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Analysis of Pandemic Closing-Reopening Cycles Using Rigorous Homotopy Continuation: A Case Study with Montreal COVID-19 Data


Abstract and Figures

Moving averages and other functional forecasting models are used to inform policy in pandemic response. In this paper, we analyze an infectious disease model in which the contact rate switches between two levels when the moving average of active cases crosses one of two thresholds. The switching mechanism naturally forces the existence of periodic orbits. In order to make unbiased comparisons between periodic orbits in this model and a traditional one where the contact rate switches based on more simplistic pointwise evaluations of active cases, we use a rigorous homotopy continuation method. We develop computer-assisted proofs that can validate the continuation and prove that the branch of periodic orbits has no folds and is isolated in the space of periodic solutions. This allows a direct, rigorous comparison between the geometric and quantitative properties of the cycles with a moving average threshold and a pointwise threshold. We demonstrate the effectiveness of the method on a sample problem modeled off of the COVID-19 pandemic in the city of Montreal.
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Abstract. Moving averages and other functional forecasting models are used to inform policy in
pandemic response. In this paper, we analyze an infectious disease model in which the contact rate
switches between two levels when the moving average of active cases crosses one of two thresholds.
The switching mechanism naturally forces the existence of periodic orbits. In order to make unbiased
comparisons between periodic orbits in this model and a traditional one where the contact rate
switches based on more simplistic pointwise evaluations of active cases, we use a rigorous homotopy
continuation method. We develop computer-assisted proofs that can validate the continuation and
prove that the branch of periodic orbits has no folds and is isolated in the space of periodic solutions.
This allows a direct, rigorous comparison between the geometric and quantitative properties of the
cycles with a moving average threshold and a pointwise threshold. We demonstrate the effectiveness
of the method on a sample problem modeled off of the COVID-19 pandemic in the City of Montreal.
Key words. rigorous numerics, periodic orbit, SIR model, moving average, COVID-19, contact
AMS subject classifications. 37M15, 37N25, 93C30
1. Introduction. Beginning in the mid-second quarter of 2020, governments
worldwide began efforts to reopen their economies [10,12,29] in response to slow-
ing of new COVID-19 cases. The various contagion management policies employed
worldwide measurably slowed the spread of the novel coronavirus [16], but threats of
a second wave [20,23,30,36] continue to loom as cases have been globally trending
upwards [3]. Until such time as a vaccine becomes in widespread use, closing and
reopening could become a regular part of the ebb and flow of daily life.
Mathematical models can provide some insight into how the closing and reopening
procedure might have an impact on the progression of a pandemic. There exist several
mathematical constructions that are suitable for describing the closing and reopening
of the economy based on active case numbers or other such metrics, but here we will
focus primarily on relays. A class of switched system — see [2,5,13] for background
— relay systems involve a partitioning of the phase space into multiple disjoint regions
with switching of the underlying vector field when an orbit enters or exits a region.
In modeling of closing and reopening, two thresholds [8] can be defined that represent
upper and lower bounds on the number of active cases. When the number of active
cases reaches the upper bound, interventions are applied (i.e., closing) that decrease
contact rates. If or when the number of active cases reaches the lower bound, the
interventions are softened or removed (i.e., reopening). The process then repeats
Delayed relays [33] involve a switching of a vector field when a delayed argument
crosses through a given threshold. In this way, the phase space can be partitioned
into two disjoint regions where stronger interventions (i.e., closing) and more lax re-
strictions (i.e., reopening) are applied [22,26]. Since the formalism involves a delayed
argument, the crossing of the threshold does not immediately cause a switching of the
vector field, and the end result is one typically observes oscillation about the thresh-
old. An advantage of this formalism is that natural time lags between collection of
McGill University, Department of Mathematics and Statistics, Montreal, Quebec, Canada.
data and enforcing distancing measures can be incorporated into the model.
In reality, epidemiological data is volatile in large part due to reporting errors
[17]. Moving averages are used to smooth out this volatility and are also used in
forecasting [15,18]. They have also been used by health officials to state targets for
control of the COVID-19 pandemic — for example, health officials in the Canadian
province of Quebec have stated [24] they want to avoid passing 20 daily new cases per
million people, where the metric of new cases is in fact a 7-day moving average. Since
the previous two relay model constructions make use of pointwise evaluations, be they
delayed (as in the delayed relay) or not, they can not fully take into account policy
changes based on moving averages (also called rolling averages) of active cases over
longer periods of time. On a related note, ARIMA models [1] and other forecasting
models have seen much application in forecasting of the COVID-19 pandemic [4,6,7,
31], and the predictions they provide make use of nonlocal properties of the time series
data, weighted and processed over long windows of observations. We therefore argue
it is more realistic and applicable to study relay models that incorporate nonlocal
functional dependence in vector field switching mechanisms.
1.1. Closing-reopening cycles. Recently, a global analysis of a relay SIR
model [8] with switching of contact rates and two thresholds was completed, and
conditions for the existence a a stable limit cycle with two switches per period were
determined. Therein, the intervention thresholds are based on pointwise numbers of
active cases (i.e., number of active cases at a single time instant). Formally, there are
two thresholds: Inat < Iint, and if I(t) denotes the number (or density) of infected
individuals, the contact rate βswitches whenever the relation
I(t)∈ {Inat, Iint }
is satisfied. Reaching the larger of the two thresholds causes the contact rate to
decrease to βint, the intervention contact rate, while reaching the lower threshold
causes it to switch to βnat, the natural contact rate.
Motivated by our interest in examining nonlocal functional dependence on epi-
demic intervention thresholds, we consider properties of limit cycles (referred to as
closing-reopening cycles in this publication) if the threshold crossing criteria is based
on a weighting between a (simple) moving average of active cases over several days
and a pointwise estimate of active cases on the current day. As in the former pub-
lication, we consider the intervention to be one that reduces the contact rate. More
formally, we consider the relation above replaced with
(1 α)I(t) + α
I(u)du ∈ {IR, IC},(1.1)
where IR< ICare reopening and closing levels, wis the time window for the rolling
average and 0 α1 is a parameter controlling the weight of the pointwise vs.
rolling average computation in the threshold. When α= 0 we get the pure pointwise
threshold of [8], while with α= 1 only the rolling average is used. The kinds of
questions we want to answer include the following.
1. (Existence) Under what conditions does a closing-reopening cycle exist for
2. (Comparison) Is there an objective way to compare closing-reopening cycles
for various weights α?
3. (Average and extrema) How does the average, maximum and minumum num-
ber of active cases of the closing-reopening cycle compare in the cases α= 0
and α= 1?
4. (Overshooting) To what extent can a closing-reopening cycle overshoot its
two thresholds if rolling averages are used (i.e., α= 1) and for how long?
We will from here on refer to the threshold condition (1.1) with α= 0 as the pointwise
threshold and α= 1 as the rolling average threshold.
To give as broad an answer as possible to these questions, we will focus exclusively
on the SIR model. We will answer these questions by computing closing-reopening
cycles, numerically continuing in the parameter α, and using a rigorous a posteriori
verification scheme to prove that the numerically computed cycles and connecting
branches exist, rather than attempting a detailed technical study of the equations us-
ing analytical means. Specifically, existence and comparison are covered in Section 3.4
through to Section 3.7, while averages will be covered in Section 3.8 and overshooting
in Section 3.9.
Before continuining, we emphasize that it is inappropriate to try and answer
these questions solely by a non-rigorous numerical exploration at distinct values of
the parameter α. For example, suppose we integrate a classical SIR model with
contact rate βthat switches whenever (1.1) is satisfied. Assume we do this at α= 0
and α= 1, and we numerically observe a limit cycle for each parameter. There is no
reason to suspect that these two limit cycles are in any way related, yet we might want
to compare such statistics as their averages or extrema. Computing other limit cycles
for intermediate values of α(0,1) might give more confidence if the cycles appear to
vary continuously with the parameter. However, we can still not be absolutely certain
that our comparison of these limit cycles is objective. Moreover, depending on the
numerical method used, it might not be provably true that the observed limit cycles
really exist. The following section outlines our idea to complete this exploration in a
mathematically rigorous way.
