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Computation of centre manifolds and some codimension-one bifurcations for

impulsive delay diﬀerential equations

Kevin E.M. Churcha,∗, Xinzhi Liub

aDepartment of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada

bDepartment of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada

1. Introduction

Without question, one of the most well-known techniques to study bifurcations in dynamical systems is the

centre manifold reduction. While certainly helpful in analyzing the dynamics near nonhyperbolic equilibria in

ﬁnite-dimensional systems of ordinary diﬀerential equations, the technique is comparatively more powerful in

the inﬁnite-dimensional case – such as when studying partial- and/or functional diﬀerential equations – since

the reduced-order equations and the centre manifold itself are often ﬁnite-dimensional in many applications.

One may consult the references [8, 12, 13, 14, 32] for background.

Of interest to us at present is the local analysis of discontinuous dynamical systems of the form

˙x=L(t)xt+f(t, xt), t 6=tk(1)

∆x=J(k)xt−+gk(xt−), t =tk.(2)

This is an impulsive delay diﬀerential equation. Equations of this type are useful for modeling systems that

exhibit both memory eﬀects and discrete jumps in state at speciﬁc times or, in a limiting sense, experience

abrupt changes in state on very short time scales. For theoretical background, one may consult the references

[2, 3, 26, 27]. Systems of the type (1)–(2) are abundant in applications. Frequently, the continuous-time

dynamics of (1) represent the evolution of a system of interest, while the discrete-time dynamics of (2) are

imposed so as to control the dynamics in a desirable way. For example, in impulsive stabilization, the goal

is to design the right-hand side of (2) so that a given state of the dynamical system — often an equilibrium

point or other reference signal — becomes stable. In large networks, impulsive pinning control is a practical

method based on this idea [21, 23, 33]. In infectious disease modeling, the impulse eﬀect may represent

an intervention such as a vaccination programme [11, 36]. Internet worm propagation models and control

strategies based on pulse quarantine have also been proposed [34, 35]. Impulsive dynamical systems have

also been used to model integrated pest control strategies; see [4] and the refernces cited therein.

Emergent behaviour such as synchronization and periodic bursting in dynamical systems featuring delays

are often born from Hopf bifurcation points. These are parameters at which an equilibrium point is on

the cusp of stability and instability; see [10, 22, 30] for some recent applications of the Hopf bifurcation to

bursting and synchronization in chemical and network models with delays. Investigations into such emergent

behaviour in delayed networks with impulses have to our knowledge not been undertaken using techniques

from bifurcation theory. There is a large body of literature concerning synchronization of impulsive complex

networks featuring delays – see [9, 15, 28] for a select few recent advances – but they are primarily concerned

with suﬃcient conditions for synchronization being imposed at a controller level, rather than organically as

a result of parameter variation as is typical of the bifurcation theory approach.

Local Hopf bifurcations in delay diﬀerential equations are typically identiﬁed after performing a centre

manifold reduction. After this has been accomplished, the Hopf bifurcation theorem of ordinary diﬀerential

equations becomes applicable by considering the dimensionally reduced dynamics on the centre manifold.

Our primary goal with this publcation is to extend the centre manifold reduction scheme to delay diﬀerential

equations with impulses that may exhibit periodic right-hand sides. With this theory, we will then study

∗Corresponding author

Email addresses: k5church@connect.uwaterloo.ca (Kevin E.M. Church ), xzliu@uwaterloo.ca (Xinzhi Liu)

Preprint submitted to Elsevier July 18, 2018

analogues of the generic Hopf and saddle-node bifurcation conditions that are typically stated in terms of

the eigenvalues of the linearization at a candidate equilibrium point.

It has been proven [6] under faily general settings that ordinary impulsive delay diﬀerential equations (1)–

(2) possess local centre manifolds at nonhyperbolic equilibria. Speciﬁcally, the centre manifold is a function

C:RCRc→ RCR,where the domain is the centre ﬁber bundle in the base space RCR of right-continuous

regulated functions, and it was proven that this centre manifold is ﬁbrewise smooth – that is, φ7→ C(t, φ) is

smooth for φ∈ RCRc(t) – with the same level of smoothness as the nonlinearities of (1)–(2). Background

information on these dynamical systems and on the centre manifold, including a new temporal smoothness

result (Theorem 2.2), are provided in Section 2.

Section 3 (outlined in Section 1.1) is devoted to the centre manifold reduction. Simple analytically

tractable examples (outlined in Section 1.3) are provided in Section 4. Our saddle-node and Hopf-type

bifurcation theorems (outlined in Section 1.4) are stated in Section 5 and examples are provided. A discussion

follows in Section 6, and Section 7 ends with a conclusion. All proofs and some calculations associated to

the examples are deferred to Appendix Appendix A.

1.1. Centre manifold reduction

Arguably, the most common version of the equation (1)–(2) that appears in applications is one that

features only discrete delays:

˙x=A(t)x(t) +

`

X

j=1

Bj(t)x(t−rj) + f(t, x(t), x(t−r1), . . . , x(t−r`)), t 6=tk(3)

∆x=C(k)x(t−) +

`

X

j=1

Ej(k)x(t−rj) + gk(x(t−), x(t−r1), . . . , x(t−r`)), t =tk.(4)

Even more common is when the system is periodic and/or the right-hand side is autonomous (time-invariant).

With this in mind, our ultimate goal with this publication is to demonstrate how to perform a centre manifold

reduction for periodic system with discrete delays of the form (3)–(4), subject to a few technical assumptions

to be introduced in Section 2. In practice, the process amounts to completing a sequence of steps.

