρ, we can use direct interval evaluations, further subdividing [sj, sj,ℓ] to get a tight enclosure.

Remark 4. If the mesh ∆mis equally spaced, ρscales linearly with respect to 1

m; see (46). However,

in the Z1bound, this inverse scaling with mis generally negated by the multiplication abs(A11 )ρin

(47), since A11 is a square matrix with (k+ 2)m+ 1 rows. Similarly, the Z1bound is generally linear

in k. Therefore, in general, the only way to control Z1is to decrease r∞. The result is that there is an

even greater need to keep Y∞and Z∞small, since decreasing r∞has a penalty on the radii polynomial

p∞; see (34). One way the bound Z1could be improved is to use the a priori bootstrap developed in

[4], since there, the Z1bound truly can be controlled by tuning mand the order of the bootstrap. To

keep the presentation simple, we have avoided doing this in the present paper. See Section 3.7 for

further discussion.

Z2bound

Here we will assume that k≥2, although a Z2bound can generally still be computed even if k= 1.

To begin, let ξ1= (u1, δ1) and ξ2= (u2, δ2). Let H(x, y ) denote the Hessian of hat (x, y)∈R×Rd.

Then

π2(DΠG(w+ξ1)−DΠG(w))ξ2=Z1

0

[u2(1) δ2]H(δ+tδ1, u(1) + tu1(1)) u1(1)

δ1dt (48)

provided k≥2. Let 1= [−1,1]d×[−r0, r0]⊂Rd×Rand 1r0= (1,1,...,1, r0)∈Rd×R. Then

|π2(DΠG(w+ξ1)−DΠG(w))ξ2| ≤ sup

b∈1r0

1⊺

r0|H(u+rb)|1r0rdef

=2.(49)

For the π1component, ﬁrst deﬁne Θ(·,·;s) : R×Rd→Rdby

Θ(x, y;s) = ˜

f(s, y, x).

13

Deﬁne 1∈Sd

m,k by

(1)(i)

j,ℓ =|sj,ℓ −sj|sup

|x|≤rr0

||y||∞≤r(Λk+r∞)

sup

t∈[sj,sj,ℓ]

rmax{Λk+r∞, r0}|D2Θ(i)(δ+x, u(t) + y;t)|12

r0,r∞,

(50)

where 1r0,r∞= (Λk+r∞,...,Λk+r∞, r0)∈Rd×R, and products with 12

r0= (1r0,1r0) and analogous;y

ones with 12

r0,r∞are interpreted as the action of the appropriate bilinear maps (second Fr´echet

derivative of Θ(i)) on these elements. Then, one can compute Z2=Z2(r) such that

Z2(r)≥ ||abs(Am,k)||Xd

m,k (51)

with = (1, 2), then (44) holds. Remarks concerning the implementation of analogous to those

for the Z1bound can also be made.

Remark 5. A similar comment can be made here as we did for the Z1bound. The bound (51) can

generally only be controlled by decreasing r. For this to be feasible, it is necessary that the Ybound

is suitably small. Thankfully, the size of the latter bound is primarily determined by the precision of

the ﬂoating point number system and the wrapping eﬀect of error from any previous computer-assisted

proofs, and can therefore be controlled. See Section 3.7 for further discussion.

3.4.4 Z∞bound

Let 1Udenote the indicator function on the set U. Once again writing ξ1= (u1, δ1) and ξ2= (u2, δ2),

the Z∞bound requires us to obtain a bound (in the norm || · ||Xd

∞) for

Π∞(DG(u+δ1, δ +δ1)ζ2−ζ2) = Π∞ t7→

m−1

X

j=0

1(sj,sj+1](t)Zt

sj

DΘ(δ+δ1, u(s) + u1;s)(δ2, u2(s))ds!.

Using Proposition 7 and Proposition 8, we can establish that if Z∞is chosen such that

Z∞≥˜

Ck,1max

i=1,...,d

max

j=0,...,m−1|sj+1 −sj|sup

|x|≤rr0

||y||∞≤r(Λk+r∞)

max

t∈[sj,sj+1]|DΘ(i)(δ+x, u(t) + y;t)1r0,r∞|

(52)

then (32) holds.

3.5 A priori maximum radius

In practice, we select r∗>0 and consider the polynomials

˜p(r) = Y+r(Z0+Z1+Z2(r∗)) −r

˜p∞(r) = Y∞+rZ∞(r∗)−rr∞

If we can ﬁnd r∈(0, r∗) such that ˜p(r)<0 and ˜p∞(r)<0, then (33) and (34) will both be

negative. This is a consequence of our bounds Z2(r) and Z∞(r) being nondecreasing functions

of r. We use this implementation for our examples, since it is less expensive than computing

roots of the full radii polynomials pand p∞directly.

14

3.6 Verifying the inclusion h(tδ, u(t)) ∈[−1,0]

Recall that the lag function is h(t, u) = t−τ(t, u). To begin, remark that by deﬁnition of the

norm on Xand the numerical enclosure aﬀorded by the radii polynomial approach, the true

zero (u, δ) of Gmust satisfy

||u−u||∞≤r(Λk+r∞),|δ−δ| ≤ rr0.(53)

Introduce an interval-valued function ˆu: [0,1] →IRd,

ˆu(t) = u(t) + r(Λk+r∞)[−1,1].