1.2. A rigorous homotopy continuation approach to closing-reopening
cycles. If a unique (up to phase shift) closing-reopening cycle exists in the base model
of [8], we first compute this orbit and represent it using truncated Chebyshev series.
For the purposes of this introduction, let us refer to the truncated orbit as PN
0and the
“true” orbit as P0. Since such a cycle P0consists of a Lipschitz (in fact, piecewise C
with a only two points where the derivative is discontinuous) curve, the representation
of PN
0exists by classical results of approximation theory [34] and the coefficients of the
series exhibit geometric decay when restricted to the two smooth segments. We then
use a Chebyshev spectral collocation procedure to transform the problem of finding a
closing-reopening cycle into one of computing a zero of a nonlinear function fin an
infinite-dimensional sequence space. Formally, we write f:X×[0,1] X, where X
is the sequence space and f(P, α) = 0 means that Pis a closing-reopening cycle for
the system with thresholds defined informally by the relation (1.1) where IR< ICare
reopening and closing levels, wis the amount of time in the rolling average and tis
time. By taking a finite-dimensional projection XNof Xby truncating the number
of modes, we can get a finite-dimensional projection fN:XN×[0,1] XNsuch that
0,0) = 0. We can then use numerical continuation to compute a discrete branch
α, α) for α∈ {α0, . . . , αM}for α0= 0 and αM= 1 such that fN(PN
αi, αi) = 0.
If the numerical method converges, then it seems likely that a closing-reopening
cycle exists for the convex parameters αisince PN
αishould be a “good approximation”
to a true closing-reopening cycle. We can then provide approximate answers to ques-
tions 2–4 by numerical quadratures and zero-finding. However, these results are not
rigorous because we have not proven that a true closing-reopening cycle exists, and
we do not have precise information about how good an approximation the numerical
branch of solutions {(PN
αi, αi), i = 1, . . . , M }is. This means that any answers to
questions 2–4 are subject to unknown numerical error. These errors are nontrivial
to quantify because although we have accurately computed the branch of closing-
reopening cycles for the finite-dimensional projection, we can only interpret this as
an approximate solution by embedding it into the infinite-dimensional space X.
To resolve this, we make use of validated continuation methods based on the radii
polynomial approach [19]. This is a computationally explicit variant of the Newton-
Kantorovich theorem that is amenable to computer-assisted proof. By performing
all of the associated computations in interval arithmetic using the INTLAB library
[32] in MATLAB, we rigorously control roundoff error and are able to prove that
the numerically computed branch is proximal to a true branch (in X) of closing-
reopening cycles, with explicitly computable error bounds. These error bounds can
then be propagated to the associated solutions of questions 2–4, giving answers with
rigorous error bounds. Uniqueness of the branch is a convenient side product of
the method, so we are able to make faithful comparisons between the orbits in the
active cases threshold and the convex combination cases in the sense that there is a
unique continuation from one to the other with respect to the convex parameter α.
More succinctly, we prove the existence of a smooth homotopy continuation from a
closing-reopening cycle at α= 0 to one at α= 1.
To summarize, we therefore propose validated numerics as a tool to directly an-
swer the questions from the previous section concerning existence and comparison of
closing-reopening cycles under the different threshold definitions, as outlined in the
previous paragraph. We will be able to answer the average and overshooting question
using post-processing of the data from this method.
From a mathematical perspective, our approach to parameterizing these periodic
orbits has elements in common with the work of Gameiro, Lessard and Ricaud [11]
for crossing periodic orbits in Filippov systems. The main difference here is that we
have a relay (i.e., switched) system, and the switching rule (1.1) involves dependence
on past values of the solution. The switching manifolds (there are two: one for each
of the thresholds ICand IR) are in fact codimension one submanifolds of an infinite-
dimensional function space rather than a subset of Rn.
1.3. Outline of the paper. The structure of the paper is as follows. In Sec-
tion 2, the model is explicitly formulated and some elementary properties of closing-
reopening cycles are stated and proven, including a boundary-value problem that
such cycles must satisfy. Section 3concerns the rigorous numerics, and it is split into
several parts. We provide necessary background on Chebyshev series in Section 3.1
before converting the boundary-value problem for closing-reopening cycles into a zero-
finding problem in Section 3.2. The associated finite-dimensional projection derived
in Section 3.3. The radii polynomial approach, which is used to prove branches of
zeroes, is stated in Section 3.4. In Section 3.5, we prove sufficient conditions — what
we call sharpness conditions — that guarantee a correspondence between zeroes of
the nonlinear map and closing-reopening cycles. The bounds for the radii polynomial
method are computed in Section 3.6, and a computational approach to the sharpness
conditions is presented in Section 3.7. The analysis of extrema, averages and over-
shooting along closing-reopening cycles is completed in Section 3.8 and Section 3.9.
In Section 4, we apply our results to the COVID-19 pandemic in the City of Montreal.
A discussion and conclusion follow in Section 5and Section 6.
1.4. Notation. For a fixed w > 0, we denote C([w, 0],R+) the vector space of
nonnegative real-valued continuous functions defined on the interval [w, 0]. We will
often write it simply as Cwhen the explicit dependence on wis understood.
If f:IRfor a closed interval I, then the function ftCis defined by
ft(θ) = f(t+θ); it exists if [tw, t]I. This is a standard convention in functional
differential equations [14].
For a normed vector space X, the symbol B(X) refers to the set of bounded linear
operators on X. If AB(X), then ||A||B(X)denotes its operator norm. The symbol
IXdenotes the identity operator on a vector space X, and 0Xdenotes the zero map.
If A:UVand B:XY, then C= diag(A, B) : U×XV×Yis defined by
C(u, x)=(A(u), B(x)). The symbol Br(x) denotes the open ball of radius rcentered
at xX. If x= [min(x),max(x)] is an interval vector with min(x)max(x) in the
usual partial order on Rnand t0, we define tx = [tmin(x), t max(x)]. If uand v
are two intervals, we write u<vif and only if sup(v)<inf(u).
2. Formulation of the model and elementary properties. As stated in the
introduction, we will focus on a relay-like SIR model with two thresholds defined by
convex relationships between pointwise and rolling average active cases. At all times,
the dynamics (in the sense of right-derivatives) follow the SIR model
S= Λ βSI µS
I=βSI (µ+γ)I.
Sand Irepresent the number of susceptible and infected humans, respectively. Λ
is a constant recruitment rate, µthe per capita death rate, βthe combined con-
tact/infection rate in units of 1/(humans ·time) and γthe combined death and recov-
ery rate. We assume immunity is permanent, so the removed (recovered or dead due
to infection) class is decoupled and has not been included. For brevity, we will refer
to βsimply as a contact rate, and γthe removal rate. All of these parameters are
assumed positive. The parameter βcan change according to a rule we will introduce
Let some w > 0 be fixed and define a parameterized functional g:C×[0,1] R
according to
g(φ, α) = (1 α)φ(0) + α
By definition, g(φ, α) is a convex combination of the value of the function φat zero and
its average. With this definition, we can introduce the relay dynamics. Let βc< βr
be two contact rates, and let 0 < IR< ICrepresent thresholds of reopening (IR)
and closing (IC). Informally, the “closing/reopening relay model” is the ordinary
differential equation (2.1) together with a rule that states when the contact rate
parameter βswitches. Specifically, the contact rate switches whenever the relationship
(1.1) is satisfied. When g(It, s) crosses (or is tangent to) IC, closing of the economy
begins (or continues if it was already in this state) and the contact rate switches to
βc. Conversely, when g(It, s) crosses (or is tangent to) IR, reopening begins and the
contact rate switches to βr. More formally, we define solutions of the relay model as
follows. Figure 1may aid in visualization.