1. Identify a nonhyperbolic equilibrium, and compute a basis matrix Φt=QteΛtof the centre ﬁber

bundle in the form of a Floquet decomposition. This amounts to determining the Floquet exponents

and computing a basis of (generalized) Floquet eigensolutions of the linearization.

2. Compute the action of the projection operator Pc(t) : RCR → RCRc(t) on the element χ0, relative to

the basis Φt.

3. Write down the evolution equation (an impulsive partial diﬀerential equation) satisﬁed by the centre

manifold.

4. Substitute a Taylor expansion ansatz for the centre manifold with time-varying coeﬃcients. Solve the

equations up to a given order nof expansion for the coeﬃcients.

5. Substitute the nth order expansion into the dynamics equation on the centre manifold.

At the end of the process, the result is a ﬁnite-dimensional system of ordinary impulsive diﬀerential equations

that captures the dynamics of all uniformly small solutions of the original inﬁnite-dimensional system. By

taking a given parameter as an additional state variable, one can study bifurcations from the nonhyperbolic

ﬁxed point.

Section 3 covers the steps of the centre manifold reduction. Speciﬁcally, Section 3.1 is devoted to the ﬁrst

step of the reduction: the computation of Floquet exponents and eigensolutions associated to a nonhyper-

bolic equilibrium. Section 3.3 discusses the monodromy operator, its resolvent, and the computation of the

projection operator Pc(t). The evolution equation for the centre manifold is derived in Section 3.4, where

we ﬁrst introduce an alternative representation of the centre manifold that is more suitable for our purposes.

Section 3.5 and Section 3.6 are devoted to approximation of the centre manifold and the truncated dynamics

on the centre manifold, and so represent the ﬁnal steps 4 and 5 of the centre manifold reduction.

2

1.2. Formal veriﬁcation of temporal diﬀerentiability of the centre manifold

Many of the calculations described earlier – in particular the derivation of the evolution equation satisﬁed

by the centre manifold – assume that the centre manifold (or at least the chosen representation) and its

derivatives in the state variable are all themselves diﬀerentiable in time. This requirement is stated precisely

in Theorem 2.2. Proving this turns out to be a fairly nontrivial matter, and the proof is deferred to Section

Appendix A.1.

1.3. Examples

The centre manifold of an impulsive functional diﬀerential equation is a ﬁbre bundle with base space RCR,

and is therefore a temporally-varying structure that can be though of as a diﬀerent embedded submanifold

of RCR for each ﬁxed moment of time. To contrast, in the case of delay diﬀerential equations without

impulses, the centre manifold of a nonhyperbolic equilibrium is a typically constructed as a ﬁnite-dimensional

embedded submanifold of the space C([−r, 0],Rn)⊂ RCR and therefore does not depend on time. However,

in an appropriate coordinate system, it is possible to visualize how the centre manifold changes when a

small impulsive perturbation is introduced. This is illustrated by way of two separate analytically tractable

examples in Section 4.

1.4. Applications to bifurcation theory

Section 5 contains two application of our results to bifurcation theory. Using centre manifold reduction,

we study analogues of the fold (saddle-node) and Hopf bifurcation for periodic impulsive delay diﬀerential

equations. As we will see, the classical saddle-node bifurcation of a single eigenvalue on the imaginary axis

generalizes naturally to an analogous one for Floquet exponents, and the generic bifurcation pattern is that

of a pair of periodic orbits persisting in a parameter half-plane. The Hopf bifurcation condition, however,

leads to a generic bifurcation pattern consisting of the birth of an invariant cylinder in the space S1×RCR.

Numerical examples are provided to reinforce the results.

2. Background

The purpose of this section is to introduce common notation present in the article and to provide back-

ground information on impulsive delay diﬀerential equations and centre manifold theory.

2.1. Notation

If f:R→Xfor a Banach space X, we denote

d+

dt f(t) = lim

→0+

f(t+)−f(t)

the right derivative of fat t.For functions of several variables we have the analogous deﬁnition for partial

derivatives. For a function f:R×X→Ywith normed spaces Xand Y, we say that fis quasi piecewise

continuous if t7→ f(t, x) is continuous except at times tkwhere it is continuous from the right and has limits

on the left for each x∈X,x7→ f(t, x) is continuous for each t∈R, and fis continuous on R\{τk:k∈Z}×X.

For f:R→X, we write f∈P Cmif fis continuous from the right except at times tkwhere it has limits on

the left and, in addition, the same is true for the right derivatives (d+)jffor j= 1, . . . , m.

We denote RCR := RCR([−r, 0],Rn) the space of functions f: [−r, 0] →Rn) that are continuous from

the right and possess limits on the left, the symbol RCR being an acronym for right-continuous-regulated,

of which these functions are; see [17]. Considered as a normed vector space with the supremum norm

||f|| = supt∈[−r,0] |f(t)|and |·| the Euclidean norm, it is complete. We will usually denote this Banach space

by RCR, supressing the domain [−r, 0] when it is clear from context. RCR1⊂ RCR consists of all functions

φ∈ RCR such that d+φexists and are themselves elements of RCR. Also, let Gdenote the set of functions

g: [−r, 0) →Rnthat posess limits on the left and right (ie. regulated functions).

For x:R→Rn, we deﬁne xt: [−r, 0] →Rnfor each t∈Rby the equation xt(θ) = x(t+θ). We then

deﬁne the uniform one-point left-limit

xt−(θ) = lims→0−xt(s), θ = 0

xt(θ), θ < 0,

3

provided the limit exists. Typically, the context will be that xt∈ RCR. Similarly, for a function t7→ xt∈

RCR, we deﬁne the regulated left-limit by

x−

t(θ) = lim

→0+xt−(θ),

provided this limit exists.

