Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
3/2026
A
GENERAL
TRANSVERSALITY
CRITERION
FOR
HOPF
BIFURCATION
IN
CHARACTERISTIC
EQUATIONS
WITH
DISTRIBUTED
DELAY
∗
Eva
Kaslik
†
Loredana
Flavia
Gabor
‡
Maria
Roxana
Matei
§
Mihaela
Neamt
¸u
Dedicated
to
the
memory
of
Professor
Mihail
Megan
DOI
10.56082/annalsarscimath.2026.3.223
Abstract
A
general
formula
is
given
for
the
Hopf
transversality
condition
associated
with
characteristic
equations
of
the
form
R
(
z
)
=
H
(
z,
τ
),
where
R
is
analytic
and
H
(
·
,
τ
)
is
the
Laplace
transform
of
a
distributed-
delay
kernel.
Our
main
result
expresses
the
crossing
direction
of
a
simple
purely
imaginary
characteristic
root
and
provides
a
geometric
interpretation
of
the
transversality
condition.
We
obtain
as
special
cases
the
classical
discrete-delay
formula
and
explicit
criteria
for
self-
similar
kernels,
including
the
exponential
and
Erlang
families.
Several
consequences
for
stability
switching
are
also
discussed
and
examples
are
given.
Keywords:
distributed
delay,
Hopf
bifurcation,
characteristic
equation,
transversality
condition,
stability
switching.
MSC:
34K18,
34K20,
34C23.
∗
Accepted
for
publication
on
August
21,
2026
†
eva.kaslik@e-uvt.ro
,
West
University
of
Timisoara,
Romania
‡
loredana.gabor@upt.ro
,
Polytechnic
University
of
Timisoara,
Romania
§
maria.matei@e-uvt.ro
,
West
University
of
Timisoara,
Romania
mihaela.neamtu@e-uvt.ro
,
West
University
of
Timisoara,
Romania
223
Transversality
criterion
for
Hopf
bifurcations
with
distributed
delay
224
1
Introduction
Time
delays
are
often
incorporated
in
mathematical
models
whenever
the
present
rate
of
change
of
a
system
depends
on
its
past
states.
However,
in
many
applications,
the
delay
is
not
concentrated
at
a
single
instant
but
spread
over
an
interval
of
past
times.
This
leads
to
equations
with
dis-
tributed
delay,
where
the
memory
term
is
weighted
by
a
kernel
that
describes
the
relative
influence
of
the
past.
Such
mathematical
models
may
arise
from
population
dynamics,
epidemiology,
physiology,
economics,
neural
systems,
and
chemical
kinetics,
among
many
other
areas
[
5
,
7
,
10
,
11
,
15
].
From
the
viewpoint
of
qualitative
theory,
retarded
functional
differen-
tial
equations
(RFDEs)
form
an
infinite-dimensional
dynamical
system
on
a
phase
space
of
history
segments.
For
a
fixed
maximal
delay
r
>
0,
the
natural
state
at
time
t
is
the
segment
x
t
∈
C
([
−
r,
0]
,
R
n
)
defined
by
x
t
(
θ
)
=
x
(
t
+
θ
).
The
general
RFDE
˙
x
(
t
)
=
F
(
x
t
,
µ
)
,
(1)
with
parameter
µ
∈
R
,
is
well
posed
under
standard
smoothness
assump-
tions,
generates
a
local
semiflow
on
the
phase
space,
and
admits
a
lineariza-
tion
principle
analogous
to
that
for
ordinary
differential
equations
[
5
,
7
].
If
x
∗
is
an
equilibrium
and
L
µ
=
D
F
(
x
∗
,
µ
)
denotes
the
Fr´
echet
derivative,
then
the
local
dynamics
near
x
∗
is
governed
by
the
linear
RFDE
˙
y
(
t
)
=
L
µ
y
t
.
For
retarded
equations,
characteristic
roots
are
isolated,
have
finite
algebraic
multiplicity,
and
the
spectral
bound
determines
local
asymptotic
stability
of
the
equilibrium
[
5
,
7
].
The
nonlinear
dynamics
near
a
critical
spectral
configuration
can
be
reduced
by
center-manifold
and
normal-form
techniques.
In
particular,
if
the
linearization
has
a
simple
pair
of
purely
imaginary
roots
±
i
ω
0
and
no
other
roots
on
the
imaginary
axis,
then
a
local
center
manifold
exists
and
the
dynamics
near
the
equilibrium
is
locally
topologically
equivalent
to
a
finite-dimensional
system.
This
reduction
makes
it
possible
to
apply
the
classical
Hopf
bifurcation
theory
to
RFDEs:
if
the
critical
pair
crosses
the
imaginary
axis
with
nonzero
velocity
as
the
parameter
varies,
then
a
branch
of
small-amplitude
periodic
solutions
bifurcates
from
the
equilibrium
[
6
–
8
].
Therefore,
in
practical
applications,
a
Hopf
bifurcation
analysis
usually
requires
two
steps:
first,
the
determination
of
the
critical
parameter
values
for
which
purely
imaginary
characteristic
roots
exist,
and
the
verification
of
the
transversality
condition,
namely
that
the
corresponding
characteristic
roots
cross
the
imaginary
axis
with
nonzero
velocity.
E.
Kaslik,
L.F.
Gabor,
M.R.
Matei,
M.
Neamt
¸u
225
For
equations
with
distributed
delay,
the
characteristic
equation
involves
Laplace
transforms
of
delay
kernels.
For
example,
for
the
linearized
equa-
tion:
˙
y
(
t
)
=
A
0
y
(
t
)
+
∞
0
K
(
s,
τ
)
y
(
t
−
s
)
d
s,
(2)
where
K
(
·
,
τ
)
is
a
matrix
kernel,
seeking
solutions
of
the
form
y
(
t
)
=
e
zt
v
leads
to
the
characteristic
matrix:
∆(
z,
τ
)
=
zI
−
A
0
−
∞
0
e
−
zs
K
(
s,
τ
)
d
s.
