Discrete Deterministic and Stochastic Dynamical Systems with Delay - Applications
491
Using the change of variables x 1 = xn− m, . . . , xm = xn−1, xm+1 = xn, the application
associated to (43) is:
⎛
⎞
⎛
⎞
x 1
x 2
⎜
⎜ . ⎟
⎜
.
⎟
.
⎜ .. ⎟ → ⎜
.
⎟
⎝
⎟
⎜
⎟
xm ⎠
⎝
xm+1
⎠ .
(44)
xm+1
kw − ak f ( x 1) + xm+1
The fixed point of (44) is ( x∗, . . . , x∗) T ∈ IRm+1, where x∗ satisfies relation w = a f ( x∗). With the translation x → x + x∗, application (44) can be written as:
⎛
⎞
⎛
⎞
x 1
x 2
⎜
⎜ . ⎟
⎜
.
⎟
.
⎜ .. ⎟ → ⎜
.
⎟
⎝
⎟
⎜
⎟
xm ⎠
⎝
xm+1
⎠
(45)
xm+1
kw − akg( x 1) + xm+1
where g( x 1) = f ( x 1 + x∗).
The following statements hold:
Proposition 3.1. ((Mircea et al., 2004)) (i) The Jacobian matrix of (45) in 0 ∈ IRm+1 is
⎛
⎞
0
1 0 . . . 0
⎜
⎜
0
0 1 . . . 0 ⎟
⎟
A = ⎜
⎜ . . . . . . . . . . . . . . . ⎟
(46)
⎝
⎟
0
0 0 . . . 1 ⎠
− akρ 1 0 0 . . . 1
where ρ 1 = g (0) .
(ii) The characteristic equation of A is:
λm+1 − λm + akρ 1 = 0.
(47)
(iii) If μ ∈ C is a root of (47), the eigenvector q ∈ IRm+1 that corresponds to the eigenvalue μ, of the
matrix A, has the components:
qi = μi−1,
i = 1, . . . , m + 1
and the components of the eigenvector p ∈ IRm+1 corresponding to μ of the matrix AT are:
p 1 = −
akρ 1
p
p
p
μ
, p
m+1 − makρ
i =
1
1, i = 2, . . . , m − 1, pm = μ 2 − μ 1, pm+1 = − μ
1.
1
μi−1
pρ 1
akρ 1
m+1
The vectors p ∈ IRm+1 , q ∈ IRm+1 satisfy the relation ∑ piqi = 1 .
i=1
The following statements hold:
492
Discrete Time Systems
Proposition 3.2. (i) If m = 2 , equation (47) becomes:
λ 3 − λ 2 + akρ 1 = 0.
(48)
Equation (48) has two complex roots with their absolute values equal to 1 and one root with the absolute
√
√
5 − 1
5 − 1
value less than 1, if and only if k =
. For k = k
, equation (48) has the roots:
2 aρ
0 =
1
2 aρ 1
λ 1,2 = exp (± iθ( k 0) i),
√
θ(
1 +
5
a 0) = arccos
.
(49)
4
(ii) With respect to the change of parameter
k = k( β) = k 0 + g( β)
where:
√5+1
g( β) =
1 + 4(1 + β)6 − (1 + β)2 −
2 k 0 ρ 1
equation (49) becomes:
λ 3 − λ 2 + ak( β) ρ 1 = 0.
(50)
The roots of equation (50) are:
μ 1,2( β) = (1 + β)exp (± iω( β)), λ( β) = − ak( β) ρ 1
(1 + β)2
where:
(
ω( β) =
1 + β)2 +
1 + 4(1 + β)6
arccos
.
4(1 + β)2
(iii) The eigenvector q ∈ IR 3 , associated to the μ = μ( β) , for the matrix A has the components:
q 1 = 1,
q 2 = μ,
q 3 = μ 2
and the eigenvector p ∈ IR 3 associated to the eigenvalue μ = μ( β) for the matrix AT has the
components:
p 1 =
akρ 1
, p
, p
.
2 akρ
2 =
μ 2
3 =
μ
1 − μ 3
μ 3 − 2 akρ 1
μ 3 − 2 akρ 1
(iv) a 0 is a Neimark-Sacker bifurcation point.