Definition 2.1. Let S0R+,φCand σ0∈ {0,1}. Let (S, I)be continuous
functions S: [0, b)R+and I: [w, b)R+with S(0) = S0and I0=φ. Define
the switching sets
ΣR={t(0, b) : g(It, α) = IR},ΣC={t(0, b) : g(It, α) = IC},
Fig. 1.Schematic drawing of a solution of the relay model in the case α= 0. The two
vector fields are drawn simultaneously, corresponding respectively to β0(green arrows) and β1(blue
arrows). The upper and lower black lines respectively represent I=ICand I=IR. On black dashed
lines the solution satisfies σ= 0 and on the dotted line it satisfies σ= 1. Between each switching,
the solution follows the vector field corresponding to the value of σ.
and switching function σ: [0, b)→ {0,1}as follows:
σ(t) =
σ0, t = 0
0,sup(ΣR[0, t]) <infC[t, b))
1,sup(ΣC[0, t]) <infR[t, b)).
(S, I)is a solution of the relay model for thresholds IR< IC, contact rates β1< β0
and convex parameter αif it is a solution of the piecewise-continuous system (2.1)
for t[0, b)with β=βσ(t). That is, it differentiable and satisfies the ODE except at
those times where σhas a discontinuity. The data (S0, φ, σ0)is the initial condition.
Remark 2.2. In Definition 2.1, we have opted to label the respective contact rates
βrand βcinstead as β0and β1. This is to avoid confusion later in Definition 2.3 and
Section 2.1. Also, since Iis continuous, gis a continuous functional and IR< IC,
the switching function is indeed well-defined and continuous from the right except
possibly at t= 0.
We will not discuss details such as existence, uniqueness and continuability of
solutions for the opening/closing relay model. The latter follows under the category
of switched systems with state-dependent [21,37] switching. Suffice it to say, given
an initial condition, a unique solution exists and can be continued to b=. That
solutions can be continued to the whole positive real line is a consequence of the
finite separation IR< ICand the fact that solutions of (2.1) are ultimately unformly
bounded for β=βσ(t)for any switching signal σ.
Definition 2.3. A solution (S, I)the relay model is a closing-reopening cycle
with period pif S,Iand σare p-periodic, and the associated switching function has
exactly two discontinuities in the interval [0, p]. A closing-reopening cycle is normal-
ized if σ(0) = 0,σ(c)=1for some c(0, p), and the discontinuities of the switching
function in [0, p]are precisely at cand p. In this case, cis the closing time and p
(equivalently, zero) is the reopening time.
By definition, if a closing-reopening cycle exists, then it can always be normalized
in a unique way. We will typically assume that our closing-reopening cycles are
normalized, and this can always be done without loss of generality by an appropriate
phase shift.
2.1. Slow closing-reopening cycles. Some closing-reopening cycles are easier
to find (and visualize) than others, and in the sections that follow we will devise a
numerical method to compute and continue branches of such cycles.
Definition 2.4. A normalized closing-reopening cycle is slow if the following are
cwand pwc.
I(pw)ICand IRI(cw).
We say the cycle is sharp if the above inequalities are strict.
The following lemma demonstrates how slow closing-reopening cycles can be construc-
ted using special solutions of (2.1) with a piecewise-constant parameter β..
Lemma 2.5. Let p, c Rand (S, I) : [0, p]R2
+be a continuous, piecewise-
differentiable function. Suppose the following are satisfied.
S.1 (S, I)satisfies (2.1)with parameter βron [0, c), and satisfies (2.1)with pa-
rameter βcon [c, p)
S.2 g(Ic1, α) = ICand g(Ip, α) = IR.
S.3 S(p) = S(0) and I(p) = I(0).
S.4 cwand pwc.
S.5 Iis monotone increasing on [0, c)and IRI(cw).
S.6 Iis monotone decreasing on [c, p)and I(pw)IC.
The image of (S, I)coincides with the image of a slow closing-reopening cycle, and
it is sharp if S.4–S.6 are satisfied with strict inequalities, Iis strictly increasing on
[0, c)and is strictly decreasing on [c, p).
Proof. Extend (S, I) to a periodic function on R. Write G(t) = g(It, α) and
consider the derivative
G0(t) = d
dtg(It, α) = (1 α)I0(t) + α
If t[0, c] then I0(t)>0. On the other hand, if t[w, c] then I(t)> I (tw), so we
can be certain that G0(t)>0 on [w, c]. As for the restriction to [0, w], we claim that
G(t)< IC. To see why, observe first that for t(w, w], we I(t)< IC; verifying
this is straightforward and uses the monotonicity properties of I. Consequently, for
t(0, w],
G(t) = (1 α)I(t) + α
I(u)du < (1 α)IC+αIC=IC,
while G(0) = IR< IC. We have therefore shown that G(t)< ICon [0, w] and Gis
strictly increasing on [w, c], from which it follows that the switching sets associated to
(S, I) satisfy ΣC(0, c] = {c}and ΣR(0, c] = . By a symmetric argument, one can
show that ΣC(c, p] = and ΣR(c, p] = {p}.Let σ0= 0 and consider the switching
function σassociated to (S, I ) with the initial point σ0. The associated switching
function σtherefore has exactly two discontinuities cand pin the interval [0, p], with
σ(c) = 1 and σ(0) = 0. (S, I) therefore defines a slow, normalized closing-reopening
2.2. A boundary-value problem for closing-reopening cycles. The defi-
nition of slow closing-reopening cycle and conditions S.1–S.3 of Lemma 2.5 define a
1.4495 1.45 1.4505 1.451
1.4605 1.461 1.4615 1.462
Fig. 2.Two cycles with different convex parameter α. On the left, α= 0 and the relay model
can be identified with a finite-dimensional switched ODE system with state-dependent switching.
On the right, α= 1 and the state space must be considered as infinite-dimensional. The segments
labeled 0,1,2,3 in equation (2.2)correspond respectively to the green, cyan (dashed), red, and blue
(dashed) curves, and they are also labeled in the figure. Arrows indicate time orientation. The black
horizontal dashed lines correspond to I∈ {IC, IR}. The cycles in this figure are the extremal points
of the branch proven in Theorem 4.1, and each of them is sharp according to Definition 2.4.
natural boundary-value problem (BVP). Introduce a family of R2
+-valued functions
(Sk, Ik) for k= 0,1,2,3. The domain of these functions are as follows:
dom(Sk, Ik) =
[0, c w], k = 0
[cw, c], k = 1
[c, p w], k = 2
[pw, c], k = 3.
These functions will define the restrictions of a candidate (S, I) for a slow closing-
reopening cycle to each of the above subintervals of [0, p]. See Figure 2for a visual-
Define β(k) = βrfor k= 1,2 and β(k) = βcfor k= 3,4. Define three boundary
L[f] = f(inf(dom(f))), R[f] = f(sup(dom(f))), G[f] = (1α)R[f]+ α
Symbolically set S4=S0and I4=I0, and consider the following boundary-value
Sk= Λ β(k)SkIkµSk, k = 0,...,3
Ik=β(k)SkIk(µ+γ)Ik, k = 0,...,3,
R[Sk] = L[Sk+1], k = 0,...,3,
R[Ik] = L[Ik+1], k = 0,...,3,
G[I1] = IC,
G[I3] = IR.
The following proposition holds.
