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Computation of centre manifolds and some codimension-one bifurcations for impulsive delay differential equations

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Based on the centre manifold theorem for impulsive delay differential equations, we derive impulsive evolution equations and boundary conditions associated to a concrete representation of the centre manifold in Euclidean space, as well as finite-dimensional impulsive differential equations associated to the evolution on these manifolds. Though the centre manifolds are not unique, their Taylor expansions agree up to prescribed order, and we present an implicit formula for the quadratic term using a variation of the method of characteristics. We use our centre manifold reduction to derive analogues of the saddle-node and Hopf bifurcation for impulsive delay differential equations, and the latter leads to a novel bifurcation pattern to an invariant cylinder. Examples are provided to illustrate the correctness of the bifurcation theorems and to visualize the geometry of centre manifolds in the presence of impulse effects.
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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: (Kevin E.M. Church ), (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) +
Bj(t)x(trj) + f(t, x(t), x(tr1), . . . , x(tr`)), t 6=tk(3)
x=C(k)x(t) +
Ej(k)x(trj) + gk(x(t), x(tr1), . . . , x(tr`)), 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
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.
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:RXfor a Banach space X, we denote
dt f(t) = lim
the right derivative of fat t.For functions of several variables we have the analogous definition for partial
derivatives. For a function f:R×XYwith 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 xX,x7→ f(t, x) is continuous for each tR, and fis continuous on R\{τk:kZX.
For f:RX, we write fP 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:RRn, we define xt: [r, 0] Rnfor each tRby the equation xt(θ) = x(t+θ). We then
define the uniform one-point left-limit
xt(θ) = lims0xt(s), θ = 0
xt(θ), θ < 0,
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
t(θ) = lim
provided this limit exists.
For ZR, 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
(ba+ 1) ×mmatrix whose rows coincide with rows athrough bof A.
For a j-dimensional multi-index ξ= (ξ1, . . . , ξj) where ξiN, we define |ξ|=Piξi. For uRjand a
j-dimensional multi-index ξwith |ξ|=m, the ξpower of uis uξ=uξ1
j. If Xis a vector space and
UXj, we similarly define UξXmby
Uξ= (U1, . . . , U1, U2, . . . , U2, . . . , Uj...,Uj)
where the factor Uiappears ξitimes. If uXand mN, we define umXmby 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 uiRkas follows:
(u1···uj)ξ= (ξ1u1)···(ξjuj).
For vectors in Rnwritten in component form, (u1, . . . , un)·(v1, . . . , vn) = Piuividenotes the standard
inner product.
If ARm×nand BMn×k(R`), we define the overloaded product ABMm×k(R`) by the equation
[AB]i,j =
Ai,uBu,j .(5)
It is readily verified that if ARm×mis invertible, then AB=Cif and only if B=A1C. Moreover,
satisfies the Leibniz law
dtA(t)B(t) = d
dtA(t)B(t) + A(t)d
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: [sr, 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 sRif the previous equality holds. There are several
conditions guaranteeing the existence and uniqueness of classical solutions satisfying given initial conditions,
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)φRnthe (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 tsby
[U(t, s)φ](θ) = X(t+θ, s)φ.
A function x: [sr, α)Rnis a mild solution of (1)(2) if
xt=U(t, s)xs+Zt
U(t, µ)χ0f(µ, xµ)+X
U(t, ti)χ0gi(xt
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) : XXbe 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 tvsand 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 suptR(||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 tsand 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 st.
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 K1such that
||Uu(t, s)|| ≤ Keb(ts), t s(8)
||Uc(t, s)|| ≤ Ke|ts|, t, s R(9)
||Us(t, s)|| ≤ Kea(ts), t s. (10)
Definition 2.3 (Definition 3.3.6 [6]).Let U(t, s) : XXbe spectrally separated. The nonautonomous sets
Xi={(t, x) : tR, x Xi(t)}
for i∈ {s, c, u}are termed respectively the stable, centre, and unstable fiber bundles associated to U(t, s).
XsXuis 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.