In
a
large
number
of
models,
after
scalar
reduction,
factorization,
or
elimi-
nation,
a
scalar
transcendental
equation
of
the
form
R
(
z
)
=
H
(
z,
τ
)
,
(3)
is
obtained,
where
R
is
rational
or,
more
generally,
analytic,
and
H
(
·
,
τ
)
is
the
Laplace
transform
of
a
delay
distribution
kernel.
Characteristic
equa-
tions
of
this
type
are
often
encountered
in
the
stability
and
Hopf
analysis
of
distributed-delay
systems
[
2
,
3
,
9
,
12
,
14
].
In
the
discrete-delay
case,
several
criteria
are
available.
Cooke
and
Grossman
[
3
]
and
Cooke
and
van
den
Driessche
[
4
]
derived
methods
for
locating
purely
imaginary
roots
and
identifying
stability
switches
by
means
of
auxiliary
scalar
equations.
For
delay-dependent
parameters,
Beretta
and
Kuang
[
1
]
developed
a
geometric
approach
which
expresses
stability
switch-
ing
in
terms
of
the
intersection
of
explicitly
computable
curves.
In
the
distributed-delay
setting,
important
results
were
obtained
by
Kuang
[
12
]
on
the
nonoccurrence
of
stability
switching
in
certain
classes
of
models,
by
Bernard,
B´
elair
and
Mackey
[
2
]
on
stability
criteria
based
on
properties
of
the
kernel,
and
by
Ruan
and
Wolkowicz
[
14
]
in
the
bifurcation
analysis
of
chemostat
models
with
distributed
delay.
Recent
work
continues
to
study
stability
and
Hopf
bifurcation
for
general
distributed
kernels
and
confirms
the
continuing
relevance
of
the
topic
[
9
].
The
purpose
of
the
present
paper
is
to
isolate
a
general
transversality
criterion
for
(
3
).
Our
starting
point
is
the
following:
write
the
characteristic
root
as
a
local
smooth
branch
z
(
τ
)
and
compute
d
z
d
τ
by
implicit
differentia-
tion.
The
main
observation
is
that,
once
R
(i
ω
)
and
H
(i
ω,
τ
)
are
written
in
polar
form,
the
sign
of
d
d
τ
Re
z
(
τ
)
can
be
expressed
in
terms
of
two
scalar
equations
in
the
variables
(
ω,
τ
),
more
precisely,
the
modulus
equation
and
the
phase
equation.
This
leads
to
a
precise
Jacobian
formula
for
the
cross-
ing
direction
and
provides
a
simple
geometric
interpretation:
the
spectral
Transversality
criterion
for
Hopf
bifurcations
with
distributed
delay
226
transversality
condition
is
equivalent
to
the
transversality
of
the
two
curves
defined
by
the
modulus
and
phase
relations.
If
we
consider
R
(i
ω
)
=
ρ
(
ω
)e
i
θ
(
ω
)
and
H
(i
ω,
τ
)
=
ρ
(
ω,
τ
)e
−
i
θ
(
ω,τ
)
,
then
purely
imaginary
characteristic
roots
are
determined
by
ρ
(
ω
)
=
ρ
(
ω,
τ
)
,
θ
(
ω
)
+
θ
(
ω,
τ
)
=
2
kπ,
k
∈
Z
.
Our
main
result
shows
that
the
crossing
direction
is
given
by
the
sign
of
the
Jacobian
determinant
of
this
system
with
respect
to
(
ω,
τ
).
When
the
phase
equation
can
be
solved
locally
as
τ
=
T
k
(
ω
),
the
criterion
reduces
to
the
sign
of
the
derivative
of
the
auxiliary
function
F
k
(
ω
)
=
ρ
(
ω
)
−
ρ
(
ω,
T
k
(
ω
))
.
For
self-similar
kernels
of
the
form
H
(
z,
τ
)
=
H
(
τz
),
this
reduction
becomes
particularly
explicit
and
includes
the
classical
discrete-delay
formula
as
a
limiting
case.
The
paper
is
organized
as
follows.
In
Section
2
we
briefly
recall
the
classical
Hopf
framework
for
RFDEs
and
formulate
the
class
of
characteristic
equations
under
consideration.
Section
3
contains
the
main
transversality
theorem,
its
proof,
and
several
equivalent
formulations.
Section
4
discusses
consequences
for
stability
switching.
In
Section
5
we
specialize
the
theory
to
the
Dirac,
exponential,
and
Erlang
kernels
and
we
work
out
a
family
of
second-order
characteristic
equations.
The
last
section
summarizes
the
main
conclusions.
2
Background
and
problem
formulation
2.1
Classical
Hopf
background
Let
r
>
0
and
let
C
([
−
r,
0]
,
R
n
)
be
endowed
with
the
supremum
norm.
For
a
continuous
function
x
:
[
t
0
−
r,
t
1
)
→
R
n
and
t
∈
[
t
0
,
t
1
),
we
define
x
t
∈
C
([
−
r,
0]
,
R
n
)
by
x
t
(
θ
)
=
x
(
t
+
θ
),
θ
∈
[
−
r,
0].
Consider
the
parameter-
dependent
delay
differential
equation
˙
x
(
t
)
=
F
(
x
t
,
µ
)
,
(4)
where
F
:
C
([
−
r,
0]
,
R
n
)
×
I
→
R
n
is
sufficiently
smooth
and
I
⊂
R
is
an
interval.
Assume
that
x
≡
0
is
an
equilibrium
for
all
µ
∈
I
,
that
is,
F
(0
,
µ
)
=
0.
Denoting
L
µ
=
D
x
F
(0
,
µ
),
The
linearized
equation
in
a
neighborhood
of
x
=
0
is
˙
y
(
t
)
=
L
µ
y
t
.
