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Rigorous continuation of periodic solutions for

impulsive delay diﬀerential equations

Kevin E. M. Church1and Gabriel W. Duchesne2

1Department of Mathematics and Statistics, McGill University

2Department of Mathematics and Statistics, McGill University

October 1, 2020

Abstract

We develop a rigorous numerical method for periodic solutions of impulsive

delay diﬀerential equations, as well as parameterized branches of periodic solutions.

We are able to compute approximate periodic solutions to high precision and with

computer-assisted proof, verify that these approximate solutions are close to true

solutions with explicitly computable error bounds. As an application, we prove the

existence of a global branch of periodic solutions in the pulse-harvested Hutchinson

equation, connecting the state at carrying capacity in the absence of harvesting to

the transcritical bifurcation at the extinction steady state.

1 Introduction

Numerical continuation is an old and ever-important topic in computational math-

ematics. Suppose we have a nonlinear equation

f(x, α) = 0,

and given some solution x∈Xfor Xa Banach space, we want to continue the

solution with respect ot the parameter α. Unless Xis ﬁnite-dimensional, some ﬁnite-

dimensional projection must be made. Once this is done, continuation can proceed

using typical predictor-corrected methods, but there then remain questions about

how ﬁne the projection must be. Such questions can be answered using so-called

rigorous continuation approaches. These ideas have been applied successfully to the

continuation of steady states of partial diﬀerential equations [8, 9, 10], invariant

manifolds [12], and periodic solutions of ordinary [3, 22], and delay [13] diﬀerential

equations among others.

Impulsive dynamical systems are characterized by a combination of continuous-

time dynamics and moments of discontinuity in state triggered by a spatiotemporal

relationship. The simplest such spatiotemporal relationship is one in which the

discontinuities – referred to as impulses – occur at ﬁxed times. The result is that

such systems are nonautonomous, and so steady states are relatively rare (in a

geometric sense) while the simplest invariant sets one can use as organizing centers

for global dynamics are the bounded trajectories. When the times at which impulses

occur are periodic, one can consider periodic solutions as these simple objects.

Bifurcation theory for periodic solutions of impulsive functional diﬀerential equa-

tions has undergone some new developments in recent years [5, 7], but such results

1

are only useful if one has computed a periodic solution to begin with. This is a

main motivation for considering here the problem of computation and continuation

of periodic solutions for impulsive delay diﬀerential equations. Our main contribu-

tion is a numerical method to compute such periodic solutions, continue them, and

rigorously prove their existence using the assistance of the computer.

Methods for computer-assisted proofs of stability for linear impulsive delay dif-

ferential equations has recently been accomplished [7] using Chebyshev spectral

collocation techniques, and the approach we take here shares some similarities. The

idea is as follows. If the period of impulse eﬀect and the discrete delay are com-

mensurate, we show that any periodic solution will be piecewise C∞. If the vector

ﬁeld is analytic, they will be piecewise-analytic. We can then show that computing

a periodic solution is equivalent to computing the solution of a higher-dimensional

boundary-value problem, where the number of extra dimensions is related to the

ratio between the delay and the periodic of impulse eﬀect. The great thing is that

this boundary-value problem can be expressed as an ordinary diﬀerential equations

without delays or impulses. To rigorously compute solutions, we then exploit the

analyticity of any such solution and expand it in uniformly convergent Chebyshev

series. This allows conversion between the problem of computing a solution of a

boundary-value problem into one of computing a zero of an inﬁnite-dimensional

nonlinear map in a sequence space. By truncating the number of modes in the

Chebyshev series, we obtain a ﬁnite-dimensional zero-ﬁnding problem. We then

apply numerical methods for the computation and continuation of such zeroes. Rig-

orous numerics can then be used to prove that such numerical zeroes (respectively,

branches of zeroes) are proximal to true zeroes (respectively, branches). The error

between the numerical and true solution can be rigorously computed.

Converting problems in nonlinear dynamics into zero-ﬁnding problems in se-

quence spaces is not a new idea. In the context of sequence spaces representing

Chebyshev series coeﬃcients, see [2, 7, 15, 20] for a few recent applications. One soli-

tary application of Chebyshev expansions in nonlinear impulsive dynamical systems

we could ﬁnd appears in [24], where it was used to generate a simpliﬁed approxima-

tion of an optimal control problem involving impulsive integrodiﬀerential equations.

The idea was that the approximate problems could be solved using existing soft-

ware. However, sequence space concepts are not used there, so we are comfortable

in asserting that the present paper is the ﬁrst such application of Chebyshev series

that is truly instrumental to solving a problem in nonlinear impulsive dynamical

systems.

We should remark especially that periodic solutions of delay equations have been

computed using a Chebyshev integrator in [15], with the period being an integer

multiple of the delay. There, the idea is based on an implicit formulation of the

integral form of the step map. The advantage of that formulation is that one

can rigorously integrate from arbitrary initial data. To contrast, here we do not

explicitly integrate the delay diﬀerential equation; rather, we carefully determine

how periodic solutions of the impulsive delay diﬀerential equations are generated by

ordinary diﬀerential equations with unﬁxed boundary conditions. The result is that

the zero-ﬁnding problem we get for periodic solutions is comparatively simpler, it is

easier to derive bounds for the computer-assisted proofs, and we can ﬁnd periodic

solutions with non-integer multiples of the delay. However, we can not integrate

from arbitrary initial data and can only ﬁnd periodic solutions.

It is perhaps initially surprising that we do not expand into Fourier series, since

our stated objective is to compute and continue periodic solutions. Fourier expan-

sions works very well for wholly continuous problems, and there are numerous such

applications in rigorous numerics [3, 13, 17, 20, 22]. However, the Fourier basis

is ill-suited for approximating functions with discontinuities. Indeed, the Fourier

series associated to a discontinuous function does not even converge pointwise and

2

the coeﬃcients decay at best linearly in the frequency. In contrast, with our setup

using piecewise Chebyshev expansions, we can represent solutions with respect to

a Schauder basis were we have geometric decay of the coeﬃcients and uniform con-

vergence of the series. The cost is that we need to increase the dimension of the

problem.

As a test case, we apply our methods to periodic orbits in the pulse-harvested

Hutchinson equation

˙x=rx(t)1−x(t−τ)

K, t /∈TZ

∆x=−hx(t−), t ∈TZ.

Several authors have studied the Hutchinson equation with impulse eﬀect [19, 25,

23], but very strong assumptions were needed to ensure existence of positive periodic

(or almost periodic) solutions. For example, with an impulse eﬀect of the form

∆x=bkxat times t=tk, it was assumed in the previously cited publications that

the function t7→ Q0<tk<t(1 + bk) is periodic or almost periodic. This is a strong

assumption, since it is then strictly necessary for the sequence bkto switch signs

inﬁnitely many times. The displayed equation above does not satisfy this condition

since we have bk=−hfor h∈(0,1), which means bk<0 for all k∈Z.

The proofs of existence of periodic/almost periodic solutions in [19, 25, 23] make

use of ﬁxed-point theory. The results are mathematically elegant because very little

is assumed apart from the sign-switching condition on the sequence bk. However,

the results are not explicit as the solutions are not explicitly constructed, and they

are not general because of this sign-switching requirement. To contrast, we here

use our rigorous numerical method to prove there is a global branch of positive

periodic solutions that exists for h∈[0, h∗], where h∗= 1 −e−rT corresponds to a

transcritical bifurcation with the steady state solution x= 0. The codes necessary

to complete the computer-assisted proofs can be found at [4].

The outline of the paper is as follows. In Section 2 we formulate the equivalent

boundary-value problem that can be used to compute periodic solutions. We de-

velop the rigorous numerical method in Section 3. Our application to the impulsive

Hutchinson equation appears in Section 4. We conclude with Section 5.

2 From periodic solutions to boundary-value

problems

The class of dynamical systems we consider in this paper are impulsive delay diﬀer-

ential equations with a single discrete delay:

˙x=f0(x(t), x(t−τ), β), t /∈TZ

∆x=g(x(t−), x(t−τ), β), t ∈TZ.

Here, β∈Ris a parameter. We will often require f0and gto be real-analytic and

that the delay τ > 0 and the period of impulse eﬀect T > 0 are commensurate

– that is, p:= T

τis rational. Under this condition, we can perform a change of

independent variable so that the delay becomes unity and the period of impulse

eﬀect is p. Speciﬁcally, we set t=τ˜

tfor a new time variable ˜

t. Completing the

change of variables and dropping the tilde, we get

˙x=f(x(t), x(t−1), β), t /∈pZ(1)

∆x=g(x(t−), x(t−1), β), t ∈pZ,(2)

with f=τf0. We will from this point work with (1)–(2). The previous hypotheses

are then equivalent to.

3

H.1 f:Rd×Rd×R→Rdand g:Rd×Rd×R→Rdare analytic.

H.2 p=p1

p2is rational, with p1and p2coprime and positive.

We will make it clear when these hypotheses are assumed. Note that it is not strictly

necessary to have p1and p2coprime, but our numerical method is more eﬃcient

when p1is small. It is therefore beneﬁcial to have the rational pexpressed in lowest

form.

2.1 Periodic solutions

The impulsive delay diﬀerential equation (1)–(2) is periodically forced, since every

ptime units there is a forced discontinuity in the state variable from the impulse

eﬀect. The ﬁrst result we prove is that except in very special (i.e. degenerate)

situations, periods of periodic solutions are constrained to be integer multiples of

the forcing period.