Denote ˆ

δ=δ+rr0[−1,1] ∈IR. It follows that

h(tδ, u(t)) = tδ −τ(t, u(t)) ∈tˆ

δ−τ(tδ, ˆu(t)).(54)

This inclusion alone will not be suﬃcient to prove that h(tδ, u(t)) ∈[−1,0] for t∈[0,1],

necessarily because where t≈0 or t≈1, the right-hand side of (54) will intersect the

complement of [−1,0]. To remedy this, we remark that due to Lemma 5, uis k+ 1 times

continuously diﬀerentiable. In particular,

d

dtu(t) = δf(u(t),Eϕ(h(tδ, u(t)))) ∈ˆ

δf ( ˆu(t),Eϕ(tˆ

δ−τ(tˆ

δ, ˆu(t)))) def

= ˆu′(t).(55)

It follows that

d

dth(tδ, u(t)) ∈ˆ

δ−D1τ(tˆ

δ, ˆu(t))ˆ

δ−D2τ(tˆ

δ, ˆu(t))ˆu′(t).(56)

If the enclosures in (53) are tight enough and the derivative (56) is strictly positive at t= 0 and

t= 1, one can use (54), (55), (56) and elementary calculus to prove the inclusion h(tδ, u(t)) ∈

[−1,0] for all but the most wildly oscillatory solutions u.

3.7 Parameter tuning for computer-assisted proofs

Selection m, k and weights r∞is crucial for a successful proof. We will not discuss them

exhaustively, but rather provide some general guidelines. In the examples we considered,

tuning r0wasn’t hugely important in obtaining proofs, and we will not discuss it here.

The Z1bound can only be controlled reliably by adjusting r∞, given the discussion of

Remark 4. As such, one should typically compute Z1ﬁrst with r∞= 1, and then adjust r∞

accordingly to ensure that Z1<1. The bound Z0will typically be machine-precision small

(or wrapping eﬀect small) and can be ignored. In practice, the Z2bound typically does not

cause much of a problem, and its contribution to the radii polynomial is quadratic. Therefore,

as long as Z0+Z1<1 and Yis suﬃciently small (we expect machine precision or wrapping

eﬀect smallness), the “ﬁnite proof” will typically succeed, which is to say that there will exist

rsuch that p(r)<0. In summary, for the ﬁnite proof to succeed, it should generally only

be necessary to adjust the r∞weight. This weight should be chosen as large as possible for

the proof to succeed (with success being weighted by how large a value of rwe are willing to

accept).

Once the r∞weight has been suitably tuned, we need to work on the Y∞and Z∞bounds.

In conditioning for the ﬁnite proof, it might have been necessary to take r∞a bit small. In

general, the smaller r∞is, the smaller we will need to make Z∞. Unfortunately, the only

way to tune Z∞is to increase mor k. The scaling of Z∞is inverse linear in m, whereas

15

for ﬁxed m, we have Z∞=O(ln(k+1)

k+1 ). It is generally better, for this reason, to adjust mif

one needs to control Z∞. As for the Y∞bound, this too beneﬁts from inverse polynomial-

order scaling with respect to m. However, khas a much larger eﬀect due to the (sj+1 −sj)k

term. The downside is that the derivatives of ˜

f(t, u(t), δ) can be quite large. This is largely

system- and data-dependent, and it is nearly impossible to predict the impact of increasing k

on the Y∞bound. Even less so when multiple steps of integration are involved. To conclude,

controlling mresults in a more reliable control of Y∞and Z∞, for ﬁxed k, than does ﬁxing

mand controlling k. However, depending on the properties of the numerical solution uand

the vector ﬁeld, Y∞might scale incredibly well with respect to k.

4 Multiple implicit steps

The Ybound of Section 3.4, needed to obtain computer-assisted proofs of solutions of DE-

SDD, is not directly implementable. Also, we have not explained how exactly the error from

one proof should be propagated forward rigorously, if one is proving a long solution using

multiple integration steps. The latter step ends up being less than trivial, and there are

several non-equivalent ways it can be done depending on how the numerical data from each

step is stored and how this interacts with the quadrature rule used to evaluate (39), for

instance. We emphasize the storage of the numerical data, since the out-of-bounds evaluation

problem makes this more complicated than merely storing a representative of each solution

on the canonical domain [0,1].

To provide a constructive answer, we will make a few simplifying assumptions. We will

assume that the function fand the delay τare polynomial. This might seem rather strong,

but it can sometimes be avoided; see Section 4.4. As for the initial condition ϕ, we will explore

in this section how it (as well as the extension Ekϕ) can be rigorously enclosed using various

polynomial-like functions. This will allow for eﬀective evaluation of the bounds needed for

computer-assisted proofs, as well as a systematic way to complete multiple steps of integration.

Before we begin, let us make a disclaimer. This section provides one way by which error

from one computer-assisted proof can be rigorously propagated to the next, while allowing

for eﬀective evaluation of subsequent Yand Zbounds. However, ours is not the only way.

Broadly, the steps we take in this section are designed to move from piecewise-polynomial

representations of numerical solutions u, into polynomial representations of subsequent initial

conditions. The advantage of this set-up is that it makes the Ybounds fairly explicit, even

for multiple steps of integration, since it reduces the calculations to polynomial integration.