(5)
E.
Kaslik,
L.F.
Gabor,
M.R.
Matei,
M.
Neamt
¸u
227
The
operator
L
µ
admits
a
Riesz
representation
in
terms
of
a
matrix-valued
measure,
and
the
associated
characteristic
roots
are
the
zeros
of
a
charac-
teristic
equation
of
the
form
∆(
λ,
µ
)
=
0
[
5
,
7
].
The
following
theorem
summarizes
the
classical
Hopf
scenario
for
delay
differential
equations.
Theorem
1
(see
[
6
–
8
])
.
Consider
(
4
)
and
assume
that
F
is
of
class
C
k
,
k
≥
3
,
in
a
neighborhood
of
(0
,
µ
0
)
.
Suppose
that
the
characteristic
equation
of
(
5
)
at
µ
=
µ
0
has
a
simple
pair
of
roots
λ
1
,
2
(
µ
0
)
=
±
i
ω
0
with
ω
0
>
0
,
that
no
other
characteristic
root
lies
on
the
imaginary
axis,
and
that
d
d
µ
Re
λ
1
(
µ
)
µ
=
µ
0
=
0
.
Then
a
Hopf
bifurcation
occurs
at
(0
,
µ
0
)
:
there
exists
a
local
branch
of
nontrivial
periodic
solutions
bifurcating
from
the
equilibrium
as
µ
passes
through
µ
0
.
In
applications,
the
main
technical
difficulty
is
usually
the
evaluation
of
the
crossing
quantity
d
d
µ
Re
λ
(
µ
0
),
which
provides
the
transversality
con-
dition.
The
aim
of
the
present
paper
is
to
derive
an
explicit
formula
for
the
crossing
quantity
when
the
characteristic
equation
can
be
written
in
the
general
form
given
in
the
following
subsection.
2.2
Characteristic
equations
generated
by
distributed
delays
We
consider
the
scalar
characteristic
equation:
R
(
z
)
=
H
(
z,
τ
)
,
(6)
where
z
∈
C
and
τ
∈
I
⊂
(0
,
∞
)
is
the
bifurcation
parameter.
In
applications
R
is
rational
and
H
(
·
,
τ
)
is
the
Laplace
transform
of
a
delay
kernel.
More
precisely,
if
g
(
·
,
τ
)
is
a
probability
density
function
on
[0
,
∞
)
with
finite
first
moment
(mean)
τ
,
then:
H
(
z,
τ
)
=
∞
0
e
−
zs
g
(
s,
τ
)
d
s,
∞
0
g
(
s,
τ
)
d
s
=
1
,
∞
0
sg
(
s,
τ
)
d
s
=
τ.
(7)
We
consider
the
following
assumptions.
Assumption
1.
There
exist
an
open
interval
J
⊂
(0
,
∞
)
,
an
open
interval
I
⊂
(0
,
∞
)
,
and
a
point
(
ω
∗
,
τ
∗
)
∈
J
×
I
such
that:
Transversality
criterion
for
Hopf
bifurcations
with
distributed
delay
228
(A1)
R
is
holomorphic
in
a
neighborhood
of
i
J
:=
{
i
ω
:
ω
∈
J
}
;
(A2)
H
is
defined
in
a
neighborhood
of
i
J
×
I
,
is
holomorphic
in
z
for
each
fixed
τ
∈
I
,
and
is
of
class
C
1
with
respect
to
τ
;
(A3)
∆(
z,
τ
)
:=
R
(
z
)
−
H
(
z,
τ
)
satisfies
∆(i
ω
∗
,
τ
∗
)
=
0
,
R
(i
ω
∗
)
=
H
(i
ω
∗
,
τ
∗
)
=
0;
(A4)
the
purely
imaginary
root
z
∗
=
i
ω
∗
is
simple,
that
is,
∆
z
(i
ω
∗
,
τ
∗
)
=
R
(i
ω
∗
)
−
H
z
(i
ω
∗
,
τ
∗
)
=
0
.
Assumption
1
(A4)
implies,
by
the
implicit
function
theorem,
that
there
exists
a
unique
C
1
branch
z
(
τ
)
of
characteristic
roots
near
τ
∗
such
that
∆(
z
(
τ
)
,
τ
)
=
0
,
z
(
τ
∗
)
=
i
ω
∗
.
The
transversality
condition
relevant
for
the
Hopf
bifurcation
is
d
d
τ
Re
z
(
τ
)
τ
=
τ
∗
=
0
.
(8)
3
A
general
transversality
theorem
By
Assumption
1
(A3),
both
R
(i
ω
∗
)
and
H
(i
ω
∗
,
τ
∗
)
are
nonzero.
We
choose
local
C
1
polar
representations
R
(i
ω
)
=
ρ
(
ω
)e
i
θ
(
ω
)
,
H
(i
ω,
τ
)
=
ρ
(
ω,
τ
)e
−
i
θ
(
ω,τ
)
,
(9)
where
ρ
(
ω
)
>
0
for
ω
∈
J
and
ρ
(
ω,
τ
)
>
0
for
(
ω,
τ
)
∈
J
×
I
.
Define
the
scalar
functions
G
1
(
ω,
τ
)
:=
ρ
(
ω
)
−
ρ
(
ω,
τ
)
,
G
2
(
ω,
τ
)
:=
θ
(
ω
)
+
θ
(
ω,
τ
)
−
2
kπ,
(10)
where
the
integer
k
is
chosen
so
that
θ
(
ω
∗
)
+
θ
(
ω
∗
,
τ
∗
)
=
2
kπ.
Then
the
condition
∆(i
ω,
τ
)
=
0
is
equivalent
to
the
system
G
1
(
ω,
τ
)
=
0
,
G
2
(
ω,
τ
)
=
0
.
(11)
E.
Kaslik,
L.F.
Gabor,
M.R.
Matei,
M.
