498
Discrete Time Systems
a = ( q 2, q 3) A 1( q 2, q 3) T,
b = ( q 1, q 3) A 2( q 1, q 3) T,
a 1 = ( q 2, q 3) A 1( q
) T
2, q 3
,
b 1 = ( q 1, q 3) A 2( q
) T
1, q 3
,
C 1 = ( q 2, q 3)( q 2 A 11 + q 3 A 12)( q
) T
) T
2, q 3
, C 2 = ( q 1, q 3)( q 1 A 21 + q 3 A 22)( q 1, q 3 ,
r 1 = (
)
) T
= (
) T
20
q 2, q 3 A 1( w 220, w 320 , r 220
q 1, q 3) A 2( w 120, w 320 ,
r 1 = (
) T
= (
) T
11
q 2, q 3) A 1( w 211, w 311 , r 211
q 1, q 3) A 2( w 111, w 311 ,
w 1 =
− p 2 a + p 3 b
20
v 20 − p 2 a + p 3 b
μ 2 − μ
μ 2 − μ ,
w 2 = μ 2
20
v 20 − p 2 a + p 3 b q
q
μ 2 − μ
2 − p 2 a + p 3 b
μ 2 − μ 2,
w 3 = a+( μ 2− a 10) μ 2 v 20 − p 2 a+ p 3 b
20
a 01
μ 2− μ q 3− p 2 a+ p 3 b
μ 2− μ q 3,
w 1 =
=
=
02
w 120, w 202
w 220, w 302
w 320
w 1 =
− p 2 a 1 + p 3 b 1
11
v 11 − p 2 a 1 + p 3 b 1
1 − μ
1 − μ
,
w 2 =
11
v 11 − p 2 a 1 + p 3 b 1
1 − μ
q 2 − p 2 a 1 + p 3 b 1
1 − μ
q 2,
w 3 = a 1 + (1 − a 10) v 11 − p 2 a 1 + p 3 b 1
11
a 01
1 − μ
q 3 − p 2 a 1 + p 3 b 1
1 − μ
q 3,
( μ − 1 − ka
p
10 )( μ − 1 − αb 10)
1 = ( μ − 1 − ka 10)( μ − 1 − kb 01) + μ(2 μ − 2 − ka 10 − αb 01)
p 2 =
p 1
μ −
, p
p
.
1 − ka
3 =
μ
1, q 1 = 1, q 2 = μ, q 3 =
αb 10
10
αb 10
μ − 1 − αb 01
(iv) The variations of the profits in the neighborhood of the fixed point ( x, y) T ∈ IR 2 , are given by: π 1 n = p( xn + yn) xn − c 1 xn − b 1, π 2 n = p( xn + yn) yn − c 2 yn − b 2.
The above model has a similar behavior as the economic models that describe the business cycles
(Kuznetsov, 1995), (Mircea et al., 2004).
The model can be analyzed in a similar way for the case m > 2 .
For m = 1 , the stochastic system associated to (53) is given by:
xn+1 = xn + k( p ( xn + yn) xn + p( xn + yn) − c (
1 xn)) + ξnb 22( xn − ¯
x)
(70)
yn+1 = yn + k( p ( xn−1 + yn) yn + p( xn−1 + yn) − c (
2 yn)) + ξnb 33( yn − ¯
y)
where ( ¯ x, ¯ y) is the solution of (55).
Discrete Deterministic and Stochastic Dynamical Systems with Delay - Applications
499
The linearized of (70) has the matrices:
⎛
⎞
⎛
⎞
0
1
0
0 0
0
A
⎝
⎠
⎝
⎠
1 =
0 1 + ka 10
ka 01
, B =
0 b 22 0
(71)
αb 10
0
1 + αb 01
0 0 b 33
Using Proposition 2.1, the characteristic polynomial of (70) is given by:
P 2( λ) = λ( λ 2 − λ( a 22 a 33 + σb 22 b 33) − a 23 a 33 a 31)( λ( λ − a 2 )( λ −
− σ ) −
)
22
a 233
b 233
a 223 a 231 , (72)
where a 22 = 1 + ka 10 , a 23 = ka 01 , a 31 = αb 10 , a 33 = 1 + αb 01 .
The analysis of the roots for the equation P 2( λ) = 0 is done for fixed values of the parameters.
The numerical simulation can be done for c 1 = 0.1 , c 2 = 0.4 , k = 0.04 , σ = 0.4 .