See Appendix A for some self-contained discussion. The disadvantage is that extremely high-

degree polynomials can be necessary. We are quite certain that there are more eﬃcient ways of

storing the data, computing the bounds Yand Z, and rigorously propagating error. Whether

or not all of this can be done simultaneously is the big question.

4.1 ϕ-mutations and interpolation of boostrapped vector ﬁelds

How to generate an enclosure of ϕwill depend on the scope in which it is considered. If ϕis

the initial condition for the ﬁrst implicit step, our view is that it could be a ﬁxed (suﬃciently

smooth) function, or perhaps it could be a theoretical object in an unstable manifold of a ﬁxed

point or periodic orbit. In both cases, ϕshould be (suﬃciently) smooth, so its derivatives and

E-extensions up to order kcan be interpolated by polynomials on arbitrarily large domains.

The error bounds enare then consequences of theoretical interpolation error. Keeping these

errors small might require a high-order interpolant and access to even higher derivatives of ϕ

on [−1,0]; see [25] for results concerning interpolation at the Chebyshev nodes.

16

In the second case, ϕcould actually be u, with its domain appropriately scaled to [−1,0],

and ubeing an output function from the implicit method of steps. In this case, ushould be

identiﬁed with uplus a C0enclosure of width rdef

=r(Λk+r∞). Our goal is therefore to obtain

an enclosure of u(or Eku) and its derivatives.

We will now overview a few ways in which an enclosure of a solution u, validated by the

radii polynomials, can be generated. First, we need a deﬁnition.

Deﬁnition 3. ACnmutation of a Ckfunction ϕ,n≤k, is a set of functions {ξ0, . . . , ξn},

with ξj:R→IRd, such that ϕ(j)∈ξj|dom(ϕ)for j= 0, . . . , n. If ξand ζare two mutations,

we write ξ⊆ζif the range of ξnis a subset of that of ζnfor n= 0,...,k.

Generally, if we say that a Cnmutation ξ={ξ0, . . . , ξn}has some property (e.g. is smooth,

is an interval-valued polynomial), this should be understood to mean that each ξjfor n=

0, . . . , n has this property. The Taylor extension Ekϕdeﬁnes a single-valued Ckmutation of

ϕin a straightforward way. Mutations will be useful later in making statements and proofs

concerning enclosures more transparent.

Lemma 10. Suppose G(u, δ) = 0, and h(tδ, u(t)) ∈[−1,0] for t∈[0,1]. Let ξbe a Cn

mutation of ϕ,n≤k, and deﬁne

˜

fξ(t, u, δ) = δf(u, ξ0(h(tδ, u))).(57)

That is, ˜

fξis the same as ˜

fexcept that we have replaced Eϕwith ξ0. Then, for j= 1, . . . , n+1,

u(j)(t)∈˜

fξ,j (t, u(t), δ), t ∈[0,1],(58)

where the sequence of vector ﬁelds ˜

fξ,j are symbolically deﬁned according to

˜

fξ,1=˜

fξ,˜

fξ,j+1 =D1˜

fξ,j + (D2˜

fξ,j )˜

fξ,

with the addendum that each (symbolic) derivative ξ(j)

0for 0≤j≤nis replaced with ξj. If ξ

is a single-valued mutation, then the inclusion in (58) becomes an equality.

We refer to the ˜

fξ,j as bootstrapped vector ﬁelds. If a mutation ξhas been selectd, we will

sometimes suppress the subscript. Recall that f, the delay functions and ϕare all assumed

to be Ck, so that these bootstrapped vector ﬁelds are indeed well-deﬁned. In what follows,

(u, δ)∈Xwill always satisfy the conditions of Lemma 10. They will admit the enclosures

u∈ˆuand δ∈ˆ

δ, where ˆuand ˆ

δare deﬁned in Section 3.6.

The basic idea of this section is that Lemma 10 allows us to enclose derivatives of uusing

the bootstrapped vector ﬁelds. The choice of mutation can make certain calculations easier.

For example, if ξis an interval-valued polynomial mutation, then the bootstrapped vector

ﬁelds are interval-valued polynomials.

4.1.1 Part 1: Bootstrapped enclosure of u

As we will see soon, having a polynomial enclosure with padding ϵ= 0 is incredibly useful.

We will show how to generate one in this section.

The following discussions will be easier if we shift the domain of uto the interval [−1,1].

Formally, this can be done by deﬁning y(t) = u((t−1)/2). If one substitutes yinto (58), the

result is that

y(j)(t) = 1

2j

˜

fξ,j t−1

2, y(t), δ.

17

However, rather than propagate this change forward, we will make a slight abuse of notation

and identify ywith u, so now we think of uas being deﬁned on [−1,1]. Inverting the transfor-

mation t7→ t−1

2has the eﬀect of inverting the derivative scaling that appears in the displayed

equation above.

Deﬁnition 4. A bootstrapped enclosure of u: [−1,1] →Rdat order k∈ {0,...,k+1}is any

interval-valued function vk:R→IRdsuch that

˜

fξ,k (t, u(t), δ)∈vk(t)

for at least one mutation ξof ϕ, and |t| ≤ 1,where we deﬁne ˜

fξ,0(t, u, δ) = u.