Proposition 2.6. If (Sk, Ik)for k= 0,1,2,3have the domains (2.2)and satisfy
the previous boundary-value problem, then the function (S, I) : [0, p]R2
+defined by
(S, I)(t) =
(S0(t), I0(t)), t [0, c w),
(S1(t), I1(t)), t [cw, c)
(S2(t), I2(t)), t [c, p w),
(S3(t), I3(t)), t [pw, p],
satisfies conditions S.1–S.4 of Lemma 2.5.
If a solution of the boundary-value problem can be computed, then one can verify
that its image is a closing-reopening cycle by subsequently checking conditions S.5
and S.6 of Lemma 2.5. This is a separate problem that we will solve on the computer
in Section 3.7. At present, we are more interested in the BVP. Before moving on, we
will perform changes of variables to transform the domains of each of (Sk, Ik) to the
interval [1,1]. This transformation will facilitate the conversion to a zero-finding
problem in Section 3.2.
To complete the change of variables, write
(S0, I0)(t)=(˜
cw1,(S1I1)(t) = ( ˜
(S2, I2)(t) = ( ˜
pwc,(S3, I3) = ( ˜
for ( ˜
Ik) : [1,1] R2
+. The, define scaling factors zk=zk(c, p) according to
2, k = 0
2, k = 2
2, k ∈ {1,3}.
If one completes the change of variables, then, dropping the tildes, we get the boundary-
value problem
R[Sk] = L[Sk+1]
R[Ik] = L[Ik+1]
0 = IC(1 α)R[I1]α
0 = IR(1 α)R[I3]α
The following lemma is a direct consequence of the previous derivation, Lemma 2.5
and Proposition 2.6.
Lemma 2.7. Suppose (Sk, Ik):[1,1] R2
+for k= 0,1,2,3is a solution of the
boundary-value problem (2.5)with zk0. If I0and I1are monotone increasing with
IRI0(1) and I2and I3are monotone decreasing with I2(1) IC, then the union
of the images of (Sk, Ik)for k= 0,1,2,3coincides with the image of a slow closing-
reopening cycle. The cycle is sharp if the inequalities are strict and the monotonicity
is strict.
Remark 2.8. One might ask why we represent a candidate closing-reopening cycle
using four smooth segments instead of two. The reason is because with four segments,
the boundary conditions are very simple since the bounds of the integrals are always
the same. However, if two segments are used, the boundary conditions involve the
variables cand pand they become non-polynomial. For example, the analogue of the
first condition is
0 = IC(1 α)R[I1]αc
for I1now representing the (infected component) segment of the cycle that runs from
time t= 0 to time t=c. When we move to computer-assisted proofs, this boundary
condition is much harder to work with. It is for this reason that we allow ourselves
to suffer the extra cost in dimension afforded by a representation in four smooth
3. Rigorous numerics for closing-reopening cycles. This section contains
the main theoretical details concerning computation, continuation and computer-
assisted validation of closing-reopening cycles. We start by converting the boundary-
value problem (2.5) into a zero-finding problem in an infinite-dimensional function
space. We then determine a finite-dimensional projection and discuss how to com-
pute its numerical zeroes. Then, we outline the radii polynomial method, which is used
for rigorously proving branches of zeroes of the infinite-dimensional problem based on
a numerical branch. Next, we determine a posteriori sharpness conditions that can
be uniformly checked along a numerically validated branch to prove that the zeroes
uniquely determine closing-reopening cycles. We then demonstrate explicitly (i.e.,
by deriving the appropriate bounds) how to implement these checks in a computer.
We conclude with an analysis of how to rigorously compute extrema, averages and
overshooting times along numerically computed closing-reopening cycles.
3.1. Chebyshev series and the space `1
ν.As suggested in the introduction, we
will be solving the boundary-value problem (2.5) using Chebyshev series expansions.
The following background is contained in [34]. Recall that any sufficiently smooth
(specifically, Lipschitz continuous) function f: [1,1] Rcan be represented as a
uniformly convergent Chebyshev series
f(t) = f0+ 2
for Tnthe nth Chebyshev polynomial of the first kind. The coefficients fncan be
computed using the formula
If fis real-analytic on [1,1] and (some C-analytic continuation) is bounded on the
Bernstein ν-ellipse in the complex plane — that is, the closed ellipse with foci at ±1
and sum of semimajor and semiminor axes equal to ν— then the quantity
||f||ν:= |f0|+ 2
is finite. If we write ωn= 1 for n= 0 and ωn= 2νnfor n1, we can write it more
compactly in the form
n=0 |fn|ωn.
If fis real-analytic on [1,1] then there necessarily exists some ν > 0 such that
the above is true. If f0(t) is written as a Chebyshev series
f0(t) = f0
0+ 2
then the coefficients fnand f0
nare related by the equation
n+1, n 1.(3.2)
Also, for n2 the Chebyshev polynomials admit the indefinite integrals
ZTn(t)dt =1
n+ 1 Tn1(t)
The Chebyshev polynomials satisfy the identities Tn(1) = 1 and Tn(1) = (1)nfor
all n0. Finally, in the scope of differentiation of Chebyshev series, the tridiagonal
operator T:`1
νdefined by
T(a)n=0, n = 0
an+1 an1, n > 0
will be quite useful.
Let `1
νdenote the normed vector space of sequences {an:nN}bounded with
respect to the ||·||νnorm. This is a Banach space. If a, b `1
ν, define their convolution
abaccording to
It is a standard exercise to check that (`1
ν,) is a Banach algebra; that is, :`1
νis a continuous bilinear map with ||ab||ν≤ ||a||ν||b||ν.
3.2. Conversion from BVP to zero-finding problem. Write each of the
functions Skand Ikfrom (2.5) as a Chebyshev series. Specifically, make the expansions
Sk(t) = ak,0+ 2
Ik(t) = bk,0+ 2
In the following, the symbols akand bkwill refer to the sequences {ak,n :nN}and
{bk,n :nZ}. The products of Chebyshev series induce convolutions at the level of
their coefficients. We have
Sk(t)Ik(t) = (akbk)0+ 2
where (akbk)n:= (ak,·bk,·)n. Substituting (3.5) and (3.6) into (2.5), using the
relations (3.2) and (3.3), and making the identification a4a0and b4b0, we get
the following set of equations.
bk,n =(TΨ1
k(ak, bk, c, p))n
k(ak, bk, c, p))n, n 1,
bk,0+ 2
n=1 ak,n
bk,n =ak+1,0
bk+1,0+ 2
0 = IC(1 α) b1,0+ 2
0 = IR(1 α) b3,0+ 2
where Ψk:`1
νis the representation of the right-hand side of the
ODE in (2.5) on the Chebyshev coefficients, depending on the unknown closing time
cand period p. In coordinates,
k(a, b, c, p)n
k(a, b, c, p)n=Λnβ(k)(ab)nµan
β(k)(ab)n(µ+γ)bnzk(c, p) := Φ1
k(a, b)n
k(a, b)nzk(c, p),
with Λ0= Λ and Λn= 0 for n > 0. To be precise, (3.7) corresponds to the differential
equations, (3.8) to the R-Lboundary conditions, and (3.9) and (3.10) the convex
threshold conditions from the BVP (2.5).
We can transform (3.7)–(3.10) into a zero-finding problem on an appropriate
Banach space. Define L:`1
ξ(for `1
ξa Banach space to be introduced in Lemma
3.1) by
L(a)n= 2nan.(3.12)
Next, define a linear map H:`1
νwith one-dimensional range by H(a, b)n= 0
for n > 0 and
H(u, v)0=u0v0+ 2
(un+ (1)n+1vn).
