Mathematics and Informatics - ICTAMI 2004, Thessaloniki, Greece
235
ON THE BAUTIN BIFURCATION FOR SYSTEMS OF DELAY DIFFERENTIAL EQUATIONS
by Anca–Veronica Ion
Abstract . For systems of delay differential equations the Hopf bifurcation was investigated
by several authors. The problem we solve here is that of the possibility of emergence of a codimension two bifurcation, namely the Bautin bifurcation, for some such systems.
Keywords
: bifurcation theory, Bautin, delay differential equation
1. Introduction
The existence of periodic solutions for evolution equations is of certain interest for both pure and applied mathematicians. Even for bidimensional
systems of differential equations the detection of limit cycles by theoretical means is difficult. The bifurcation theory offers a strong tool for finding limit
cycles, namely the theory concerning the Hopf bifurcation when there is a varying parameter[2], [6]. Several authors studied the Hopf bifurcation for
delay differential equations e.g. [4], [7], [1], [5] . We are interested to find sufficient conditions for the Bautin bifurcation for a class of such systems.
2.
Setting of the problem, theoretical frame
Consider a system of the form α
α α
, ,
r t
x t
x f
r t
x B
t x
A t
x −
+ −
+ =
•
, 1
[ ]
, ,
r s
s s
x −
∈ =
φ , 2
where
n n
x x
x R
∈ = ,...,
1
,
2
R ∈
=
2 1
, α
α α
, ,
α α B
A are nxn matrices
over R ,
n
f f
f ...,
,
1
= is continuously differentiable on its domain of
definition,
1 2
+
⊂
n
D R
. Moreover, ,
, =
α f
and the differential of f in the first two vectorial variables, calculated at
α ,
, is equal to zero.
φ is an element of the Banach space
[ ]
n
r C
R ,
, B
− =
column vectors. In order to write eq. 1 as a differential equation in a Banach space, the space
[ ] [
{ }
n n
R R
∈ ∃
→ −
=
→
lim and
r, -
on continuous
is ,
, :
B
-
s
s r
ψ ψ
ψ
differential equations
236 is considered in section 8.2 of [4]. Its elements are
σ ϕ
ψ X
+ =
, with
n
R ∈
∈ σ
ϕ B, column vector and
=
≤ −
= ,
, ,
s I
s r
s X
n
where 0,
n
I are the zero and, respectively, the unity nxn matrix. The norm of
ψ is defined as the sum of the norm of ϕ in B and the norm of σ in
n
R . The
complexifications
C
B ,
C
B of B , respectively
B , are used below. Consider
⋅
δ - the Dirac function, and the nxn matrix valued function
n
I s
s δ
= ∆
. Also consider the bounded linear operator
n
R →
B :
α
L
,
∫
−
=
r
s s
d L
ϕ η
ϕ
α α
, with
r s
B s
A s
+ ∆
− ∆
= α
α η
α
. By denoting
[ ]
. ,
, r
s s
t x
s x
t
− ∈
+ =
, we have in the spirit of [4], the following relations, equivalent with 1, 2:
, ,
,
α
α
r x
x f
x L
dt dx
t t
t t
− +
=
3
, s
ds dx
s dt
dx
t t
=
4 .
ϕ =
x 5
Define see also [4] the linear operator,
,
~
− +
=
• •
ϕ ϕ
ϕ ϕ
α α
L X
A
[ ]
BC BC
r C
A
n
→ ⊂
− R
, ,
:
1 ~
α
. Now we can rewrite the above problem as
, ,
,
~
α
α
r x
x f
X x
A dt
dx
t t
t t
− +
=
, 6 .
ϕ =
x 7
The last term of 6 may be written as
∫ ∫
− −
+ ∆
∆ ,
,
r t
r t
s x
r s
d s
x s
d f
X α .