Neamt
¸u
229
Theorem
2.
Assume
that
Assumption
1
holds.
Let
z
(
τ
)
be
the
unique
local
root
branch
satisfying
z
(
τ
∗
)
=
i
ω
∗
.
Then
d
d
τ
Re
z
(
τ
)
τ
=
τ
∗
=
ρ
(
ω
∗
,
τ
∗
)
|
∆
z
(i
ω
∗
,
τ
∗
)
|
2
det
∂
ω
G
1
∂
τ
G
1
∂
ω
G
2
∂
τ
G
2
(
ω
∗
,τ
∗
)
.
(12)
In
particular,
sgn
d
d
τ
Re
z
(
τ
)
τ
=
τ
∗
=
sgn
det
DG
(
ω
∗
,
τ
∗
)
,
G
=
(
G
1
,
G
2
)
.
(13)
Therefore
the
spectral
transversality
condition
is
equivalent
to
the
transversal
intersection
of
the
modulus
and
phase
curves
in
the
(
ω,
τ
)
-plane.
Proof.
Since
∆(
z
(
τ
)
,
τ
)
=
0,
differentiation
with
respect
to
τ
yields
∆
z
(
z
(
τ
)
,
τ
)
z
(
τ
)
+
∆
τ
(
z
(
τ
)
,
τ
)
=
0
.
Because
∆
τ
=
−
H
τ
,
we
obtain
at
τ
=
τ
∗
z
(
τ
∗
)
=
H
τ
(i
ω
∗
,
τ
∗
)
R
(i
ω
∗
)
−
H
z
(i
ω
∗
,
τ
∗
)
=
H
τ
(i
ω
∗
,
τ
∗
)
∆
z
(i
ω
∗
,
τ
∗
)
.
(14)
Taking
real
parts
gives
d
d
τ
Re
z
(
τ
)
τ
=
τ
∗
=
Re
H
τ
(i
ω
∗
,
τ
∗
)
∆
z
(i
ω
∗
,
τ
∗
)
|
∆
z
(i
ω
∗
,
τ
∗
)
|
2
.
(15)
We
now
compute
the
derivatives
in
terms
of
the
polar
representation
(
9
).
Since
d
d
ω
R
(i
ω
)
=
i
R
(i
ω
)
,
∂
∂ω
H
(i
ω,
τ
)
=
i
H
z
(i
ω,
τ
)
,
a
direct
calculation
yields
R
(i
ω
)
=
ρ
(
ω
)
θ
(
ω
)
−
i
ρ
(
ω
)
e
i
θ
(
ω
)
,
(16)
H
z
(i
ω,
τ
)
=
−
ρ
(
ω,
τ
)
θ
ω
(
ω,
τ
)
+
i
ρ
ω
(
ω,
τ
)
e
−
i
θ
(
ω,τ
)
,
(17)
H
τ
(i
ω,
τ
)
=
ρ
τ
(
ω,
τ
)
−
i
ρ
(
ω,
τ
)
θ
τ
(
ω,
τ
)
e
−
i
θ
(
ω,τ
)
.
(18)
At
the
critical
pair
(
ω
∗
,
τ
∗
),
the
identity
R
(i
ω
∗
)
=
H
(i
ω
∗
,
τ
∗
)
implies
ρ
(
ω
∗
)
=
ρ
(
ω
∗
,
τ
∗
)
,
θ
(
ω
∗
)
+
θ
(
ω
∗
,
τ
∗
)
=
2
kπ.
(19)
Transversality
criterion
for
Hopf
bifurcations
with
distributed
delay
230
Hence
e
i
θ
(
ω
∗
)
=
e
−
i
θ
(
ω
∗
,τ
∗
)
,
and
therefore
H
τ
∆
z
=
ρ
τ
+
i
ρθ
τ
e
i
θ
ρ
θ
−
i
ρ
e
i
θ
+
ρθ
ω
+
i
ρ
ω
e
−
i
θ
(
ω
∗
,τ
∗
)
=
ρ
τ
+
i
ρθ
τ
ρ
(
θ
+
θ
ω
)
+
i(
ρ
ω
−
ρ
)
(
ω
∗
,τ
∗
)
.
Taking
real
parts,
we
obtain
Re
H
τ
∆
z
=
ρ
ρ
τ
(
θ
+
θ
ω
)
+
θ
τ
(
ρ
−
ρ
ω
)
(
ω
∗
,τ
∗
)
.
(20)
On
the
other
hand,
from
(
10
)
we
have
∂
ω
G
1
=
ρ
−
ρ
ω
,
∂
τ
G
1
=
−
ρ
τ
,
∂
ω
G
2
=
θ
+
θ
ω
,
∂
τ
G
2
=
θ
τ
.
Therefore
det
DG
=
(
ρ
−
ρ
ω
)
θ
τ
+
ρ
τ
(
θ
+
θ
ω
)
.
(21)
Combining
(
15
),
(
20
),
and
(
21
)
proves
(
12
).
Since
ρ
(
ω
∗
,
τ
∗
)
>
0,
the
sign
identity
(
13
)
follows
immediately.
Remark
1.
The
Jacobian
determinant
in
(
12
)
is
independent
of
the
par-
ticular
local
branches
chosen
for
the
arguments
in
(
9
)
.
Indeed,
replacing
θ
or
θ
by
another
local
branch
only
adds
an
integer
multiple
of
2
π
,
which
does
not
affect
derivatives.
Theorem
2
is
especially
convenient
when
one
of
the
two
equations
in
(
11
)
can
be
solved
locally
with
respect
to
τ
.
Corollary
1.
Assume
that
Assumption
1
holds
and
that
θ
τ
(
ω
∗
,
τ
∗
)
=
0
.
(22)
Then
there
exist
a
neighborhood
U
of
ω
∗
and
a
unique
C
1
function
T
k
:
U
→
I
such
that
T
k
(
ω
∗
)
=
τ
∗
and
θ
(
ω
)
+
θ
(
ω,
T
k
(
ω
))
=
2
kπ,
ω
∈
U.