Finally, we set X= (`1
ν)8×R×R. We will sometimes write it as Xνwhen we want to
emphasize the choice of ν(see in particular the proof of Theorem 3.5). The norm on
Xmust be chosen carefully, as it is typical for the Sand Icomponents of solutions
to the BVP (2.5) to differ by several orders of magnitude, which can result in poor
conditioning. To facilitate this, we let W= (W1, W2, W3)R3
+denote a weight
vector. This will be explicitly chosen whenever we want to do a computer-assisted
proof. For φ= (a0, b0, . . . , a3, b3, c, p)X, we define the norm
||φ||X= max{W1max{||a0||ν,...,||a3||ν}, W2max{||b0||ν,...,||b3||ν}, W3|c|, W3|p|}.
The role of the weights will be further elaborated upon in Section 4.
We consider a formal nonlinear map F:X×[0,1] Ywith Ya Banach space
that we will subsequently identify:
F(a0, b0, . . . , a3, b3, c, p, α) =
L(a0) + H(a0, a1) + TΨ1
0(a0, b0, c, p)
L(b0) + H(b0, b1) + TΨ2
0(a0, b0, c, p)
L(a1) + H(a1, a2) + TΨ1
1(a1, b1, c, p)
L(b1) + H(b1, b2) + TΨ2
1(a1, b1, c, p)
L(a2) + H(a2, a3) + TΨ1
2(a2, b2, c, p)
L(b2) + H(b2, b3) + TΨ2
2(a2, b2, c, p)
L(a3) + H(a3, a1) + TΨ1
3(a3, b3, c, p)
L(b3) + H(b3, b1) + TΨ2
3(a3, b3, c, p)
IC(1 α)(b1,0+ 2 Pn1b1,n )αb1,0Pn2b1,n 1+(1)n
IR(1 α)(b3,0+ 2 Pn1b3,n )αb3,0Pn2b3,n 1+(1)n
The boundary conditions (3.8) have been encoded into the linear map H. By con-
struction, if it so happens that (a0, b0, . . . , a3, b3, c, p) is a zero of Ffor parameter α,
then this (a0, b0, . . . , a3, b3) will satisfy (3.7)–(3.10) for closing time cand period p.
Subsequently, if the monotonicity requirements of Lemma 2.7 can be checked, this will
uniquely define a slow closing-reopening cycle through the identifications (3.5)–(3.6)
and Lemma 2.7. It can also be verified that Fis also smooth. The following lemma
summarizes this fact and characterizes an appropriate codomain for F. The proof is
simple and is omitted.
Lemma 3.1. Let ν > 1be fixed and consider the norm || · ||ωon real-valued se-
quences defined as follows:
||a||ξ=|a0|+ 2
Let `1
ξdenote the vector space of real-valued sequences for which the norm ||·||ξis finite.
This is a Banach space, and with Y= (`1
ξ)8×R×Requipped with the induced max
norm, the map F:X×[0,1] Yis well-defined and C. If F(a0, b0, . . . , c, p, α) = 0,
then this data defines a solution of the boundary-value problem (2.5)by way of the
equivalence (3.5)(3.6), and vice-versa.
Remark 3.2. A zero of F(equivalently, a solution of the BVP (2.5)) does not
necessarily define a closing-reopening cycle. Indeed, up to the identification with the
Chebyshev series (3.5)–(3.6), it only defines a solution that satisfies S.1–S.3 of Lemma
2.5. The conditions S.4–S.6 need to be checked after the fact. We show how this can be
accomplished numerically with rigorous error bounds in Section 3.7. Unsurprisingly,
S.4 requires the least effort.
3.3. Finite-dimensional pro jection and numerical continuation. Let N >
0 be a fixed integer. Define a projection map πN:`1
νaccording to
πN(a)n=an, n N
0n > N,
Then, define a projection πN: (`1
πN(a0, b0, . . . , a3, b3, c, p) = (πNa0, πNb0, . . . , πNa3, πNb3)
and a complementary projector π: (`1
ν)8via π=I(`1
ν)8πN. Define also
˜πN:XXto be the projection operator ˜πN= diag(πN, IR2), with ˜π=IX˜πN.
Let XN=πN(X)×R2. We will write XN
νif we want to emphasize the value of
ν. Introduce the “computational isomorphism” iNon π(`1
ν) by
iN(a) =
and extend this to an isomorphism iN:XNR8(N+1)+2 via
iN(a0, b0, . . . , a3, b3, c, p) =
Whenever we want to think of an element of Xwith zero tail (i.e. all zero Chebyshev
ceofficients above mode N) as being a vector in some finite-dimensional space, we can
apply the isomorphism iN. Similarly, we can apply the inverse
N:XNXN, XN:= R8(N+1)+2
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 or computed with finite-dimensional vectors) while
quantities without bars will typically be analytical. Define the maps FN:X×RX
and FN:XN×RXNby
FN(x, s) = ˜πNF(˜πNx, α), F N(x, s) = iNFN(i1
Nx, α).(3.15)
FNis the nonlinear map Ffrom (3.14) truncated to NChebyshev modes, while
FNis its representation in the finite-dimensional space XN. By Lemma 3.1, each of
these maps is C. Since FNis a nonlinear map on XN=R8(N+1)+2 and FNF
pointwise, it should be expected that numerical zeroes of FNwill, when embedded in
XN, generate approximate zeroes of F.
3.3.1. Numerical computation of zeroes. In practice, we compute such nu-
merical zeroes first at convex parameter α= 0 by implementing Newton’s method
for FNin double arithmetic. We initialize the method at a random guess with coeffi-
cients uniformly distributed over an appropriate hypercube and run for 300 iterations
or until blowup (defect greater than 103) or numerical convergence (defect less than
1010). If a solution is not found, another random guess is taken. This process of
random searches generally converges to a numerical zero in less than a second on
modern hardware. We then manually inspect the output closing time and period c
and p, ensuring that they satisfy cwand pwc. If this is true, the numerical
zero is held as a candidate and refined further with Newton’s method until the defect
is less than 1015.
3.3.2. Numerical continuation of zeroes. Once a candidate zero for α= 0
has been computed and refined, we implement natural parameter continuation in α
over a mesh of size 102until α= 1. The previous zero is used as the predictor for
the next zero, and we use Newton’s method to correct it until the defect is less than
1015. Tangent predictors might yield faster convergence, but even with this crude
implementation the branch computation is not too lengthy. We do not expect folds
in the solution branch, and so do not implement a pseudo-arclength continuation.
3.4. The radii polynomial approach. Once a branch (xαi, αi) of zeroes for
FNhas been computed for a mesh {αi:i= 1, . . . , M }with α0= 0 and αM= 1,
we would like to obtain rigorous results about the embedded branch in XN. We
accomplish this with validated continuation and a method that is sometimes called
the radii polynomial approach. The relevant theorem quoted below is a summary of
the validated continuation result in [19].
Theorem 3.3. Let Xand Ybe Banach spaces. Let x0, x1∈ X and λ0, λ1R.
Define the predictors
xs= (1 s)x0+sx1, λs= (1 s)λ0+1.
Let FCk(X ×R,Y)for some k1and assume there exist bounded linear operators
AB(X,Y)and AB(Y,X), with Ainjective, and Y0,Z0,Z1and Z2(r)0such
||AF (xs, λs)k|XY0,s[0,1](3.16)
||A[DxF(x0, λ0)A]||B(X)Z1
||A[DxF(xs+δ, λs)DxF(x0, λ0)]||B(X)Z2(r),s[0,1], δ Br(0) ⊂ X.
Define the radii polynomial
p(r) = Z2(r)r+ (Z1+Z01)r+Y0.(3.20)
If there exists r0>0such that p(r0)<0, then there exists a Ckfunction
˜x: [0,1] [
such that Fx(s), λs)=0. Furthermore, these are the only zeroes of Fin the tube
Ss[0,1] Br0(xs).
