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Discrete Time Systems
2.4 The Neimark-Sacker bifurcation for a class of discrete-time dynamical systems with
delay
A two dimensional discrete-time dynamical system with delay is defined by the equations
xn+1 = xn + f 1( xn, yn, α)
yn+1 = yn + f 2( xn− m, yn, α)
(13)
where α ∈ IR, f 1, f 2 : IR 3 → IR are seamless functions, so that for any | α| small enough, the system f 1( x, y, α) = 0, f 2( x, y, α) = 0, admits a solution ( x, y) T ∈ IR 2.
Using the translation xn → xn + x, yn → yn + y, and denoting the new variables with the
same notations xn, yn, system (13) becomes:
xn+1 = xn + f ( xn, yn, α)
yn+1 = yn + g( xn− m, yn, α)
(14)
where:
f ( xn, yn, α) = f 1( xn + x, yn + y, α); g( xn− m, yn, α) = f 2( xn− m + x, yn + y, α).
With the change of variables x 1 = xn− m, x 2 = xn−( m−1), . . . , xm = xn−1, xm+1 = xn, xm+2 =
yn, application (14) associated to the system is:
⎛
⎞
x 1
⎛
⎞
⎜
x 2
⎜ x 2 ⎟
⎜
⎟
⎜
.
⎟
.
⎜ . ⎟ → ⎜
.
⎟
⎜ .. ⎟
⎜
⎟
⎝
⎟
⎝ xm+1 + f( xm+1, xm+2, α) ⎠ .
(15)
xm+1 ⎠
xm+2 + g( x 1, xm+2, α)
xm+2
We use the notations:
a 10 = ∂ f (
(
∂
0, 0, α), a
0, 0, α),
xm+1
01 =
∂ f
∂xm+2
b 10 = ∂g (
(
∂
0, 0, α), b
0, 0, α)
x 1
01 =
∂g
∂xm+2
a 20 =
∂ 2 f
(
(
∂
0, 0, α),
a
0, 0, α),
xm+1 ∂xm+1
11 =
∂ 2 f
∂xm+1 ∂xm+2
a 02 =
∂ 2 f
(
(
∂
0, 0, α),
a
0, 0, α),
xm+2 ∂xm+2
30 =
∂ 3 f
∂xm+1 ∂xm+1 ∂xm+1
(16)
a 21 =
∂ 3 f
(
(
∂
0, 0, α), a
0, 0, α),
xm+1 ∂xm+1 ∂xm+2
12 =
∂ 3 f
∂xm+1 ∂xm+2 ∂xm+2
a 03 =
∂ 3 f
(
∂
0, 0, α)
xm+2 ∂xm+2 ∂xm+2
Discrete Deterministic and Stochastic Dynamical Systems with Delay - Applications
485
b 20 =
∂ 2 g (
(
∂
0, 0, α),
b
0, 0, α),
x 1 ∂x 1
11 =
∂ 2 g
∂x 1 ∂xm+2
b 02 =
∂ 2 g
(
(
∂
0, 0, α),
b
0, 0, α),
xm+2 ∂xm+2
30 =
∂ 3 g
∂x 1 ∂x 1 ∂x 1
(17)
b 21 =
∂ 3 g
(
(
∂
0, 0, α),
b
0, 0, α),
x 1 ∂x 1 ∂xm+2
12 =
∂ 3 g
∂x 1 ∂xm+2 ∂xm+2
b 03 =
∂ 3 g
(
∂
0, 0, α).
xm+2 ∂xm+2 ∂xm+2
With (16) and (17) from (15) we have:
Proposition 2.4. ((Mircea et al., 2004)) (i) The Jacobian matrix associated to (15) in (0, 0) T is:
⎛
⎞
0 1 . . .
0
0
⎜
⎜ 0 0 . . .
0
0
⎟
⎜
⎟
⎟
A = ⎜ . .
.
.
⎜ .. ..
..
..
⎟ .