(23)
Define
F
k
(
ω
)
:=
ρ
(
ω
)
−
ρ
(
ω,
T
k
(
ω
))
.
(24)
Then
F
k
(
ω
∗
)
=
0
and
d
d
τ
Re
z
(
τ
)
τ
=
τ
∗
=
ρ
(
ω
∗
,
τ
∗
)
θ
τ
(
ω
∗
,
τ
∗
)
F
k
(
ω
∗
)
|
∆
z
(i
ω
∗
,
τ
∗
)
|
2
.
(25)
E.
Kaslik,
L.F.
Gabor,
M.R.
Matei,
M.
Neamt
¸u
231
In
particular,
sgn
d
d
τ
Re
z
(
τ
)
τ
=
τ
∗
=
sgn
θ
τ
(
ω
∗
,
τ
∗
)
sgn
F
k
(
ω
∗
)
.
(26)
Proof.
Condition
(
22
)
allows
us
to
solve
(
23
)
locally
for
τ
=
T
k
(
ω
)
by
the
implicit
function
theorem.
Differentiating
(
23
)
with
respect
to
ω
gives
T
k
(
ω
∗
)
=
−
θ
(
ω
∗
)
+
θ
ω
(
ω
∗
,
τ
∗
)
θ
τ
(
ω
∗
,
τ
∗
)
.
(27)
Differentiating
(
24
)
and
evaluating
at
ω
∗
,
we
obtain
F
k
(
ω
∗
)
=
ρ
(
ω
∗
)
−
ρ
ω
(
ω
∗
,
τ
∗
)
−
ρ
τ
(
ω
∗
,
τ
∗
)
T
k
(
ω
∗
)
.
Using
(
27
),
we
find
θ
τ
(
ω
∗
,
τ
∗
)
F
k
(
ω
∗
)
=
θ
τ
(
ω
∗
,
τ
∗
)(
ρ
−
ρ
ω
)
+
ρ
τ
(
θ
+
θ
ω
)
,
all
quantities
being
evaluated
at
(
ω
∗
,
τ
∗
).
The
conclusion
follows
from
(
12
).
Corollary
2.
Assume
that
Assumption
1
holds
and
that
ρ
τ
(
ω
∗
,
τ
∗
)
=
0
.
(28)
Then
there
exist
a
neighborhood
U
of
ω
∗
and
a
unique
C
1
function
S
k
:
U
→
I
such
that
S
k
(
ω
∗
)
=
τ
∗
and
ρ
(
ω
)
−
ρ
(
ω,
S
k
(
ω
))
=
0
,
ω
∈
U.
(29)
Define
P
k
(
ω
)
:=
θ
(
ω
)
+
θ
(
ω,
S
k
(
ω
))
−
2
kπ.
(30)
Then
P
k
(
ω
∗
)
=
0
and
d
d
τ
Re
z
(
τ
)
τ
=
τ
∗
=
ρ
(
ω
∗
,
τ
∗
)
ρ
τ
(
ω
∗
,
τ
∗
)
P
k
(
ω
∗
)
|
∆
z
(i
ω
∗
,
τ
∗
)
|
2
.
(31)
Hence
sgn
d
d
τ
Re
z
(
τ
)
τ
=
τ
∗
=
sgn
ρ
τ
(
ω
∗
,
τ
∗
)
sgn
P
k
(
ω
∗
)
.
(32)
Transversality
criterion
for
Hopf
bifurcations
with
distributed
delay
232
Proof.
The
proof
is
analogous
to
that
of
Corollary
1
.
Differentiating
(
29
)
yields
S
k
(
ω
∗
)
=
ρ
(
ω
∗
)
−
ρ
ω
(
ω
∗
,
τ
∗
)
ρ
τ
(
ω
∗
,
τ
∗
)
,
and
therefore
ρ
τ
(
ω
∗
,
τ
∗
)
P
k
(
ω
∗
)
=
ρ
τ
(
θ
+
θ
ω
)
+
θ
τ
(
ρ
−
ρ
ω
)
.
The
claim
now
follows
from
(
12
).
A
particularly
important
class
of
kernels
is
given
by
self-similar
families.
Remark
2
(Self-similar
kernels)
.
Assume
that
H
(
z,
τ
)
=
H
(
τz
)
(33)
for
some
analytic
function
H
,
and
let
H
(i
ξ
)
=
ρ
(
ξ
)e
−
i
θ
(
ξ
)
.
(34)
Then
ρ
(
ω,
τ
)
=
ρ
(
ωτ
)
,
θ
(
ω,
τ
)
=
θ
(
ωτ
)
,
(35)
and
ρ
ω
(
ω,
τ
)
=
τ
ρ
(
ωτ
)
,
ρ
τ
(
ω,
τ
)
=
ω
ρ
(
ωτ
)
,
θ
ω
(
ω,
τ
)
=
τ
θ
(
ωτ
)
,
θ
τ
(
ω,
τ
)
=
ω
θ
(
ωτ
)
.
(36)
Moreover,
if
θ
(
ω
∗
τ
∗
)
=
0
,
from
Corollary
1
we
have
θ
τ
(
ω
∗
,
τ
∗
)
=
ω
∗
θ
(
ω
∗
τ
∗
)
=
0
,
and
hence:
d
d
τ
Re
z
(
τ
)
τ
=
τ
∗
=
ρ
(
ω
∗
,
τ
∗
)
ω
∗
θ
(
ω
∗
τ
∗
)
F
k
(
ω
∗
)
|
∆
z
(i
ω
∗
,
τ
∗
)
|
2
.
(37)
Hence
sgn
d
d
τ
Re
z
(
τ
)
τ
=
τ
∗
=
sgn
θ
(
ω
∗
τ
∗
)
sgn
F
k
(
ω
∗
)
.