A ﬁrst step is to obtain a bootstrapped enclosure of u– not even its derivatives – on

[−1,1]. We will do this with Chebyshev interpolation. To get a theoretical interpolation

error, we compute an upper bound for supt∈[−1,1] |u(k+1)(t)|.

Lemma 11. Suppose a real V > 0is computed such that

V≥sup

t∈[−1,1] |˜

fξ,k+1(t, ˆu(t),ˆ

δ)|.(59)

for some Ckmutation ξof ϕ. Then V≥supt∈[−1,1] |u(k+1)(t)|.

Proof. Let ξbe the mutation of ϕsuch that ξnis single-valued on [−1,0] and coincides with

ξnotherwise. Then ξ⊆ξ.

|u(k+1)(t)|=|˜

fξ,k+1(t, u(t), δ )| ∈ {|˜

fξ,k+1(t, x, y )|: (x, y)∈ˆu(t)×ˆ

δ, h(ty, x)∈dom(ϕ)=[−1,0]}

⊆ {| ˜

fξ,k+1(t, x, y )|: (x, y)∈ˆu(t)×ˆ

δ}=|˜

fξ,k+1(t, ˆu(t),ˆ

δ)|

⊆ | ˜

fξ,k+1(t, ˆu(t),ˆ

δ)|.

Every element of this set is bounded by V.

The bound Vis indeed implementable once an implementable mutation of ϕhas been

selected. This can be with a Taylor extension, or something completely diﬀerent. Once it

has been computed, consider the following well-known [25] error bound associated to the

Chebyshev interpolant yN=PN

n=0 cnTnof a function yon the domain [−1,1]:

sup

t∈[−1,1] |y(t)−yN(t)| ≤ 4W

πk(N−k)k., (60)

where Wis a uniform bound for y(k+1). On the other hand, recall the Lebesgue constant

ΛNassociated to the degree NChebyshev interpolation on [−1,1], which satisﬁes ΛN≤

1 + 1

2ln(N+ 1). We generally advocate the following approach to interpolating uon [−1,1].

Lemma 12. Let uNdenote the degree NChebyshev interpolant of (the piecewise-polynomial

function) u. Then

sup

t∈[−1,1] |u(t)−uN(t)| ≤ 4V

πk(N−k)k+ ΛNrdef

=e0,

provided N > k, for Vbeing the constant from Lemma 11. As consequence, u(t)∈uN(t) +

e0[−1,1].

18

Proof. Let PNdenote the degree NChebyshev interpolation operator on C0([−1,1],Rd). Since

PNis linear and ||PN|| ≤ ΛN, with ||·|| being the operator norm associated to the supremum

norm || · || on C0([−1,1],Rd), we have

||u−PNu|| ≤ ||u−PNu|| +||PN(u−u)||,

which admits the bound stated in the lemma.

Remark 6. One could instead directly interpolate the enclosure ˆuof u. However, this results

in wrapping eﬀect of intervals for which the interpolant has C0radius ≈Nr. To compare, ΛN

is sub-linear in N. Consequently, Lemma 12 gives a tighter bound than naively interpolating

the enclosure ˆu.

Now that we have a polynomial enclosure of uon [−1,1], we can directly generate a poly-

nomial enclosure of u(j), for j= 1,...,k+ 1. Indeed, the following is a direct consequence of

Lemma 12 and Lemma 10.

Corollary 13. For n≤kand any Cnmutation ξof ϕ, we have u(j)(t)∈˜

fξ,j (t, uN(t) +

e0[−1,1],ˆ

δ)for j= 1, . . . , n + 1. Consequently,

dj

dtju(t)∈pj(t)def

=uN(t) + e0[−1,1], j = 0

˜

fξ,j (t, uN(t) + e0[−1,1],ˆ

δ), j ≥1.(61)

Each pj(t)is a bootstrapped enclosure of uat order n. In particular, if ξis a (interval-valued)

polynomial mutation, then pjis a composition of (interval-valued) polynomials.

The primary utility of Corollary 13 will be in the computation of the bound (59), since this

bound can be computed with respect to any mutation of the initial data ϕ. It is not useful for

obtaining tight enclosures of derivatives because of the wrapping eﬀect, and it does not handle

out-of-bounds evaluations. In the next section, we will resolve this issue by interpolating the

boostrapped vector ﬁelds themselves.

4.1.2 Part 2: Bootstrapped enclosures for derivatives of Eku

Consider the CkTaylor extension Ekuassociated to u. Once again, we interpret uas being

deﬁned on [−1,1]. What we will do is interpolate dn

dtnEkuon [−1−2ϵ, 1 + 2ϵ], for some ϵ≥0.

Note that after inverting the transformation to return uto the domain [−1,0], this will result

in a polynomial enclosure on [−1−ϵ, ϵ]. See the ﬁrst paragraph of Section 4.1.1. In this

section, a given Ckmutation ξof ϕwill be ﬁxed, and we will suppress the subscript ξon the

bootstrapped vector ﬁelds.