Proposition 1. If ψis a periodic solution of (1)–(2) with period P, then exactly

one of the following must occur.

•P=mp for some m∈N,

•P /∈pNand ψis continuous; in particular, g(ψ(kp−), ψ (kp −1)) = 0 for all

k∈Z.

Proof. Suppose P /∈pN. Since ψis a solution it satisﬁes (2) at time t=kp for

k∈Z, so

ψ(kp) = ψ(kp−) + g(ψ(kp−), ψ(kp −1)).

By periodicity, ψ(kp) = ψ(kp +P) and ψ(kp−) = ψ((kp +P)−). Since P /∈pN, we

have kp +P /∈pZ, so ψis continuous at kp +Pand therefore ψ(kp +P) = ψ((kp +

P)−). Combining the previous two observations on periodicity and continuity, it

follows that ψ(kp) = ψ(kp−), which means that ψis continuous at t=kp. As k∈Z

was arbitrary and the discontinuities of ψare a subset of pZ, we conclude that ψis

continuous. This also implies g(ψ(kp−), ψ(kp −1)) = 0.

Proposition 1 indicates that if we are interested in discontinuous periodic solu-

tions – that is, solutions of (1)–(2) that explicitly do not solve the delay diﬀerential

equation (1) in isolation – we need only concern ourselves with solutions whose

period is an integer multiple of p.

2.2 Splitting into solution segments

To make the process of ﬁnding such periodic solutions more concrete (although

perhaps a bit indirect), we make an argument that can be considered a twist on

the classical method of steps for delay diﬀerential equations. Let ψbe a periodic

solution of (1)–(2) with period mp. For t∈[0, mp) we can write

ψ(t) =

m−1

X

k=0

p1−1

X

q=0

1kp+q

p2,kp+q+1

p2(t)zk,q t−kp −q

p2(3)

where we deﬁne zk,q :h0,1

p2i→Rdby zk,q(t) = ψt+kp +q

p2for t < 1

p2, and

extend to the closure by continuity. In this way, a periodic solution ψis uniquely

associated to the family {zk,q :k= 0,...,m −1, q = 0,...,p1−1}of continuous

functions. We will call these functions the solution segments. See Figure 1 for a

visualization.

4

Figure 1: To visualize the segments, we can instead deﬁne ˜zk,q (t) = ψ(t) for kp +q

p2≤

t < kp +q+1

p2. Then, the restriction of ψto such intervals of tcorrespond precisely to the

˜zk,q . Above is a visual depiction of these shifted segments; the ones in blue correspond

to k= 0 and the ones in red to k= 1. The vertical lines are used to delineate the

boundaries of the domains. Since the boundary between ˜z0,p1−1and ˜z1,0corresponds to

an impulse time, namely t=p, we have made the curve discontinuous there. We obtain

the segments by pulling the domains of the ˜zk,q to the interval [0,1

p2) by translation and

taking a continuous closure at the endpoint.

2.3 Ordinary diﬀerential equations for the solution seg-

ments

The solution segments zk,q themselves satisfy diﬀerential equations. To establish

this correspondence, we will need the following construction. See Figure 2 for a

more intuitive diagram.

Deﬁnition 1. For (k, q)∈ {0,...,m−1} × {0,...,p1−1}, the delay shift is the

unique pair η(k, q) = (η1, η2)∈N×Nwith 0≤η1< m and 0≤η2< p1such that

kp −1 + q

p2

∈η1p+η2

p2

+mpZ.(4)

The delay shift is indeed well-deﬁned. Also, if t∈hkp +q

p2, kp +q+1

p2,

t−1∈kp −1 + q

p2

, kp −1 + q

p2

+1

p2⊂η1p+η2

p2

, η1p+η2+ 1

p2+mpZ.

Since ψis periodic with period mp, this means that if we denote zη(k,q)=zη1,η2for

η(k, q) = (η1, η2), then

˙

ψ(t) = f(ψ(t), ψ(t−1), β ) = fψ(t), ψ η1p+η2

p2

+s, β

where s=t−kp −q

p2∈[0,1

p2). But ψη1p+η2

p2+s=zη(k,q)(s) by (3), while

ψ(t) = zk,q (s). It follows that the functions zk,q satisfy the ordinary diﬀerential

equation

˙zk,q =f(zk,q , zη(k,q), β ).(5)

Remark 1. We previously said that what we do here is a “twist” on the method of

steps for delay diﬀerential equations. Indeed, the method of steps exploits the fact

that in solving an initial-value problem

˙y=F(y(t), y(t−1)), y0(θ) = h(θ)

5

Figure 2: The delay shift deﬁnes a bijection from the set of solution segments to itself. The

segments zk,q are identiﬁed by their indices (k, q), and the delay shift η(k, q) corresponds

to the segment that would need to be evaluated if we wanted to compute ψ(t−1) for

(k+mN)p+q

p2≤t < (k+mN)p+q+1

p2and arbitrary N∈Z, in the sense of the

representation (3) for ψ. In the ﬁgure above, we have p=2

7=p1

p2and m= 2, so

the periodic solution has period mp =4

7. The segments are identiﬁed with the indices

(0,0), (0,1), (1,0) and (1,1). Since 1 = 7

7, shifting back by one time unit results in

“wrapping backwards” by seven solution segments, each of length 1

7. The result is that

η(0,1) = (1,0). This “wrapping backwards” is illustrated by the arrow diagram above

the ﬁgure. The vertical lines delineate boundaries between solution segments, and the

colours and line styles are formally analogous to what we have in Figure 1.

for some initial function h: [−1,0] →Rd, one can compute y(t)for t∈[0,1] by

solving the nonautonomous ordinary diﬀerential equation

˙y=F(y(t), h(t−1))

subject to the initial condition y(0) = h(0). The idea is that the solution history (in

this case the function h) informs the future evolution and allows one to temporarily

forget that one is solving a delay diﬀerential equation. We do a very similar thing

here, this time exploiting that our history is can be identiﬁed with some solution

segment after applying the correct delay shift.

2.4 Boundary condtions for the solution segments

The solution segments zk,q also satisfy some boundary conditions. First, since ψ(t)

is continuous whenever t /∈pZ, the segments must satisfy

zk,q+1 (0) = zk,q 1

p2(6)

whenever 0 ≤q < p1−1. Indeed, one can verify from (3) that in this case,

lim

s→0−

ψkp +q+ 1

p2

+s=zk,q 1

p2

ψkp +q+ 1

p2=zk,q+1 (0),

6

and requiring continuity forces (6). On the other hand, at impulse times, (2) implies

ψ(kp) = ψ((kp)−) + g(ψ(kp−), ψ(kp −1), β )

=ψ((kp)−) + g(ψ((kp)−), ψ (η1p+η2/p2), β)

=ψ((kp)−) + g(ψ((kp)−), zη(k,0) (0), β)

where η=η(k, 0). Since ψ(kp) = zk,0(0), we can write

zk,0(0) = ψ((kp)−) + g(ψ(kp−), zη(k ,0)(0), β ).

To proceed further, we need to express ψ(kp−) in terms of the solution segments.

This can be done as follows:

ψ(kp−) =

zk−1,p1−11

p2, k > 0

zn−1,p1−11

p2, k = 0.

We can, however, simplify this formula if we introduce the convention z−1,q =

zm−1,q. Doing this, we can fully express the impulse condition (2) at the level of

the solution segments by

zk,0(0) = zk−1,p1−11

p2+gzk−1,p1−11

p2, zη(k,0)(0), β .(7)

2.5 The boundary-value problem

Combining (5), (6) and (7), we can write down the boundary-value problem for the

solution segments. It is given by

˙zk,q =f(zk,q , zη(k,q), β ),(8)

zk,q+1 (0) =

zk,q 1

p2, q < p1−1

zk−1,q 1

p2+gzk−1,q 1

p2, zη(k,0)(0), β , q =p1−1,(9)

where we have introduced a similar convention to z−1,q =zm−1,q from the previous

section: we deﬁne zk,p1≡zk,0so that the case q=p1−1 has a sensible meaning.

Since fis analytic, we may conclude by the Cauchy-Kovalevskaya theorem that

Lemma 2. Assume H.1 and H.2. Every solution of the boundary-value problem

(8)–(9) is analytic.

Conversely, given a solution of the boundary-value problem, one can reverse the

argument from Section 2.2 and obtain a periodic solution ψof the impulsive delay

diﬀerential equation (1)–(2) by extending (3) to a periodic function on R. The

argument is straightforward, and the following lemma is therefore proven.

Lemma 3. Assume H.1 and H.2. Every periodic solution of (1)–(2) of period mp

for m∈Nis uniquely associated with an analytic solution z: [0,1/p2]→(Rd)mp1

of the boundary-value problem (8)–(9).

Remark 2. The construction we have completed in this section crucially relies on

the period pbeing rational. In fact, if the period is not rational then not only is

our solution segment idea no longer applicable, but periodic solutions can be highly

irregular. See for example Proposition 2.3.1 of [7].

As a ﬁnal bit of preparation for what is to come, we will perform a change

of variables so that the domain of the solution zof the bondary-value problem

becomes [−1,1]. The reason for this is because we will eventually expand solutions

of the BVP as Chebyshev series, and these have very nice convergence properties

on [−1,1].

