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Rigorous verification of Hopf bifurcations in functional differential
equations of mixed type
Kevin E. M. Church
*
Jean-Philippe Lessard
August 24, 2021
Abstract
We propose a rigorously validated numerical method to prove the existence of Hopf bifurcations
in functional differential equations of mixed type. The eigenvalue transversality and steady state
conditions are verified using the Newton-Kantorovich theorem. The non-resonance condition and
simplicity of the critical eigenvalues are verified by either computing a pair of generalized Morse
indices of the step map, or by applying the argument principle to the characteristic equation and a
suitable contour in the complex plane, computing the contour integral using the equivalence with a
winding number. As a first application and test problem, we prove the existence of Hopf bifurcations
in the Lasota-Wazewska-Czyzewska model and a pair of two such coupled equations. We then use
our method to prove the existence of periodic traveling waves in the Fisher equation with nonlocal
reaction. These periodic traveling waves are solutions of an ill-posed functional differential equation
of mixed type.
1 Introduction
The Hopf bifurcation is a fundamental pathway to oscillations in nonlinear dynamical systems. Since
the pioneering work of Henri Poincar´e in the late 1800s as applied to ordinary differential equations,
this bifurcation has been studied in myriad scenarios, including differential equations in Banach spaces
[16, 38], partial differential equations [3, 31, 32, 50, 51], stochastic differential equations [6, 14], functional
differential equations [21, 48], and piecewise-smooth systems [27, 52]. Our interest here is in the verification
of Hopf bifurcations at equilibrium solutions of functional differential equations, including those of mixed-
type. Recall that a functional differential equation of mixed type is an equation of the form
˙x=f(xt),
where f:C([−a, b],Rn)→Rnis a functional, a, b ≥0 and xt(θ)def
=x(t+θ) for θ∈[−a, b]. For example,
differential difference equations with forward and backward arguments (sometimes called advance-delay
equations) such as
˙x=g(x(t), x(t−a), x(t+b))
for g:Rn×Rn×Rn→Rnare of this class. The functional representation for this equation is f(φ)def
=
g(φ(0), φ(−a), φ(b)) for φ∈C([−a, b],Rn).
For f:C([−a, b],Rn)×R→Rn, let D1fdenote the Fr´echet derivative with respect to the first
variable. Recall the Hopf bifurcation theorem of Rustichini [48] for functional differential equations of
*
McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montreal, QC, H3A 0B9,
Canada (kevin.church@mcgill.ca)
McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montreal, QC, H3A 0B9,
Canada (jp.lessard@mcgill.ca)
1
mixed-type, which we paraphrase here with a trivial modification concerning the non-stationarity of the
equilibrium with respect to parameter variation and the definition of the state space.
Theorem 1. Let I ⊂ Rbe a compact interval containing zero. Suppose f:C(I,Rn)×R→Rnis C2.
Let α7→ x0(α)be a C2branch of zeroes of fdefined on an open interval containing some α0∈R, so that
f(x0(α), α)=0for |α−α0|sufficiently small. Let σ(α)denote the set of eigenvalues of the linear system
˙y=D1f(x0(α), α)yt.
Suppose the following conditions are met.
1. σ(α0)contains a pair ±iω0for ω0>0;
2. there is a C1eigenvalue branch λ(α)∈σ(α)such that λ(α0) = iω0and Re(λ0(α0)) 6= 0;
3. σ(α0)∩iR={ω0,−ω0}and the pair ±iω0is simple.
Then, a Hopf bifurcation occurs at α=α0in the functional differential equation ˙x=f(xt, α)at the
equilibrium x0(α0).
Aside from the smoothness requirement, the conditions of the theorem can be summarized as follows.
We have (1.) a simple complex-conjugate imaginary pair of eigenvalues that (2.) cross the imaginary axis
transversally (3.) without resonance. We briefly recall that the eigenvalues λof the linearized equation
satisfy
∆(λ)vdef
=D1f(x0(α), α)vexpλ−λv = 0 (1)
for some nonzero eigenvector v∈Cn, where expλ(θ)def
=eλθ and by an abuse of notation, (vexpλ)(θ)def
=
veλθ , and by analogy, we define expλvdef
=vexpλ. In this way, we can identify ∆(λ) for each λ∈Cwith an
operator ∆(λ) : Cn→Cn, so ∆(λ) can be identified with an n×nmatrix. It is called the characteristic
matrix. The domain of expλwill depend on the class of problem: namely, it will be the same as the
interval Ifrom the statement of the theorem. For systems of retarded type it will be I= [−τ , 0] for some
τ > 0, while for mixed-type equations it can be taken to be I= [−a, b]. The eigenvalues also satisfy the
characteristic equation
det∆(λ)=0.(2)
This equation is transcendental and has infinitely many solutions, and it is here that the difficulties in
rigorous bifurcation verification arise.
As an example, suppose f(φ, α) = αφ(0) −φ(−1)2+φ(1)2for φ∈C([−1,1],R). The associated
functional differential equation (of mixed type) can then be written
x0(t) = αx(t)−x(t−1)2+x(t+ 1)2.
The Fr´echet derivative of fis then Dxf(φ, α)h=αh(0) −2φ(−1)h(−1) + 2φ(1)h(1), so equation (1)
becomes
αv −2x0(α)ve−λ+ 2x0(α)veλ= 0.
Since vmust be nonzero, the characteristic matrix associated with this functional differential equation is
∆(λ) = α−x0(α)e−λ+x0(α)eλ.Here we see the clear transcendence of the equation det ∆(λ) = 0.
Since the eigenvalues of functional differential equations satisfy transcendental equations, it is generally
impossible to compute them exactly. In applications, numerical methods are often necessary. There exist
several software packages that can test for the existence of bifurcations [1, 19, 54, 20] in delay differential
equations, but they are suitable only for mathematically non-rigorous numerical exploration. That is,
2
they can not prove the existence of bifurcations. In a recent preprint [18], numerical Hopf bifurcation
in retarded functional differential equations was studied using a pseudospectral approach. The approach
therein is broadly applicable, but non-resonance and simplicity conditions analogous to (3.) from Theorem
1 could not be rigorously checked.
Functional differential equations of mixed type frequently come up in the analysis of traveling wave
solutions of lattice differential equations [2, 39, 49], and computational approaches have been proposed to
solve boundary-value problems and propagate solutions on half-lines [17, 22, 23, 44, 56], as well as prove
traveling wave solutions with computer assistance [5]. However, there has been little work done on proving
the existence of Hopf bifurcations using the aid of the computer. The Cauchy problem of such equations is
generally ill-posed [28], so numerical computation of the eigenvalues based on the characteristic equation
seems to be the only available option. The eigenvalues themselves can, in this case, be distributed in the
complex plane in such a way that they accumulate at both positive and negative real infinity. This is in
contrast to equations of advanced or retarded type, where the step map (either in forward or backward
time) is eventually compact. One could envision making use of the holomorphic factorization [40, 41] and
the associated semigroups on the “forward” and “backward” space for a functional differential equation
of mixed type to compute eigenvalues using a discretization approach. To our knowledge this has not
been done.
1.1 Computable conditions for the rigorous verification of Hopf bifurcations
The techniques introduced in the present paper belong to the field of rigorously validated numerics. In a
broad sense, this field aims at developing numerical methods which can lead to computer-assisted proofs
of existence of different type of dynamical objects arising in the study of differential equations. This
rather new area of mathematics lies at the intersection of mathematical analysis, scientific computing,
approximation theory, topology and numerical analysis. In a nutshell, the goal of rigorously validated
numerics is to construct algorithms that provide an approximate solution to a problem together with
precise and possibly efficient bounds within which the exact solution is guaranteed to exist in the mathe-
matically rigorous sense. As already mentioned in more details in [8], this requires an a priori setup that
allows analysis and numerics to go hand in hand: the choice of function spaces, the choice of the basis
functions and Galerkin projections, the analytic estimates, and the computational parameters must all
work together to bound the errors due to approximation, rounding, and truncation sufficiently tightly for
the verification proof to go through. We encourage the interested reader to consult the books [9, 46, 57]
and the survey articles [8, 25, 33, 47] for an introduction to the field.