To begin, let us construct a few polynomials. Deﬁne

pL(t) =

k

X

n=0

1

n!˜

fn(−1, u(−1), δ)(t+ 1)n,

pR(t) =

k

X

n=0

1

n!˜

fn(+1, u(+1), δ)(t−1)n,

where ˜

f0(t, y, δ) = y. Deﬁne also the interval polynomials

pL(t) =

k

X

n=0

1

n!˜

fn(−1, u(−1) + r[−1,1],ˆ

δ)(t+ 1)n,

19

pR(t) =

k

X

n=0

1

n!˜

fn(+1, u(+1) + r[−1,1],ˆ

δ)(t−1)n,

By construction, the inclusions

dn

dtn(Eku)|(−∞,−1] ∈dn

dtnpL=p(n)

L,dn

dtn(Eku)|[1,∞)∈dn

dtnpR=p(n)

R(62)

are satisﬁed for n= 0,...,k. We also have the bound

sup

|t|≤1+2ϵ

dk

dtkEku≤V, (63)

where Vis any number satisfying (59). This is a direct consequence of the deﬁnition of

Ek, Lemma 10 and Lemma 11. We can ﬁnally prove our enclosure result which, due to its

importance in rigorously proving multiple-step integration of DE-SDD, we give the title of

theorem.

Theorem 14. For n= 1,...,k, suppose

|˜

fn(t, u(t) + η1, δ +η2)−˜

fn(t, u(t), δ)| ≤ rn(64)

is satisﬁed for some rn>0, whenever |η1| ≤ r(Λk+r∞) = rand |η2| ≤ rr0. Denote r0=r.

Let pndenote the degree NChebyshev interpolant on the interval [−1−2ϵ, 1+2ϵ]of the

function

wn(t) =

˜

fn(t, u(t), δ), t ∈[−1,1]

p(n)

L(t), t < −1

p(n)

R(t), t > 1.

If 0≤n < k< N, then, for |t| ≤ 1+2ϵ,

|(Eku)(n)(t)−pn(t)| ≤ en(t)def

=4V(1 + 2ϵ)k+1−n

π(k−n)(N−k+n)k−n+ ΛN(rn,|t| ≤ 1

Pk−n

j=0

rj+nϵj

j!,|t|>1

(65)

Proof. Deﬁne y(t) = Eku((1 + 2ϵ)t) for t∈[−1,1]. Then

sup

t∈[−1,1] |y(k+1)(t)|= (1 + 2ϵ)k+1 sup

|t|≤1+2ϵ|(Eku)k+1(t)| ≤ (1 + 2ϵ)k+1 V.

Also, it follows that pn(t) = ˜pn((1 + 2ϵ)−1t), where ˜pnis the Chebyshev interpolant of t7→

˜wn(t) = wn((1 + 2ϵ)t), for t∈[−1,1]. Then, using (Theorem 7.2, [25]), we have

|y(n)(t)−(1 + 2ϵ)n˜pn(t)|≤|y(n)(t)−PNy(n)(t)|+|PN(y(n)−(1 + 2ϵ)n˜wn)(t)|

≤4||y(k+1)||

π(k−n)(N−k+n)k−n+ ΛNsup

|t|≤1|y(n)(t)−(1 + 2ϵ)n˜wn(t)|.

We have ||y(k+1)|| ≤ (1 + 2ϵ)k+1 V. As for the other diﬀerence, let s= (1 + 2ϵ)tfor t∈[−1,1].

Then

|y(n)(t)−(1 + 2ϵ)n˜wn(t)|= (1 + 2ϵ)n|(Eku)(n)(s)−wn(s)| ≤ (1 + 2ϵ)n

rn,

k−n

X

j=0

rj+nϵj

j!

20

for all |s| ≤ 1+2ϵ. This follows from the Lipschitz-like bound (64), the inclusions (62), and

the deﬁnition of polynomials pL, pRand pL, pR. We get the desired result by combining these

two bounds with the previous one for |y(n)(t)−(1 + 2ϵ)n˜pn(t)|and dividing both sides by

(1 + 2ϵ)n.

Remark 7. Similar to Remark 6, the reason for introducing the Lipschitz-like constant rn

in (64) is to allow for some amount of control of the wrapping eﬀect. We want to avoid

interpolating an interval-valued function as much as possible. Further discussion will appear

in Section 4.2.

By (65), we have the enclosure

dn

dtnEku(t)∈pn(t) + en(t),|t| ≤ 1+2ϵ, n = 0,...,k−1

Remark 8. When n= 0 and ϵ= 0 in (65), we see that Theorem 14 directly generalizes Lemma

12. Indeed, in that case, e0(t) = e0for |t| ≤ 1. Also, we see that only the Lipschitz-like bounds

r0,...,rnare needed to enclose the derivative u(n)(t)for |t| ≤ 1.

4.1.3 Part 3: Hybrid enclosures

It can be challenging to compute tight Lipschitz-like constants rnneeded for Theorem 14.

Indeed, directly evaluating the left-hand side of (64) generally results in a poor bound, while

using the mean-value inequality requires explicitly computing derivatives. This can quickly

become technical; see Section 5 for an example. It is therefore of interest to avoid these

calculations when possible. To this end, we can recycle our inclusions (62) from the previous

section to deﬁne hybrid enclosures.