7

Corollary 4. Assume H.1 and H.2. With the reparameterization of time t7→ t+1

2p2,

solutions zk,q of the boundary-value problem (8)–(9) are transformed to solutions

φk,q : [−1,1] →Rdof

˙

φk,q =1

2p2

f(φk,q , φη(k,q), β ),(10)

φk,q+1 (−1) = φk,q (0), q < p1−1

φk−1,q(0) + gφk−1,q (0), φη(k ,0)(−1), β , q =p1−1,(11)

3 Rigorous numerics setup

In this section we review some basics of Chebyshev series before converting the

boundary-value problem (10)–(11) into the problem of ﬁnding a zero of a function

Fon a weighted sequence space. We then show how Fcan be discretized so that

approximate solutions can be found with Newton’s method. Then, we state the

general-purpose a-posteriori solution validation tool that can be used to validate

branches of F= 0. We will then explain some subtleties of the method as it applies

to the BVP (10)–(11) before proceeding to an example.

3.1 Preliminaries on Chebyshev series

In what follows, sequences of vectors in Rdwill be indicated with curly braces:

{a}n∈N. The Chebyshev polynomials are the polynomials Tnfor n∈Ndeﬁned by

the recursion

Tn+1(x) = 2xTn(x)−Tn−1(x), n ≥1

T1(x) = x,

T0(x) = 1.

If h: [−1,1] →Rdis analytic, then it can be uniquely extended to a Chebyshev

series

h(t) = h0+ 2 X

n≥1

hnTn(t) (12)

that is uniformly convergent on the Bernstein ν-ellipse (in the complex plane) for

some ν > 1 [21], for some sequence {h}n∈N. Moreover, there exists some ν > 1 such

that the quantity

||h||ν=

∞

X

n=1

|hn|ωn(13)

is ﬁnite, where the sequence of weights ωis deﬁned by ω0= 1 and ωn= 2νnfor

n≥1, and | · | is any norm on Rd. The symbol 1

νwill denote the normed vector

space of all sequences {h}n∈Nfor which the norm || · ||νis ﬁnite. This is a Banach

space. We will sometimes write 1

ν(Rd) when we want to emphasize the dimension

of terms of the sequence.

Let {a}n∈Nand {b}n∈Nbe the coeﬃcients of two Chebyshev series. The product

of those series can be written such that

a0+ 2 X

n≥1

anTn(t)

b0+ 2 X

n≥1

bnTn(t)

=a0b0+ 2 X

n≥1

(a∗b)nTn(t),(14)

where we deﬁne the convolution by

(a∗b)n

def

=X

k1+k2=n

ak1bk2.

8

Likewise, the convolution operator deﬁnes a bilinear map on 1

νand one can show

that ||(a∗b)||ν≤ ||a||ν||b||ν, so (1

ν,∗) is in fact a Banach algebra.

Let h(t) be an analytic function that can be extended to a Chebyshev series such

that

h(t) = a0+ 2 X

n≥1

anTn(t),

h0(t) = b0+ 2 X

n≥1

bnTn(t).

Note that for n≥2

ZTn(t)dt =1

2Tn+1(t)

n+ 1 −Tn−1(t)

n−1.

Then,

h(t) = a0+ 2 X

n≥1

anTn(t) = Zh0(t)dt

=Zb0+ 2 X

n≥1

bnTn(t)dt

=Zb0+ 2b1t+ 2 X

n≥2

bnTn(t)dt

=C+b0t+b1t2+X

n≥2

bnTn+1(t)

n+ 1 −Tn−1(t)

n−1

=C+b0T1(t) + b1T2(t) + 1

2+X

n≥3

bn−1

nTn(t)−X

n≥1

bn+1

nTn(t)

=C+b1

2+X

n≥1

bn−1−bn+1

nTn(t).

Since this is true for all t∈[−1,1], for n≥1 we have

2nan=bn−1−bn+1.(15)

The above equation precisely relates the Chebyshev series coeﬃcients of a func-

tion and those of its derivative. Finally, we should mention that the Chebyshev

polynomials satisfy

Tn(1) = 1, Tn(−1) = (−1)n(16)

for all n≥0.

3.2 Boundary-value problem to zero-ﬁnding problem

In what follows, we will assume hypotheses H.1 and H.2. Denote X= (1

ν)mp1where

p1is as in H.2 and we seek periodic solutions of period mp. We will sometimes write

instead X=Xνwhen we want to emphasize the base νof the weight ωn= 2νn

from the space 1

ν. Also we will let Y= (Ω)mp1where the norm in Ω is

||a||Ω=|a0|+

∞

X

n=0

1

nνn|an|.

9

Equipped with the || · ||Ωnorm, Ω is a Banach space and so is Ywhen given the

induced max norm. Elements aof Xwill be identiﬁed as follows:

a={ai,j :i= 0,...,m−1, j = 0,...,p1−1},

where each aij ∈1

ν. In this case, ai,j,n ≡(ai,j )nwill denote the nth element of the

sequence. Equip Xwith the norm ||a||X:= maxi,j ||ai,j ||ν.Xis a Banach space

and the following inclusion property is elementary.

Proposition 5. There is a continuous embedding Xν2→Xν1whenever ν1≤ν2.

There is also a continuous embedding Xν→Y.

Remark 3. If a∈Xν0and ν0>1, then a∈Xνfor |ν−ν0|suﬃciently small.

This fact will be important later when we consider continuation of zeroes.

Since solutions of the boundary-value problem (10)–(11) are analytic (Lemma

3), we are free to make the Chebyshev expansion

φk,q (t) = ak,q,0+ 2

∞

X

n=1

ak,q,n Tn(t),(17)

for a∈Xand some ν > 1 (recall, the weight sequence ωdepends on ν). The next

step is to substitute this expansion into the boundary-value problem. To facilitate

this, assume that we can make the convergent Chebyshev expansion

f(φk,q (t), φη(k,q)(t), β ) = fk,q,0(a, β) + 2

∞

X

n=1

fk,q,n (a, β)Tn(t).(18)

This is not truly an assumption since we have required fto be analytic, although

there are some subtleties. For example, if fhas poles in the complex plane then it

might be that if a∈Xνthen f(a, β)/∈Xνfor the same value of ν. This will be

discussed further in Section 3.4. For the nonlinearity g, we write

g(φk−1,p1−1(0), φη(k,0)(−1)) = gk(a, β ).(19)

Remark 4. The coeﬃcients fk,q,n and gkwill generally depend nonlinearly on the

coeﬃcients of aand on β, which is why we have included the explicit dependence. If

fis polynomial, then the coeﬃcients fk,q,n are expressible in terms of convolutions

(see (14) for the scalar case). On the other hand, if fis expressible in terms of

elementary functions then one can embed it in a polynomial vector ﬁeld using auto-

matic diﬀerentiation [11, 14]. Whether gis polynomial or not is of no consequence

(although polynomials do make things easier) since the evaluations in (19) are at

+1 and −1, which leads to an explicit expression for gkin terms of all coeﬃcients

of ak−1,p1−1and aη(k,0) due to (16).

We can now substitute the Chebyshev expansion of φinto the boundary-value

problem. Substituting into the ODE (10), we get

˙

φk,q (t) = 1

2p2

f(φk,q (t), φη(k,q)(t), β ) = 1

2p2 fk,q,0(a, β ) + 2

∞

X

n=1

fk,q,n (a, β)Tn(t)!

(20)

On the other hand, from (15) we get

2nak,q,n =bk,q,n−1−bk ,q,n+1

for n≥1, where bare the Chebyshev series coeﬃcients of ˙

φk,q . Since these can be

directly extracted from (20), we get

2nak,q,n =1

2p2

(fk,q,n−1(a, β )−fk,q,n+1(a, β)) , n ≥1.(21)

10

To get an equation for the order n= 0 modes, we need to use the boundary

conditions. Substituting the Chebyshev expansion (17) into (11) and using (16)

and (19), we get

ak,q+1,0+ 2

∞

X

n=1

(−1)nak,q+1,n =ak,q,0+ 2 P∞

n=1 ak,q,n , q < p1−1

gk(a, β) + ak−1,q,0+ 2 P∞

n=1 ak−1,q,n, q =p1−1.

(22)

Note that in the above equation, we must remember that our cyclic variable con-

vention means that ak,p1,n =ak,0,n and a−1,q,n =am−1,q,n for all n≥0.

To conclude, if φis a solution of the BVP (10)–(11) then its sequence of Cheby-

shev coeﬃcients must satisfy (21) and (22), and a∈Xfor some ν > 1. One can

similarly reverse the argument; any sequence a∈Xfor some ν > 1 that satisﬁes

both of those equations generates a solution φof the boundary-value problem by

(17). Motivated by this, we formally deﬁne a nonlinear map Fas follows:

F(a, β)k,q ,n =

ak,q,0−ak,q+1,0+ 2 P∞

n=1 ak,q,n −(−1)nak,q+1,n , n = 0, q < p1−1

gk(a, β) + ak−1,q −ak,q +1,0+ 2 P∞

n=1 ak−1,q,n −(−1)nak,q+1,n , n = 0, q =p1−1

2nak,q,n −1

2p2(fk,q,n−1(a, β )−fk,q,n+1(a, β)) , n ≥1

(23)

Observe that if F(a, β) = 0 for some a∈X, then asatisﬁes both (21) and (17).

The converse also holds, and we have the following lemma.

