Nr 1
Peter MALIČKÝ*, Rudolf ZIMKA**
CAN TORI ARISE IN A TWO-REGIONAL MODEL WITH
FIXED EXCHANGE RATES?***
A two-regional, five dimensional model describing the development of income, capital stock and money stock, which was introduced by T. Asada in [1], is analysed. Sufficient conditions are found for the existence of two pairs of purely imaginary eigenvalues and a fifth negative one for the linear approximation matrix of the model. A theorem on the existence of invariant tori is presented. Keywords: dynamic model, equilibrium, linear approximatiomatrix, eigenvalues, normal form of
differ-ential equations on invariant surface, bifurcation equation, torus
1. Introduction
In [2] T. Asada, T. Inaba and T. Misawa developed and studied a two regional model of business cycles with fixed exchange rates, which consists of a five dimensional dis-crete time system. In [1] T. Asada introduced and analysed a continuous time version of this model, describing the dynamic interaction of two regions which are connected through interregional trade and capital movement. Throughout the paper we adopt the notation used in [1] with minor changes. The analysed model is of the form
0, > ), ( = i i i i i i i i C I G J Y Y& α + + + − α , = i i I K& (1) 1,2, = , = 1 1 1 pA i M&
* Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica, Slovakia, e-mail: malicky@fpv.umb.sk
** Department of Quantitative Methods and Informatics, Faculty of Economics, Matej Bel University, Tajovského 10, 975 50 Banská Bystrica, Slovakia, e-mail: rudolf.zimka@umb.sk
where: 0, 1, < < 0 , ) ( = i i− i + 0i i 0i ≥ i c Y T C c C C 0, 1, < < 0 , = i i− 0i i 0i ≥ i Y T T T τ τ 0, < 0, < 0, > ), , , ( = i i i i i i i i i i i r I K I Y I r K Y I I ∂ ∂ ∂ ∂ ∂ ∂ 0, < 0, > ), , ( = i i i i i i i i i r L Y L r Y L p M ∂ ∂ ∂ ∂ 0, > 0, > 0, < ), , , ( = 1 2 1 1 1 2 1 1 1 E J Y J Y J E Y Y J J ∂ ∂ ∂ ∂ ∂ ∂ (2) 0, > , = 1 2 1 β ⎟⎟ β ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − − E E E r r Q e , = 1 1 1 J Q A + , = 0 p1J1+Ep2J2 , = 0 p1Q1+Ep2Q2 , =M1 EM2 M +
and the subscript i, i = 1, 2, is the index of a region. The meanings of the symbols in (1) and (2) are as follows: Yi – real regional income, Ki – real physical capital stock,
Mi – nominal money stock, Li – demand for money, Ci – consumption, Ti – taxes, Ii –
net real private investment expenditure on physical capital, Gi - real government
ex-penditure (fixed), pi – price level, ri – nominal rate of interest, E – exchange rate, Ee –
expected exchange rate, Ji – current account balance (net exports) in real terms, Qi –
capital account in real terms (net capital inflow), Ai – total balance of payments in real
terms, αi – adjustment speed in goods market and β – degree of capital mobility.
As Asada in [1], in this paper we assume fixed price economy with fixed exchange rates. Therefore, normalizing the price levels in the two regions, we can suppose that p1 = p2 = 1, E = Ee. Furthermore we suppose that the nominal interest rate ri, i = 1, 2,
adjusts instantaneously to keep the money stock Mi and demand for money Li in
equilib-rium. Under these assumptions, taking into account (2) and supposing that ri is implicitly
determined by the relations Mi=Li(Yi,ri), i=1,2, model (1) takes the form + + + + − 1 1 1 01 01 1 1 1 1= [c(1 )Y cT C G Y& α τ I1(Y1,K1,r1(Y1,M1))+J1(Y1,Y2,E)−Y1], )), , ( , , ( = 1 1 1 1 1 1 1 I Y K r Y M K&
⎢ ⎣ ⎡ + + + − 2 2 2 02 02 2 2 2 2= c (1 )Y cT C G Y& α τ (3) , ) , , ( 1 , , , 1 1 1 2 2 2 2 2 2 2 ⎥ ⎥ ⎦ ⎤ − − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − + J Y Y E Y E E M M Y r K Y I , , , , = 1 2 2 2 2 2 2 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − E M M Y r K Y I K& . , ) , ( ) , , ( = 1 2 2 1 1 1 2 1 1 1 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − + E M M Y r M Y r E Y Y J M& β
In the whole article we suppose that:
1. Model (3) has a unique equilibrium point (Y10,K10,Y20,K20,M10) with positive coordinates for an arbitrary triple of positive parameters (α1,α2,β).
