Bifurcation analysis of Bogdanov-Takens bifurcations in delay differential equations_2

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Bifurcation analysis of Bogdanov-Takens bifurcations in delay

diﬀerential equations∗

M.M. Bosschaert†Yu.A. Kuznetsov‡

October 7, 2022

Abstract: In this paper, we will perform the parameter-dependent center manifold reduction

near the generic and transcritical codimension two Bogdanov–Takens bifurcation in classical delay

diﬀerential equations (DDEs). Using a generalization of the Lindstedt-Poincaré method to approx-

imate the homoclinic solution allows us to initialize the continuation of the homoclinic bifurcation

curves emanating from these points. The normal form transformation is derived in the functional

analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization

technique based on the Fredholm alternative. The obtained expressions give explicit formulas,

which have been implemented in the freely available bifurcation software package DDE-BifTool.

The eﬀectiveness is demonstrated on various models.

Keywords: generic Bogdanov–Takens bifurcation, transcritical Bogdanov–Takens bifurcation, ho-

moclinic solutions, delay diﬀerential equations, sun-star calculus, strongly continuous semi-groups,

Center Manifold Theorem, DDE-BifTool

2020 MSC: 37G05, 37G10, 65P30, 34K16, 34K18, 34K19

1 Introduction

The Bogdanov–Takens bifurcation caused by the presence of an equilibrium with a double zero eigen-

value is a well-studied singularity in dynamical systems. It implies existence of saddle homoclinic

orbits nearby, which is a global phenomenon. In particular, the codimension two Bogdanov–Takens

bifurcation in ﬁnite-dimensional ordinary diﬀerential equations (ODEs) has been studied theoretically

and applied in numerous reserach publications. The same is true for the inﬁnite-dimensional dynamical

systems generated by delay diﬀerential equations (DDEs). In the simplest case, often encountered in

applications, such DDEs have the form

˙x(t) = f(x(t), x(t−τ1), . . . , x(t−τm), α), t ≥0,(1)

where x(t)∈Rn, α ∈Rp, while 0< τ1< τ2<··· < τmare constant delays, and f:Rn×(m+1) ×Rp→

Rnis a smooth mapping. These are known as discrete DDEs.

Up to date, the standard available parameter-dependent center manifold theorem for DDEs in [13]

assumed that the equilibrium persists for all nearby parameter values. This was a serious limitation,

since in generic unfoldings of the codimension two Bogdanov–Takens singularity it is not the case.

∗Submitted to the editors DATE.

†Department of Mathematics, Hasselt University, Diepenbeek Campus, Agoralaan Gebouw D, 3590 Diepenbeek,

Belgium (maikel.bosschaert@uhasselt.be).

‡Department of Mathematics, Utrecht University, Budapestlaan 6, 3508 TA Utrecht, The Netherlands and De-

partment of Applied Mathematics, University of Twente, Zilverling Building, 7500AE Enschede, The Netherlands

(I.A.Kouznetsov@uu.nl).

arXiv:2210.02560v1 [math.DS] 5 Oct 2022

However, recently, in [2], this obstruction has been removed and the existence of ﬁnite-dimensional

smooth parameter-dependent local center manifolds has been rigorously established in the functional

analytic perturbation framework for dual semigroups (sun-star calculus) developed in [5,6,7,8]. Once

the existence of these invariant manifolds is proved, the normalization technique for local bifurcations

of ODEs developed in [27] can be lifted rather easily to the inﬁnite dimensional setting of DDEs. The

advantages of this normalization technique are that the center manifold reduction and the calculation of

the normal form coeﬃcients are performed simultaneously by solving the so-called homological equation.

This method gives explicit expressions for the coeﬃcients rather than a procedure as developed in

[17,18]. The explicit expressions make them particularly suitable for both symbolic and numerical

evaluation.