Definition 2.4. The system (3)(4) satisfies the overlap condition if Ej(k)=0and
gk(x0, xj1, xj+ξ, xj+1 , . . . , x`) = gk(x0, . . . , xj1, xj, . . . , x`)
for all ξRn, whenever tkrj=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 tkrjand 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 cN
such that J(k)and gk(·)are c-periodic, and the sequence of impulses satisfies tk+c=T+tkfor all kZ.
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 Tr, 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, Corollary and Theorem 8.3.1 therein.
Theorem 2.1. Consider the T-periodic model system, let 0Rnbe an equilibrium point and assume the
nonlinearities f(t, ·)and gk(·)are Cm(uniformly in tR) with vanishing first Fr´echet derivatives at 0.
Also, let the first mFechet 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) +
Bj(t)φ(rj), J(k)φ=C(k)φ(0) +
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. WcR×RCR is locally attracting near 0∈ RCR provided the unstable fiber bundle of the linearization
(6)(7) is trivial.
5. Every small solution t7→ xt∈ RCR of the semilinear equation (1)(2) in Wccan be written in the
xt=C(t, xt) = w(t)+(IPc(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
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) = lim0+w(tk)(θ)is the regulated left-limit at time tkand tw(τk) := w(τk)
k)is the regulated jump at time τk,C(t, w(t))(θ) := lim0+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
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
Theorem 2.2. Suppose the nonlinearities of the T-periodic model system (3)(4) are Cm+1 for some m0
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.
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
Bj(t)y(trj), t 6=tk(15)
y=C(k)y(t) +
Ej(k)y(trj), 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+
eλrjBj(t)p(trj), t 6=tk(17)
p=C(k)p(t) +
eλrjEj(k)p(trj), 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:
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.
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=jwith λj+1 =j
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 ±j, there is similarly a complex-valued periodic eigensolution qj(t) with
exponent λ=j. 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
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 (VµI)kξ= 0 for some natural number k1 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
y(t) = eλt
where k1 is the rank of the associated generalized eigenvector ξ, and each pj(t) is a complex-valued
periodic column vector of period 1 with pk16= 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, . . . , pk1) 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 Tr. This operator is compact and the projection Pc(t) : RCR →
RCRc(t) can be represented by the Dunford integral
Pc(t) = 1
2πi X
λσ(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(tt= Φt, I =1
2πi X
λσ(Λ) ZΓλ
(zI eΛT)1dz (22)
for all tR.
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 ξRn. 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) = eΛTY(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 tR.
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=IPuPcas 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, ·) = (IPc(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:RRαis 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).
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)+
and the following proposition is clear.
Proposition 3.3. Let x:RRnbe continuous except at times tk, where it is right-continuous and has
limits on the left. Then, +
t(θ)=∆txt(θ)for θ < 0and all tR. If the overlap condition is satisfied,
tk, gk(xt
k) = gk(x
Let x:RRnbe a complete solution in the centre manifold. Such a function satisfies an evolution
equation that is similar to (12)–(13):
dt xt=L(t)xt+χ0f(t, xt), t 6=tk(27)
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)[ ˙uΛu] + 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
Qt(0)[ ˙uΛu] + 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
where we denote u=u(t)when t6=tkand u=u(t)when t=tk,u=u(tk)u(t
k), we define by
Ω(t, h, u, θ) = Z1
uh(t, u +su, θ)ds, t =tk,
and all derivatives in tand θare the right-derivatives +
∂t and +
∂θ .
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=tkθ /∈ {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)
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
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||k1) 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)iRndefined 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
3! D3gk(0)[Q
tu]3+ 3D2gk(0)[Q
tu, h2(t, θ)u2]+O(||u4||), t =tk,
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] =
11(t, θ)u1v1+···c1
1α(t, θ)u1vα+c1
21(t, θ)u2v1+c1
22(t, θ)u2v2+· ·· +c1
αα(t, θ)uαvα
11(t, θ)u1v1+···cn
1α(t, θ)u1vα+cn
21(t, θ)u1v1+cn
22(t, θ)u2v2+· ·· +cn
αα(t, θ)uαvα
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
cξ(t, θ)(u1···up)ξ(37)
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] =
2f(t, 0)
∂x(trj)x(trk)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