(18)
⎝
⎟
0 0 . . . 1 + a
⎠
10
a 01
b 10 0 . . .
0
1 + b 01
(ii) The characteristic equation of A is:
λm+2 − (2 + a 10 + b 01) λm+1 + (1 + a 10)(1 + b 01) λ − a 01 b 10 = 0.
(19)
(iii) If μ = μ( α) is an eigenvalue of (19), then the eigenvector q ∈ C m+2, solution of the system Aq = μq, has the components:
q 1 = 1, qi = μi−1, i = 2, . . . , m + 1, qm+2 =
b 10
μ −
.
(20)
1 − b 01
The eigenvector p ∈ C m+2 defined by AT p = μp has the components
( μ − 1 − a
p
10 )( μ − 1 − b 01)
1 =
p
m( μ − 1 − a
1, i = 2, . . . , m,
10)( μ − 1 − b 01) + μ(2 μ − 2 − a 10 − b 01) , pi =
1
μi−1
(21)
pm+1 =
1
p
μm−1( μ − 1 − a
1.
10) p 1, pm+2 = μ
b 10
The vectors q, p satisfy the condition:
m+2
< q, p >= ∑ qip =
i
1.
i=1
The proof is obtained by straight calculation from (15) and (18).
The following hypotheses are taken into account:
H1. The characteristic equation (19) has one pair of conjugate eigenvalues μ( α), μ( α) with
their absolute values equal to one, and the other eigenvalues have their absolute values less
than one.
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Discrete Time Systems
H2. The eigenvalues μ( α), μ( α) intersect the unit circle for α = 0, and satisfy the transversality
condition
d | μ( α)|
dα
α=0 = 0.
H3. If arg( μ( α)) = θ( α), and θ 0 = θ(0), then eiθ 0 k = 1, k = 1, 2, 3, 4.
From H2 we notice that for all | α| small enough, μ( α) is given by:
μ( α) = r( α) eiθ( α)
with r(0) = 1, θ(0) = θ 0, r (0) = 0. Thus r( α) = 1 + β( α) where β(0) = 0 and β (0) = 0.
Taking β as a new parameter, we have:
μ( β) = (1 + β) eiθ( β)
(22)
with θ(0) = θ 0. From (22) for β < 0 small enough, the eigenvalues of the characteristic
equation (19) have their absolute values less than one, and for β > 0 small enough, the
characteristic equation has an eigenvalue with its absolute value greater than one. Using
the center manifold Theorem (Kuznetsov, 1995), application (15) has a family of invariant
manifolds of two dimension depending on the parameter β. The restriction of application (15)
to this manifold contains the essential properties of the dynamics for (13). The restriction of
application (15) is obtained using the expansion in Taylor series until the third order of the
right side of application (15).
2.5 The center manifold, the normal form
Consider the matrices:
A 1 =
a 20 a 11 , C
, D
a
1 =
a 30 a 21
1 =
a 21 a 12
11 a 02
a 21 a 12
a 12 a 03
A 2 =
b 20 b 11 , C
, D
b
2 =
b 30 b 21
2 =
b 21 b 12
11 b 02
b 21 b 12
b 12 b 03
with the coefficients given by (16) and (17).
Denoting by x = ( x 1, . . . , xm+2) ∈ IRm+2, application (15), is written as x → F( x), where F( x)= ( x 2, . . . , xm, xm+1 + f ( xm+1, xm+2, α), xm+2 + g( x 1, xm+2, α)).
The following statements hold:
Proposition 2.5. (i) The expansion in Taylor series until the third order of function F( x) is:
F( x) = Ax + 1 B( x, x) + 1 C( x, x, x) + O(|§| ),
(23)
2
6
where A is the matrix (18), and
B( x, x)
= (0, . . . , 0, B 1( x, x), B 2( x, x)) T,
C( x, x, x) = (0, . . . , 0, C 1( x, x, x), C 2( x, x, x)) T,