In the above theorem, the idea is that x0and x1are numerical zeroes of Fat param-
eters λ0and λ1. Here, the term “numerical” is somewhat arbitrary. If the assertions
of the theorem hold, the line segment (xs, λs) for s[0,1] is an approximate branch
of zeroes of Fin the sense that the tube of radius r0around (xs, λs) contains exactly
one zero for each value of s. In this way, the number r0gives a precise statement
about how far away the numerical zeroes can be from a true zero.
Remark 3.4. Theorem 3.3 is a rigorous continuation result for two numerical ze-
roes x0and x1. The theory in [19] guarantees that if the radii polynomial theorem
proves a branch from (x0, λ0) to (x1, λ1), and then separately from (x1, λ1) to (x2, λ2),
then then there is a Ckbranch from parameters λ0to λ2that is r0-close to the union
of the two predictor branches, with r0being the smallest radius from the two proofs.
In this way, one can obtain global continuation of branches of zeroes.
To apply the theorem we will need to first construct the operators Aand Afor our
map Ffrom (3.14). After that, we will construct the Yand Zbounds analytically. In
practice, they will be verified along a numerical branch of zeroes by computing them
with the interval arithmetic package INTLAB. This will allow for a rigorous control
of roundoff error. If all of this can be done, then we will have proven the existence of
a numerical branch of zeroes of F, which in turn corresponds to a branch of solutions
(Sα, Iα) of (2.1) for a switching signal σα, such that conditions S.1 through S.3 of
Lemma 2.5 are satisfied. The operators Aand Aand the bound will be constructed
in Section 3.6. To check that these truly coincide with slow closing-reopening cycles,
conditions S.4–S.6 must also be checked. We show how to do this in Section 3.7.
3.5. A posteriori sharpness conditions and hyperbolicity. Here we show
that if the radii polynomial proves the existence of a branch of zeroes of Fand
some a posteriori checks can be successfully completed, then the associated branch
of closing-reopening cycles is hyperbolic in the sense of static bifurcation theory. We
briefly mention a definition: we say the inequality f(s)< g(s) for real-valued functions
f, g : [0,1] Rholds uniformly if there exists h > 0 such that h < g(s)f(s) for all
Theorem 3.5. Suppose Theorem 3.3 successfully proves the existence of a branch
of zeroes (xs, αs)of F:X×[0,1] Yfor the map in (3.14)parameterized by the
continuation parameter s[0,1]. Let t7→ Ik(t;s)for k= 0,1,2,3denote func-
tions generated by the identification (3.6)for each continuation parameter s, and let
(cs, ps)be the crossing time and period. Suppose the following sharpness conditions
are satisfied.
S.4 cs> w and psw > csuniformly for s[0,1].
S.5 IR< I0(1; s)and I2(1; s)< ICuniformly for s[0,1].
S.6 d
dt I0(t;s)>0,d
dt I1(t;s)>0,d
dt I2(t;s)<0and d
dt I3(t;s)<0for t(1,1),
uniformly for s[0,1].
Let t7→ Φ(t;s) := (S(t;s), I(t;s)) denote the branch of functions for convex parameter
αsdefined by Proposition 2.6 after inverting the transformation (2.3), and extend it
to t[w, )by periodicity. For each s,t7→ Φ(t;s)is a sharp closing-reopening
cycle for convex parameter αs. Also, the following isolation properties hold.
Isolation in X: There exists r > 0such that if t7→ Θ(t;s)is another normal-
ized, sharp closing-reopening cycle for convex parameter αs, its representative
θsXsatisfies ||xsθs||X> r.
Isolation in C: If t7→ Θ(t;s)is any other branch of (not necessarily slow or
sharp) closing-reopening cycles for convex parameter αs, there exists δ > 0
such that sups[0,1] ||Φ(t;s)Θ(t;s)||> δ.
Proof. Clearly, the sharpness conditions imply S.4–S.6 from Lemma 2.5, which
together with Theorem 3.3 prove t7→ Φ(t;s) defines a sharp closing-reopening cycle
for convex parameter αsfor each s. We will prove only isolation in C, since the
result on isolation in Xis similar (and easier). By way of contradiction, suppose
Θn: [w, )R2is a sequence of closing-reopening cycles and sn[0,1] is a
sequence with limn→∞ ||ΘnΦ(·;sn)||= 0. Without loss of generality, we may
assume Θnis normalized. Since Φ(·;s) is sharp and slow uniformly in sby the
assumptions of the theorem, it follows that Θnis sharp for nsufficiently large, say
nm. Decomposing Θnfor nmaccording to Proposition 2.6 and applying the
change of variables (2.3), let θnXdenote the associated coordinates in the Banach
space X. It follows that F(θn, αs) = 0. From the radii polynomial, we know that for
some r > 0, the ball Br(xs) contains exactly one zero in Xνfor some ν > 1. However,
limn→∞ θn=xsin Xν. To see this, first observe that each segment k= 0,1,2,3
of t7→ Φ(t;s) and t7→ Θn(t) can be identified with the solution of a polynomial
ordinary differential equation. In particular, since Θn(t)Ψ(t;s) uniformly, there
exist positive constants Zand sequences ∆unand ∆znsatisfying limn→∞ un= 0,
limn→∞ zn= 0, such that the norms un(t) of the segments (defined for t[1,1])
for k= 0,1,2,3 of the difference Ψ(t;s)Θn(t) satisfy the integral inequality
un(t)un+ (tt0)∆znΛ + ZZt
2βrun(s)2+ (2µ+γ)un(s)ds
for 1t0t1. Recall that βc< βr. Extending to complex arguments, let
ω(t) be a path in the complex plane (parameterized by real t) satisfying ω(0) = t0.
The solutions of the complexified ODE that coincide on the real axis with one of the
aforementioned solutions then satisfy the inequality
un(ω(t)) un+ (1 + |ω(t)|)∆znΛ + ZZt
2βrun(ω(s))2+ (2µ+γ)un(ω(s))ds.
Since ∆un0 and ∆zn0 as n→ ∞, Gronwall’s inequality can be used to obtain
uniform bounds (in kand t0[1,1]) for z7→ un(z) on the Bernstein ν-ellipse, for
any ν>1 and nlarge enough. It follows [34] that the segments of Φ(t;s)Θn(t)
are elements of Xνand that (θnxs)0 in Xν. Since xsXν, we must have
θnXνfor nlarge enough, and limn→∞ θn=xsin Xν. This is a contradiction, since
it implies the existence of two distinct zeroes of F(·, αs) in the ball Br(xs).
Remark 3.6. Analogously to Remark 3.4, this result can be globalized. If the
sharpness conditions of Theorem 3.5 are satisfied separately along each segment of
a global branch of zeroes of F, then the global branch of closing-reopening cycles is
If the conditions of Theorem 3.5 are satisfied, the closing-reopening cycle is iso-
lated in a tube of radius r > 0 with respect to the topology on X, up to identification
by the map (which we have implicitly defined) that sends a cycle from Cinto Xby
way of normalization (Definition 2.4) and the segment decomposition. In the space
C, we have the slightly weaker result that no other branch of closing-reopening cycles
can ever intersect the branch, but we do not have a uniform tube enclosure.
If the entire branch is hyperbolic for the range α[0,1], then there are no folds
along the branch nor are there solution-crossings from α= 0 through to α= 1. As a
consequence, we will have a unique, continuous correspondence between the closing-
reopening cycle at α= 0 and α= 1 by way of the rigorous continuation, thereby
giving a conditional (based on the success of the computer-assisted proof) answer to
the questions of existence and comparison from the introductory section.
3.6. Implementation of the radii polynomial approach. In this section we
define the operators Aand A, as well as construct the Yand Zbounds necessary to
apply Theorem 3.3.