In this paper, we focus our attention on the rigorous verification of Hopf bifurcations in functional
differential equations. While to the best of our knowledge, this has never been achieved before in the
field of rigorous numerics, the rigorous verification of bifurcations in ODEs, PDEs and discrete dynamical
systems is not new. Using a Krawczyk-based interval validation method, a computer-assisted approach is
proposed in [30] to study turning points, symmetry breaking bifurcation points and hysteresis points in
ODEs. Still in the context of finite dimensional dynamical systems, rigorous methods to verify existence
of double turning points [43, 55], period doubling bifurcations [58], saddle-node bifurcations [36] and
cocoon bifurcations [34] have also been developed. More recently, a method based on desingularization
and continuation was proposed in [10] to study Hopf bifurcations in ODEs. Techniques for infinite
dimensional dissipative PDEs also started to appear. More explicitly, computational methods for the
rigorous verifications of bifurcations of steady states of PDEs are presented in [4, ?, 59] and the recent
preprint [11] presents proofs of Hopf bifurcations in the Kuramoto-Sivashinsky equation. An approach
to prove rigorously a weaker (topological) notion of bifurcations for steady states of nonlinear partial
differential equations is also proposed in [42].
The conditions (1.) and (2.) of Theorem 1, in addition to the branch of steady states, can be made
equivalent to the existence of a zero for particular nonlinear map of dimension 6n. This map is derived
in Section 2, and therein we review how a twist on the standard Newton-Kanrotovich theorem, namely
3
the radii polynomial approach, can be applied to prove the existence of zeroes.
The non-resonance and simplicity condition (3.) of Theorem 1 is more subtle. We must count the
number of eigenvalues on the imaginary axis or, having successfully proven the existence of at least one
complex-conjugate pair using the method of Section 2, we must find a neighbourhood of the imaginary
axis that contains at most two eigenvalues. To this end, we propose two approaches.
Use the argument principle in conjunction with a winding number argument to rigorously count the
number of zeroes of (1) in a strip containing the imaginary axis.
Use the Chebyshev spectral method from [37] to compute generalized Morse indices at radii r= 1±δ
for some δ > 0 small, and use these indices to determine an upper bound on the number of zeroes
of (2) on the imaginary axis.
The advantage of the first method is that it is general and can be set up for any functional differential
equation, including those that involve advanced, delayed or mixed-type arguments, distributed arguments,
or combinations thereof. The implementation of the method in code is also fairly straightforward. How-
ever, we require a priori bounds on the absolute value of the potential Hopf frequencies to select the
contour over which to compute. To compare, the Chebyshev spectral method is based on discretizing the
step map, and has thus far only been developed rigorously for differential equations with a single discrete
delay. However, general-purpose code is available and no a priori bounds concerning the Hopf frequencies
are needed. For systems with a single discrete delay, the code at [13] based on the methods of this paper
is general-purpose and can be used to prove the existence of a Hopf bifurcation. Both methods – based
on the argument principle and Morse indices – will be discussed in Section 3.
1.2 Application: Hopf bifurcation in the Lasota-Wazewska-Czyzewska model
In [53], Hopf bifurcation in the Lasota-Wazewska-Czyzewska model of red blood cell survival was proven.
The rescaled model is given by
˙x(t) = −σx(t) + e−x(t−τ),(3)
for σ > 0 and τ > 0. Since the equilibria x∗are solutions of the transcendental equation
−σx∗+e−x∗= 0,
this is a good test problem as the solutions will need to be approximated numerically. Using the methods
of this paper, we will prove the following.
Theorem 2. The Lasota-Wazewska-Czyzewska model has a Hopf bifurcation with respect to the delay τ
at the following parameter values and equilibria:
σ= 0.3,x∈1.104542018324 + [−1,1]7.2×10−13,τ∈19.208854104207 + [−1,1]7.2×10−13 .
σ= 0.35,x∈1.025065556445 + [−1.1]9.8×10−13,τ= 37.030171112739 + [−1,1]9.8×10−13 .
1.3 Application: Coupled Lasota-Wazewska-Czyzewska model
Once again using the techniques of this paper, we will prove some Hopf bifurcations in a system of two
coupled Lasota-Wazewska-Czyzewska equations.
Theorem 3. Consider the coupled Lasota-Wazewska-Czyzewska equations
˙x(t) = −σ1x(t) + e−x(t−τ)−ξx(t) + ρξy(t)
˙y(t) = −σ2y(t) + e−y(t−τ)+ξx(t)−ρξy(t),
4
where ξ≥0is the coupling strength and ρ∈ {0,1}determines whether coupling is unidirectional (ρ= 0)
or bidirectional (ρ= 1). For σ1= 0.1,σ2= 0.5, and τ= 17, there is a Hopf bifurcation with respect to
the parameter ξat the following parameter values and equilibria:
(x, y)∈(1.14466886800860,1.08412201968491)+[−1,1]2·7.5×10−14,ρ= 0,ξ∈0.02809728931663+
[−1,1]7.2×10−14.
(x, y)∈(1.5858236138052,0.9030535509805) + [−1,1]2·2.1×10−13,ρ= 1,ξ∈0.1780972893166 +
[−1,1]2.2×10−13.
1.4 Application: Hopf bifurcation of periodic traveling waves in a nonlocal
Fisher equation
It has been suggested [7, 26] that the nonlocal Fisher equation:
ut=Duxx +µu(σ−φ∗u) (4)
might have, for large µand some diffusion D, wave train (periodic traveling wave) solutions. Wave trains
are spatially periodic solutions of the form
u(t, x) = ψ(x+ct)
for ψnontrivial and periodic, with cthe wave velocity. Here, φ≥0 is an integrable function with
RRφ(x)dx = 1, and ∗is spatial convolution:
(φ∗u)(x) = Z∞
−∞
φ(y)u(x−y)dy.
Periodic traveling waves of (4) can be identified with periodic solutions of the functional differential
equation
cψ0(x) = Dψ00(x) + µψ(x)σ−Z∞
−∞
φ(y)ψ(x−y)dy.(5)
Depending on the support of the kernel φ, this equation can be of retarded, advanced, or mixed-type.
In [24], numerical results suggested that model (4) with φ=φ0,
φ0(y) = 1/N, 0≤y≤N
0 otherwise
might have stable periodic traveling wave solutions. With the kernel φ=φ1,
φ1(y) = 1/(2N),|y| ≤ N
0 otherwise.
pulsating fronts were observed numerically [45]. These are solutions of the form u(t, x) = U(x+ct, x) for
Uperiodic in its second variable. Symmetry of the kernel φseems to be a precursor to the existence of
pulsating fronts [15], and analytical sufficient conditions for the existence of such fronts have been proven.
We will consider equation (4) with piecewise-constant kernel
φ(y) =
h/N1−N1< y < 0
(1 −h)/N20< y < N2
0 otherwise.
(6)
for N1, N2>0 and h∈[0,1]. Equation (5) with the above kernel is of mixed-type whenever h /∈ {0,1},
and in this case the Cauchy problem is ill-posed. We will prove the following theorem.
5
Theorem 4. The functional differential equation (5) for traveling wave solutions of the nonlocal Fisher
equation with kernel φhas, for σ= 1 and µN 2
1D−1= 100, Hopf bifurcations from the steady state ψ= 1
with respect to the wave velocity parameter cat the following parameter configurations:
(W1) For N2/N1= 1 and h= 0.55: two Hopf bifurcations at c=c∗
j,j= 1,2, with c∗
1∈ −0.499960441060187+
[−1,1]2.6×10−15 and c∗
2∈ −1.407518070559178 + [−1,1]5.7×10−15.