Let a Ckmutation ξof ϕbe ﬁxed. Let p0and e0be computed according to Theorem 14,

with ϵ= 0, so that Eku(t)∈p0(t) + e0[−1,1] for |t| ≤ 1. See Remark 8. Then, deﬁne the

sequence of hybrid enclosures

hn(t) =

p0(t) + e0[−1,1], n = 0,|t| ≤ 1

˜

fn(t, p0(t) + e0[−1,1],ˆ

δ), n = 1,...,k,|t| ≤ 1

Pk−n

j=0 1

j!˜

fj+n(−1, p0(−1) + e0[−1,1],ˆ

δ)(t+ 1)j, n = 1,...,k, t < −1

Pk−n

j=0 1

j!˜

fj+n(+1, p0(+1) + e0[−1,1],ˆ

δ)(t−1)j, n = 1,...,k, t > 1

It is then straightforward to prove that dn

dtn(Eku)(t)∈hn(t) for t∈Rand n= 0,...,k.

4.2 Discussion on the interpolation of piecewise polynomials

In Section 4.1, we introduced three strategies to compute enclosures of a solution uof the

implicit method of steps, together with its derivatives. These all require one to make a choice

of mutation for the initial condition ϕ. It might be unclear at this point which strategy should

be used. We ﬁnd it is much more instructive to explain this with an example, so we will

postpone this to Section 5. However, we can make some general comments.

The polynomial interpolants pnof (61), and pnof Theorem 14, are not themselves inter-

polants of diﬀerentiable functions. This can result in a slow decay of the coeﬃcients of the

interpolant, since the function being interpolated could even fail to be continuous at some

of the mesh points in ∆m. Despite this fact, the theoretical enclosures aﬀorded by (61) and

Theorem 14 can be rather tight. However, the slow decay of the coeﬃcients can result in bad

wrapping interval wrapping when only 64-bit ﬂoating point arithmetic is used. This is the pri-

mary reason we have opted to use Julia for the implementation, since the interval arithmetic

21

package IntervalArithmetic.jl, a package of JuliaIntervals [1], supports the BigFloat type for

extended precision arithmetic.

4.3 Mutations, multiple steps, and the out-of-bounds problem

The point of introducing mutations is that they can used to connect one round of the implicit

method of steps to another, with rigorous error propagation. They also provide a formal

mechanism for handling the out-of-bounds evaluation problem. We will outline one way this

can be done here. In what follows, Ekdenotes the Taylor extension.

Suppose without loss of generality that at the ﬁrst implicit step, the initial condition ϕ

is an interval-valued polynomial deﬁned on the real line, or can be theoretically extended to

an arbitrary compact subset of the real line. We formally write ˜

ϕ=Ekϕ. The ﬁrst solution,

denoted u, is then proven using the radii polynomials, and we check the monotonicity of the

associated lag argument. We now store three representations of u.

A polynomial interpolant of the solution and its ﬁrst derivative on [0,1], with rigorous

error bounds for both. Denote these by ˜u(j)for j= 0,1.

Hybrid enclosures of its derivatives up to order k. Denote these by (Hu)(j)for j=

0,...,k.

The restrictions of the hybrid enclosures (Hu)(j)to [0,1], extended to the real line by

polynomial evaluation. Denote these by by (Hu)(j).

Note that the latter representation exists because ϕand ˜u(0) are (interval) polynomial. To

compute ˜u(j), we use Theorem 14 with ϵ= 0, computing Vusing the fact that ˜

ϕis a mutation

of ϕ. Being (interval) polynomials, ˜u(j)can be stored on the computer. As the the Hybrid

enclosures, they can be stored using a copy of ˜u(0) and evaluations of the the bootstrapped

derivatives up to order kat the time arguments t∈ {0,1}. The third class, Hu, are equivalent

to (interval) polynomials and are therefore representable as interval vectors.

For the second (and subsequent) implicit step, we now think of ϕ=uprevious step. There-

fore, Ekϕadmits two “computable” mutations.

AC1polynomial mutation, deﬁned by aﬃnely shifting the domain of ˜u(j)(from the

previous step) for j= 0,1 to the interval [−1,0], and extending to the real line by

polynomial evaluation.

ACkmutation, deﬁned by aﬃnely shifting the domain of Hu (from the previous step).

Let it be denoted Hϕ.

ACkmutation, deﬁned by aﬃnely shifting the domain of Hϕ(from the previous step)

The ﬁrst of these mutations will have far tighter error bounds (for most interval-based cal-

culations) than the second, and is therefore well-suited to computations of the Y,Z0and Z1

bounds, since these only require access to the derivatives of Ekϕon the domain [−1,0]. See

Section 4.5 for some related discussion. The second mutation in fact encloses Ekϕand its

derivatives up to order k, and can therefore be used in computations of the Y∞,Z2and Z∞

bound.

Having proven the new solution – which we also denote by u– we proceed to the computable

representations of u. We generate the same three classes as in the previous step, with some

changes. To compute ˜u(j), we use the fact that Hϕdeﬁnes a mutation of ϕto compute V,

and use Theorem 14 with ϵ= 0. For the hybrid enclosures Hu, we store a copy of ˜u(0) and

compute the bootstrapped derivatives as before.

The setup we have presented above results in a lot of recursion. Indeed, each hybrid enclo-

sure depends on all previously computed hybrid enclosures. This results in heavy computation

22

overhead, and at present, we do not know if there is a way to avoid it. Also, although Hϕis

a polynomial, it is itself a nested composition of every previous iterate, and can therefore be

extremely high-degree.