Lemma 6. Every solution of the boundary-value problem (10)–(11) is uniquely

associated to some a∈Xνsatisfying F(a, β) = 0, for some ν > 1.

While a given solution aof F(·, β) = 0 might be an element of Xν, the map

F(·, β) generally does not map Xνinto itself. However, it does map into Yprovided

the nonlinearity fis entire.

Lemma 7. Suppose fis entire. The linear map L:Xν→Ywith L(a)k,q,n =

2nak,q,n is well-deﬁned, as is the map F:Xν×R→Y. Also, this map is C∞.

Proof. Deﬁne ˜

F=F−L. We will show that ˜

Fhas range in Xνand Lis well-deﬁned,

which will give the required result for F. If a∈Xν, then

||L(a)k,q ||Ω=

∞

X

n=1

1

nνn·2n|ak,q,n = 2

∞

X

n=1

νn|ak,q,n | ≤ ||ak,q ||ν,

from which the well-deﬁnition of Lfollows. As for that of ˜

F, observe that it is

suﬃcient to prove that f(a, β) = {fk,q ,n(a, β )} ∈ Xν. From here on we will suppress

the input βfor brevity. Since fis entire, we have

f(x) =

∞

X

|j|=0

cjxj

for cj∈Rd,x= (x, y) and xjthe standard scalar-valued multiindex power. At the

level of the Chebyshev coeﬃcients,

fk,q,n =

∞

X

|j|=0

cj(ak,q )∗j

n

where ak,q = (ak,q , aη(k,q)) the convolution power is deﬁned componentwise (i.e.

in the components in (Rd)2) such that it is compatible with the previous power

series. However, since ak,q and aη(k,q)are elements of 1

ν, each of their components

11

is summable with respect to the weight sequence ω. Write a= (a(1),...,a(2d)) for

these components, so that each a(i)∈1

ν(R). Then deﬁne r∈R2das follows:

r(i)=

∞

X

n=0

ωn|a(i)

n|=||a(i)||ν.

Using the Banach algebra and the assumption that fis entire one can then verify

that

||fk,q ||ν≤

∞

X

n=0

ωn

∞

X

|j|=0

|cj| · |(ak,q )∗j

n|

=

∞

X

|j|=0

|cj|

∞

X

n=0

ωn|(ak,q )∗j

n|

≤

∞

X

|j|=0

|cj|rj<∞.

Since each fk,q is an element of 1

ν, it follows that f(a, β)∈Xνas required. Smooth-

ness of ˜

Fis a consequence of the regularity of fand g.

The above lemma covers the cases where fis polynomial, of course. By Remark

4, many non-polynomial nonlinearities can be handled using polynomial embed-

dings.

3.3 Finite-dimensional projection and numerical zeroes

Let N∈Nbe ﬁxed. Deﬁne the projection map πN:X→Xaccording to

πN(a)k,q,n =ak,q,n , n ≤N

0n > N,

Then, deﬁne the complementary projector π∞:Xν→Xνby π∞=IXν−πN.

Let XN=πN(X) (we will write XN

νif we want to emphasize the value of ν) and

introduce the “computational isomorphism” iN:XN→((Rd)N+1)m·p1as follows.

First, for a given x∈((Rd)N+1)m·p1we make the indexing convention xk,q,n ∈Rd

for k∈ {0,...,m−1},q∈ {0,...,p1−1}and n∈ {0,...,N}. We can then deﬁne

iN(a)k,q,n =ak,q,n .

Whenever we want to think of an element of Xwith zero tail (i.e. an= 0 for n > N)

as being a vector in some ﬁnite-dimensional space, we can apply the isomorphism

iNto it. Similarly, we can apply the inverse

i−1

N:XN→XN, XN:= ((Rd)N+1 )mp1(24)

to embed a ﬁnite-dimensional vector object of appopriate dimension into XN.

In what follows, we will use bars to denote “numerical” objects (i.e. objects

that in practice will be represented as ﬁnite matrices or vectors) while quantities

without bars will typically be analytical. Deﬁne the maps FN:X×R→Xand

FN:XN×R→XNby

FN(a, β) = πNF(πNa, β ), F N(a, β) = iNFN(i−1

Na, β).(25)

Intuitively, FNis the restriction of Fto Chebyshev series with Nnonzero modes,

with the output truncated to Nmodes. FNis the representation of this nonlinear

map on the Euclidean space XN. By Lemma 7, each of these maps is C∞.

12

Since FN→Fpointwise, it seems reasonable that approximate zeroes of FN(·, β)

in XNfor Nlarge enough should yield good approximations to zeroes of F(·, β ).

Equivalently, approximate zeroes of FN(·, β) on embedding by i−1

Nshould be good

approximations to zeroes of F(·, β). In practice (and, indeed, in our application in

Section 4), such approximate solutions can be computed by implementing FN(and

its derivative) in a computer and applying Newton’s method.

3.4 A-posteri analysis for branches of zeroes

If a numerical branch of zeroes has been computed – that is, one has a discrete

set of parameters β0,...,βMand a corresponding discrete set of numerical zeroes

a0, . . . , aMsuch that FN(aj, βj)≈0 – we want a means of proving that there

is a unique, continuous branch of true zeroes (i.e. of F) nearby. The primary

theoretical tool we will use for the a-posteriori analysis of numerical branches of

zeroes is often called the radii polynomial approach. It is a twist on the standard

Newton-Kantorovich theorem.

We will present the result in general and subsequently apply it to our problem.

To begin, let X,Ybe Banach spaces, let F:X × R→ Y be a nonlinear map

depending on a real parameter (note: Fis distinct from the objects of the previous

sections) and suppose x0and x1satisfy F(x0, λ0)≈0 and F(x1, λ1)≈0 for some

λ0, λ1∈R. The meaning of the symbol ≈is essentially arbitrary, although the

intuition is of course that x0and x1are approximate zeroes of Fat the parameters

λ0and λ1. Deﬁne the convex predictors

xs= (1 −s)x0+sx1, λs= (1 −s)λ0+sλ1.(26)

The following is a more explicit (and in some sense, slightly more general) version

of a theorem from [16], although the proof is identical and as such, will be omitted.

Theorem 8. Let Xand Ybe Banach spaces with a continuous embedding X→ Y.

Let F∈Ck(X × R,Y)for some k≥1and assume there exist bounded linear

operators A†∈B(X,Y)and A∈B(Y,Y)such that the following range hypotheses

are satisﬁed:

•AF has range in Xand A:Y → Y is injective,

•AA†has range in X,

•ADxF(x, λ)has range in Xfor all x∈ X and λ∈R.

Suppose there exist Y0,Z0,Z1and Z2≥0such that

||AF (xs, λs)k|X≤Y0,∀s∈[0,1] (27)

||IX−AA†||B(X,X)≤Z0(28)

||A[DxF(x0, λ0)−A†]||B(X,X)≤Z1(29)

||A[DxF(xs+b, λs)−DxF(x0, λ0)]||B(X,X)≤Z2(r),∀s∈[0,1], b ∈Br(0).

(30)

Deﬁne the radii polynomial

p(r) = Z2(r)r+ (Z1+Z0−1)r+Y0.(31)

If there exists r0>0such that p(r0)<0, then there exists a Ckfunction

˜x: [0,1] →[

s∈[0,1]

Br0(xs)

such that F(˜x(s), λs) = 0. Furthermore, these are the only zeroes of Fin the tube

Ss∈[0,1] Br0(xs).

13

Since such computer-assisted a-posteriori error analysis is somewhat new in the

impulsive dynamical systems literature, we would do well to explain how this the-

orem works and what the individual pieces (e.g. A†and A) represent. The basic

idea is that we would like to apply a uniform Newton-Kantorovich argument; that

is, we would like to prove convergence of the iterated map

x7→ x−DxF(x0, λ0)−1F(x, λs)

for all s∈[0,1]. If this could be done, then we would be guaranteed the existence

of ˜x(s) such that F(˜x(s), λs) = 0 for all s∈[0,1] as desired. The problem is that

proving the convergence of this method is complicated by the fact that the inverse

of DxF(x0, λ0) is diﬃcult to impossible to express analytically. However, there is

no reason why we should need to use the exact inverse of this operator; we could

instead make use of an approximation. In practice, we therefore take A†∈B(X,Y)

to be an approximate derivative:A†≈DxF(x0, λ0). Similarly, A∈B(Y,Y) is

thought of as an approximate inverse of DxF(x0, λ0). We explicitly construct A†

so that it is easy to invert.

There is a bit of a subtle point here: the derivative DxF(x0, λ0) is a bounded

linear map from Xto Y, so DxF(x0, λ0)−1∈B(Y,X). In practice, it might be

diﬃcult to construct an approximate inverse A≈DxF(x0, λ0)−1that maps into

Xor explicitly prove that the latter is true. However, to prove the theorem it

is suﬃcient to require A∈B(Y,Y) and that the compositions of Awith each of

F,A†and DxF(x, λ) have range in X. The bound inequalities (27)–(30) will fail

automatically if these range conditions are not satisﬁed anyway, while in most cases,

establishing the bounds will indirectly verify the range conditions.

With these constructions complete, the theorem is proven by showing that if

p(r0)<0, the linear operator T:X × [0,1] → X deﬁned by

T(x, s) = x−AF (x, λs)

is a uniform (in s) contraction on the tube ∪s∈[0,1]Br0(xs). Note that the conditions

that AF and ADxF(x, λ) have range in Xensures that both Tand the derivative

DxT(x, s) : X → X are well-deﬁned. For details, the reader should consult [16].