For Z⊂R, the symbol χZwill always denote the identity-valued indicator function:

χZ(θ) = 0, θ /∈Z

I, θ ∈Z,

with Ithe identity on Rn. The domain of χZwill be either stated or implied. The evaluation fuctional

evx:RCR → Rfor x∈[−r, 0] is deﬁned by evxφ=φ(x).

The set Mn×m(Rk) denotes the set of n×mmatrices with entries in the vector space Rk. If A∈

Mn×m(Rk), we Ai,j denotes the entry in its ith row and jth column. The notation [A]a:bdenotes the

(b−a+ 1) ×mmatrix whose rows coincide with rows athrough bof A.

For a j-dimensional multi-index ξ= (ξ1, . . . , ξj) where ξi∈N, we deﬁne |ξ|=Piξi. For u∈Rjand a

j-dimensional multi-index ξwith |ξ|=m, the ξpower of uis uξ=uξ1

1···uξj

j. If Xis a vector space and

U∈Xj, we similarly deﬁne Uξ∈Xmby

Uξ= (U1, . . . , U1, U2, . . . , U2, . . . , Uj...,Uj)

where the factor Uiappears ξitimes. If u∈Xand m∈N, we deﬁne um∈Xmby um= (u, . . . , u).

For a vector multi-index ξ= (ξ1, . . . , ξj) where each ξi∈ {e0

1, . . . , e0

k}for {e0

i:i= 1, . . . , k}the standard

ordered basis of Rk∗, we write |ξ|=jand deﬁne (u1·· · uj)ξfor ui∈Rkas follows:

(u1···uj)ξ= (ξ1u1)···(ξjuj).

For vectors in Rnwritten in component form, (u1, . . . , un)·(v1, . . . , vn) = Piuividenotes the standard

inner product.

If A∈Rm×nand B∈Mn×k(R`), we deﬁne the overloaded product A∗B∈Mm×k(R`) by the equation

[A∗B]i,j =

n

X

u=1

Ai,uBu,j .(5)

It is readily veriﬁed that if A∈Rm×mis invertible, then A∗B=Cif and only if B=A−1∗C. Moreover, ∗

satisﬁes the Leibniz law

d

dtA(t)∗B(t) = d

dtA(t)∗B(t) + A(t)∗d

dtB(t)

whenever t7→ A(t) and t7→ B(t) are diﬀerentiable. Clearly, when `= 1 the overloaded product reduces to

the standard matrix product.

2.2. Impulsive delay diﬀerential equations

Consider the impulsive delay diﬀerential equation

˙x=L(t)xt+f(t, xt), t 6=tk

∆x=J(k)xt−+gk(xt−), t =tk,

with L(t), f (t, ·), J(k), gk(·) : RCR → Rn. A function x: [s−r, α)→Rnis said to be a classical solution

of the impulsive delay diﬀerential equation if it satisﬁes the diﬀerential equation at all but ﬁnitely-many

points in any compact subset of [s, ∞), and satisﬁes the jump condition (the second equation) at all times

tk∈(s, α). Note that this equation reads as ∆x=x(tk)−x(t−

k). Such a function satisﬁes the initial

condition xs=φ∈ RCR for some given φ∈ RCR and s∈Rif the previous equality holds. There are several

conditions guaranteeing the existence and uniqueness of classical solutions satisfying given initial conditions,

4

but to discuss centre manifolds the most applicable notion is the mild solution [6]. These are deﬁned in terms

of the linear part:

˙x=L(t)xt, t 6=tk(6)

∆x=J(k)xt−, t =tk.(7)

Deﬁnition 2.1. Denote t7→ X(t, s)φ∈Rnthe (unique) solution in the extended sense of the linear system

(6)–(7) satisfying the intial condition xs=φ. The evolution family is the family of bounded linear operators

U(t, s) : RCR → RCR deﬁned for t≥sby

[U(t, s)φ](θ) = X(t+θ, s)φ.

A function x: [s−r, α)→Rnis a mild solution of (1)–(2) if

xt=U(t, s)xs+Zt

s

U(t, µ)χ0f(µ, xµ)dµ +X

s<ti≤t

U(t, ti)χ0gi(xt−

i)

for all t∈[s, α), where the integral is interpreted in the weak (Pettis) sense.

The following notions of spectral separation and invariant ﬁbre bundles are necessary for certain state-

ments concerning centre manifolds. They are taken verbatim from [6].

Deﬁnition 2.2 (Deﬁnition 3.3.5 [6]).Let U(t, s) : X→Xbe a family of bounded linear operators deﬁning a

forward process on a Banach space X— that is, U(t, s) = U(t, v)U(v, s)for all t≥v≥sand U(t, t) = IX.

We say that Uis spectrally separated if there exists a triple (Ps, Pc, Pu)of bounded projection-valued functions

Pi:R→ L(X)with Ps+Pc+Pu=Isuch that the following hold.

1. There exists a constant Nsuch that supt∈R(||Ps(t)|| +||Pc(t)|| +||Pu(t)||) = N < ∞.

2. The projectors are mutually orthogonal; Pi(t)Pj(t) = 0 for i6=j.

3. U(t, s)Pi(s) = Pi(t)U(t, s)for all t≥sand i∈ {s, c, u}.

4. Deﬁne Ui(t, s)as the restriction of U(t, s)to Xi(s) = R(Pi(s)). The operators Uc(t, s) : Xc(s)→

Xc(t)and Uu(t, s) : Xu(s)→Xu(t)are invertible and we denote Uc(s, t) = Uc(t, s)−1and Uu(s, t) =

Uu(t, s)−1for s≤t.