We define
∫ ∫
− −
+ ∆
∆ =
, ,
,
r t
r t
t
s x
r s
d s
x s
d f
x F
α α
. Thus 6 and 7 take the form
, ,
~
α
α
t t
t
x F
X x
A dt
dx +
=
8
differential equations
237 ϕ
= x
, 9 this being the abstract problem in
B equivalent to 1,2. The eigenvalues of
α
~
A
are see [4] the roots of the equation det
= −
−
−
α α
λ
λ
B e
A I
r
. We assume the following hypothesis, that we denote H1.
H1. An open set U exists in the parameter plane such that for every
U ∈
α ,
there is a pair of complex conjugated simple eigenvalues α
ω α
µ α
λ
i ±
=
2 ,
1
, with the property that there is a U
∈ α
such that
2 ,
1
ω α
ω α
λ
i i
± =
± =
, with ω
and for every
U ∈
α , all other
eigenvalues have strictly negative real parts, uniformly bounded from above by a negative number.
By a simple eigenvalue we mean an eigenvalue having the algebraic multiplicity equal to 1.
We remark that H1 implies the existence of a neighborhood of α such that
each eigenvalue different from α
λ
2 ,
1
has real part strictly less than α
µ . The eigenvectors corresponding to
α λ
i
, i=1,2, are elements of
C
B -the complexification of B , namely
[ ]
, ,
r s
e s
i s
i
i
− ∈
=
α ϕ
α ϕ
α λ
, where
α ϕ
i
is a solution of
= −
−
−
ϕ α
α α
λ
α λ
B e
A I
r i
i
. Obviously,
1 2
α ϕ
α ϕ
= .
Denote by
{ }
α λ
2 ,
1
M the linear subspace of
C
B , spanned by {
α ϕ
α ϕ
2 1
, } and
α Φ the matrix having as columns the vectors
α ϕ
α ϕ
2 1
, . Let {
α ψ
α ψ
2 1
, } be two eigenvectors for the adjoint problem
[3], [4], corresponding to the eigenvalues α
λ
2 ,
1
−
of the infinitesimal generator of the adjoint problem. They are elements of
C
B
- the complexification of
[ ]
n
r C
R ,
, B
= -row vectors, and we assume that they
are selected such that, α
Ψ being the matrix having as rows the vectors
α ψ
α ψ
2 1
, , the relation
2
, I
= Φ
Ψ α
α holds, where
C →
× ⋅
⋅
C C
B B
: ,
is defined by
differential equations
238 .
B ,
B ,
,
C C
∈ ∈
− −
=
∫∫
−
ϕ ψ
ξ ξ
ϕ θ
η θ
ξ ψ
ϕ ψ
ϕ ψ
θ α
r
d d
In [4]
a projection
{ }
α λ
π
2 ,
1
0C
B :
M →
, is defined by
[ ]
α ϕ
α α
α ϕ
π ,
Ψ +
Ψ Φ
= + X
. With this projection the space
0C
B is decomposed as
{ }
π
α λ
Ker B
2 ,
1
0C
⊕ = M
. Since the solution
t
x of 8 and 9
belongs to ],
, [
1
n
R r
C −
, it is decomposed as t
y t
u x
t
+ Φ
= α
, 10 with
t
x t
u ,
α Ψ
= and
t
x I
t y
π −
= , where
,
_
t z
t z
t u
= -column
vector,
C ∈
t z
. Let us define
=
1 1
α λ
α λ
α
B
. The projection of eq. 8 on
{ }
α λ
2 ,
1
M is
, α
α α
α α
α α
t y
t u
F u
B u
+ Φ
Ψ Φ
+ Φ
= Φ
•
, and since
α
Φ
is invertible, this is equivalent to ,
α α
α α
t y
t u
F u
B u
+ Φ
Ψ +
=
•
. 11 By projecting the initial condition we find
ϕ α,
Ψ =
u .
3. Existence of the invariant manifold and the restricted equation If