(38)
In
particular,
if
θ
is
positive
on
the
relevant
interval,
then
the
crossing
direction
is
determined
solely
by
the
sign
of
F
k
(
ω
∗
)
.
E.
Kaslik,
L.F.
Gabor,
M.R.
Matei,
M.
Neamt
¸u
233
In
the
particular
case
of
a
discrete
time
delay,
we
consider
the
Dirac
kernel,
and
we
obtain:
Corollary
3
(Discrete
delay)
.
Suppose
that
H
(
z,
τ
)
=
e
−
τz
.
Then
d
d
τ
Re
z
(
τ
)
τ
=
τ
∗
=
ω
∗
ρ
(
ω
∗
)
|
∆
z
(i
ω
∗
,
τ
∗
)
|
2
,
sgn
d
d
τ
Re
z
(
τ
)
τ
=
τ
∗
=
sgn
ρ
(
ω
∗
)
.
(39)
If,
moreover,
R
=
P/Q
with
polynomials
P
and
Q
,
then
at
a
critical
fre-
quency
ω
∗
the
crossing
direction
satisfies
sgn
d
d
τ
Re
z
(
τ
)
τ
=
τ
∗
=
sgn
d
d
ω
|
P
(i
ω
)
|
2
−
|
Q
(i
ω
)
|
2
ω
=
ω
∗
.
(40)
Proof.
In
this
case
H
(
ξ
)
=
e
−
ξ
,
so
ρ
≡
1
and
θ
(
ξ
)
=
ξ
.
Hence
F
k
(
ω
)
=
ρ
(
ω
)
−
1,
and
(
39
)
follows
from
Remark
2
.
If
R
=
P/Q
,
then
ρ
(
ω
)
=
|
P
(i
ω
)
|
|
Q
(i
ω
)
|
.
Therefore
ρ
(
ω
)
=
|
Q
(i
ω
)
|
d
d
ω
|
P
(i
ω
)
|
−
|
P
(i
ω
)
|
d
d
ω
|
Q
(i
ω
)
|
|
Q
(i
ω
)
|
2
.
At
a
critical
frequency
we
have
ρ
(
ω
∗
)
=
1,
hence
|
P
(i
ω
∗
)
|
=
|
Q
(i
ω
∗
)
|
,
and
consequently
sgn
ρ
(
ω
∗
)
=
sgn
|
P
(i
ω
∗
)
|
d
d
ω
|
P
(i
ω
)
|
ω
=
ω
∗
−
|
Q
(i
ω
∗
)
|
d
d
ω
|
Q
(i
ω
)
|
ω
=
ω
∗
.
This
is
equivalent
to
(
40
).
4
Consequences
for
stability
switching
The
previous
section
provides
a
local
formula
for
the
crossing
direction
of
a
simple
pair
of
imaginary
roots.
We
now
provide
a
standard
consequence
for
the
variation
of
the
unstable
dimension.
Proposition
1.
Let
ν
(
τ
)
denote
the
number
of
characteristic
roots
of
(
6
)
in
the
open
right
half-plane,
counted
with
algebraic
multiplicity.
Assume
that
at
τ
=
τ
∗
the
only
characteristic
roots
on
the
imaginary
axis
are
the
simple
Transversality
criterion
for
Hopf
bifurcations
with
distributed
delay
234
conjugate
pair
±
i
ω
∗
and
that
Assumption
1
holds.
Then
there
exists
ε
>
0
such
that
ν
(
τ
)
=
ν
(
τ
−
∗
)
+
2
sgn
d
d
τ
Re
z
(
τ
)
τ
=
τ
∗
for
τ
∈
(
τ
∗
,
τ
∗
+
ε
)
,
while
ν
(
τ
)
=
ν
(
τ
−
∗
)
for
τ
∈
(
τ
∗
−
ε,
τ
∗
)
.
In
other
words,
the
unstable
dimen-
sion
changes
by
+2
or
−
2
according
to
the
sign
of
the
crossing
direction.
Proof.
The
simplicity
of
the
roots
±
i
ω
∗
implies
that
they
admit
local
con-
tinuations
as
a
conjugate
pair
of
characteristic
roots.
Since
all
other
roots
remain
a
positive
distance
away
from
the
imaginary
axis,
the
number
of
roots
in
the
right
half-plane
can
change
only
through
this
pair.
The
sign
of
d
d
τ
Re
z
(
τ
)
τ
=
τ
∗
determines
whether
the
pair
moves
from
left
to
right
or
from
right
to
left
as
τ
increases.
This
provides
the
stated
jump
in
ν
(
τ
).
An
immediate
consequence
of
Proposition
1
is:
Corollary
4
(Monotone
crossing
criterion)
.
Assume
that
all
critical
values
of
τ
are
simple
Hopf
points
and
that
no
two
distinct
pairs
of
characteristic
roots
cross
the
imaginary
axis
simultaneously.
If
at
every
critical
pair
(
ω
j
,
τ
j
)
one
has
sgn
d
d
τ
Re
z
(
τ
)
τ
=
τ
j
=
+1
,
then
ν
(
τ
)
is
increasing.
In
particular,
an
equilibrium
which
has
become
unstable
cannot
regain
stability
as
τ
increases.
The
self-similar
case
gives
a
convenient
sufficient
condition
for
the
avoid-
ance
of
stability
switching.
Corollary
5.
Assume
that
the
hypotheses
of
Remark
2
hold,
and
suppose
that
θ
(
ξ
)
>
0
for
every
admissible
critical
value
ξ
=
ωτ
.
If,
in
addition,
each
critical
frequency
is
a
simple
zero
of
the
corresponding
auxiliary
function
F
k
and
satisfies
F
k
(
ω
∗
)
>
0
,
then
every
simple
Hopf
crossing
occurs
from
left
to
right
as
τ
increases.