5. The operators Ucand Uudeﬁne all-time processes on the family of Banach spaces Xc(·)and Xu(·).

Speciﬁcally, the following holds for all t, s, v ∈R.

Uc(t, s) = Uc(t, v)Uc(v, s), Uu(t, s) = Uu(t, v )Uu(v, s).

6. There exist real numbers a < 0< b such that for all > 0, there exists K≥1such that

||Uu(t, s)|| ≤ Keb(t−s), t ≤s(8)

||Uc(t, s)|| ≤ Ke|t−s|, t, s ∈R(9)

||Us(t, s)|| ≤ Kea(t−s), t ≥s. (10)

Deﬁnition 2.3 (Deﬁnition 3.3.6 [6]).Let U(t, s) : X→Xbe spectrally separated. The nonautonomous sets

Xi={(t, x) : t∈R, x ∈Xi(t)}

for i∈ {s, c, u}are termed respectively the stable, centre, and unstable ﬁber bundles associated to U(t, s).

Xs⊕Xuis the hyperbolic ﬁber bundle.

In the following, our Banach space will typically be RCR, so RCRs,RCRcand RCRuwill symbolically

represent the stable, centre, and unstable ﬁber bundles of our evolutionary system. These ﬁber bundles play

the roles of the stable, centre and unstable subspaces associated to linear system of autonomous ordinary

diﬀerential equations or delay diﬀerential equations.

The following requirement will be assumed throughout most of the paper. This requirement is not merely

technical: see Section 6.

5

Deﬁnition 2.4. The system (3)–(4) satisﬁes the overlap condition if Ej(k)=0and

gk(x0, xj−1, xj+ξ, xj+1 , . . . , x`) = gk(x0, . . . , xj−1, xj, . . . , x`)

for all ξ∈Rn, whenever tk−rj=tifor some i<k.

Qualitatively, the overlap condition states that the jump functionals do not have “memory” of the state

of the system at previous jump times. The condition fails when a lagged impulse time tk−rjand another

impulse tioverlap, and the system’s jump response at time tkis nontrivial with respect to the overlapping

times. We will also need a deﬁnition of periodicity for the model system.

Deﬁnition 2.5. The model system (3)–(4) is T-periodic if L(t)and f(t, ·)are T-periodic, there exists c∈N

such that J(k)and gk(·)are c-periodic, and the sequence of impulses satisﬁes tk+c=T+tkfor all k∈Z.

The number cis the number of impulses per period.

A ﬁnal preliminary result about the evolution family U(t, s) when the linear part (6)–(7) is periodic is

provided below. The result is taken from [Lemma 7.2.1, Theorem 7.2.5 [6]].

Proposition 2.1. If (6)–(7) is T-periodic, then the evolution family U(t, s)is spectrally separated. Moreover,

if T≥r, then RCRc(t+T) = RCRc(t)and Pc(t+T) = Pc(t).

2.3. Centre manifolds

In [6] the existence and smoothness of centre manifolds was proven for impulsive retarded functional

diﬀerential equations, as was a reduction principle. We specialize the result to our model system (3)–(4),

with a very minor modiﬁcation that will later be more suited to our needs. One may refer to Theorem 5.4.1,

Theorem 5.4.2, Theorem 6.1.1, Corollary 8.2.1.1, Corollary 8.2.1.2 and Theorem 8.3.1 therein.

Theorem 2.1. Consider the T-periodic model system, let 0∈Rnbe an equilibrium point and assume the

nonlinearities f(t, ·)and gk(·)are Cm(uniformly in t∈R) with vanishing ﬁrst Fr´echet derivatives at 0.

Also, let the ﬁrst mFr´echet derivatives of f(t, ·)be right-continuous in tfor other variables ﬁxed, t7→ Bj(t)

and t7→ A(t)be right-continuous. Deﬁne the functionals L(t) : RCR → Rnand Jk:RCR → Rnby

L(t)φ=A(t)φ(0) +

`

X

j=1

Bj(t)φ(−rj), J(k)φ=C(k)φ(0) +

`

X

j=1

Ej(k)φ(−rj).

By an abuse of notation, write also

f(t, xt) = f(t, xt(0), xt(−r1), . . . , xt(−r`)),

gk(xt) = gk(xt(0), xt(−r1), . . . , xt(−r`)).

There exists a function C:R× RCR → RCR — the centre manifold — with the folllwing properties.

1. C(t, ·) : RCR → RCR is Cmand each of its derivatives Dj

2C(t, ·)is uniformly (in t) Lipschitz contin-

uous in some neighbourhood of 0. If (3)–(4) is T-periodic, the same is true of t7→ C(t, φ).

2. Let Pc(t) : RCR → RCRc(t)denote the projection onto the d-dimensional centre ﬁber bundle associated

to the linearization (6)–(7).C(t, Pc(t)φ) = C(t, φ)for all (t, φ)∈R× RCR and D2C(t, 0) = Pc(t).

3. The nonautonomous set Wc={(t, C(t, φ)) : φ∈ RCR} is d-dimensional and locally positively invariant

near 0∈ RCR under the nonautonomous process generated by (1)–(2) and contains all of its small

complete solutions.

4. Wc⊂R×RCR is locally attracting near 0∈ RCR provided the unstable ﬁber bundle of the linearization

(6)–(7) is trivial.