As for the bounds Y0,Z0,Z1and Z2, they each have a fairly straightforward

interpretation. Y0is a bound on the numerical defect across the convex predictor.

Since Ais basically an approximate left-inverse of A†, the bound Z0measures the

quality of this approximate inverse. Z1is a bound on the error between the deriva-

tive DxF(x0, λ0) and the approximate derivative A†. As for Z2, it is essentially

a uniform (in s) local bound on the second derivative of F(or in the C1case, a

uniform local Lipschitz constant for the ﬁrst derivative).

Remark 5. We have not elaborated on how the numerical branch of zeroes can

be extended globally. Theorem 8 only applies to one segment of the branch –

that is, between two numerical zeroes x0and x1for parameters λ0and λ1. In

fact, one can show (see [16]) that if Theorem 8 is successfully applied to the zeroes

{(x0, λ0),{x1, λ1}and to the zeroes {(x1, λ1),(x2, λ2)}so that the Theorem grants

the existence of two Ckcurves of zeroes, then the curve obtained by “gluing” them

together is Ck.

3.5 Construction of A†and Afor the boundary-value

problem

For the boundary-value problem (10)–(11), there is a straightforward way to con-

struct A†and Asuch that the range hypotheses of Theorem 8 are satisﬁed for Fthe

nonlinear map (23) with domain and codomain as stated in Lemma 7. To begin,

14

suppose x0∈XNis an approximate (i.e. numerical) zero of FN(·, λ0); this might be

computed using Newton’s method applied to FN(·, λ0). Denote x0=i−1

N(x0)∈XN;

this object has the same interpretation as x0from Theorem 8. Deﬁne the ﬁnite-

dimensional linear map (representable by a matrix)

A†=DxFN(x0, λ0) (32)

Deﬁne the linear map L:X→Yby

L(a)k,q,n = 2nak,q,n ,(33)

and deﬁne the linear map A†:X→Yby

A†=i−1

N◦A†◦iN◦πN+Lπ∞(34)

Formally, A†“applies the ﬁnite-dimensional truncation A†” to the ﬁrst Nmodes,

and applies the diagonal operator Lto the tail (the modes N+ 1 onward). This

structure results in A†being diagonally dominant.

To construct A, we ﬁrst let Abe a numerical inverse of A†. That is, Ais a

matrix of the same dimenesion as A†such that ||I−A A†|| ≈ 0. Let L+:Y→X

be the linear map

L+(a)k,q,n =(0, n = 0

1

2nak,q,n , n > 0.

Note that L+behaves like a Moore-Penrose pseoduinverse of L, hence the use of

the symbol L+. We can now deﬁne A:Y→Xas follows:

A=i−1

N◦A◦iN◦πN+L+π∞.(35)

Lemma 9. A†and Aare well-deﬁned and bounded, and the range hypotheses are

satisﬁed provided provided Ais maximal rank.

Proof. It is easy to check that L+:Y→Xνis bounded. Since Ais bounded and

πNhas range in XN, boundedness of Afollows. Checking that A†is well-deﬁned

and bounded is similar. To prove that Ais injective, one makes use of the fact

that Ais injective (since it is maximal rank) and the restriction of L+to π∞(X)

is injective. There is no need to verify the range hypotheses because Aalready has

range in X.

Remark 6. The choice of Yis not unique and we could just have easily used the

space (∞)mp1instead, since there is an embedding Ω→∞. However, if this were

done, then we would have A:Y→Yrather than A:Y→Xand we would have

had to explicitly verify the range hypotheses. This is not diﬃcult, but our choice for

the space Yis in some sense “ minimal”. The inclusion of the range hypotheses in

Theorem 8 is a reﬂection of this non-uniqueness of the space Yand the consequential

“roughness” or robustness of the radii polynomial approach.

With these choices of A†and Aand the condition that Ais maximal rank, the

range hypotheses of Theorem 8 are satisﬁed for the boundary-value problem (10)–

(11). In fact, the range hypotheses and at least C2smoothness of fand gimply that

the bounds Y0and Z0,Z1and Z2from (27)–(30) do indeed exist; they just may not

be small enough to ensure p(r0)<0 for some positive r0.Computing these bounds

is another matter entirely, but as it has become a somewhat standard routine in

the ﬁeld of rigorous numerics we will not compute the fully general bounds for the

map Ffrom (23). Rather, we will instead compute them in Section 4 for a speciﬁc

example.

15

3.6 Implementation

Here we will comment on the role of νand the standard approach to implementation

of Theorem 8 in the computer. νserves a few diﬀerent purposes. First, it char-

acterizes the regularity of the functions that can be represented by the Chebyshev

series in coeﬃcients of X: higher values of νcorrespond to more regular functions

since the coeﬃcients of the Chebyshev series decay geometrically with a faster rate.

Second, if a zero of Fis very irregular it might require a very large number Nof

modes to represent accurately. However, if νis ﬁxed and Nis increased, some of the

technical bounds (27)–(30) might explode (in practice) because of large numerical

roundoﬀ errors near the limits of the ﬂoating point number system, as these bounds

always require computing powers νnfor n≤N. While we will always rigorously

track numerical roundoﬀ by using interval arithmetic in our implementation, even-

tually a limit might be reached that can only be overcome by decreasing ν. The

good news is that the embedding property of Proposition 5 guarantees we can not

lose zeroes of Fby decreasing ν. However, we can lose zeroes by increasing ν. For

example, if a solution of (10)–(11) has its Chebyshev coeﬃcients in Xνif and only

if ν < 1.1 (for example, based on precise location of the poles of the solution), then

the computer-assisted proof will always fail to validate a representative numerical

zero with ν= 1.11 unless there is a (human) error in the implementation of the

bounds.

In practice, we implement Theorem 8 in a computer by ﬁrst determining ex-

plicit formulas for the bounds Y0,Z0,Z1and Z2using a combination of analytical

estimates and ﬁnite-dimensional computations such as matrix norms. We then im-

plement these bounds on the computer using interval arithmetic. In MATLAB,

we use the package INTLAB [18] to accomplish this. After computing the numer-

ical branch of zeroes of FNusing a double arithmetic implementation of Newton’s

method, we feed all of the data into our bounds. The result is we obtain rigorously

veriﬁed over-estimates of our explicit bounds. We can then rigorously enclose the

roots of the radii polynomial (if Z2(r) is a polynomial in r) or, at worst, reliably

compute the sign of p(r0) for a given r0(if Z2(r) is not a polynomial), thereby

checking all conditions of the theorem.

4 Transcritical branch in the pulse-harvested

Hutchinson equation

The Hutchinson equation, sometimes called the delay logistic equation, is the scalar

delay diﬀerential equation

˙x=rx(t)1−x(t−τ)

K.

xrepresents a single-species population. The parameters rand Kare positive, and

are called respectively the intrinsic growth rate and carrying capacity. τ > 0 is a

delay that takes into account such factors as maturation or gestation time. If a

linear impulsive harvesting is introduced such that the population is reduced to a

proportion β∈[0,1] each T > 0 time units, we get the impulsive delay diﬀerential

equation

˙x=rx(t)1−x(t−τ)

K, t /∈TZ(36)

∆x= (β−1)x(t−), t ∈TZ.(37)

16

The equation is referred to as the Hutchinson equation with pulse harvesting because

we can also interpret h= 1 −β∈[0,1] as being the proportion of the population

that is removed (i.e. harvested) at times kT .

We can perform a few changes of variables to eliminate some of the parameters.

If we deﬁne

y(t) = 1

Kx(tτ), α =rτ, u =T

τ,

then the Hutchinson equation with pulse harvesting becomes

˙y=αy(t)[1 −y(t−1)], t /∈uZ(38)

∆y= (β−1)y(t−), t ∈uZ.(39)

Doing this, we eliminate two parameters and have re-scaled the delay to unity. Using

our rigorous numerical method for continuation of periodic solutions in additional

to analytical results on the pulse-harvested Hutchinson equation (which we will

develop in Section 4.1), we will ultimately prove the following theorem.

Theorem 10. Let K > 0be arbitrary and let the parameters r,τ,Tbe one of the

sets from (the rows of) Table 1, deﬁne β∗=e−rT and consider the pulse-harvested

Hutchinson equation (36)–(37). The solution x= 0 undergoes a transcritical bifur-

cation with a nontrivial periodic solution of period Tat β=β∗. Let this branch be

denoted xβ. The following are true.

•For 0< β < β∗, there are no nontrivial non-negative periodic solutions, x= 0

is the global attractor on R+, and any nontrivial periodic solution xin this

parameter regime satisﬁes

inf

t∈Ry(t)≤K1 + log(β)

rT .

•β7→ xβis continuous 1for β∈[β∗−βtol,1], with βtol = 0.001. This branch

has no folds, x1=Kand this is the only branch of periodic solutions that

crosses through the trivial solution x= 0.

If β= 0, it is easy to see that every solution converges to zero uniformly in ﬁnite

time, so the dynamics are trivial and we do not mention this in the theorem. The

existence of the transcritical bifurcation and the ﬁrst conclusion of the theorem will

be established by analytical means. The nonlocal part concerning the branch yβ

will be proven using our rigorous numerical method.