3.6.1. Definition of the operators Aand A.Suppose x0XNis an ap-
proximate (numerical) zero of FN(·, α0). Denote x0=i1
N(x0); this symbol has the
same interpretation as x0in Theorem 3.3. Define the finite-dimensional linear map
(interpreted as a matrix)
A=DxFN(x0, α0).
Next, by an abuse of notation we define L: (`1
ξ)8according to
L(a0, b0, . . . , a3, b3)=(La0, Lb0, . . . , La3, Lb3),
where on the right-hand side Lis the operator from (3.12). Then define A:XY
NAiN˜πN+ diag(L, 0R2π.(3.21)
It is straightforward to verify that Ais well-defined.
To construct A:YX, we first define a linear map L+:`1
ν, with
L+(a)n=0, n = 0
2nan, n > 0.
Then, extend it to a map L+: (`1
L+(a0, b0, . . . , a3, b3)=(L+a0, L+b0, . . . , L+a3, L+b3).
Next, let Abe a numerical inverse of A; that is, a matrix such that ||IAA|| ≈ 0.
We can now define A:YXas follows:
NAiN˜πN+ diag(L+,0R2π.(3.23)
The following lemma is now a straightforward consequence of the definitions of A
and A.
Lemma 3.7. Aand Aare well-defined and bounded, and Ais injective provided
Ahas maximal rank.
3.6.2. The Yand Zbounds: preparation. With Lemma 3.7 at our disposal,
we can move on to the computation of the bounds Yand Z. First, some preparation.
In what follows, we denote (xs, αs)XN×Rthe predictors and parameters
xs= (1 s)i1
Nx1, αs= (1 s)α0+1
for s[0,1], given numerical zeroes (x0, α0) and (x1, α1) of FN. Whenever we need
to extract individual components, we will write
xs= (as,0, bs,0, . . . , as,3, bs,3, cs, ps).(3.24)
We will now suppress the use of bars on the xterms; it will be understood that
xsrepresents both the infinite-dimensional object in XNwith zeroes in its tail, and
the finite-dimensional object in XNobtained by cutting off the zero modes with the
operator iN:XNXN. We also set
so that xs=x0+sxand αs=α0+sα.
We will define a norm on XNby way of the computational isomorphism: ||x||XN=
||iNx||X. The induced operator norm can then be evaluated as needed using one of
the technical lemmas that appears in Appendix A.
To facilitate the computation of partial derivatives of the map F, we will make a
few additional definitions now. By an abuse of notation define T: (`1
T(a0, b0, . . . , a3, b3) = (T a0, T b0, . . . , T a3, T b3), where on the right-hand side Tis the
usual tridiagonal operator from (3.4). Then, define H:X(`1
H(x)=(H(a0, a1), H(b0, b1), H(a1, a2), H (b1, b2), H(a2, a3)H(b2, b3), H (a3, a1), H(b3, b1)).
The range of this operator is 8-dimensional, as each component has range in π0(`1
R. Next, define G:X×[0,1] ×[0,1] R2by
G(x, q1, q2) =
(1 q1)(b1,0+ 2 Pn1b1,n ) + q2b1,0Pn2b1,n 1+(1)n
(1 q1)(b3,0+ 2 Pn1b3,n ) + q2b3,0Pn2b3,n 1+(1)n
and write I= [ ICIR]|. The reason for introducing the parameters q1and q2will
be apparent soon. Define Ψ : X(`1
Ψ(x) = (Ψ1
0(a0, b0, c, p),Ψ2
0(a0, b0, c, p),...,Ψ1
3(a3, b3, c, p),Ψ2
3(a3, b3, c, p)),(3.25)
where we recall the components Ψj
kare defined in (3.11). We can then compactly
write F:X×[0,1] Yin the form
F(x, α) = L(x) + H(x) + TΨ(x)
I − G(x, α, α).(3.26)
The expression (3.26) allows us to simplify the expression for the differential
DxF(x, α). Each of L,Hand Gis linear in x(for αfixed). In block operator form,
we get
DxF(x, α) = L+H+T DxΨ(x)
G(·, α, α).(3.27)
Similarly, we can write the differential DαF(x, α) in block form:
DαF(x, α) = 0
G(x, 2,1) .(3.28)
We will also compute terms of Fup to order three (note that Fis indeed cubic). The
only terms that require a detailed look are those involving the functions Ψ. Because
of the structure (3.25) of the operator Ψ, it suffices to compute the differentials of the
individual Ψj
k, which only involve four variables. Let h= (u, v)(`1
ν)2. For brevity
we will define (λ1, λ2) = (µ, µ +γ). Then
k(a, b, c, p)h=zk(c, p)β(k)(akvk+ukbk)λjuk, j = 1
β(k)(akvk+ukbk)λjvk, j = 2
k(a, b, c, p)=Φj
k(a, b, c, p)zk
k(a, b, c, p)[h1, h2]=(1)jzk(c, p)β(k)(u1
k(a, b, c, p)h=zkβ(k)(akvk+ukbk)λjuk, j = 1
β(k)(akvk+ukbk)λjvk, j = 2
k(a, b, c, p)[h1, h2]=(1)jzkβ(k)(u1
while all partial derivatives with respect to (c, p) of order 2 and above are zero. The
gradient zkcan be computed directly from (2.4).
3.6.3. Computation of the Y0bound. The first bound, Y0, is a proxy for the
numerical defect. Using the definition of FNand A, we can write
AF (xs, αs) = i1
NAF N(xs, αs) + diag(L+,0R2)˜πF(xs, αs)≡ Y(1)
To control the Y(1)
0term along the numerical branch, we expand FN(xs, αs) with
respect to sas a third-order Taylor expansion at s= 0. Since (xs, αs) = (x0, α0) +
s(∆x, α) and |s| ≤ 1, we get
ADmFN(x0, α0)x
For implementation, these are computed using the the explicit formulas from (3.27),
(3.28) and (3.29)–(3.33).
For Y2
0, many of the terms coming from Fcancel because xsXNand the post-
multiplication with πkills all modes lower than N. The result is that in the norm
on X,
0= max
k,j Wj||L+TΨj
k(as,k, bs,k , cs, ps)||νmax
k,j Wj
k(as,k, bs,k , cs, ps))n|,
where the max runs for j= 1,2 and k= 0,1,2,3. Ψj
kcontains convolutions and linear
terms, and since their inputs are elements of πN(`1
ν), all modes above n= 2N+ 1
vanish. Thus, Ψj
k(as,k, bs,k , cs, ps)n= 0 for n2N+ 2. Taking into account the
tridiagonal operator, the infinite sum above terminates after n= 2N+ 2. If we define
the intervals
ak=a0,k + [1,0](a1,k a0,k), bk=b0,k + [1,0](b1,k b0,k ),(3.35)
c=c0+ [1,0](c1c0), p =p0+ [1,0](p1p0),
then using the computational isomorphisms and the above discussion,
ADmFN(x0, α0)x
+ max
k,j Wj
Nak, iNbk, c, p))n|
is a bound satisfying (3.16). The expression for (3.36) requires only a finite number
of computations and can be rigorously bounded above using interval arithmetic.
Remark 3.8. For our particular parameter choices, it was not necessary to use the
third-order expansion for the Y(1)
0term. Using the mean-value theorem for integrals
in Banach space, we can also get
0≤ ||AF N(x0, α0)||XN+Z1
ADF N(x0+tsx, x0+tsα)sx
≤ ||AF N(x0, α0)||XN+ sup
ADF N(x0+zx, α0+zα)x
In our implementation, this bound was used instead of the one in (3.34).
3.6.4. Computation of the Z0bound. Z0measures the quality of Aas an
approximate inverse of A. Because of the structure of Aand A, the error here is
entirely due to numerical inversion of A. The calculation of IAAis straightforward
and the following bound on the operator norm is tight:
An explicit formula for the operator norm || · ||B(XN)is provided in Appendix A.