4.4 On the generality of polynomial vector ﬁelds

At the beginning of Section 4, we introduced the assumption that the vector ﬁeld fand lag

function τare polynomial. This was done primarily to allow use to compute any integrals

reliably without the use of an approximate quadrature rule. When non-polynomial terms

in the vector ﬁeld and delay are themselves solutions of ordinary diﬀerential equations, it is

sometimes possible to construct a [16]. To illustrate the point, we will brieﬂy present two

such examples now, and how the machinery of the present paper can be adapted to handle

non-polynomial nonlinearities. However, we have not implemented the rigorous integrator for

these examples.

4.4.1 Ikeda equation with state-dependent delay

Consider an initial-value problem for a state-dependent delayed perturbation of the Ikeda

equation:

˙x(t) = −ax(t) + bsin(x(t−c−px(t))), x0=ϕ

for constants a, b, c, p. Deﬁne y(t) = sin(x(t)) and z(t) = cos(x(t)). Then the triple (x, y, z)

solves the initial-value problem

˙x(t) = −ax(t) + by(t−c−px(t)), x0=ϕ

˙y(t) = z(t)(−ax(t) + by(t−c−px(t))), y0= sin(ϕ)

˙z(t) = −y(t)(−ax(t) + by(t−c−px(t))), z0= cos(ϕ),

and vice-versa. The vector ﬁeld and delay function are now polynomials.

4.4.2 Cubic Ikeda equation with state-dependent delay

Consider a state-dependent delay perturbation of the cubic Ikeda equation:

˙x(t) = −x(t−cepx(t)) + x(t−cepx(t))3, x0=ϕ,

for constants c, p. Deﬁne y(t) = epx(t). Then (x, y) solves the initial-value problem

˙x(t) = −x(t−cy(t)) + x(t−cy(t))3, x0=ϕ

˙y(t) = py(t)(−x(t−cy(t)) + x(t−cy(t))3), y0=epϕ ,

and vice-versa. The vector ﬁeld and delay function are now polynomials.

4.5 Centering of numerical data for the ﬁnite bounds

In Section 4.3, we discussed using a bootstrapped enclosure with ϵ= 0 for the initial data ϕ

to compute the Y,Z0and Z1bounds. This, however, can only be done if t7→ h(tδ, u(t)) is

contained in [−1,0] for t∈[0,1]. For a typical candidate numerical solution (u, δ), this will

not be the case. However, if (u, δ) is a good numerical approximation, it might be possible to

perform near-trivial modiﬁcations to the numerical data uthat results in the inclusion being

achieved.

23

Let us try to ﬁnd small ϵ1, ϵ2such that the modiﬁed data (v, ∆) = ((1 + ϵ1)u, (1 + ϵ2)δ)

satisﬁes h(t∆, v(t)) ∈[−1,0] for t∈[0,1]. If (u, δ) is a good numerical approximation, we

should have h(0, u(0)) ≈ −1 and h(δ, u(1)) ≈0, but they may be outside of [−1,0]. Therefore,

let η1, η2>0 be small, and solve the equations

h(0,(1 + ϵ1)u(0)) = −1 + η1(66)

h((1 + ϵ2)δ, (1 + ϵ1)u(1)) = −η2(67)

for (ϵ1, ϵ2) using Newton’s method, initialized at (0,0). If t7→ h(tδ, u(t)) is strictly monotone

and (ϵ1, ϵ2) is small, then the time lag associated to the modiﬁed data should have the same

property, and therefore be contained in the interval [−1 + η1,−η2]⊂[−1,0].

It is possible that (66)–(67) does not have a solution for any η1, η2>0. As a contrived

example, take uscalar-valued, h(t, u) = t−1−u, and numerical data such that u(0) = 0. Then

(66) has a solution if and only if η1= 0. In such a case, and in general, it might be necessary

to consider more sophisticated modiﬁcations to u. For example, if uis stored as a vector

in Rmd(k+1)+1 (see Section 3.3), then one could try near-unit scaling of suitable components

individually.

5 Example

This section concerns an application of our validated integrator to the following initial-value

problem:

˙u(t) = −γu(t)−κu(t−α−cu(t)), x0=ϕ(68)

with γ, κ, α, c all being non-negative. In particular, we will require κ, c positive so as not to

trivialize the state-dependent delay. A similar equation with two delays was studied in [5],

where Hopf bifurcations and periodic orbits were analyzed using delay expansions and normal

forms. In particular, (68) is a restricted version of that equation. We have implemented our

rigorous integrator for this equation using a few initial functions ϕand sets of parameters.

Despite being similar to a linear delay diﬀerential equation, the composition of u(t) with the

lag t−α−cu(t) can result in nonlinear phenomena including attracting periodic orbits.

5.1 Set-up

For (68), we have ˜

f(t, u, δ) = (−γu −κϕ(tδ −α−cu))δ.

We computed the bootstrapped vector ﬁelds ˜

fkfor k= 1,...,8 using symbolic algebra. We will

not list all of them here, since the higher derivatives are quite lengthy. They can be viewed

in our Julia implementation; see the functions Dx,D2x,...,D8xwithin vector fields.jl.

However, to illustrate the computation, we will at least calculate ˜

f2explicitly. Recall that

˜

f1=˜

f.