(W2) For N2/N1= 1 and h= 0.45: two Hopf bifurcations at c=c∗
j,j= 1,2, with c∗
1∈0.499960441060186+
[−1,1]3.3×10−15 and c∗
2∈1.407518070559176 + [−1,1]7.6×10−15.
(W3) For N2/N1= 1.05 and h= 0.47: two Hopf bifurcations at c=c∗
j,j= 1,2, with c∗
1∈ −0.363754795740408+
[−1,1]4.1×10−15 and c∗
2∈0.225343700115205 + [−1,1]5.9×10−15.
(W4) For N2/N1= 2 and h= 0: two Hopf bifurcations at c=c∗
j,j= 1,2,c∗
1∈0.743510960061904 +
[−1,1]2.6×10−15 and c∗
2∈38.377317897727600 + [−1,1]1.4×10−14.
The statement of the previous theorem is in terms of the quantity ρdef
=µN2
1D−1which, after a
change of variables (see Section 4.3), plays the role of the diffusion coefficient. This change of variables
also leads us to consider the ratio Mdef
=N2
N1. In this way, several families of Hopf bifurcations are validated
simultaneously.
Corollary 5. For the parameter configurations of Theorem 4, the nonlocal Fisher equation with kernel φ
has periodic traveling waves of the form
u(t, x) = ψx+cD
N1
t(7)
with ψperiodic, small amplitude and mean close to 1, for cin an interval of the form (c∗
j−, c∗
j)or
(c∗
j, c∗
j+), for > 0small, with the form of the interval depending on the specific parameter configuration
and bifurcation point c∗
j,j= 1,2.
We prove the theorem and its corollary in Section 4.3. The final configuration coincides (N1= 1 and
N2= 2) with the parameter set from the numerical explorations of Genieys, Volpert and Auger [24],
in which stable periodic traveling waves were observed; see Figure 8 from that publication. Since the
theorem above is proven by way of Hopf bifurcation from ψ= 1 in the traveling wave equation (5), this
suggests the numerically observed waves in [24] in fact come from a Hopf bifurcation.
In Theorem 4, we do not state the direction of the Hopf bifurcation. It can generally be computed
from the first Lyapunov coefficient [35] of the dynamics restricted to the centre manifold. In the same
way, we do not specify the exact form of the interval of existence of periodic traveling waves in Corollary
5. We also do not investigate the stability of the traveling wave solutions in the PDE (4) itself, as this is
far beyond the scope of our work here.
We observe that with the first two (W1 and W2) conclusions of Theorem 4, the critical wave velocities
c∗
jare seemingly related by velocity reversal: namely, cj7→ −cj. There is a reason for this:
Proposition 6. Let ρ=µN2
1D−1. Suppose N2/N1= 1, and consider the linearization at v=σof the
functional differential equation of mixed-type
cv0(x) = v00 (x) + ρv(x)σ−hZ0
−1
v(x−y)dy + (1 −h)Z1
0
v(x−z)dz.(8)
If h=1
2, there are imaginary eigenvalues λ=iω,ω6= 0 if and only if c= 0. The frequencies ω
satisfy
b(ω)def
=ω2+ρsin(ω)
ω= 0.
6
If h6=1
2and b(ω)=0for some ω6= 0, then λ=iω is an eigenvalue provided the wave velocity c
satisfies
c=c∗(h)def
=ρ(cos(ω)−1)
ω2(2h−1) .
The proof of this proposition is straightforward and is omitted. Since the frequencies ωof some
potential Hopf bifurcations for h6=1
2are fixed by the equation b(ω) = 0 and the critical wave velocity
c∗(h) satifies c∗(1
2+) = −c∗(1
2−), this explains the velocity reversal observed in the theorem. When
h=1
2there are generally two pairs of imaginary eigenvalues, so it is possible a double Hopf bifurcation
occurs here.
Remark 1. Proposition 6 does not guarantee the existence of a Hopf bifurcation at wave speed c∗(h)
when h6=1
2, since that proposition does not rule out the existence of other resonant eigenvalues on the
imaginary axis. It merely characterizes those with frequencies ωsatisfying b(ω)=0.
2 Zero-isolation conditions for Hopf bifurcation and zero-finding
problem
Before we begin, it is necessary to introduce some notation. For brevity, let X=C(I,Rn). For a
functional f:X×R→Rn, we denote D1fand D2fthe partial Fr´echet derivatives with respect to the
first and second variables, respectively. Mixed and multiple Fr´echet derivatives will be denoted by such
symbols as D2D1and D2
1. We also associate to such a functional fafunction, denoted f0:Rn×R→Rn
and defined by f0(x, α) = f(cx, α), where cx:I → Rnis defined by cx(θ) = x. The notation for Fr´echet
derivatives will be the same for f0.
From this point on, the symbol θwill denote the function θ:I → Rdefined by θ(s) = s. In this way,
for any φ∈X, we have φ=φ(θ). This will be useful later, since we will need to write the action of a
linear map L:X→Rnon functions such as s7→ seλsvfor λ∈C,v∈Cnand s∈ I. The latter function
can be identified with θeλθ v, and the action of Lon this function as L(θeλθ v).
If B:X×X→Rnis a bilinear map, we will write the action on a pair x1, x2using braces: B[x1, x2].
Finally, if x∈Rn, we will sometimes treat xas an element of Xby way of the identification x≡cx, with
cx(s) = xfor all s∈ I. This will usually be in the scope of an evaluation of a linear or bilinear map.
2.1 Steady state, eigenvalue and transversality conditions
The baseline hypothesis of Theorem 1 is that we have a branch of steady states parameterized for αnear
some α0. In fact, to have a Hopf bifurcation it is necessary for this branch of steady states to be isolated
near α0, since otherwise we would generally expect a violation of the non-resonance condition. We will
therefore need to investigate the solvability of the equation
f0(x, α) = 0 (9)
for (x, α)∈Rn×R. Note the use of the map f0:Rn×R→Rn; this is because a steady state solution is
precisely a zero of f0(·, α). From the implicit function theorem, the existence of a unique curve of zeroes
through some (x, α) will depend on whether one can uniquely solve the equation
D1f0(x, α)x0+D2f0(x, α) = 0 (10)
for x0∈Rn(note: x0is the derivative of the implicit function xat α). The equations (9) and (10) will be
used to define the steady state portion of our zero-finding problem.
7
Next, we require ±iω to be a pair of eigenvalues of the linearized equation. From (1), this requires
solving the equation
D1f(x, α)vexpiω −iωv= 0.(11)
The eigenvector v∈Cncorresponding to this imaginary eigenvalue will also need to be computed.
However, since we will ultimately want to identify Hopf bifurcations by computing and verifying data
including the eigenvector v, the lack of uniqueness of the eigenvector is a problem. To circumvent this issue,
we assume we have already computed a numerical candidate eigenvector v, fix an integer m∈ {1, . . . , n},
and introduce a function hm:Cn−1→Cndefined by
hm(z) = z1· · · zm−1vmzm· · · zn−1|.(12)
We then replace equation (11) with
D1f(x, α) expiω hm(v)−iωhm(v)=0,(13)
where now v∈Cn−1. A solution (ω, v ) of (11) then uniquely defines a complex eigenvalue-eigenvector
pair (iω, hm(v)). Related to this equation is
D1f(x, α)hm(v) expλ−λhm(v)=0,(14)
which is merely what we would get had we not assumed an imaginary eigenvalue but still fixed the mth
component of the eigenvector using the hmfunction.
Finally, we need to deal with the eigenvalue transversality condition. We will implicitly differentiate
(14) with respect to α. To accomplish this, first recall by the Riesz representation theorem that a
functional L:X→Rncan be represented in the form
Lφ =ZI
[dK(s)]φ(s)
for K(s) matrix-valued and of bounded variation. Then,
Ld
dα expλ(α)=ZI
[dK(s)] d
dαeλ(α)θ=ZI
[dK(s)]eλ(α)θλ0(α)θ=λ0(α)Lθeλ(α)θ.