4.1 Analytical results

Since our focus with this publication is on the rigorous numerical method for periodic

solution continuation rather than the analysis of the pulse-harvested Hutchinson

equation, we will skip many of the details for the proofs of the analytical parts and

refer the reader to relevant background to ﬁll in the gaps. The ﬁrst result concerns

the hyperbolicity of the equilibrium solution y= 0 in (38)–(39). The following can

be proven by linearizing (38)–(39) about y= 0 and using the theory from [5, 6].

Lemma 11. Deﬁne β∗=e−αu . The equilibrium solution y= 0 is hyperbolic if

β6=β∗. When β=β∗the centre ﬁbre bundle is one-dimensional.

Lemma 12. Let {βn:n∈N}be a convergent sequence of parameters and suppose

yβnis a sequence of nontrivial periodic solutions of period ufor parameter βn. If

yβn→0then βn→β∗.

1Here, continuity is with respect to the supremum norm on the function space consisting of bounded

φ:R→Rthat are continuous from the right with limits on the left.

17

Proof # r τ T Proof runtime (seconds) u1β∗

1 3/10 1/10 1 79.7391 10 0.7408

2 1 1/2 1 22.9848 2 0.6065

3 1 1/4 1 63.2405 4 0.6065

4 2 1/2 1 460.1527 2 0.1353

5 3/10 7/5 2 157.4058 10 0.5488

6 3/10 7/5 3 327.9192 15 0.4066

7 1 3/2 1 20.3223 2 0.6065

8 1 5/3 1 33.3345 3 0.6065

Table 1: Parameters (r, τ, T ) for the rigorous branch continuation of the pulse-harvested

Hutchinson equation for Theorem 10. The function run proofs.m uses Proof # refer-

ences. Runtimes are for a machine with an AMD Ryzen 5 3600XT CPU and 16gb of

DDR4 memory. Larger values of u1result in generally slower proofs because the dimen-

sion of the system being handled is larger: for example, proofs far away from β∗involve

matrices of size u1(N+ 1) ×u1(N+ 1). Smaller values of β∗also require more computa-

tion time because a longer section of the branch of periodic solutions must be computed

and validated, namely for β∈[β∗−βtol,1].

Proof. Suppose not; let limn→∞ βn=b6=β∗. Let S(·, β ) : X→Xdenote the

time umap starting from initial time t0= 0 at parameter β, where Xis the set of

right-continuous functions with limits on the left with domain [−1,0] and codomain

R. This map is C1[5] and its diﬀerential at y= 0 is the monodromy operator.

Since b6=β∗, the equilibrium y= 0 is hyperbolic so 1 is not an eigenvalue of

DS(0, b). Since the latter is compact, it follows that DS(0, b)−Iis a Banach space

isomorphism, which implies that the equation S(y, β) = yhas a unique C1solution

curve (yβ, β) deﬁned on a neighbourhood of β=b. This is a contradiction, since

S(0, β) = 0 for all β∈Rbut we know that S(yβn, βn) = yβn.

The next result concerns the (essential) sign-constancy of solutions. Its proof is

simple and omitted.

Lemma 13. If β∈[0,1], every solution of (38)–(39) is eventually either strictly

positive, strictly negative, or zero. In particular, every periodic solution is either

strictly positive, strictly negative, or identically zero.

Lemma 14. If β < e−αu then:

•y= 0 is the global attractor in R+;

•any nontrivial periodic solution ysatisﬁes inft∈Ry(t)≤1 + (αu)−1log(β).

Proof. If yis a nonnegative solution, then ysatisﬁes the impulsive integral inequality

y(t)≤y(0) + Zt

0

αy(s)ds, t /∈uZ

∆y(t) = (β−1)y(t−), t ∈uZ.

Solving this with the impulsive Gronwall-Bellman inequality [1], we get y(u)≤

eαuβy(0) < y(0). It follows that for any initial condition y(0) >0, we have

limn→∞ y(nu) = 0 for integer n. On the other hand, y(nu +t)≤eαuy(nu) for

t∈[0, u). These two facts together imply limt→∞ y(t) = 0, so y= 0 is the global

attractor as claimed.

To prove the other claim, ﬁrst observe that by Proposition 1, any nontrivial

periodic solution must have its period be an integer multiple of u. Suppose the

period of such a solution is ku for some k∈N. Let −C= inf t∈[0,ku]y(t). Since we

18

have already proven that y= 0 is the global attractor in R+, any nontrivial periodic

solution must be strictly negative by the previous lemma. It follows that |y(t)| ≤ C.

Let w(t) = −y(t). Then

w(t)≤w(0) + Zt

0

αw(s)(1 + C)ds, t /∈uZ

∆w= (β−1)w(t−), t ∈uZ.

Applying the Gronwall-Bellman inequality again, we get w(ku)≤w(0)βkeαku(1+C).

But since w(ku) = w(0) by periodicity, this implies 1 ≤(βeαu(1+C))kand, con-

sequently, C≥ −1−(αu)−1log β. Since |y(t)| ≤ Cand yis negative, we get

y(t)≤ −C≤1 + (αu)−1log β, as claimed.

Lemma 15. The solution y= 0 undergoes a transcritical bifurcation with a branch

of nontrivial periodic solutions yβof period uat β=β∗. There is a neighbourhood

U⊂Rof zero such that for |β−β∗|small, the only solutions y:R→Rthat are

deﬁned for all time and contained in Uare the trivial solution and yβ.

Proof. At β=β∗, we know the centre ﬁbre bundle is one-dimensional. The dy-

namics on the centre manifold can be computed using the theory from [6]; they are

given to quadratic order by

˙z=−αφ(t−1,0)z2+O(z3), φ(t) = e−αu[bt

uc− t

u].

Since the quadratic term is strictly negative, the associated quadratic coeﬃcient

of the time umap restricted to the parameter-dependent centre manifold does not

vanish. On the other hand, the dominant Floquet multiplier of y= 0 (in fact,

the only Floquet multiplier) is given precisely by µ(β) = βeαu. This crosses the

unit circle transversally at β=β∗. It follows that a transcritical bifurcation occurs

at β=β∗in the parameter-dependent centre manifold at the level of the time u

(Poincar´e) map, which in turn forces [6] a transcritical bifurcation with a nontrivial

periodic solution in the impulsive delay diﬀerential equation.

4.2 Rigorous branch continuation far away from β=β∗

Here we apply the theory from Section 3 to the rescaled impulsive Hutchinson

equation (38)–(39). We will set up the rigorous continuation scheme for periodic

solutions of period u, so in the notation of the aforementioned section this means

that m= 1.

4.2.1 The F= 0 map

The boundary-value problem (10)–(11) for the impulsive Hutchinson equation can

be written

˙

φq=α

2u2φq(1 −φq+δ) for q= 0,1, ..., u1−1

φq(1) = φq+1(−1) for q= 0,1, ..., u1−2

βφu1−1(1) = φ0(−1),

(40)

where δis deﬁned according to

δ= min ({ku1−u2:k∈N} ∩ R+) (41)

and the indices on the φare subject to the cyclic condition φk≡φ[k]u1for [·]u1

the remainder modulo u1. Note that the delay shift function ηsatisﬁes for k=

0,...,u1−1 the equation

η(k, 0) = [k+δ]u1,

hence our decision to impose this cyclic variable convention.

19

Transforming now to the F= 0 map, we get

Fq,n(a, β ) = (aq)0−(aq+1)0+ 2 Pm≥1(aq)m−(aq+1 )m(−1)mn= 0,

2n(aq)n+ (γq)n+1 −(γq)n−1n≥1,

for q= 0,...,u1−2, and

Fu1−1,n(a, β ) = β(au1−1)0−(a0)0+ 2 Pm≥1β(au1−1)m−(a0)m(−1)mn= 0,

2n(au1−1)n+ (γu1−1)n+1 −(γu1−1)n−1n≥1,

where the nonlinear function f(x, y) = α

2u2x(1 −y) that deﬁnes the right-hand side

of the diﬀerential equation in (40) has been converted into Chebyshev form and

stored in the sequence γ, which is deﬁned according to

(γq)n

def

=α

2u2

[(aq)n−(aq∗aq+δ)n].(42)

Note that the multiplications in φhave turned into convolutions in a. Deﬁne T:

X→X(or T:Y→Y) as follows:

(T a)q,n =0, n = 0

aq,n+1 −aq,n−1, n ≥1.(43)

This operator allows us to express the operator Fa bit more cleanly and will be

useful in computing the bounds (27) through (30). Note that for this example,

X= (1

ν)u1. The operator Tis a componentwise tridiagonal operator.

4.2.2 A few technical estimates

The following will be beneﬁcial in the computation of the bounds Y0,Z0,Z1and

Z2. First, the tridiagonal operator T:X→Xsatisﬁes

||T||X≤2ν+1

ν.(44)

The proof of this bound is a standard exercise and is omitted. The next result

essentially allows for the computation of the norm of the “ﬁnite part” of an operator

on X. Its proof can be accomplished using Fubini’s theorem and some careful

bookkeeping, and is also omitted. We will state it speciﬁcally as it applies to the

present situation where X= (1

ν)u1with 1

νconsisting of scalar sequences, but of

course there are more general versions; see for example [7, 15].

Lemma 16. Suppose L:XN→XNis represented in the form

[L(a)]q,n =

u1−1

X

j=0

N

X

m=0

(Lq,j )n,maj,m

for reals Lq,j . Deﬁne the norm || · ||XNon XNaccording to ||a||XN=||iNa||X.