˜

f2(t, u, δ) = D1˜

f1+ (D2˜

f1)˜

f

= (−κ˙

ϕ(tδ −α−cu)δ)δ+ (−γ−κ˙

ϕ(tδ −α−cu)(−c))δ˜

f1

=−κ˙

ϕ(tδ −α−cu)+(−γ+cκ ˙

ϕ(tδ −α−cu))(−γu −κϕ(tδ −α−cu))δ2

We follow Section 4.3 to handle out-of-bounds evaluations and multiple steps of integration.

This requires the Lipschitz-like bound r1of Theorem 14 to be computed analytically. Using

24

the mean-value inequality, one can choose any

r1≥sup

t,η1,η2

η1(|δ|+η2)| − γ+κc ˙

ϕ|+η2|γ(u(t) + η1) + κϕ|+|(δ+η2)κt ˙

ϕ|,

where the supremum is over t∈[0,1], |η1| ≤ r(Λk+r∞) and |η2| ≤ rr0, and with an abuse of

notation (to keep the expression compact), we write ϕ=Ekϕ(t(δ+η2)−α−c(u(t) + η1)) and

similarly, ˙

ϕ= (Ekϕ)(1)(t(δ+η2)−α−c(u(t) + η1)).

5.2 Model parameter and initial conditions

We have elected to present proofs for three sets of parameters.

P.1 (γ , κ, α, c) = (1,2,1/2,1): The equilibrium at x= 0 is locally attracting.

P.2 (γ , κ, α, c) = (1/3,2,1,1): There is a nontrivial periodic orbit, and the equilibrium at

x= 0 is unstable.

P.3 (γ , κ, α, c) = (1/3,2,1,2): Same as the previous case, with larger cparameter.

The ﬁrst set of parameters results in the easiest proofs, and the ﬁnal set the hardest.

We consider two classes of initial conditions. For the simplest example, we have the constant

initial condition ϕ0(θ) = 1

4. For our second initial condition, we select a degree two polynomial

ϕ1(θ) = η0+η1θ+η2θ2with the following properties.

ϕ1is on the solution manifold; see [29].

The lag at time t= 0 is equal to −1.

The derivative of the time lag is equal to 2 at time t= 0.

In order, these three conditions result in a system of three algebraic equations:

η1=−γη0−κ(η0+η1(−α−cη0) + η2(−α−cη0)2)

−1 = −α−cη0

2=1−cη1,

which can be solved directly, yielding the unique polynomial

ϕ1(θ) = 1−α

c−1

cθ+1+(α−1)γ+ (α−2)κ

κc θ2.(69)

Being on the solution manifold, the solution of the initial-value problem from ϕ1will be

globally C1-smooth.

5.3 Proof parameters and implementation details

Our initial conditions will be polynomials, but we formally treat them as elements of Ck([−1,0],R)

for k= 8. The piecewise polynomial interpolation order k, number of subdivisions m, and

the weight r∞were chosen at each step to ensure proofs that resulted in near-optimal C0

enclosures, but we did not make a serious, systematic eﬀort to also optimize them for memory

usage. In all cases, the subdivision mesh was equally-spaced, and we used r∗= 10−4and

r0= 1 unless explicitly noted. After each proof, the solution uwas interpolated at an order

Nthat ensured the contribution of ΛNr0in (65) was no smaller than the other term of that

equation. Succinctly, we chose Nsuch that the interpolation error was no worse than the

wrapping eﬀect.

25

In all cases, we completed four steps of integration. The time required to complete a

proof obviously depends on the interpolation parameters kand m, but is also exponential

in the number of proofs due to the nested calls to previously computed solutions. This is

main reason we have not attempted to prove more than four implicit steps. To illustrate the

point, consider that on our machine, the ﬁrst three implicit steps of the proof with parameters

P.3 and initial condition ϕ0=1

4took a total of ﬁfteen minutes, while the proof of the ﬁnal

implicit step required almost an hour. Figure 3 includes plots of the solutions from the two

initial conditions ϕ0and ϕ1, for the three parameter sets. The technical results are presented

in Section 5.4 and Section 5.5.

Figure 3: Solutions of the initial-value

problem (68) plotted for four implicit

steps. Top row: parameter sets P.1 and

P.2. Bottom row: parameter set P.3. The

line styles indicate the initial condition of

the associated plot, be it ϕ0or ϕ1. Dots in-

dicate terminal integration points for each

of the four implicit steps. Note that solu-

tions from ϕ1are globally C1, since ϕ1is

on the solution manifold.

All proofs except for the ﬁrst step required 512-bit ﬂoating point precision. Anything less

resulted in massive error propagation during the computation of the Ybound in the second

and subsequent steps. We suspect this is partly due to our implementation of the Ybound,

which requires computing compositions of high-order polynomials and reliably computing their

coeﬃcients in the Chebyshev basis. As discussed at the beginning of Section 4, we suspect some

improvements can be made in this direction. Our implementation computes several objects

(e.g. Am,k) with lower precision when possible, always with appropriate directed rounding

when we move from one level of precision to another.

The function main from our Julia implementation [6] takes as its input a data ﬁle (all of

which are included) consisting of pre-set proof parameters and numerically computed solutions,

and does the following for each implicit step:

reﬁne the numerical solution with Newton’s method;

center the numerical solution (see Section 4.5);

26