Making use of this calculation and the chain rule, we complete the implicit differentiation of (14) and
ultimately evaluate at λ=iω, obtaining
0 = D2
1f(x, α)[x0, hm(v) expiω] + D2D1f(x, α)hm(v) expiω +λ0(D1f(x, α)θeiωθ −I)hm(v)
+D1f(x, α)jm(v0) expiω −iωjm(v0)
≡F4(x, x0, α, ω, v, λ0, v 0),(15)
where jmis defined by
jm(z) = z1· · · zm−10zm· · · zn−1|.(16)
Here, the interpretation is that v0∈Cn−1is the (implicit) derivative of vwith respect to α, evaluated at
the bifurcation parameter (namely, such that λ=iω is an eigenvalue), and λ0is the (implicit) derivative
of the eigenvalue. We can then prove the following.
8
Theorem 7 (Zero-isolation condition for Hopf bifurcation).Fix some m∈ {1, . . . , n}and vm∈C. Let
u0= (x0, x0
0, α0, ω0, v0, λ0
0, v0
0)∈Rn×Rn×R×R×Cn−1×C×Cn−1be an isolated zero of the nonlinear
map Fdefined as follows:
(x, x0, α, ω, v, λ0, v 0)7→ F(u) =
f0(x, α)
D1f0(x, α)x0+D2f0(x, α)
D1f(x, α)hm(v) expiω −iωhm(v)
F4(x, x0, α, ω, v, λ0, v 0)
=
F1(x, α)
F2(x, x0, α)
F3(x, α, ω, v)
F4(x, x0, α, ω, v, λ0, v 0)
.
There exists a unique C2curve x0: (α0−, α0+)→Rnfor some > 0such that x0(α0) = x0.
Furthermore, the conditions (1.) and (2.) of Theorem 1 are satisfied provided v0and vmare not both
zero, and Re(λ0
0)6= 0.
Proof. Since the zero u0is isolated, the kernel of D1f0(x0, α0) must be trivial, from which the implicit
function theorem guarantees the existence of the unique steady state curve. The existence of the imaginary
eigenvalues ±iω follows from the equation D1f(x0, α0)hm(v) expiω −iωhm(v) = 0, the assumption on v0
and vmwhich guarantees hm(v0)6= 0, and the complex-conjugate parity of eigenvalues for real systems.
The existence of the C1eigenvalue branch α7→ λ(α) in σ(α) with λ(α0) = iω0follows once again from the
implicit function theorem and the isolation of u0, this time applied to F4(u0) = 0. We get transversality
from the assumption Re(λ0
0)6= 0 and the equality λ0
0=d
dα λ(α0).
2.2 Verifying zeroes of Fusing the radii polynomial approach
The nonlinear map Fdefined in Theorem 7 can be more formally described as F:U→V, where
U=Rn×Rn×R×R×Cn−1×C×Cn−1,
V=Rn×Rn×Cn×Cn.
Remark that as real vector space, Uand Vare 6n-dimensional. If an approximate zero u0of Fhas been
computed, we can verify the existence of a nearby isolated zero using the radii polynomial approach. In
what follows, we denote Br(u) = {x∈U:||x−u|| < r} ⊂ Uthe ball of radius r > 0centered at u∈U.
Its closure is denoted Br(u). The following theorem appears in various forms in the literature under
name radii polynomial approach; see for instance the book [9] and references therein. To keep the paper
self-contained, we will provide a brief proof.
Theorem 8. Let F:U→Vbe continuously differentiable and U, V be finite-dimensional vector spaces.
Let u0∈Ube given, and suppose there exists A:V→Uinjective and constants Y0≥0,Z0≥0and
Z2: [0,∞)→[0,∞)such that
||AF (u0)||U≤Y0
||IU−ADF (u0)||B(U)≤Z0
sup
δ∈Br(0)
||A[DF (u0+δ)−DF (u0)]||B(U)≤Z2(r).
where || · ||Uis a norm on Uand || · ||B(U)is the induced operator norm. Define the radii polynomial
p(r)=(Z2(r) + Z0−1)r+Y0.
If there exists r0>0such that p(r0)<0, then Fhas a unique zero in Br0(u0).
Proof. Define a Newton-like operator T:U→Uby T u =u−AF (u). We will show that Tis a contraction
on the closed ball Br0(u0), and therefore has a unique fixed point therein. Since Ais injective, this will
9
imply Fhas a unique zero in this ball. Since Fis continuously differentiable, the same is true of T. For
any u∈Br0(u0),
||DT (u)||B(U)=||I−ADF (u)||B(U)≤ ||I−AD F (u0)||B(U)+||A[DF (u)−DF (u0)]||B(U)
≤Z0+Z2(r0),
which is strictly less than 1 because p(r0)<0. It remains to show that the image of Br0(u0) under Tis
contained in Br0(u0), since the previous derivation has proven Tis contractive. We have
||T u −u0|| =||T u −T u0+ (T u0−u0)|| ≤ ||T u −T u0|| +||AF (u0)||
≤(Z0+Z2(r0))||u−u0|| +Y0≤(Z0+Z2(r0))r0+Y0
and since p(r0)<0, it follows that ||T u −u0|| < r0. Therefore, Tmaps Br0(u0) into itself, while any
fixed point must be strictly in the interior since ||T u −u0|| < r0.
In practice, Awill be a machine computed inverse of DF (u0) and we will use of Taylor’s theorem to
determine the Z2(r) bound. To be precise, given δ∈Br(0), define δj:U→Uj+1 by δju= (δ, . . . , δ, u)∈
Uj+1. For any δ∈Br(0) and integer order k≥0, we have the following bound due to the Lagrange
remainder:
||A[DF (u0+δ)−DF (u0)]||B(U)≤
k
X
j=1
1
j!ADj+1F(u0)δj
B(U)
+1
(k+ 1)!||ADk+2 F(u0+sδ)δk||B(U)
for some s∈[0,1]. If k= 0, the sum is treated as an empty sum and vanishes. In this way, since
Dj+1F(u0) : Uj+1 →Vis (j+ 1)-linear, each of ADj+1F(u0)δjis a linear operator on Udefined by
u7→ ADj+1F(u0)δju=ADj+1f(u0)[δ,...,δ,u].
It follows that we can take
Z2(r) = sup
δ∈Br(0)
k
X
j=1
1
j!ADj+1F(u0)δj
B(U)
+ sup
s∈[0,1]
sup
ξ∈Br(0)
1
(k+ 1)!||ADk+2 F(u0+sξ)ξk||B(U)(17)
as the Z2bound. It is straightforward to get an upper bound for this quantity using interval arithmetic
once the Fr´echet derivatives have been computed.
2.3 Implementation for the case of a single discrete delay
Here we will assume the functional fonly has a single discrete delay. That is, we restrict to delay
differential equations of the form
x0(t) = f(x(t), x(t−τ), α),(18)
where now, f:Rn×Rn×R→Rnis assumed C∞in a neighbourhood of (x0, x0, α0) for the candidate
zero u0of F, and expressible in terms of elementary functions. We can formally identify the functional
that defines the right-hand side of (18). It is
C([−τ, 0],Rn)×R3(φ, α)7→ f(φ(0), φ(−τ), α)≡˜
f(φ, α).
10
We want to express the nonlinear map Ffrom Theorem 7 in terms of the discrete-delay functional ˜
f.