Using Proposition 3.2, we obtain:
Proposition 3.3. The solution of equation (43) in the neighborhood of the fixed point x∗ ∈ IR is:
un = x∗ + zn + zn + 1 w 1
w 1
2 20 z 2 n + w 111 znzn + 12 02 z 2 n
xn = x∗ + q 3 zn + q 3 zn + 1 w 3
w 3
2 20 z 2 n + w 311 znzn + 12 02 z 2 n
Discrete Deterministic and Stochastic Dynamical Systems with Delay - Applications
493
where:
μ 2( μ 2 − 1) h 1 − ( μ 2 − 1) h 2 + h 3
w 1 =
20
20
20
= μ 2
−
= μ 4
− μ 2
−
20
w 1
h 1
w 1
h 1
h 2
μ
, w 2
6 − μ 4 + akρ
20
20
20, w 320
20
20
20
1
w 1 = h 311 , w 2 = w 1 − h 1
= w 1 − h 1 − h 2
11
akρ
11
11
11, w 311
11
11
11
1
h 1 =
)
=
)
=
)
20
4 akρ 2( p 3 + p 3 , h 220
akρ 1( p 3 q 2 + p 3 q 2 , h 320
4 akρ 2(1 + p 3 q 3 + p 3 q 3
h 1 = h 1
= h 2
= h 3
11
20, h 211
20, h 311
20
and zn ∈ C is a solution of equation:
zn+1 = μzn − 1 p
p
− kaρ
+ ρ
2 3 akρ 2( z 2 n + 2 znzn + z 2 n) + 12 3(− kaρ 1 w 120
1 w 111
3 ),
rho 1 = f (0),
ρ 2 = f (0), ρ 3 = f (0) .
Let
( μ − 3 − 2 μ)
| p
C
3|2
1( β) = − p 3 a 2 k 2 ρ 22
+ a 2 k 2 ρ 22
+ ak| ρ 2 p 3| + p
− akρ
+ ρ
2( μ 2 − μ)( μ − 1)
1 − μ
2( μ 2 − μ)
3 (− akρ 1 w 120
1 w 111
3 )
and
l(0) = Re(exp (− iθ( a 0)) C 1(0)).
If l(0) < 0 , the Neimark–Sacker bifurcation is supercritical (stable).
The model of an Internet network with r links and a single source, can be analyzed in a similar
way.
The perturbed stochastic equation of (43) is:
xn+1 = xn − αk f ( xn− m) + kw + ξnb( xn − x∗)
(51)
and x∗ satisfies the relation w = a f ( x∗), where E( ξn) = 0, E( ξ 2 n) = σ > 0.
We study the case m = 2. Using (46) the linearized equation of (51) has the matrices:
⎛
⎞
⎛
⎞
0
1 0
0 0 0
A
⎝
⎠
⎝
⎠
1 =
0
0 1
, B =
0 0 0
− akρ 1 0 1
0 0 b
Using Proposition 2.1, the characteristic polynomial of the linearized system of (51) is given
by:
P 2( λ) = ( λ 3 − (1 + σb 2) λ 2 − a 2 k 2 ρ 2)( λ 3 +
)
1
akρ 1 λ + a 2 k 2 ρ 21 .
If the roots of P 2( λ) have their absolute values less than 1, then the square mean values of the
solutions for the linearized system of (51) are asymptotically stable. The analysis of the roots
for the equation P 2( λ) = 0 can be done for fixed values of the parameters.
The numerical simulation can be done for: w = 0.1, a = 8 and f ( x) = x 2/(20 − 3 x).
494
Discrete Time Systems
4. A discrete economic game with delay
The economic game is described by a number of firms that enter the market with a
homogeneous consumption product at different moments n, where n ∈ IN. In what follows
we consider two firms F 1, F 2 and x, y the state variables of the model that represent the
firms’outputs. The price function of the product (the inverse demand function) is p : IR+ →
IR+, derivable function with lim p( x) = 0, lim p( x) = ∞ and p ( x) < 0. The cost functions x→∞
x→0+
are Ci : IR+ → IR+, i = 1, 2, derivable functions with C ( x) = 0, C ( x) ≥ 0. The profit i
i
functions of the firms, πi : IR 2+ → IR+, i = 1, 2, are given by:
π 1( x, y) = p( x + y) x − C 1( x), π 2( x, y) = p( x + y) y − C 2( y).
The non-cooperative game F 1, F 2, denoted by Γ = ( IR 2+, π 1, π 2) is called deterministic economic
game. The Nash solution of Γ is called the solution of the det