Consequently,
the
unstable
dimension
is
increasing
and
restabilization
can-
not
occur.
Proof.
By
(
38
),
the
sign
of
the
crossing
direction
is
the
sign
of
F
k
(
ω
∗
).
The
conclusion
follows
from
Corollary
4
.
E.
Kaslik,
L.F.
Gabor,
M.R.
Matei,
M.
Neamt
¸u
235
5
Examples
5.1
Erlang
kernels
For
n
∈
N
,
consider
the
Erlang
(gamma)
family
g
n
(
s,
τ
)
=
n
n
(
n
−
1)!
τ
n
s
n
−
1
e
−
ns/τ
,
s
≥
0
,
(41)
which
is
a
probability
density
on
[0
,
∞
)
with
mean
τ
.
Its
Laplace
transform
is
H
n
(
z,
τ
)
=
∞
0
e
−
zs
g
n
(
s,
τ
)
d
s
=
1
+
τz
n
−
n
.
(42)
The
cases
n
=
1
and
n
=
2
correspond
to
the
classical
weak
and
strong
gamma
kernels,
respectively.
Then,
we
can
write
the
polar
form:
H
n
(i
ω,
τ
)
=
ρ
n
(
ωτ
)e
−
i
θ
n
(
ωτ
)
,
(43)
where
ρ
n
(
ξ
)
=
1
+
ξ
2
n
2
−
n/
2
,
θ
n
(
ξ
)
=
n
arctan
ξ
n
.
(44)
Moreover,
θ
n
(
ξ
)
=
1
1
+
ξ
2
/n
2
>
0
.
(45)
Consequently,
Remark
2
applies.
Solving
θ
n
(
ωT
k,n
(
ω
))
=
2
kπ
−
θ
(
ω
)
gives
T
k,n
(
ω
)
=
n
ω
tan
2
kπ
−
θ
(
ω
)
n
(46)
and
F
k,n
(
ω
)
=
ρ
(
ω
)
−
cos
n
2
kπ
−
θ
(
ω
)
n
.
(47)
At
any
simple
critical
frequency
ω
∗
satisfying
F
k,n
(
ω
∗
)
=
0
one
has
sgn
d
d
τ
Re
z
(
τ
)
τ
=
τ
∗
=
sgn
F
k,n
(
ω
∗
)
.
(48)
Transversality
criterion
for
Hopf
bifurcations
with
distributed
delay
236
Remark
3.
For
n
=
1
one
obtains
the
exponential
or
weak
kernel,
H
1
(
z,
τ
)
=
1
1
+
τz
,
ρ
1
(
ξ
)
=
1
1
+
ξ
2
,
θ
1
(
ξ
)
=
arctan
ξ.
For
n
=
2
one
obtains
the
strong
kernel,
H
2
(
z,
τ
)
=
1
+
τz
2
−
2
,
ρ
2
(
ξ
)
=
1
+
ξ
2
4
−
1
,
θ
2
(
ξ
)
=
2
arctan
ξ
2
.
Both
kernels
satisfy
θ
n
>
0
,
so
the
crossing
direction
is
determined
by
F
k,n
.
5.2
A
second-order
characteristic
equation
with
weak
kernel
Consider
the
family
λ
2
+
aλ
+
b
+
c
1
+
τλ
=
0
,
a,
b,
c
>
0
.
(49)
This
equation
appears,
for
instance,
after
linearization
of
second-order
sys-
tems
with
exponentially
distributed
delay.
The
second-order
terms
capture
inertial
or
oscillatory
behavior,
which
can
be
found
in
mechanical
systems,
biological
and
economic
processes.
The
rational
term
1
1+
τλ
to
the
Laplace
transform
of
the
weak,
or
exponential,
delay
kernel
(see,
for
example,
the
references
[
5
],
[
10
],
[
11
],
[
15
]).
Equation
(
49
)
can
be
rewritten
as
(
6
)
with
R
(
λ
)
=
−
λ
2
+
aλ
+
b
c
,
H
(
λ,
τ
)
=
1
1
+
τλ
.
(50)
Let
d
(
ω
)
:=
(
ω
2
−
b
)
2
+
a
2
ω
2
.
(51)
Then
R
(i
ω
)
=
ω
2
−
b
−
i
aω
c
,
ρ
(
ω
)
=
d
(
ω
)
c
.
(52)
Choosing
the
local
argument
branch
in
the
lower
half-plane,
we
also
have
cos
θ
(
ω
)
=
ω
2
−
b
d
(
ω
)
,
sin
θ
(
ω
)
=
−
aω
d
(
ω
)
.
(53)
E.
Kaslik,
L.F.
Gabor,
M.R.
Matei,
M.
Neamt
¸u
237
Proposition
2.
For
equation
(
49
)
,
the
purely
imaginary
roots
±
i
ω
are
de-
termined
by
the
relations
τ
=
a
ω
2
−
b
,
ω
2
>
b,
(54)
and
p
(
ω
2
)
=
0
,
p
(
x
)
:=
x
2
+
(
a
2
−
2
b
−
c
)
x
+
b
2
+
bc.
(55)
Moreover,
if
ω
∗
>
0
is
a
simple
positive
root
of
(
55
)
,
then
the
corresponding
crossing
direction
is
given
by
sgn
d
d
τ
Re
λ
(
τ
)
τ
=
τ
∗
=
sgn
2
ω
2
∗
+
a
2
−
2
b
−
c
.
(56)
Proof.
Since
the
weak
kernel
is
self-similar
with
ρ
1
(
ξ
)
=
1
1
+
ξ
2
,
θ
1
(
ξ
)
=
arctan
ξ,
Remark
2
applies.
The
phase
condition
is
θ
(
ω
)
+
arctan(
ωτ
)
=
0
for
the
relevant
local
branch.
Taking
the
tangent
and
using
(
53
),
we
obtain
ωτ
=
−
tan
θ
(
ω
)
=
aω
ω
2
−
b
,
which
yields
(
54
).