3.6.5. Computation of the Z1bound. Making use of the identity A=
iN˜πNDxFπNx0, α0), we can carefully compute
DxF(x0, α0)A= ˜πNH+T DxΨ(x0)
G(·, α0)π+ ˜πT DxΨ(x0)
We have deliberately partitioned the result as a sum of a map with range in XNand
another with range in ˜π(X). Applying Aon the left, we get
A[DxF(x0, α0)A] = i1
NAiN˜πNH+T DxΨ(x0)
G(·, α0)π+ ˜πL+T DxΨ(x0)
0R2:= Z(1)
To proceed further we will need to compute the differential DxΨ(x). For cleanli-
ness of presentation, it suffices for us to calculate partial derivatives of the functions
k. Let h= (u, v)(`1
ν)2. Then
k(a, b, c, p)h=zk(c, p)β(k)(akvk+ukbk)λjuk, j = 1
β(k)(akvk+ukbk)λjvk, j = 2
k(a, b, c, p)=Φj
k(a, b, c, p)zk
The gradient zkcan be computed directly from (2.4). With this done, we will first
compute a bound for Z(2)
1. Let hXsatisfy ||h|| ≤ 1. Define for k= 0,...,3 and
j= 1,2
1) + λjW1
j+||∇zk||1· ||Φj
Na0,k, i1
Nb0,k, c0, p0)||ν.
k,j Wj
n=N+1 T DxΨj
k(a0,k, b0,k , c0, p0)hn
2(N+ 1) max
k,j Wj||D(a,b)Ψj
k(a0,k, b0,k , c0, p0)h||ν+||D(c,p)Ψj
k(a0,k, b0,k , c0, p0)h||ν
2(N+ 1) max
k,j Wjβ(k)(||a0,k ||νW1
1) + λjW1
j+||∇zk||1· ||Φj
k(a0,k, b0,k , c0, p0)||ν
2(N+ 1) max
k,j Z(2,k,j)
To understand the meaning of the subscripts, recall the indexing convention for our
points xsalong the predictor branch (3.24). To get the bound we used the Banach
algebra and the bound ||T||B(X)2ν+ν1, which can be inferred from Appendix A
and the definition of the norm on X.
Next, we need a uniform (in ||h||X1) bound for Z(1)
1h. To facilitate this, we will
further decompose the partial derivative D(a,b)Ψ as follows: with h= (u0, v0, . . . , u3, v3, d, q),
k(a, b, c, p)h=ˆ
k(a, b, c, p)h+rj
k(c, p)h
k(a, b, c, p)h= (1)jzk(c, p)β(k)(akvk+ukbk),
k(c, p)h=zk(c, p)µuk, j = 1
(µ+γ)vk, j = 2
ψ(a, b, c, p)h= ( ˆ
0(a0, b0, c, p)h, ˆ
0(a0, b0, c, p)h, . . . , ˆ
3(a3, b3, c, p)h, ˆ
3(a3, b3, c, p)h,
r(c, p)h= (r1
0(c, p)h, r2
0(c, p)h, . . . , r1
3(c, p)h, r2
3(c, p)h).
We can therefore write
D(a,b)Ψ(a, b, c, p)h=ˆ
ψ(a, b, c, p)h+r(c, p)h.(3.39)
For brevity, set ˜πh=h= (u
0, v
0, . . . , u
3, v
3,0,0). We can then decompose
1further as
ψ(x0)h+T D(c,p)Ψ(x0)h
NAiN˜πNH(h) + πNT r(c0, p0)h
G(h, α0)
:= Z(1,1)
The additional factor of πNwe have added in front of the T r(c0, p0) term is not
superfluous (and is valid, because of the post-composition with ˜πN) and will be used
later. For k= 0,1,2,3 and m= 0, . . . , N + 1, define
ak(m) = max
2νn|(iNa0,k)nm|,bk(m) = max
ck(m) = |zk(c0, p0)β(k)|(W1
2ak(m) + W1
1bk(m)),h= (c0,c0,...,c3,c3,0,0).
Using the technical estimates from Appendix A, the (weighted) norm on X, the
triangle inequality and the fact D(c,p)Ψ(x0)h= 0, we can get the bound
To get a bound for Z(1,2)
1h, we first define a few interval vectors in XN. Let en
denote the nth standard basis vector in RN+1 and set
H= [ e|
1[1,1] ·· · e|
1[1,1] 0 0 ]|, G = [ 0··· 0 [1,1] [1,1] ]|
r= [ e|
N+1[1,1] · ·· e|
N+1[1,1] 0 0 ]|.
Let W: (`1
ν)8×R2be the diagonal operator defined by
W(a0, b0, . . . , a3, b3, c, p) = (W1a0, W2, b0, . . . , W1a3, W2, b3, W3c, W3p).
One can then verify
νN+1 W1i1
NH, πNT r(c0, p0)h
νN+1 W1i1
G(h, α0)1
νN+1 W1i1
where ξ= (µ+γ) maxk|zk(c0, p0)|/2. It follows that
νN+1 ||AW1H||XN+||AW1G||XN+ξ||AW1r||XN,
where we have abused notation and identified W1with iNW1i1
N. Combining the
previous results, we conclude that
Z1= (2ν+ν1)||abs(A)h||XN+2ν+ν1
2(N+ 1) max
k,j Z(2,k,j)
νN+1 ||AW1H||XN+||AW1G||XN+ξ||AW1r||XN
is a suitable Z1bound.
3.6.6. Computation of the Z2bound. Let δBr(0) X. Then
DxF(xs+δ, αs)DxF(x0, α0) = T(DxΨ(xs+δ)DxΨ(x0))
G(·, αs) + G(·, α0):= "TZ(1)
To avoid excessive use of indices, we will abuse notation and refer to each component
(in `1
νor R) of δby also using the symbol δ. This should not cause too much confusion.
We bound the difference of the DxΨ first, as this is the most tedious. Let hX
satisfy ||h||X1, and set Z(1)
2h= (ξ0,1, ξ0,2, . . . , ξ3,1, ξ3,2) for ξk,j `1
ν. If we set
h= ((hk,j ), u, v) for u, v Rand hk,j `1
νfor indices k= 0,...,3 and j= 1,2, we
can write
ξk,j = (D(a,b)Ψj
k(as,k +δ, bs,k +δ, cs+δ, ps+δ)D(a,b)Ψj
k(a0,k, b0,k , c0, p0))[hk,1, hk,2]
+ (Φj
k(as,k +δ, bs,k +δ)Φj
k(a0, b0))zk[u v ]|
k,j +ξ(2)
k,j zk[u v ]|.
We can bound each of these quantities in turn. Taking into account ||h||X1, ||δ||X
rand the weights in the space X, we ultimately get |∇zk[u v ]|| ≤ W1
k,j ||νβ(k)h||akbk||ν+||a0,k bk+b0,k ak||ν+rW1
1(||b0,k||ν+||bk||ν)· ··
k,j (r),
k,j ||ν≤ |zk(c0, p0)|β(k)W1
2(rW 1
1+||ak||ν) + W1
1(rW 1
+ abs(zk)[ c+W1
1r). . .
k,j (r).
Observe that each of the ˆ
k,j (r) for q= 1,2 can be interpreted as degree two polynomi-
als in r. As for Z(2)
2hR2, the following bound is straightforward and its derivation
is omitted: for hXwith ||h|| ≤ 1,
Now, let
µ=A[DxF(xs+δ, αs)DxF(x0, α0)]h= ((µk,j ), µ)
for µ= (µ,1, µ,2)R2. Combining the previous two estimates, the definition of
Aand using the technical bounds from Appendix A, we can get
||µk,j ||νX
m,` +W1
m,`) + A(1,2)
k,j W1
2(N+ 1) (ˆ
k,j +W1