5. The discrete deterministic and stochastic Kaldor model
The discrete Kaldor model describes the business cycle for the state variables characterized by
the income (national income) Yn and the capital stock Kn, where n ∈ IN. For the description
of the model’s equations we use the investment function I : IR+ × IR+ → IR denoted by
I = I( Y, K) and the savings function S : IR+ × IR+ → IR, denoted by S = S( Y, K) both considered as being differentiable functions (Dobrescu & Opri¸s, 2009), (Dobrescu & Opri¸s,
2009).
The discrete Kaldor model describes the income and capital stock variations using the
functions I and S and it is described by:
Yn+1 = Yn + s( I( Yn, Kn) − S( Yn, Kn))
(73)
Kn+1 = Kn + I( Yn, Kn) − qKn.
In (73), s > 0 is an adjustment parameter, which measures the reaction of the system to the
difference between investment and saving.
We admit Keynes’s hypothesis which states that the saving function is proportional to income,
meaning that
S( Y, K) = pY,
(74)
where p ∈ (0, 1) is the propensity to save with the respect to the income.
The investment function I is defined by taking into account a certain normal level of income
pu
u and a normal level of capital stock
, where u ∈ IR, u > 0. The coefficient q ∈ (0, 1)
q
represents the capital depreciation.
In what follows we admit Rodano’s hypothesis and consider the form of the investment
function as follows:
pu
I( Y, K) = pu + r
− K + f ( Y − u)
(75)
q
where r > 0 and f : IR → IR is a differentiable function with f (0) = 0, f (0) = 0 and
f (0) = 0.
System (73) with conditions (74) and (75) is written as:
500
Discrete Time Systems
Yn+1 = (1 − sp) Yn − rsKn + s f ( Yn − u) + spu 1 + rq
(76)
Kn+1 = (1 − r − q) Kn + f ( Yn − u) + pu 1 + rq
with s > 0, q ∈ (0, 1), p ∈ (0, 1), r > 0, u > 0.
The application associated to system (76) is:
⎛
⎞
(1 − sp) y − rsk + s f ( y − u) + spu 1 + r
y
⎜
⎟
→ ⎜
q
⎟
k
⎝
⎠ .
(77)
(1 − r − q) k + f ( y − u) + pu 1 + rq
The fixed points of the application (77) with respect to the model’s parameters s, q, p, r are the
solutions of the following system:
py + rk − f ( y − u) − pu 1 + r
= 0
q
( r + q) k − f ( y − u) − pu 1 + r = 0
q
that is equivalent to:
qk − py = 0,
p 1 + r
( y − u) = f ( y − u).
(78)
q
Taking into account that f satisfies f (0) = 0, by analyzing (78) we have:
pu
Proposition 5.1. (i) The point of the coordinates P u,
is the fixed point of the application (77).
q
(ii) If f ( x) = arctan x, and p 1 + r
≥ 1 then application (77) has an unique fixed point
q
pu
given by P u,
.
q
(iii) If f ( x) = arctan x and p 1 + r
< 1 then the application (77) has the fixed points
q
pu
py
p
P u,
, R y
r
, Q y
y
q
r,
q
q, q q , where yq = 2 u − yr and yr is the solution of the
following equation:
arctan( y − u) = p 1 + r
( y − u)
q
Let ( y 0, k 0) be a fixed point of the application (77). We use the following notations: ρ 1 =
f ( y 0 − u), ρ 2 = f ( y 0 − u), ρ 3 = f ( y 0 − u) and
a 10 = s( ρ 1 − p), a 01 = − rs, b 10 = ρ 1, b 01 = − q − r.
Discrete Deterministic and Stochastic Dynamical Systems with Delay - Applications
501
Proposition 5.2. (i) The Jacobian matrix of (77) in the fixed point ( y 0, k 0) is:
A =
1 + a 10
a 01
.
(79)
b 10
1 + b 01
(ii) The characteristic equation of A given by (79) is:
λ 2 − aλ + b = 0
(80)
where a = 2 + a 10 + b 01,
b = 1 + a 10 + b 01 − a 01 b 10.
(iii) If q + r < 1, ρ 1 < 1 + r( q + r − 4)
(
and s = s
q + r − 2)2
0, where:
s 0 =
q + r
(1 − q − r)( ρ 1 − p) + r
then equation (80) has the roots with their absolute values equal to 1.
(iv) With respect to the change of variable:
s( β) = (1 + β)2 − 1 + q + r
(1 − q − r)( ρ 1 − p) + r
equation (80) becomes:
λ 2 − a 1( β) λ + b 1( β) = 0
(81)
where
( ρ
a
1 −