Next,
the
amplitude
equation
is
ρ
(
ω
)
=
ρ
1
(
ωτ
)
=
cos
arctan(
ωτ
)
=
cos
θ
(
ω
)
=
ω
2
−
b
d
(
ω
)
.
Hence
d
(
ω
)
c
=
ω
2
−
b
d
(
ω
)
,
i.e.
d
(
ω
)
2
=
c
(
ω
2
−
b
)
.
Using
(
51
),
this
becomes
exactly
(
55
).
Finally,
F
(
ω
)
=
ρ
(
ω
)
−
ρ
1
(
ωT
(
ω
))
=
d
(
ω
)
c
−
ω
2
−
b
d
(
ω
)
=
p
(
ω
2
)
c
d
(
ω
)
.
If
p
(
ω
2
∗
)
=
0
and
ω
∗
>
0,
then
sgn
F
(
ω
∗
)
=
sgn
d
d
ω
p
(
ω
2
)
ω
=
ω
∗
=
sgn
2
ω
2
∗
+
a
2
−
2
b
−
c
.
Since
θ
1
>
0,
formula
(
56
)
follows
from
Remark
2
.
Transversality
criterion
for
Hopf
bifurcations
with
distributed
delay
238
Remark
4.
For
the
discrete-delay
we
have:
λ
2
+
aλ
+
b
+
c
e
−
τλ
=
0
.
(57)
This
equation
relies
on
the
Dirac
kernel,
meaning
the
system’s
memory
is
localized
at
a
single
past
moment.
The
exponential
term
e
−
τλ
can
model
a
strict
transport
lag,
a
fixed
processing
time
in
neural
signal
propagation,
or
a
specific
maturation
period
in
population
models,
where
the
historical
influence
is
strictly
delayed
rather
than
continuously
distributed
across
an
interval.
Corollary
3
gives
the
critical-frequency
equation
q
(
ω
2
)
=
0
,
q
(
x
)
:=
x
2
+
(
a
2
−
2
b
)
x
+
b
2
−
c
2
,
(58)
and
the
crossing
direction
sgn
d
d
τ
Re
λ
(
τ
)
τ
=
τ
∗
=
sgn
2
ω
2
∗
+
a
2
−
2
b
.
(59)
For
instance,
if
a
=
1,
b
=
3,
and
c
=
2,
then
q
(
x
)
=
x
2
−
5
x
+
5,
so
the
critical
values
are
ω
2
1
,
2
=
5
±
√
5
2
.
The
corresponding
crossings
have
opposite
directions
because
2
ω
2
1
−
5
<
0
and
2
ω
2
2
−
5
>
0.
Thus,
the
characteristic
equation
admits
two
simple
crossings
with
different
orientation,
a
typical
mechanism
behind
stability
switching.
Using
the
formula
(
54
)
we
calculate
the
critical
value
of
the
time
delay.
In
Proposition
2
we
have
the
condition
ω
2
>
b
so
that
τ
is
positive.
In
our
example
b
=
3,
the
frequency
ω
2
(
ω
2
2
≈
3
.
618)
leads
to
a
valid
critical
value
τ
∗
=
1
.
618.
6
Concluding
remarks
A
general
formula
for
the
Hopf
transversality
condition
has
been
derived,
associated
with
characteristic
equations
of
the
form
R
(
z
)
=
H
(
z,
τ
),
where
H
is
the
Laplace
transform
of
a
distributed-delay
kernel.
The
main
contri-
bution
is
the
Jacobian
identity
(
12
),
which
shows
that
the
spectral
crossing
direction
is
exactly
the
Jacobian
determinant
of
the
modulus
and
phase
equations
in
the
(
ω,
τ
)-plane.
This
gives
a
direct
geometric
interpretation
of
the
transversality
condition
and
places
a
number
of
existing
criteria
into
a
single
framework.
E.
Kaslik,
L.F.
Gabor,
M.R.
Matei,
M.
Neamt
¸u
239
In
addition,
several
particular
cases
have
been
discussed.
When
the
phase
equation
can
be
solved
locally
for
the
delay
parameter,
the
cross-
ing
direction
is
reduced
to
the
derivative
of
an
auxiliary
scalar
function
F
k
multiplied
by
the
sign
of
θ
τ
.
An
analogous
reduction
is
available
when
the
modulus
equation
is
solvable.
For
self-similar
kernels,
the
formula
becomes
particularly
simple
and
immediately
provides
the
classical
discrete-delay
cri-
terion
as
a
special
case.
From
an
application
perspective,
the
results
are
useful
for
two
reasons.
First,
they
provide
a
unified
formula
for
d
d
τ
Re
z
(
τ
).
Second,
they
make
it
possible
to
formulate
monotonicity
criteria
for
stability
switching
directly
in
terms
of
the
functions
governing
the
amplitude
and
phase
relations
on
the
imaginary
axis.
Several
examples,
including
the
Dirac,
exponential,
and
Erlang
kernels
are
given
to
illustrate
the
theoretical
results.
Directions
for
future
research
include
considering
determinant-type
char-
acteristic
equations
associated
with
higher-dimensional
systems,
kernels
de-
pending
on
several
shape
parameters,
and
situations
in
which
the
charac-
teristic
equation
cannot
be
reduced
globally
to
a
single
branch
of
the
phase
relation.
Combining
the
present
criterion
with
center-manifold
and
normal-
form
computations
can
also
be
investigated,
in
order
to
obtain
a
unified
treatment
of
existence,
direction,
and
stability
of
Hopf
bifurcations
in
gen-
eral
classes
of
distributed-delay
models.
Acknowledgements
.
This
work
received
support
from
a
joint
NSF
(USA)
-
UEFISCDI
(Romania)
grant:
Project
No.
ROSUA-2024-0002
(E.
Kaslik,
M.
Neamt
¸u).
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