It suffices to write down the components F1through F4. The first one is obvious:
F1(x, α) = f(x, x, α).(19)
Let Djdenote the partial Fr´echet derivative with respect to the jth variable. Similarly, write Dzfor the
derivative with respect to the variable z. Then F2is
F2(x, x0, α) = D1˜
f0(x, α)x0+D2˜
f0(x, α) = Dx˜
f0(x, α)x0+Dα˜
f0(x, α)
=Dx(f(x, x, α))x0+Dαf(x, x, α)
= (D1f(x, x, α) + D2f(x, x, α))x0+D3f(x, x, α).(20)
To compute F3, we will need a representation for the Fr´echet derivative D1˜
f(x, α). From its definition,
we have for φ∈C([−τ, 0],Rn),
D1˜
f(x, α)φ=D1f(x, x, α)φ(0) + D2f(x, x, α)φ(−τ).
This together with linearity of D2f(x, x, α) implies the representation
F3(x, α, ω, v) = (D1f(x, x, α) + e−iωτ D2f(x, x, α)−iωI)hm(v).(21)
To compute F4, we need expressions for D2
1˜
fand D2D1˜
f. For φ, ψ ∈C([−τ, 0],Rn),
D2D1˜
f(x, α)φ=D3D1f(x, x, α)φ(0) + D3D2f(x, x, α)φ(−τ),
D2
1˜
f(x, α)[φ, ψ] = D2
1f(x, x, α)[φ(0), ψ(0)] + D2D1f(x, x, α)[φ(0), ψ(−τ)]
+D1D2f(x, x, α)[φ(−τ), ψ(0)] + D2
2f(x, x, α)[ψ(−τ), ψ(−τ)].
Using these in (15) and simplifying with bilinearity, we get, for u= (x, x0, α, ω, v, λ0, v0) as in Theorem 7,
F4(u) = D2
1f(x, x, α)[x0, hm(v)] + e−iωτ D2D1f(x, x, α)[x0, hm(v)] + D1D2f(x, x, α)[x0, hm(v)]
+e−iωτ D2
2f(x, x, α)[x0, hm(v)] + D3D1f(x, x, α)hm(v) + e−iωτ D3D2f(x, x, α)hm(v)
−λ0(τe−iωτ D2f(x, x, α) + I)hm(v)+(D1f(x, x, α) + e−iωτ D2f(x, x, α)−iωI )jm(v0).(22)
To build the radii polynomial, we first require several partial derivatives of F= (F1, F2, F3, F4) from
(19)–(22). Rather than explicitly write down the various partial derivatives (of which there are many),
we rely instead on MATLAB’s symbolic algebra toolbox. First, we use the toolbox to build a realified
version of the map F:U→V, which we also denote F:R6n→R6n, based on a user-inputted anonymous
function of the form
@(x,xtau,alpha,para)f(x,xtau,a,para).
The interpretation of xand xtau are x(t) and x(t−τ). The variable ais the bifurcation parameter α, and
para is a vector of any other possible parameters or numerical constants that appear in the vector field.
This is done so that when a computer-assisted proof is done, these numerical parameters can be treated
as intervals (note: the symbolic algebra toolbox is not compatible with INTLAB). This step is necessary
because any floating point numbers included in fwill be operated on as such by the symbolic algebra
system, and if rounding occurs the resulting computer-assisted proof be incorrect. See the documentation
of the function files at [13] for further details.
The resulting symbolic function version of Fis saved, and a multivariable Taylor expansion of this
symbolic function is computed to a prescribed order kusing the taylor function. The expansion is
symbolically scaled by k! before being converted to a MATLAB function file. This step is done to
11
prevent MATLAB from saving a function file that contains string expressions of doubles that can not be
represented as a 64-bit double precision floating point numbers due to the Taylor expansion having terms
of the form 1/j! for j= 1, . . . , k. The coefficients of the re-scaled Taylor expansion will be representable
as 64-bit doubles so long as the order ksatisfies k!≤253, so practically speaking this is not a limitation.
In fact, in the applications of this paper we only use k= 2.
Once these function files are saved, the radii polynomial is constructed. We undo the scaling by k!
using interval arithmetic. We get the Z2bound using the Taylor expansion with Lagrange remainder
from (17). The system constants para and the delay tau are passed as machine precision thin intervals.
The result is an interval representation of the radii polynomial. By choosing an a-priori maximum radius
r∗, we can take the bound Z2(r)≤Z2(r∗) and compute the zero r0of
p∗(r)=(Z2(r∗) + Z0−1)r+Y0.
If r0< r∗, we then explicitly check that p(r0)<0 using interval arithmetic. The result is that given an
approximate zero u0of F, the zero-isolation conditions of Theorem 7 are automatically checked for the
true zero u∈Bu0(r0). This makes up a large part of the Hopf bifurcation conditions for systems of delay
differential equations. The condition at the end of the theorem concerning the eigenvector being nonzero
can be accomplished by explicitly requiring vm6= 0 to machine precision. As for Re(λ0)6= 0, it is enough
to find a radius r0such that the radii polynomial satisfies p(r0)<0 and 0 /∈Re(λ0
0) + r0[−1,1]. The
MATLAB function prove Hopf isolation.m is a complete implementation of this proof process.
3 Verification of non-resonance and simplicity
Here we outline two approaches to verify the non-resonance condition and the simplicity of the imaginary
eigenpair iω0that is required for condition (3.) of Theorem 1. The first is based on computation of
generalized Morse indices for the step map, and is applicable to equations with delayed arguments. The
second one is based on contour integration and the argument principle, and can be applied to general
functional differential equations.
3.1 Difference of generalized Morse indices and step map for delay equations
For discrete delay equations, we use a Chebyshev spectral method for the discretization of the step map
[37]. Using this method, one can rigorously prove (under certain conditions) that the step map and its
discretization have the same number of eigenvalues outside of a given closed ball of radius rcentered at
zero. This number of eigenvalues outside a given ball of radius ris called the generalized Morse index
and it is denoted µr. If this number of eigenvalues is proven to be equal for both the discretization and
the full, infinite-dimensional operator, we will say µrhas been validated. Since an imaginary eigenvalue
λ=eiω corresponds to an eigenvalue of the step map on the unit circle, we can obtain an upper bound for
the number of imaginary eigenvalues by computing the difference µ1−δ1−µ1+δ2for two positive offsets
δ1, δ2>0.
To accomplish the proof, there are three technical bounds denoted C1,C2and C3that must be
computed with interval arithmetic. If their product Cdef
=C1C2C3satisfies C < 1, then µris theoretically
validated. We encourage readers interested in the details to consult the paper [37], as we will not discuss
any details here. The final step that needs to be done is to rigorously compute the eigenvalues of the
discretization of the step map. This can be done with verifyeig of INTLAB or with the radii polynomial
approach [12].
Assume we have a delay differential equation
˙x=f(x(t), x(t−τ), α)
12
depending on the parameter α, we have located a candidate isolated zero u0of the map Ffrom Theorem
7, and that the radii polynomial implemented in Section 2.3 is negative at some r0>0. We compute the
linearization
˙y=D1f(x0, x0, α0)y(t) + D2f(x0, x0, α0)y(t−τ),
and replace each of x0and α0with intervals [x0−r0, x0+r0] and [α0−r0, α0+r0]. We then choose some
δ1, δ2>0 and rigorously compute the difference of Morse indices D(δ1, δ2)def
=µ1−δ1−µ1+δ2. We compute
and validate each one (as described in the previous paragraph) and take the difference. If D(δ1, δ2) = 2,
then we have proven the simplicity and non-resonance conditions for the Hopf bifurcation. Indeed, we
already know that i[ω0−r0, ω0+r0] contains an imaginary eigenvalue and this interval does not intersect
zero, so its complex conjugate defines precisely the second eigenvalue in the count D(δ1, δ2) = 2. This
proves the non-resonance, and since the indices count multiplicities, we can conclude that the eigenvalues
±iω0are simple.