Indeed, utilizing the normalization method, the authors in [2] obtained asymptotics to initialize the

continuation of codimension one bifurcation curves of nonhyperbolic equilibria and cycles emanating

from the codimension two generalized Hopf, fold-Hopf, Hopf-Hopf and transcritical-Hopf bifurcation

points in DDEs of the form (1). These asymptotics have been implemented into the fully GNU Octave

compatible MATLAB package DDE-BifTool [16,35].

Another recent developent is the rigorous derivation of higher-order asymptotics for the codimen-

sion one homoclinic bifurcation curve emanating from the generic codimension two Bogdanov–Takens

bifurcation point in in two-parameter ODEs [3]. Thus, by combining the results of the parameter-

dependent center manifolds in DDEs from [2] and the homoclinic asymptotics in ODEs from [3], we

are in the position to perform the parameter-dependent center manifold reduction and normalization

for the generic and transcritical codimension two Bogdanov–Takens bifurcations. This will allow us

to initialize the continuation of codimension one bifurcation curves of nonhyperbolic equilibria and

homoclinic solutions emanating from these codimension two points. Hopefully, our results and their

software implementation will make the numerical analysis of Bogdanov–Takens bifurcations in DDEs

from applications rather routine.

This paper is organized as follows. We begin in Section 2 with a short summary from [2] on

parameter-dependent center manifolds for classical DDEs and we state various results needed for the

normalization technique. In Section 3 we describe the general technique that we use to derive the trans-

formation from the orbital normal form on the parameter-dependent center manifold in the inﬁnite-

dimensional setting of DDEs. In Section 4 the method is then applied to the generic and transcritical

codimension two Bogdanov–Takens bifurcations. We provide explicit transformations necessary for the

predictors of codimension one bifurcation curves. We do this in a form suitable for classical DDEs,

covering cases that are more general than (1). It is here where we see the true beneﬁt of allowing for

orbital normal forms on the center manifold. Indeed, we do not need to derive homoclinic asymptotics

for the transcritical codimension two Bogdanov–Takens bifurcation separately. Instead, we only need

to derive the center manifold transformation for the transcritical codimension two Bogdanov–Takens

bifurcation. Then, using the blow-up transformations (82), we obtain the same perturbed Hamiltonian

system (up to order three) as in the generic Bogdanov–Takens bifurcation.

We employ our implementation in DDE-BifTool to illustrate the accuracy of the codimension one

bifurcation curve predictors through various example models, displaying the generic and transcritical

codimension two Bogdanov–Takens bifurcations in Section 5. An in-depth treatment of the examples,

including the MATLAB and Julia source code to reproduce the obtained results, as well as a more in-depth

analysis of the examples, are provided in the Supplement.

2 Parameter-dependent center manifolds for DDEs

Here we summarize those results from [2] on parameter-dependent center manifold for classical DDEs,

which are required for the normalization technique in Section 4. For a general introduction on pertur-

bation theory for dual semigroups (also known as sun-star calculus) we refer to [13].

Consider the classical parameter-dependent DDE

˙x(t) = F(xt, α), t ≥0,(DDE)

where F:X×Rp→Rnis Ck-smooth for some k≥1with F(0,0) = 0 and X:=C([−h, 0],Rn). Here

for each t≥0, the history function xt: [−h, 0] →Rndeﬁned by

xt(θ):=x(t+θ),∀θ∈[−h, 0].

It is convenient to split the right hand-side into its linear and nonlinear parts and write

F(φ, α) = hζ, φi+D2F(0,0)α+G(xt, α).(2)

Here ζ: [0, h]→Rn×nis a matrix-valued function of bounded variation, normalized by the requirement

that ζ(0) = 0 and is right-continuous on the open interval (0, h), and G:X→Rnis a Ck-smooth

nonlinear operator with G(0,0) = G1(0,0) = G2(0,0) = 0. The pairing is deﬁned by

hζ, φi:=Zh

dζ(θ)φ(−θ),(3)

where the integral is of the Riemann–Stieltjes type.