6

5. Every small solution t7→ xt∈ RCR of the semilinear equation (1)–(2) in Wccan be written in the

form

xt=C(t, xt) = w(t)+(I−Pc(t))C(t, w(t)),(11)

for the function w(t) = Pc(t)xtlying in the centre ﬁber bundle. Moreover, if the overlap conditions are

satisﬁed, the function w:R→ RCR1is pointwise C1and satisﬁes the impulsive diﬀerential equation

d+

dt w(t) = L(t)w(t) + Pc(t)χ0f(t, C(t, w (t))), t 6=τk(12)

∆tw(τk) = J(k)w(τ−

k) + Pc(τk)χ0gk(C(τ−

k, w(τ−

k))), t =τk,(13)

where w(τ−

k)(θ) = lim→0+w(tk−)(θ)is the regulated left-limit at time tkand ∆tw(τk) := w(τk)−

w(τ−

k)is the regulated jump at time τk,C(t−, w(t−))(θ) := lim→0+C(t−, w(t−))(θ), and the linear

operators Land Jare

L(t)φ=L(t)φ, θ = 0

d+φ(θ), θ < 0,J(k)φ(θ) = J(k)φ, θ = 0

φ(θ+)−φ(θ), θ < 0.(14)

To be clear, the centre manifold as deﬁned in [6] is a function c:RCRc→ RCR satisfying D2c(t, 0) =

IRCRc(t), where RCRcis the centre ﬁber bundle and RCRc(t) is the associated t-ﬁber. The map C:

R×RCR → RCR is deﬁned by C(t, φ) = c(t, Pc(t)φ). Also, the local centre manifold depends on the choice

of a cutoﬀ function, and is generally not unique.

As stated in the theorem, the centre manifold is Cmin the state space RCR. In the sequel, we will need

additional regularity; namely, we need conditions guaranteeing piecewise diﬀerentiability in the time variable.

The following deﬁnition makes this precise.

Deﬁnition 2.6. A function F:R× RCR → RCR is eﬀectively P C 1,m at zero if it satisﬁes the following

conditions.

•x7→ F(t, x)is Cmin a neighbourhood of 0∈ RCR, uniformly in t;

•for j= 0, . . . , m,t7→ Dj

2F(t, x)is pointwise right-diﬀerentiable at zero: for all φ1, . . . , φj∈ RCR and

θ∈[−r, 0], the function t7→ Dj

2F(t, 0)[φ1, . . . , φj](θ)is right-diﬀerentiable with left limits.

The choices of projections Pc(t) and Pu(t) onto the centre and unstable ﬁber bundles have an eﬀect on

the centre manifold, and can in principle aﬀect how temporally smooth it is. The following theorem provides

an aﬃrmative answer as to whether there is a choice of projections that guarantees the associated centre

manifold is eﬀectively P C1,m at zero. The proof is somewhat technical and is defered to Section Appendix

A.1.

Theorem 2.2. Suppose the nonlinearities of the T-periodic model system (3)–(4) are Cm+1 for some m≥0

and the hypotheses of Theorem 2.1 are satisﬁed, in addition to the overlap condition. There exists a choice

of projections Pc(t)and Pu(t)such that the resulting center manifold is eﬀectively P C1,m at zero.

From this point forward, we will assume that the projectors are chosen according to Theorem 2.2 to

guarantee the centre manifold is temporally smooth. Appropriate choices are those constructed according to

Section 3.3.

3. The centre manifold reduction

As outlined in Section 1.1, there are several steps that must be completed in order to obtain a centre man-

ifold reduction. We proceed through these steps methodically here, stating and proving necessary theorems

where appropriate.

7

3.1. Basis calculation for the centre ﬁber bundle RCRc

This section concerns the calculation of a basis for RCRc, the centre ﬁber bundle associated to the

linearization

˙y=A(t)y+

`

X

j=1

Bj(t)y(t−rj), t 6=tk(15)

∆y=C(k)y(t−) +

`

X

j=1

Ej(k)y(t−rj), t =tk.(16)

Ultimately, our goal is to obtain a matrix Φ(t) whose columns φ(t) are solutions of the linearization such

that φt∈ RCRc(t). Functions t7→ φt∈ RCRc(t) are characterized by having subexponential growth in both

forward and backward time [6]. Due to the periodicity of the model system, a Floquet theory is available.

3.1.1. Floquet theory

Recall the model system (1)–(2), and let the period Tof the system be given. By rescaling the time

variable (eg. setting s=t/T ; note that this will rescale the delays as well), we will assume from this point

onward that the period is T= 1. Let us assume also that the origin is an equilibrium point. If another

equilibrium point exists, one may translate it to the origin.

The Floquet theory for impulsive delay diﬀerential equations [6] implies the existence of a set Λ ⊂C, the

Floquet exponents, such that the origin is linearly stable provided Λ is conﬁned to the open left half-plane.

Moreover, λis a Floquet exponent if and only if there exists a solution of the linearization that can be written

in the form y(t) = p(t)eλt for p(t) a periodic function. The function y(t) will be referred to as a Floquet

eigensolution with exponent λ. The function p(t) will be called a periodic eigensolution associated to the

Floquet exponent λ. The period of pcan be taken to be one (assuming time is rescaled) by considering pto

be complex-valued. Clearly, the periodic solution pmust satisfy the equation

˙p+λp =A(t)p+

`

X

j=1

e−λrjBj(t)p(t−rj), t 6=tk(17)

∆p=C(k)p(t−) +

`

X

j=1

e−λrjEj(k)p(t−rj), t =tk.(18)

The centre ﬁber bundle (time-varying centre subspace) RCRc(t) consists of all Floquet eigensolutions

associated to Floquet exponents with zero real part, in addition to the generalized Floquet eigensolutions

of rank kwhich can also be written in the form y(t) = eλtq(t) for q(t) a polynomial of degree kwith

periodic coeﬃcients, which we will refer to as rank kgeneralized periodic eigensolutions. Generalized Floquet

eigensolutions will be brieﬂy discussed in Section 3.2. However, what will be important later is the following

proposition. The proof is similar to the proof of [[13] Lemma 1.2], but we include a proof in Section Appendix

A.2 to help keep the paper self-contained.