The function prove non resonance.m completes this calculation. It takes as its inputs the matrices
K1=D1f(x0, x0, α0), K2=D2f(x0, x0, α0), the delay τ, the dimension nof the system, a number N
of modes to use in the Chebyshev spectral method, a weight ν > 1 in the sequence space (see [37] for
details), the offsets δ1and δ2, as well as some data relating to the mesh needed to verify the Morse indices
(m, the number of mesh points, a flag for uniform or adaptive step size, and a parameter that specifies
if the eigenvalues of interval matrices should be validated by the radii polynomial approach [12] or using
verifyeig). The function F Hopf build.m that is called during a run of prove Hopf isolation.m will
output functions Jacobian 0.mat and Jacobian tau.mat that can be respectively used to compute K1
and K2.
3.2 Argument principle and validated winding number computation
If we have a more complicated functional differential equation — for example, one with multiple discrete
delays, distributed delays, or an equation of mixed-type — we must use a different method to verify
non-resonance and simplicity of the critical eigenvalues. This is because the computer-assisted validation
of Morse indices has not been developed for functional differential equations in full generality. In this
section, we develop an approach based on contour integration.
If we have proven the existence of a candidate Hopf bifurcation — that is, one for which conditions
(1.) and (2.) of Theorem 1 have been verified — we are left with proving that the characteristic equation
(2) has at most two zeroes on the imaginary axis (or at most one with positive imaginary part). In what
follows, we will assume that we are able to prove an a priori, analytical upper bound ˆωsuch that any
imaginary eigenvalue λ=iω must satisfy |ω| ≤ ˆω. Recalling the characteristic matrix ∆(λ) defined in
(1), it suffices to compute the number of zeroes of the nonlinear function
g(z)def
= det ∆(z)
contained in some rectangle of the form
Rδ,,ˆω={z∈C:−δ≤Re(z)≤δ, −≤Im(z)≤ˆω+},
where δ > 0 is chosen small enough so that any other zeroes of gthat are not on the imaginary axis
are excluded from the rectangle, and 0 < |ω0|. Note that we can choose > 0 as small as we want
because eigenvalues come in complex-conjugate pairs, so those in the bottom half-plane do not need to
be counted. For the applications in this paper, the functions ∆(z) are
∆(z) = τ(−σz −e−x0−z)−z
for the Lasota-Wazewska-Czyzewska model (Section 1.2), with x0∈R(see Section 4.1; a change of
variables is used to treat the delay τas a bifurcation parameter),
∆(z) = z−σ1−ξ ρξ
ξ−σ2−ρξ +e−zτ e−x00
0e−y0−zI
13
for the coupled Lasota-Wazewska-Czyzewska equations (Section 1.3), with x0, y0∈R, and
∆(z) = 0 1
ρσ h
z(ez−1) + 1−h
zM (1 −e−zM )c−zI
for the traveling waves of the nonlocal Fisher equation (Section 1.4), where ρand Mare some bulk
parameters resulting from a change of variables; see Section 4.3 for details. The function(s) gcan then
be computed by taking the determinant.
Denote Γ = Γδ,, ˆωthe positively oriented simple closed curve in Cconsisting of the boundary of the
above rectangle (see Figure 1 for an example). As long as fis continuously differentiable, the characteristic
-0.15 -0.1 -0.05 0 0.05 0.1 0.15
0
1
2
3
4
5
Figure 1: Example of a contour Γ = Γδ,,ˆω. Here we have δ= 0.09, = 0.2 and ˆω= 5.
matrix is continuous in its argument and gwill have no poles inside Γ. If fis analytic, gis meromorphic
– in fact, analytic – in Γ. Since any strip in Ccontains finitely-many eigenvalues – see Lemma 4.3 from
[29] – the number of zeroes of ginside of Γ is finite.
By Cauchy’s argument principle
Nzeros −Npoles =1
2πi IΓ
g0(z)
g(z)dz,
where Nzeros and Npoles denote respectively the number of zeros and poles of g(z) inside the contour Γ.
Since the function gdoes not have any poles inside Γ, the number of zeros inside Γ is simply given by the
formula
Nzeros =1
2πi IΓ
g0(z)
g(z)dz =1
2πi Ig(Γ)
1
wdw,
where the second equality follows by the substitution w=f(z). Therefore, Nzeros is precisely equal to
the winding number of the path g(Γ) around 0 ∈C.
To rigorously compute the winding number of g(Γ), we will use a straightforward path-covering ar-
gument in conjunction with interval arithmetic. To begin, let `be a distinguished ray in Cemanating
from 0. Decompose Γ into four arcs Γjfor j= 1,2,3,4 corresponding to each side of the rectangle
Rδ,,ˆω. Suppose one of these arcs is denoted γ. We abuse notation and treat γas a parameterization,
γ: [0,1] →Γj⊂C, for some j. The following algorithm determines how many (signed) times the curve
g(γ) traverses the ray `in the counterclockwise direction. See Figure 2 for a visual aid.
14
1. Select equally-spaced gridpoints γ1, . . . , γM(for some M > 0) along the arc γ, such that γ1=γ(0)
and γM=γ(1), with spacing |γk+1 −γk|=h.
2. Initialize complex intervals γkfor k= 1, . . . , M, centered at the gridpoints γkand having radius
h
2+ξfor some ξ > 0 small. By construction, the intersections γk∩γk+1 are nonempty.
3. Compute gk=g(γk) for k= 1, . . . , M using interval arithmetic, and verify that 0 /∈gk. If there is
an index ksuch that 0 ∈gk, the spacing h(and/or ξ) is too large. In that case, halt the algorithm
and restart with different parameters.
4. Compute the set K={k= 1, . . . , m :gk∩`6=∅}.
5. Partition Kas a disjoint union K=SJ
j=1 Kjsuch that Kj={κj, κj+ 1, . . . , κj+nj}for some
nj>0. In other words, determine all sequential runs of integers in K.
6. For each partition Kj, compute g0
j={g0(γu)γ0
u:u∈ Kj}, where γ0
u=γ0(su) such that γ(su) = γu.
In other words, the derivative is computed along the composition g◦γ.
7. Compute the (interval) angle θj⊂[−π, π] counterclockwise from g0
jto `for each j. If the angle
contains zero, then either the spacing h(and/or ξ) is too large or g(γ) is tangent to `, and a different
ray should be selected. In that case, halt the algorithm and re-start with different parameters.
8. A strictly positive angle θjcounts as +1 for the number of traversals of `, while a strictly negative
angle counts as −1. For j= 1, . . . , J , let cj= 1 if θj>0 and cj=−1 if θj<0. The number of
signed traversals of the ray `by g(γ) is therefore w(γ) = PJ
j=1 cj.
If this algorithm terminates at step 8 for all four arcs Γ1,Γ2,Γ3and Γ4, then the winding number is given
by w=P4
n=1 w(Γn). By previous discussions, this is precisely the number of zeros of gin the rectangle
Rδ,,ˆω.
4 Computer-assisted proofs
This section contains outlines of the computer-assisted proofs of Theorem 2, Theorem 3 and Theorem
4. The enclosures of zeroes provided in those theorems are computed from the numerical outputs of the
proofs, which appear in Table 1, Table 2 and Table 3. The codes to complete the proofs are available at
[13].
4.1 Proof of Theorem 2
First, we scale time to make the delay a smooth parameter. Let ˆ
t=tτ. Completing the change of
variables and dropping the hats, we get the delay equation
˙y=τ(−σy(t) + e−y(t−1)).
We now treat τas the parameter and σ∈ {0.3,0.35}as a fixed constant, since the delay is now equal to 1.
We automatically prove the theorem by running the MATLAB function prove single delay examples.m
with the input NUMBER set to 1 for the first proof (σ= 0.3) and 2 for the second (σ= 0.35). This function
calls prove Hopf isolation.m as outlined in Section 2.3 to verify the hypotheses of Theorem 7. To
check the non-resonance and simplicity, we use the generalized Morse index approach, making use of
prove non resonance.m for the verification of µ1−δ1−µ1+δ2= 2. The parameters used in the proof and
associated outputs appear in Table 1.