Deﬁne the quantities

Bm,j (L) = max

n=0,...,N

1

ωn

N

X

r=0

|(Lm,j )r,n|ωr,

Then

||L||BXN≤max

m=0,...,u1−1

u1−1

X

j=0

Bm,j (L).

20

We will typically not write down the explicit bound appearing in Lemma 16, but

will rather use the symbol || · ||B(XN)as a proxy for the associated upper bound.

The following is a consequence of (or can be used to prove) the previous lemma. Its

proof is also omitted.

Lemma 17. Let L:XN→XNbe represented as in Lemma 16. With the same

notation as in that lemma, for h= (h0,...,hu1−1)∈XN, we have the bound

||Lh||XN≤max

m=0,...,u1−1

u1−1

X

j=0

Bm,j (L)||hj||ν.

Next, we need a result concerning the dual of XN. Its proof will be omitted.

Lemma 18. For a linear functional U:XN→Rwith

Uh =

u1−1

X

j=0

N

X

m=0

Uj,mhj,m

for reals Uj,m, deﬁne B∗

j(U) := maxm≥0ω−1

m|Uj,m|. Then

|Uh| ≤

u1−1

X

j=0

B∗

j(U)||hj||ν

The next result will be useful in Z2bound calculation. Its proof is straightfor-

ward and is omitted.

Lemma 19. Let L:Y→Xbe an operator of the form L=i−1

NLiNπN+L+π∞.

Let h= (h0,...,hu1−1)∈Y. If L:XN→XNis represented as in Lemma 16,

then with the notation from that lemma, we have

||Lh||X≤max

m=0,...,u1−1(u1−1

X

j=0

Bm,j (L)||hj||ν+1

2(N+ 1) ||hm||ν)

The ﬁnal result we will need concerns bounds for convolutions of sequences with

particular structure.

Lemma 20. Suppose a∈πN(1

ν)and h∈π∞(1

ν)with ||h||ν≤1. Then for

k∈ {0,...,N + 1},

|(a∗h)k| ≤ max

n=N+1,...,N+k|an−k|1

ωn

:= ak,

where the right-hand side is treated as zero when k= 0.

Proof. By deﬁnition,

(a∗h)k=X

n∈Z

a|n−k|h|k|=

−N−1

X

n−∞

a|n−k|h|n|+

∞

X

m=N+1

a|m−k|h|m|=

N+k

X

m=N+1

a|m−k|hm,

where the second equality is due to h∈π∞(1

ν) and the third because a∈πN(1

ν).

We can then make the estimate

|(a∗h)k| ≤

N+k

X

n=N+1

|a|n−k||hn≤

N+k

X

n=N+1

akhnωn≤ak.

21

4.2.3 The operators Aand A†

To apply Theorem 8 and compute the bounds (27)-(30), we need to deﬁne the

linear operators A†and Aas in Section 3.5. To do so, we ﬁrst need to compute two

numerical approximations of the solution (a0, β0) and (a1, β1) such that F(ai, βi)≈

0 for i= 0,1.

We deﬁne the linear operators A†and Asuch as (34) and (35) respectively from

Section 3.5 using the approximate solution (a0, β0). We now have everything to

compute the bounds from Theorem 8.

4.2.4 The bound Y0

For the bound Y0, we have

AF (as, βs) = i−1

NAF N(as, βs) + L+π∞F(as, βs)≡Y(1)

0+Y(2)

0,

where asand βsare convex predictors deﬁne in the same way as (26). Using the

mean-value inequality, we can bound the ﬁrst term by

||Y(1)

0||X=||AF N(as, βs)||XN

≤ ||AF N(a0, β0)||XN+||A[FN(as, βs)−FN(a0, β0)]||XN

≤ ||AF N(a0, β0)||XN+Z1

0

AD(a,β)FN(a0+ts∆a, β0+ts∆β)s∆a

s∆βXN

dt

≤ ||AF N(a0, β0)||XN+ sup

s∈[0,1]

AD(a,β)FN(as, βs)s∆a

s∆βXN

,(45)

with ∆a=a1−a0, where the norm on XNis simply deﬁne as

||a||XN=||iN(a)||X

and Dis the Frchet derivative. For the second term, we have

Y(2)

0= max

q=0,1,...,u1−1

∞

X

n=N+1

1

2n|(T γq(as))n|2νn

≤max

q=0,1,...,u1−1sup

s∈[0,1]

1

N+ 1

2N+1

X

n=N+1

|(T γq(as))n|νn.(46)

This last inequality came from the fact that for n > 2N+ 1, we have T γq(as) = 0.

As mention in Section 3.6, we can rigorously compute (45) and (46) using interval

arithmetic and we can deﬁne

Y0

def

=||AF N(a0, β0)||XN+ sup

s∈[0,1]

AD(a,β)FN(as, βs)∆a

∆βXN

+ max

q=0,1,...,u1−1sup

s∈[0,1]

1

N+ 1

2N+1

X

n=N+1

|(T γq(as))n|νn.

which satisﬁes the bound (27).

4.2.5 The bound Z0

For the bound Z0, we have

IX−AA†=πN−iNAA†i−1

N.

22

Using Lemma 16, we can rigorously compute a numerical bound using interval

arithmetic such that

Z0

def

=||I−AA†||BXN.(47)

4.2.6 The bound Z1

To simplify the notation, let zdef

=DF (a0, β0)−A†hwith ||hx|| ≤ 1, then

(zq)n=

Pk≥N+1[1 + (β0−1)δq,u1−1](hu1−1)k−(h1)kif n= 0,

−α

2u2T(hI

q∗(a0)q+δ+ (a0)q∗hI

q+δ)nif 1 ≤n≤N,

α

2u2T(hq−h∗

q(a0)q+δ−(a0)q∗hq+δ)nif n≥N+ 1,

with hI

i=hi−πN(hi) for i= 0,...,u1−1 and where δq,u1−1is the Kronecker

delta. Then, we can write

Az =i−1

NA z0+i−1

NA(zN−z0) + L+π∞z≡Z(1)

1+Z(2)

1+Z(3)

1

where zk=iNπkzfor k= 0, N . For the ﬁrst term, we need to ﬁnd a bound for the

terms (zq)0. To achieve this, we see that

1≥ |(hq)0|+ 2

∞

X

k=1

|(hq)k|νk≥2

∞

X

k=N+1

|(hq)k|νk≥2νN+1

∞

X

k=N+1

|(hq)k|.

Hence,

∞

X

k=N+1

|(hq)k| ≤ 1

2νN+1 .

Using this result, we get

|(zq)0|=X

k≥N+1

[1 + (β0−1)δq,u1−1](hu1−1)k−(h1)k

≤[2 + (β0−1)δq,u1−1]

2νN+1

≤1

νN+1 ,

where the last inequality came from the fact that β0∈[0,1]. Now, let e=

(e0,...,eu1−1)∈X, such that

(eq)n=1 if n= 0

0 if n≥1

for q= 0,...,u1−1, then we can bound the ﬁrst term by

||Z(1)

1||X=||A z0||XN≤||Ae||XN

νN+1 .

Using Lemma 20, we deﬁne aqand aq+δsuch that

(aq)k

def

= max

n=N+1,...,N+k

|(aq)n−k|

2νn≥ |(aq∗hI

q+δ)|,

(aq+δ)k

def

= max

n=N+1,...,N+k

|(aq+δ)n−k|

2νn≥ |(aq+δ∗hI

q)|,

23

for q= 0,...,u1−1. Then, for 1 ≤n≤Nwe have

|(zq)n|=

α

2u2

T(hI

q∗(a0)q+δ+ (a0)q∗hI

q+δ)n

≤|α|

2u2

T(aq+aq+δ)n

and we deﬁne z= (z0,...,zu1−1)∈XNsuch that

(zq)n=(0n= 0

|α|

2u2T(aq+aq+δ)n1≤n≤N.

Now using these results, we get

||Z(2)

1||X=||A(zN−z0)||XN

≤ ||(AA)1

2|zN−z0| ||XN

≤ ||(AA)1

2z||XN.

For the last term, we have

||Z(3)

1||X= max

q=0,...,u1−1

∞

X

n=N+1

|(zq)n|2νn

= max

q=0,...,u1−1

∞

X

n=N+1

1

2n

α

2u2

T(hq−hq∗(a0)q+δ−(a0)q∗hq+δ)n

2νn

≤|α|

2(N+ 1)u2ν+ν−1max

q=0,...,u1−1{||hq||ν+||hq∗(a0)q+δ||ν+||(a0)q∗hq+δ||ν}

≤|α|

2(N+ 1)u2ν+ν−1max

q=0,...,u1−1{1 + ||(a0)q+δ||ν+||(a0)q||ν}

We deﬁne the bound Z1satisfying (29) by

Z1

def

=||Ae||XN

νN+1 +||(AA)1

2z||XN

+|α|

2(N+ 1)u2ν+ν−1max

q=0,...,u1−1{1 + ||(a0)q+δ||ν+||(a0)q||ν}

and we use interval arithmetic to do the numerical computations.

4.2.7 The bound Z2

For Z2, we have

||A[DF (as+b, βs)−DF (a0, β0)]||B(X)≤ ||A||B(X)||T||B(X)max

q=0,...,u1−1sup

||h||X≤1

||Z(q)

2h||ν,

where

(Z(q)

2h)n=0 if n= 0,

α

2u2(hq∗[s(∆a)q+δ+b] + [s(∆a)q+b]∗hq+δ)nif n≥1.