Proposition 3.1. Let φ1, . . . , φαbe a real basis for the α-dimensional 0-ﬁber RCRc(0) of the periodic linear

system (15)–(16), and deﬁne the matrix

Φt=Uc(t, 0)[ φ1··· φα].

Then, the columns of Φtare a basis for RCRc(t)and there exists a real α×αmatrix Λwith σ(Λ) ⊂iRand

a2T-periodic matrix t7→ Qtwhose columns are in RCR, such that the Floquet decomposition is satisﬁed:

Φt=QteΛt.(19)

Speciﬁcally, there is a nonsingular matrix Msuch that ΦT= Φ0M, and Λis given by the principal logarithm

Λ = 1

2Tlog(M2). If Mhas a real logarithm, then one can take Λ = 1

Tlog M, and t7→ Qtwill be T-periodic.

8

Assume the linearized system posesses exactly αdistinct Floquet exponents λ1, . . . , λkon the imaginary

axis. We can order them so that λ0= 0 (if the zero Floquet exponent exists) and λj=iωjwith λj+1 =−iωj

for some ωj∈(0,2π), and j= 1, . . . , κ. For simplicity, assume that there is only one generalized eigensolution

associated to each Floquet exponent, of rank 1.

To the zero Floquet exponent λ= 0, if it exists, there exists a rank 1 periodic eigensolution q0. Also,

for each complex conjugate pair ±iωj, there is similarly a complex-valued periodic eigensolution qj(t) with

exponent λ=iωj. From this solution, deﬁne the n×2 matrix

Pj(t)=[ Re(qj(t)) Im(qj(t)) ].

If one deﬁnes the α×αmatrix Λ by Λ = diag(01×1, W1, . . . , Wκ) where the 2 ×2 matrix Wjis deﬁned via

Wj=0ωj

−ωj0,

and n×αmatrix Q(t) = [ q0(t)P1(t). . . Pκ(t)], then Φt:= QteΛtis a matrix whose columns form a

basis for RCRc(t) and satisfy the impulsive diﬀerential equation (15)–(16). Moreover, it is written as a real

Floquet decomposition of period T= 1, so in this case we do not need to double the period to guarantee a

real decomposition.

3.2. Generalized Floquet eigensolutions

Assuming the period has been normalized to 1, RCRc(0) can be understood more fully as the generalized

eigenspace of the monodromy operator V:= U(1,0). This eigenspace consists of elements ξ∈ RCR such

that (V−µI)kξ= 0 for some natural number k≥1 referred to as the rank, and Floquet multiplier µ=eλ

with |µ|= 1. A generalized Floquet eigensolution is the solution through such a generalized eigenvector, and

as such they can be written in the form q(t)=[U(t, 0)ξ](0).Speciﬁcally, every Floquet eigensolution is of the

form

y(t) = eλt

k−1

X

m=0

pm(t)tm,(20)

where k≥1 is the rank of the associated generalized eigenvector ξ, and each pj(t) is a complex-valued

periodic column vector of period 1 with pk−16= 0. This can be proven by appealing to Proposition 3.1.

Computing generalized eigensolutions of rank 2 or higher is obviously more diﬃcult than in the rank 1

case, but the idea is the same. One may consult the Section Appendix A.3, where we derive a triangular

homogeneous system for the vector ~p = (p0, . . . , pk−1) of periodic solutions. See the discussion (Section 6)

for further comments.

3.3. Monodromy operator, resolvent and action of the projection on Φt

Associated to the evolution family U(t, s) : RCR → RCR, the monodromy operator is Vt:RCR → RCR

deﬁned by Vtξ=U(t+T, t)ξ, assuming T≥r. This operator is compact and the projection Pc(t) : RCR →

RCRc(t) can be represented by the Dunford integral

Pc(t) = 1

2πi X

λ∈σ(Λ) ZΓλ

(zI −Vt)−1dz, (21)

where Γλis a positively-oriented closed curve in Csuch that the only singularity of (zI −Vt)−1inside

Γλis eλT [6], and Λ is the matrix appearing in the Floquet decomposition Φt=QteΛt. The resolvent is

R(z;Vt)=(zI −Vt)−1. Assuming Φtadmits a real Floquet decomposition for the period T, Proposition 3.1

implies Φt+T= ΦteΛTand, consequently, VtΦt= ΦteΛT. One can easily verify that

(zI −Vt)−1Φt= Φt(zI −eΛT)−1.

As Λ is a real Jordan matrix, it follows that we have the identity

Pc(t)Φt= Φt, I =1

2πi X

λ∈σ(Λ) ZΓλ

(zI −eΛT)−1dz (22)

9

for all t∈R.

If the Floquet decomposition Φt=QteΛtwith t7→ Qtbeing 2T-periodic must be used, then we can

artiﬁcially consider the linear system (15)–(16) to be 2T-periodic. Then, equation (21) remains correct if one

replaces Vtwith V2

t=U(t+ 2T, t).

Of importance later will be the action of the projection on functions of the form χ0ξfor ξ∈Rn. To this

end, Proposition 2.1 guarantees we can uniquely deﬁne a matrix Y(t)∈Rα×nsatisfying the identity

Pc(t)χ0= ΦtY(t).(23)

Note that t7→ Y(t) need not be periodic. Indeed, one can easily verify Y(t+T) = e−ΛTY(t). However,

t7→ eΛtY(t) is periodic. This fact will be useful in Section 3.4.2.

We will also need the following result, which is a consequence of adjoint theory for evolution systems and

backward evolutions systems. Some details are covered in Section Appendix A.1.3.