15
0 100 200 300 400 500
-100
-80
-60
-40
-20
0
20
40
60
-40 -30 -20 -10 0 10 20 30 40 50
-50
-40
-30
-20
-10
0
10
Figure 2: Left: plot of the curve g(Γ) in blue, with segments in yellow (very small) indicating traversals
of the ray [0,∞). Right: zoomed in section with intervals plotted (rings) from the twisted section on
the left, featuring two such traversals with the ray [0,∞). Over each yellow-coded group of interval, the
derivative g0is computed and if the imaginary part is strictly bounded away from zero, there is a unique
traversal of the ray. In this example, gis the function appearing in (29) with parameters coming from
the proof W4 with wave velocity c2of Theorem 4. There are four traversals of the ray, three of which are
clockwise and one anticlockwise, yielding a winding number of two.
σ= 0.3σ= 0.35
x01.104542018324 1.025065556445
x0
00 0
α019.208854104207 37.030171112739
ω0-2.703005650033 -2.919994153135
Re(λ0) 0.007171184698 0.001131897009
Im(λ0) -0.017941603965 -0.005411625903
r07.1870665 ×10−13 9.7779366 ×10−13
δ10.1 0.094
δ20.1 0.3
ν1.15 1.06
N100 490
C(1 −δ1) 0.91823 0.98829
C(1 + δ2) 0.89056 0.16294
µ1−δ12 2
µ1+δ20 0
Table 1: Top portion: Candidate zero (x0,...,Im(λ0)) and smallest validation radius r0for the Hopf
bifurcation isolation proofs of the Lasota-Wazewska-Czyzewska equation. Note that the eigenvector vis
fixed to v= 1, and system bifurcation parameter τhas label αin the coordinates of the zero (hence in
the table), so the bifurcation occurs at τ=α0. Bottom portion: Parameters (δ1, δ2, ν, N ) used for the
Morse index computations, product Cof the technical bounds (less than 1 indicates a successful proof )
for the indices µ1−δ1and µ1+δ2, and these indices.
16
ρ= 0 ρ= 1
x01.144668868008599 1.585823613805185
y01.084122019684906 0.903053550980479
x0
0-1.919211635684326 -1.727579568874282
y0
00.957842795546628 0.581589081774491
α00.178097289316629 0.067660704405282
ω0-0.154904637991545 -0.142261377213083
Re(v2) 0.868760285342032 0.208874995011447
Im(v2) -0.041065942913387 -0.099748188056958
Re(λ0) -0.047501861871352 -0.008458308451281
Im(λ0) -0.073744885404130 -0.149652361211147
Re(v0
2) 4.238789580398924 2.225857970381196
Im(v0
2) -6.020231755626770 -2.762306284945626
r07.0027385 ×10−14 1.8492517 ×10−13
δ10.05 0.15
δ20.1 0.5
ν1.12 1.15
N280 150
C(1 −δ1) 0.90069 0.95435
C(1 + δ2) 0.29771 0.19048
µ1−δ12 2
µ1+δ20 0
Table 2: Top portion: Candidate zero (x0, y0,...,Re(v0
2),Im(v0
2)) and uniqueness radius r0(rounded
up for readability) for the Hopf bifurcation isolation proofs of the coupled Lasota-Wazewska-Czyzewska
equations. Note that the first component of the eigenvector vis fixed to v1= 1, and the bifurcation
parameter ξcorresponds to the component αin the coordinates of the zero. Bottom portion: Parameters
(δ1, δ2, ν, N ) used for the Morse index computations, product Cof the technical bounds (less than 1
indicates a successful proof), and the indices µ1−δ1and µ1+δ2.
4.2 Proof of Theorem 3
The setup for Theorem 3 is even more straightforward than the previous one. The only important differ-
ence is that since the system is two-dimensional, we fix the first component of the eigenvector vto v1= 1.
The coupling ξis treated as the bifurcation parameter and we run prove single delay examples.m with
the input NUMBER set to 3 (for ρ= 0) or 4 (for ρ= 1). This function calls prove Hopf isolation.m as
outlined in Section 2.3 to verify the hypotheses of Theorem 7. To check the non-resonance and simplicity,
we use the generalized Morse index approach, making use of prove non resonance.m for the verification
of µ1−δ1−µ1+δ2= 2. The parameters used in the proof and associated outputs appear in Table 2, and
the eigenvalues at the bifurcation point can be visualized with Figure 3.
4.3 Proof of Theorem 4
If we substitute the kernel φfrom (6) into the PDE (4) we get
ut=Duxx +µu σ−h
N1Z0
−N1
u(t, x −y)dy −1−h
N2ZN2
0
u(t, x −z)dz!.
17
-1 0 1
-1
-0.5
0
0.5
1
-1 0 1
-1.5
-1
-0.5
0
0.5
1
1.5
Figure 3: Eigenvalues of the step map at the bifurcation point for the coupled Lasota-Wazewska-
Czyzewska equations. The circles in red and green have radius 1 −δ1and 1 + δ2respectively, and
the two eigenvalues in the gap correspond to the imaginary eigenvalues ±iω0from Table 2. Left: ρ= 0.
Right: ρ= 1. The proof for ρ= 0 is more computationally expensive because, compared to ρ= 1, there
are more eigenvalues closer to the unit circle.
We make the change of variables u(t, x) = w(DN−2
1t, N −1
1x) and define new parameters ρ=µN2
1D−1
and M=N2
N1. We get
wt=wxx +ρw σ−hZ0
−1
w(t, x −y)dy −1−h
MZM
0
w(t, x −z)dz!.(23)
Suppose w(t, x) = ψ(x+ct) for a wave profile ψand speed c. Substituting this into (23), we get the
functional differential equation
cψ0(t) = ψ00 (t) + ρψ(t) σ−hZ0
−1
ψ(t−y)dy −1−h
MZM
0
ψ(t−z)dz!(24)
for the wave profile. Observe that σis a fixed point. If ψis a periodic solution of this equation, then
inverting the first change of variables (from uto w) reveals
u(t, x) = ψ1
N1
x+cD
N2
1
t
is a solution of the nonlocal PDE. To obtain the form of (7) from Corollary 5, we simply scale out by
N−1
1: define ˜
ψ(z) = ψ(N−1
1z) so u(t, x) = ˜
ψ(x+cDN −1
1t).
The remaining steps of the proof are completed by running the MATLAB function prove Twave Hopf.m.
This function takes two inputs (j, k) and completes the proof Wjfrom Theorem 4 for the wave velocity
ck. We will now outline the steps that this function completes and state the numerical outputs of the
proof.
18
4.3.1 Hopf isolation and transersality conditions
From this point forward, we let σ= 1 as in the statement of the theorem. Define f:C([−M, 1],R2)×R→
R2by
f(q, c) =
q2(0)
cq2(0) −ρq1(0) 1−hZ0
−1
q1(−s)ds −1−h
MZM
0
q1(−s)ds!
.
Defining (q1, q2)=(ψ, ψ0), the equation (24) is equivalent to ˙q=f(qt, c) for qt(θ) = q(t+θ) for θ∈[−M, 1],
and therefore has a fixed point at (σ, 0) = (1,0), since σis a fixed point of (24). We use Theorem 7 and
Theorem 8 to verify the isolation and transversality conditions (1.) and (2.) from Theorem 1. To do this,
we need to compute a few partial differentials. Letting x, y , w ∈C([−M , 1],R2),
D1f(x, c)y=
y2(0)
cy2(0) −ρy1(0) σ−hR0
−1x1(−s)ds −(1−h)
MRM
0x1(−s)ds
+ρx1(0) hR0
−1y1(−s)ds +(1−h)
MRM
0y1(−s)ds
(25)
D2f(x, c) = 0
x2(0)
D2
1f(x, c)[y, w] =
0
ρy1(0) hR0
−1w1(−s)ds +(1−h)
MRM
0w1(−s)ds
+ρw1(0) hR0
−1y1(−s)ds +(1−h)
MRM
0y1(−s)ds
D2D1f(x, c)y=0
y2(0) .