Let Tbe the C0-semigroup on Xcorresponding to the linearization of (DDE) at 0∈Xfor the

critical parameter value α= 0. Suppose that the generator

D(A) = {φ|˙

φ∈X, ˙

φ(0) = hζ, φi}, Aφ =˙

φ,

of Thas 1≤n0<∞purely imaginary eigenvalues with corresponding n0-dimensional real center

eigenspace X0. Then by [2, Corollary 20] there exists a Ck-smooth map C:U×Up→Xdeﬁned in a

neighborhood of the origin in X0×Rpand such that for every suﬃciently small α∈Rpthe manifold

loc(α):=C(U, α)is locally positively invariant for the semiﬂow generated by (DDE) at parameter

value α.

Since Xis not reﬂexive, i.e. does not isomorphic to its dual space X?, the adjoint semigroup T?is

only weak?continuous on X?and A?generates T?only in the weak?sense. The maximal subspace of

strong continuity

X:={x?∈X?:t7→ T?(t)x?is norm-continuous on R+}

is invariant under T?, and we have the representation

X=Rn? ×L1([0, h],Rn?).(4)

The duality pairing between φ= (c, g)∈Xand φ∈Xis

hφ, φi=cφ(0) + Zh

g(θ)φ(−θ)dθ. (5)

At this stage, we again have a C0-semigroup Twith generator Aon a Banach space Xso we

can iterate the above construction once more. On the dual space

X?=Rn×L∞([−h, 0],Rn),

we obtain the adjoint semigroup T?with the weak?generator

D(A?) = {(α, φ)∈X?|φ∈Lip(α)}, A?(α, φ)=(hζ, φi,˙

φ).(6)

The duality pairing between φ?= (a, ψ)∈X?and φ= (c, g)∈Xis

hφ?, φi=ca +Zh

g(θ)ψ(−θ)dθ. (7)

By restriction to the maximal subspace of strong continuity X =D(A?), we end up with the

C-semigroup T. Its generator A is the part of A?in X. The canonical injection j:X→X?

is given by

j(φ)=(φ(0), φ),(8)

mapping Xonto X. Therefore, Xis sun-reﬂexive with respect to the shift semigroup T.

We are now in the position to state the second part of [2, Corollary 20]. That is, if the history xt

associated with a solution of (DDE) exists on some nondegenerate interval Iand xt∈ Wc

loc(α)for all

t∈I, then u:I→Xdeﬁned by u(t):=xtis diﬀerentiable and satisﬁes

j˙u(t) = A?ju(t)+(D2F(0,0)α)r?+G(u(t), α)r?,∀t∈I.

Here, for i= 1, . . . , n, we denote r?

i:= (ei,0), where eiis the ith standard basis vector of Rnand

wr?:=

i=1

wir?

i,∀w= (w1, . . . , wn)∈Rn.

2.1 Spectral computations for classical DDEs in case of multiple eigenvalues

It is well known that for classical DDEs all spectral information about the generator Ais contained in

a holomorphic characteristic matrix function ∆ : C→Cn×ndeﬁned by

∆(z):=zI −ˆ

ζ(z)with ˆ

ζ(z):=Zh

e−zθ dζ(θ),(9)

where ζis the real kernel from (3), see [13, Sections IV.4 and IV.5]. In particular, the eigenvalues of A

are the roots of the characteristic equation

det ∆(z) = 0,(10)

and the algebraic multiplicity of an eigenvalue equals its order as a root of (10).

For the normalization technique in Section 4, we will need normalized representations for the (gener-

alized) eigenfunctions and adjoint (generalized) eigenfunctions of the generator Aand A?, respectively.

In this section, we will consider the more general case where λis an eigenvalue of algebraic multiplicity

k∈Nand geometric multiplicity 1. Although in Section 4 we will only need the special case where

λ= 0 is a double eigenvalue of A, the expressions are useful when considering for example the 1:1

resonant Hopf bifurcation and the triple zero bifurcation.