Proposition 3.2. There exists a matrix Ψtwhose rows are elements of RCR∗, the topological dual of RCR,

such that ΨtΦt=Iα×αfor all t∈R.

Finally, we should remark that Dunford integral construction (21) also applies for generating a projection

Pu(t) : RCR → RCRu(t) onto the unstable ﬁber bundle. Moreover, choosing Ps=I−Pu−Pcas the

complementary projector, the triple (Ps, Pc, Pu) of projectors results in the associated local centre manifold

being eﬀectively P C1,m at zero; see the proof of Theorem 2.2. From this point onward, the local centre

manifold will always be one that is computed using these projectors.

3.4. An evolution equation for the centre manifold

The centre manifold is shown in this section to satisfy a nonlinear impulsive evolution equation. However,

it is necessary to recast the centre manifold in a more familiar Euclidean space setting. This is done ﬁrst,

before deriving the evolution equation.

3.4.1. Euclidean space representation of the centre manifold

From Theorem 2.1, we can reinterpret solutions in the centre manifold Wcas being written in the form

xt(θ) = wt(θ) + H(t, wt, θ),(24)

where H(t, w, ·) = (I−Pc(t))C(t, w) and wt=Pc(t)xt∈ RCRc(t) is the component of the centre manifold

in the centre ﬁber bundle. It follows that wt= Φtz(t) for coordinate function z(t)∈Rα. If we introduce the

time-dependent change of variables u=eΛtz, then wt=Qtu(t). Finally, if we deﬁne h:R×Rα×[−r, 0] →Rn

by the equation h(t, u, θ) = H(t, Qtu, θ), then in the variable u, equation (24) becomes

xt=Qtu(t) + h(t, u(t),·).(25)

Note that u:R→Rαis diﬀerentiable except at times tkwhere it is right-continuous and has limits on the

left, as implied by Theorem 2.1. The function hwill be referred to as the Euclidean space representation of

the centre manifold. We also defne the left limit

h(t−, u, θ) = lim

→0+h(t−, u, θ).

The Euclidean space representation of the centre manifold satisﬁes a number of important smoothness prop-

erties. They are summarized in the following theorem, which is a consequence of Theorem 2.1, Theorem 2.2,

the periodicity of t7→ Qtand Taylor’s theorem with remainder. The proof is omitted.

Theorem 3.1. Under the assumptions of Theorem 2.1, the Euclidean representation h:R×Rα×[−r, 0] of

any local centre manifold that is eﬀectively P C1,m at zero enjoys the following properties.

1. hadmits a Taylor expansion near u= 0:

h(t, u, θ) = 1

2!h2(t, θ)u2+1

3!h3(t, θ)u3+· · · +1

m!hm(t, θ)um+O(um+1),

with hi(t, θ) = Di

2h(t, 0, θ).

10

2. t7→ hi(t, ·)is periodic for i= 2, . . . , m.

3. Pc(t)hi(t, ·)=0for i= 2, . . . , m.

Finally, we provide some assurance that even though local centre manifolds depend non-canonically on a

choice of cutoﬀ function, the Taylor expansion described by Theorem 3.1 is indeed unique. This is a direct

result of the local centre manifolds (and consequently their derivatives) being constructed as solutions of

ﬁxed-point equations in appropriate Banach spaces, for which the evaluation at zero does not depend on the

cutoﬀ. See the proof of [Theorem 8.2.1, [6]].

Corollary 3.1. Every local centre manifold posesses the same degree mTaylor expansion at zero. In par-

ticular, any Euclidean representation of a local centre maniﬂd that is eﬀectively P C 1,m at zero posesses the

same degree mTaylor expansion at zero.

3.4.2. The evolution equation

At this stage, we should deﬁne the right-jump operator on RCR. Speciﬁcally, we deﬁne ∆+

θ:RCR →

G([−r, 0),Rn) by ∆+

θφ(θ) = φ(θ+)−φ(θ).This operator permits a decomposition of J(k) into

J(k) = χ0J(k) + χ[−r,0)∆+

θ,

and the following proposition is clear.

Proposition 3.3. Let x:R→Rnbe continuous except at times tk, where it is right-continuous and has

limits on the left. Then, ∆+

θx−

t(θ)=∆txt(θ)for θ < 0and all t∈R. If the overlap condition is satisﬁed,

then

J(k)xt−

k=J(k)x−

tk, gk(xt−

k) = gk(x−

tk).(26)

Let x:R→Rnbe a complete solution in the centre manifold. Such a function satisﬁes an evolution

equation that is similar to (12)–(13):

d+

dt xt=L(t)xt+χ0f(t, xt), t 6=tk(27)

∆txt=J(k)x−

t+χ0gk(x−

t), t =tk,(28)

with Land Jas deﬁned in (14). Note that all left-limits are now regulated left-limits because we have used

equation (26) of Proposition 3.3. Moreover, we have implicitly assumed the overlap condition; see Section 6

for a discussion on this assumption and its impact on the above equation. It is at this stage that we make the

substitution (25) to write xtin terms of the Euclidean representation of the centre manifold. The following

theorem contains our ﬁrst important result. The proof is deferred to Section Appendix A.4

Theorem 3.2. The Euclidean space representation of the centre manifold is a solution of the following

boundary-value problem:

Qt(θ)[ ˙u−Λu] + ∂th(t, u, θ) + ∂uh(t, u, θ) ˙u=∂θh(t, u, θ), θ < 0,t 6=tk

Qt(θ)∆u+ ∆th(t, u + ∆u, θ) + Ω(t, h, u, θ)∆u= ∆+

θh(t−, u, θ), θ < 0,t =tkor t+θ=tk

(29)