The MATLAB function prove Twave Hopf isolation.m generates (if not yet done by the user) a symbolic
representation of F, a symbolic representation of the Fr´echet derivative DF , and Taylor expansions using
a routine formally analogous to the one in Section 2.3 for the case of a single discrete delay. It then
implements the radii polynomial from Theorem 8 using the bound Z2from (17) and automatically checks
the transversality condition Re(λ0)6= 0. The symbolic calculations are a bit slow and result in inefficient
function files that must then be passed interval data, so the code takes a few seconds to run rather than
being nearly instant. This code automatically checks the zero-isolation conditions of Theorem 7. The
approximate zeroes that are passed as the inputs to the proof and the resulting outputs are provided in
Table 3.
4.3.2 Non-resonance and simplicity
Before we begin, we emphasize that the radius r0from the previous Hopf isolation and transversality
section are propagated into this proof. The result is that we replace cwith the interval [c−r0, c +r0] in
any subsequent computations. This will not be explicitly written in what follows.
Making use of (25), we can calculate the characteristic matrix as defined by (1). We find
∆(λ) = −λ1
ρh
λ(eλ−1) + 1−h
λM (1 −e−λM )c−λ
Therefore,
det ∆(λ) = λ2−cλ +ρ
λh(1 −eλ) + 1−h
M(e−Mλ −1),(26)
19
W1 W2 W3 W4
c1-0.499960441060187 0.499960441060186 -0.363754795740408 0.743510960061904
ω04.637215079560793 4.637215079560793 4.592098499884637 2.889763462936531
Re(v2) 0 0 0 0
Im(v2) 4.637215079560793 4.637215079560793 4.592098499884637 2.889763462936531
Re(λ0) 0.070132958310718 0.048464565698432 0.040085041058377 -0.057440819946235
Im(λ0) 0.216934012587930 0.183223000908009 0.196850686215880 0.052484450151143
Re(v0
2) 1.076102632753096 0.898109028431434 0.944042781931582 0.094226826472853
Im(v0
2) -0.108287599264741 -0.041517613973126 0.012776229303893 0.218474832912888
r02.5496855 ×10−15 3.2214306 ×10−15 4.0559742 ×10−15 2.5866136 ×10−15
W1 W2 W3 W4
c2-1.407518070559178 1.407518070559176 0.225343700115205 38.377317897727600
ω03.648913186685016 3.648913186685016 3.513777373551947 1.612796341206019
Re(v2) 0 0 0 0
Im(v2) 3.648913186685016 3.648913186685016 3.513777373551947 1.612796341206019
Re(λ0) 0.111230975589598 0.103152903698957 0.094953411313056 0.025059472072250
Im(λ0) 0.048288669938503 0.054733236581360 0.050485616225114 0.003084428687102
Re(v0
2) 0.287432140095682 0.302869732410634 0.272348627294689 0.030034027373518
Im(v0
2) -0.357583503658218 -0.321662753970615 -0.283159531988272 -0.037331396183577
r05.6381454 ×10−15 7.5708092 ×10−15 5.8092945 ×10−15 1.3536766 ×10−15
Table 3: Candidate zeroes for the Hopf isolation and transversality part of the proof of Theorem 4 and
radius from the radii polynomial. The parameter is α=c1, and the steady state is fixed at x0= (1,0),
so x0
0= (0,0) in terms of the variables of Theorem 7. The first component of the eigenvector vis fixed to
v1= 1. Top: the wave with velocity near c1. Bottom: the wave with velocity near c2. We have rounded
up the expressions of r0for readability.
20
and eigenvalues λsatisfy det ∆(λ) = 0.
If λ=iω, then det ∆(λ) = 0 reduces to
0 = −ω2+iωc −1
ωiρ h(1 −cos ω−isin ω) + 1−h
M(cos(ωM )−isin(ωM )−1),
which we can equivalently write as
ω3=iω2c−iρ h(1 −cos ω−isin ω) + 1−h
M(cos(ωM )−isin(ωM )−1)(27)
Define
ˆω=3
sρh+1−h
M.
Taking real parts and absolute values in (27), it follows that
|ω|3=ρh|sin ω|+1−h
M|sin(ωM )|≤ρh+1−h
M= ˆω3,(28)
from which we conclude that |ω| ≤ ˆω.
Let use define a function
g(z)def
=z∆(z) = z3−cz2+ρh(1 −ez) + 1−h
M(e−Mz −1).(29)
Observe that the multiplication by zpreserves zeroes of ∆(z), in the sense that if ∆(z) = 0 then g(z) = 0.
However, it does two other things: it introduces a zero of order 1 at z= 0, and removes the reciprocal 1
z
term. This helps with the polynomial embedding.
We use winding number approach of Section 3.2 to compute the number of zeros of gin the rectangle
Rδ, ={z∈C:−δ≤Re(z)≤δ, −≤Im(z)≤ˆω+},
where ˆωis the bound defined by (28) and δ= 0.09 and = 0.05 are two positive parameters that we have
selected to ensure the contour both strictly includes (by our choice of ) 0 ∈Cand iˆω, as well as ensuring
that no extraneous zeroes close to the imaginary axis are included (by choice of δ).
Note that the rectangle Rδ, must enclose any z=iω for which iω,ω > 0 is a solution of the
characteristic equation (26), due to the inequality (28). Our goal is to prove that this contour contains
exactly two zeroes of g: one at z= 0 that comes from the multiplication we made by zwhen defining
g(z) = z∆(z), and the other being z=iω0for ω0>0 coming from the Hopf bifurcation. This will prove
that ∆(z) = 0 has exactly two solutions on the imaginary axis, namely ±iω0. Doing this, we will have
proven all the sufficient conditions of Theorem 1, and our Hopf bifurcation will br proven.
The MATLAB function compute winding number.m is an implementation of the rigorous winding
number counter from Section 3.2 specified to the function gfrom (29). We used the ray `= [0,∞) for all
computations. The mesh width subdivision (i.e. gridpoint spacing) on the “long” sides of the rectangle
was chosen to be 3 ×10−3, padded with an extra gridpoint at the end of each arc to accomodate for any
remainder from the division of the arclength divided by 3×10−3. The subdivision on the “short” arcs was
scaled to the relative lengths of the long vs. short arcs to ensure that the radii of the intervals covering
the curve was uniform. We used ξ= 0.05 for the interval generation along the arcs.
Figure 4 provides plots of all of the curves g(Γ) for the proofs W1–W4 for both velocities, c1and c2.
For visual clarity, we have not plotted the intervals. The winding number is proven to be exactly 2 in all
cases, completing the proof.
21
0 10 20
-100
-80
-60
-40
-20
0
-20 -10 0
-100
-80
-60
-40
-20
0
-15 -10 -5 0 5
-100
-80
-60
-40
-20
0
-100 -50 0
-100
-50
0
-10 0 10 20
-100
-80
-60
-40
-20
0
-20 -10 0 10
-100
-80
-60
-40
-20
0
-10 0 10
-100
-80
-60
-40
-20
0
0 200 400
-100
-50
0
50
Figure 4: Plots of g(Γ) for all fo the proofs of Theorem 4. The regions where the interval subdivision of
g(Γ) intersect the ray [0, `) are displayed in yellow, although these are fairly small and difficult to resolve.
Top row: velocity c1with proofs W1–W4 (left to right). Bottom row: velocity c2with proofs W1–W4
(left to right).
Acknowledgments
The authors thank the two anonymous referees for their helpful comments that lead to significant im-
provements to the paper. Kevin E. M. Church acknowledges the support of NSERC through the NSERC
Postdoctoral Fellowships Program. Jean-Philippe Lessard is supported by an NSERC Discovery Grant.
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