Proposition 1. Let λbe an eigenvalue of the generator Awith algebraic multiplicity k∈Nand

geometric multiplicity one, then there are (generalized )eigenfunctions φisuch that

Aφ0=λφ0, Aφi=λφi+φi−1, i ∈ {1, . . . , k −1},(11)

and adjoint (generalized )eigenfunctions ψisuch that

A?ψk−1=λψk−1, A?ψk−i=λψk−i+ψk−i+1, i ∈ {2, . . . , k}.(12)

Let the ordered set (q0, . . . , qk−1)of vectors be a Jordan chain for ∆(λ), i.e. q06= 0 and

∆(z)[q0+ (z−λ)q1+··· + (z−λ)kqk−1] = O((z−λ)k).

Similarly, let (pk−1, . . . , p0)be a Jordan chain for ∆T(λ). Then the (generalized )) eigenfunctions and

adjoint (generalized )eigenfunctions are given by

φi: [−h, 0] →Rn:θ7→ eλθ

l=0

qi−l

θl

l!,

ψi: [0, h]→Rn:θ7→ pi+

k−1−i

l=0

pi+lZθ

0Zh

eλ(σ−s)(σ−s)l

l!dζ(s)dσ,

(13)

for i∈ {0, . . . , k −1}, respectively. Furthermore, the following identities hold

hψi, φji=hψi+1, φj+1i, i, j ∈ {0, . . . , k −2},

hψk−1, φk−1i=pk−1

k−1

l=0

∆(l+1)(λ)

(l+ 1)! qk−1−l,

hψi, φji= 0, i > j,

hψ0, φji=

k−1

l=0

m=0

∆(l+m+1)(λ)

(l+m+ 1)! qj−m, j > 0,

(14)

which can be normalized to satisfy

hψi, φji=δij .(15)

Proof. The (generalized) eigenspace at an eigenvalue λof Aof algebraic multiplicity kand geometric

multiplicity 1 is given by

N((A−λ)k),

which leads to the expressions in (11) and similarly for (12). The representations of the (generalized)

eigenfunctions and adjoint (generalized) eigenfunctions can be found in Theorem IV.5.5 and IV.5.9

in [13], respectively.

The ﬁrst identity in (14) follows directly from

hψi+1, φj+1i=h(λ−A?)ψi, φj+1i=hψi,(λ−A)φj+1i=hψi, φji,

where i, j ∈ {0, . . . , k −2}. For the second identity in (14) we notice that

hψk−1, φk−1i=Zh

dψk−1(θ)φk−1(−θ) = pk−1q0+Zh

ψ0

k−1(θ)φk−1(−θ)dθ

=pk−1q0+Zh

pk−iZh

eλ(θ−s)dζ(s)e−λθ

k−1

l=0

qi−l

(−1)lθl

l!dθ

=pk−1q0+pk−i

k−1

l=0 Zh

0Zh

e−λs (−1)lθl

l!dζ(s)dθqi−l

=pk−1q0+pk−i

k−1

l=0 Zh

0Zs

e−λs (−1)lθl

l!dθdζ(s)qi−l

=pk−1q0+pk−i

k−1

l=0 Zh

e−λs (−1)lsl+1

(l+ 1)! dζ(s)qi−l=pk−1

k−1

l=0

∆(l)(λ)

(l+ 1)! qi−l,

where we used Fubini’s theorem to change the order of integration. The last equality holds since

∆0(z) = zI +Zh

θe−zθ dζ(θ)

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BifurcationanalysisofBogdanov-Takensbifurcationsindelaydierentialequations*M.M.BosschaertYu.A.KuznetsovOctober7,2022Abstract:Inthispaper,wewillperformtheparameter-dependentcentermanifoldreductionnearthegenericandtranscriticalcodimensiontwoBogdanovTakensbifurcationinclassicaldelaydierentialequat...

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Bifurcation analysis of Bogdanov-Takens bifurcations in delay differential equations_2

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