Qt(0)[ ˙u−Λu] + ∂th(t, u, 0) + ∂uh(t, u, 0) ˙u=L(t)h(t, u, ·) + f(t, Qtu+h(t, u, ·)), θ = 0, t 6=tk

Qt(0)∆u+ ∆th(t, u + ∆u, 0) + Ω(t, h, u, 0)∆u=J(k)h(t−, u, ·) + gk(Qtu+h(t−, u, ·)), θ = 0, t =tk

(30)

where we denote u=u(t)when t6=tkand u=u(t−)when t=tk,∆u=u(tk)−u(t−

k), we deﬁne Ωby

Ω(t, h, u, θ) = Z1

0

∂uh(t−, u +s∆u, θ)ds, t =tk,

and all derivatives in tand θare the right-derivatives ∂+

∂t and ∂+

∂θ .

11

Remark 3.1. Note that h(t, θ, u)must, by deﬁnition, possess discontinuities along the lines t+θ=tkfor u

ﬁxed – see equation (25). These discontinuities are captured by the second equation of (29) when θ < 0, and

in the second equation of (30) when θ= 0. Where t=tk−θ /∈ {tj:j < k}, we have ∆u= ∆u(t)=0in the

second equation of (29), and the result is the constraint h(t, u, θ) = h(t−, u, θ+).

As stated, Theorem 3.2 provides an implicit equation for the centre manifold in terms of the derivative

( ˙u) and jumps (∆u) of the coordinate u(t) for the component in the centre ﬁber bundle. The derivative and

jumps can be explicitly related to the coordinate with the following lemma. A short proof is available in

Section Appendix A.5.

Lemma 3.1. Let the local centre manifold be deﬁned by the projectons Pc(t)and Pu(t)from Section 3.3.

For a solution xt=Qtu(t) + h(t, u(t),·)on the centre manifold, the coordinate u(t)satisﬁes the ordinary

impulsive diﬀerential equation

˙u= Λu+eΛtY(t)f(t, Qtu+h(t, u, ·)), t 6=tk(31)

∆u=eΛtY(t)gk(Q−

tu+h(t−, u, ·)), t =tk,(32)

where the notation for uis as in Theorem 3.2, and Λis the real α×αJordan matrix in the Floquet decom-

position Φt=QteΛt, and Y(t)is deﬁned in equation (23).

Remark 3.2. It is worth reminding the reader that, by construction, t7→ eΛtY(t)is periodic. Combined

with the periodicity of t7→ Qtand t7→ h(t, ·,·), this means that the dynamics of the coordinate u(t)are

periodically driven, which is to be expected.

3.5. Approximation of the centre manifold by Taylor expansion

Equations (29)–(30) and (31)–(32) of Theorem 3.2 and Lemma 3.1 yield a system of impulsive partial

delay diﬀerential equations and boundary conditions for the Euclidean space representation of the centre

manifold.

In the ucoordinates, the dynamics on the centre manifold are given by (31)–(32). If one seeks to obtain

the O(||u||k) dynamics on the center manifold, it is necessary to compute the terms of order O(||u||k−1) of

the center manifold h. For example, Theorem 3.1 implies we can take

h(t, u, θ) = 1

2!h2(t, θ)u2+1

3!h3(t, θ)u3+· · · (33)

for symmetric multilinear mappings hi(t, θ):(Rα)i→Rndeﬁned by hi(t, θ) = Di

2h(t, 0, θ). It is easily

veriﬁed that

˙u= Λu+eΛtY(t)1

2!D2f(t, 0)[Qtu]2+1

3! D3f(t, 0)[Qtu]3+ 3D2f(t, 0)[Qtu, h2(t, θ)u2]+O(||u4||), t 6=tk

(34)

∆u=eΛtY(t)1

2!D2gk(0)[Q−

tu]2+1

3! D3gk(0)[Q−

tu]3+ 3D2gk(0)[Q−

tu, h2(t−, θ)u2]+O(||u4||), t =tk,

(35)

so we need only compute the second-order term h2if the cubic-order dynamics are suﬃcient for applications.

Note that h2(t, θ) can be represented in the form

h2(t, θ)[u, v] =

c1

11(t, θ)u1v1+···c1

1α(t, θ)u1vα+c1

21(t, θ)u2v1+c1

22(t, θ)u2v2+· ·· +c1

αα(t, θ)uαvα

.

.

.

cn

11(t, θ)u1v1+···cn

1α(t, θ)u1vα+cn

21(t, θ)u1v1+cn

22(t, θ)u2v2+· ·· +cn

αα(t, θ)uαvα

,

(36)

and similarly for the higher-order terms, where symmetrically, cij =cji . In terms of vector multi-indices, we

can write

hp(t, θ)[u1, . . . , up] = X

|ξ|=p

cξ(t, θ)(u1···up)ξ(37)

12

for multiindex ξ= (ξ1, . . . , ξp) and ξi∈ {∅, e0

1, . . . , e0

α}. Also, the Fr´echet derivative D2f(0) can be repre-

sented, for xi∈ RCR, in the form

D2f(t, 0)[x1, x2] =

`

X

j,k=0

∂2f(t, 0)

∂x(t−rj)∂x(t−rk)x1(−rj)x2(−rk) (38)

and similarly for the higher-order derivatives and the jump map g.

As a consequence of the above observations, one can substitute an appropriate order O(||u||k) expansion

of the impulsive diﬀerential equation (31)–(32) into the evolution equation and boundary conditions (29)–(30)

to obtain a O(||u||k) impulsive evolution equation for the center manifold.

3.5.1. Quadratic order dynamics