24

Then, for ||h||X≤1, we have

||Z(q)

2h||X=

∞

X

n=1

|α|

2u2(hq∗[s(∆a)q+δ+b] + [s(∆a)q+b]∗hq+δ)n2νn

≤|α|

2u2

||hq∗[s(∆a)q+δ+b] + [s(∆a)q+b]∗hq+δ||ν

≤|α|

2u2

(||∆aq||ν+||∆aq+δ||ν+ 2r).

Using Lemma 19 and (44), we can deﬁne the bound Z2by

Z2(r)def

=|α|

u22ν+ν−1 max

m=0,...,u1−1(u1−1

X

j=0

Bm,j (A) + 1

2(N+ 1) )!r

+|α|

2u22ν+ν−1 max

m=0,...,u1−1((||∆am||ν+||∆am+δ||ν) u1−1

X

j=0

Bm,j (A) + 1

2(N+ 1) !)!.

This bound satisﬁes (30) and once again all the computations are ﬁnite, meaning

that we can use interval arithmetic to rigorously compute this bounds.

4.3 Rigorous branch continuation near β=β∗

At β=e−αu, we have from Lemma 15 that the trivial solution y= 0 of the

impulsive DDE (38)–(39) undergoes a transcritical bifurcation. We can use the

rigorous continuation from Section 4.2 to do continuation in β∈[β∗+δ, 1] for δ > 0

large enough, but as β→e−αu the transcritical branch gets O(|β−β∗|) close to

the solution y= 0, resulting in a lack of isolation of zeroes of F. Since the radii

polynomial approach is based on the contraction mapping principle, the computer-

assisted proof will eventually fail when βgets close to β∗. In this section we explore

a way to resolve this issue.

4.3.1 Interlude: desingularizing the bifurcation

To handle the lack of isolation of zeroes near β=e−αu, we will use desingularization

to quotient out the known trivial solution a= 0. To do this, we ﬁrst perform a re-

scaling at the level of the boundary-value problem (40). Introduce a quasi-amplitude

parameter and deﬁne {z0,...,zu1−1}by the equation φq=zq. The boundary-

value problem becomes

˙zq=α

2u2zq(1 −zq+δ) for q= 0,1, ..., u1−1

zq(1) = zq+1(−1) for q= 0,1, ..., u1−2

βzu1−1(1) = z0(−1)

.(48)

There are now two parameters: βand .

4.3.2 The F= 0 map

Our nonlinear map that encodes the solutions of the boundary-value problem as

zeroes is very similar to the previous one from Section 4.2. The main diﬀerence is

in the γcoeﬃcient from (42). The required modiﬁcation is

(γq(a, ))n=α

2u2

[(aq)n−(aq∗aq+δ)n].(49)

25

Then, {z0,...,zu1−1}is a solution of the BVP (48) if and only if its Chebyshev

coeﬃcients a∈Xsatisﬁes F(a, , β) = 0 where

Fq,n(a, , β ) =

(aq)0−(aq+1)0+ 2 Pm≥1(aq)m−(aq+1 )m(−1)mn= 0, q 6=u1−1

β(au1−1)0−(a0)0+ 2 Pm≥1β(au1−1)m−(a0)m(−1)mn= 0, q =u1−1

2n(aq)n+ (γq(a, ))n+1 −(γq(a, ))n−1n≥1,

(50)

In other words, zeroes of a7→ F(a, , β) are uniquely associated to periodic solutions

with speciﬁc quasi-amplitude .

Since zeroes of F(·,·, β) are not isolated, we will introduce a phase condition.

The function F: (X×R)×R→(Y×R) deﬁned by

F(a, , β) =

F(a, , β)

1−(a0)0−2

∞

X

m=1

(a0)m

(51)

will be called the desingularized periodic solution map. Note that if F(a, , β )=0

then F(a, , β) = 0, which by previous discussion means ais uniquely associated

with a periodic solution with quasi-amplitude . In terms the original map F, this

implies F(a, β) = 0.

4.3.3 Properties of the map F

Before we proceed with the rigorous numerics, we will develop a few properties of

the map F. These will be useful later in the more analytical aspects of the branch

continuation proof.

Proposition 21. If F(a, β ) = 0 and a6= 0, then g(a) := (a0)0+2 P∞

m=1(a0)m6= 0.

Proof. Zeroes of F(·, β) correspond uniquely to solutions of (40). Suppose by way of

contradiction that (a0)0+ 2 P∞

m=1(a0)m= 0. By properties of Chebyshev polyno-

mials, this implies z0(1) = 0. Since the diﬀerential equations of the boundary-value

problem are smooth, this implies φ0≡0 and consequently, a0= 0. The bound-

ary condition then implies that φ1(−1) = 0 and by the same argument, we get

φ1≡0 and therefore a1= 0. A simple inductive argument then gives aq= 0 for

q= 0,...,u1−1. But a6= 0, which is a contradiction.

The following proposition guarantees that nontrivial zeroes of F(·, β) can be

transformed into zeroes of F(·, , β) for some . Speciﬁcally, we have the following.

Lemma 22. The transformation

d:a7→ 1

g(a)a, g(a)(52)

maps nontrivial zeroes aof F(·, β)to zeroes of F(·,·, β).

Proof. Let =g(a). By the previous proposition, 6= 0. The second component

of F(−1a, , β) is zero by design since it is precisely 1 −g(−1a) = 0. For the ﬁrst

26

component, it is simple to check that Fq,0(−1a, , β) = 0, while for n≥1,

Fq,n(−1a, , β ) = 2n−1(aq)n+ (γq(−1a, ))n+1 −(γq(−1a, ))n−1

=−12n(aq)n+α

2u2(−1aq)n+1 −(−1aq∗−1aq+δ)n+1

−α

2u2(−1aq)n−1−(−1aq∗−1aq+δ)n−1

=−12n(aq)n+α

2u2

[(aq)n+1 −(aq∗aq+δ)n+1]

−α

2[(aq)n−1−(aq∗aq+δ)n−1]

=−1F(a, β)

= 0.

We conclude F(−1a, , β) = 0 as claimed.

As we will see in the computer-assisted proof, in the bifurcation regime β≈e−αu

the map F(·,·, β) has an isolated zero with small quasi-amplitude ≈0. Also

of importance, we have a result that guarantees folds in a branch of nontrivial

zeroes of the map Fmust induce a fold in the analogous branch of zeroes of F.

To be more precise, we will say a map G:U×R→Ufor Ua metric space

has a branch point at (y, z) if there exist sequences y1,n , y2,n ∈Uand zn∈R

for n∈Nsuch that the following hold: y1,n 6=y2,n for all n, limn→∞ yj,n =y

for j= 1,2, and limn→∞ zn=z. Branch points include folds but also higher

codimension singularities.

Lemma 23. Let b∗∈Rbe given, and let F(a∗, b∗)=0for a∗6= 0. If the map

F:X×R→Yhas a branch point at (a∗, b∗), then F: (X×R)×R→(Y×R)

has a branch point at (d(a∗), b∗).

Proof. Let there be sequences aj,n for j= 1,2 such that a1,n 6=a2,n,aj,n →a∗and

a parameter sequence bn→b∗with F(aj,n, bn) = 0. Since g(a∗)6= 0 and ker(g) is

closed, the sequences aj,n are eventually (i.e. for large ﬁnite n) contained in the open

set X\ker(g). The map (52) is continuous and injective on X\ker(g), from which

it follows that Fhas a branch point at (d(a∗), b∗) as evidenced by the sequences

d(aj,n) and bn.

Remark 7. The requirement a∗6= 0 of the lemma is crucial. We know from Lemma

15 that Fhas a branch point at (0, e−αu)because there is a transcritical bifurcation

of periodic solutions there. The contrapositive to Lemma 23 is the result we will

make the most use of; any point that is not a branch point of Fis either not a

branch point of F, or corresponds to the trivial zero of F(·, β)by way of the map

(a, )7→ a.

4.3.4 Discretization of Fand the operators A†and A

Now that we have deﬁned the desingularized periodic solution map F: (X×R)×R→

(Y×R), we will need to perform the setup for Theorem 8 and compute the bounds

from (27)–(30). Strictly speaking, since the structure of Fis diﬀerent from our

general map Fof (23) from Section 3, we can not use the deﬁnition of A†and A

from Section 3.5 and will have to construct them from scratch. Thankfully, since

the structure of the map Fis so similar, the changes are not too dramatic. Before

we continue though, we should emphasize that now, the space (X×R) plays the

role of the space Xfrom Theorem 8, while the space (Y×R) plays the role of the

space Yfrom the theorem. We will use script for the operators A†and Ato avoid

confusion with A†and Afrom Section 4.2.3.

27

To begin, we deﬁne the ﬁnite-dimensional projections FN: (X×R)×R→(Y×R)

and FN: (XN×R)×R→(XN×R) much like (25):

FN(a, , β) = πN0

0 1 F(πNa, , β),FN(a, β) = iN0

0 1 FN(i−1

Na, , β).

We then deﬁne A†as the diﬀerential

A†=D(x,)FN(x0, 0, β0),

where x0∈XNand 0∈Rare such that

FN(x0, 0, β0)≈0.

That is, they are an approximate (i.e. numerical) zero of the ﬁnite-dimensional

projection at parameter β0.