2. All functions in model (3) are linear with respect to their variables, except the functions Ii and ri, i=1,2, which are nonlinear in Yi of type
6
C in a small neigh-bourhood of the equilibrium point.
Remark 1. The analysis of the existence of an equilibrium for model (3) was per-formed by Asada in [1]. The requirement on the functions in (3) to be of class C6 with
respect to Yi enables the transformation of model (3) to its partial normal form on an
invariant surface and to use a theorem on the existence of tori (see e.g. [3]).
In [1] Asada found sufficient conditions for local stability of the equilibrium point. The question of the existence of business cycles around the equilibrium is analysed in [1]. In the present paper we are interested in the existence of tori in a small neigh-bourhood of the equilibrium point. Tori can arise only in the case when the linear approximation matrix of model (3) at the equilibrium point has two pairs of purely imaginary eigenvalues. Model (3) is analysed in section 2. Theorem 1 gives sufficient conditions for the existence of two pairs of purely imaginary eigenvalues with the remaining one being negative. Theorem 2 comments on the existence of tori in a small neighbourhood of the equilibrium point.
2. The analysis of model (3)
Let us write model (3) in the abbreviated form ), , , ; ( = ξ α1α2 β ξ& Ξ
where )ξ =(Y1,K1,Y2,K2,M1 . After translating the equilibrium point ξ0 = ) , , , ,
(Y10 K10 Y20 K20 M10 to the origin via the coordinate shift y=ξ−ξ0, model (3) becomes ). , , ; ( =Ξ y+ξ0 α1α2 β y&
Its Taylor expansion at y=0 gives
), , , ; ( ) , , ( = Aα1α2 β y Y y α1 α2 β y& + (4) where ( ; , , )= (|| ||2) 2 1 O y y
Y α α β and the linear approximation matrix A(α1,α2,β) is
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ) ( 0 ) ( 0 ) ( 0 0 0 0 0 0 = ) , , ( 55 53 51 35 34 43 35 2 34 2 33 2 31 2 15 12 21 15 1 13 1 12 1 11 1 2 1 β β β α α α α α α α α β α α F F F G G F G G G G G G F G G G G A , (5) where: , ) (1 1 = 1 1 1 1 1 1 1 1 1 1 11 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ − − − − ∂ ∂ ∂ ∂ + ∂ ∂ Y J c Y r r I Y I G τ 0, < = 1 1 12 K I G ∂ ∂ 0, > = 2 1 13 Y J G ∂ ∂ 0, > = 1 1 1 1 15 M r r I G ∂ ∂ ∂ ∂ , = 1 1 1 1 1 1 21 Y r r I Y I F ∂ ∂ ∂ ∂ + ∂ ∂ 0, > 1 = 1 1 31 Y J E G ∂ ∂ −
, 1 ) (1 1 = 2 1 2 2 2 2 2 2 2 2 33 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ + − − − ∂ ∂ ∂ ∂ + ∂ ∂ Y J E c Y r r I Y I G τ 0, < = 2 2 34 K I G ∂ ∂ 0, < 1 = 2 2 2 2 35 M r r I E G ∂ ∂ ∂ ∂ − , = 2 2 2 2 2 2 43 Y r r I Y I F ∂ ∂ ∂ ∂ + ∂ ∂ , = ) ( 1 1 1 1 51 Y r Y J F ∂ ∂ + ∂ ∂ β β , = ) ( 2 2 2 1 53 Y r Y J F ∂ ∂ − ∂ ∂ β β 0, < 1 = ) ( 2 2 1 1 55 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ + ∂ ∂ M r E M r F β β
while all the derivatives are evaluated at the equilibrium point ξ0.
Assumption 1. As in [1], we asume that the values of
1 0 1( ) Y I ∂ ∂ ξ and 2 0 2( ) Y I ∂ ∂ ξ are sufficiently large that G11>0 and G33>0 at the equilibrium point ξ0.
Assumption 2. The value of the parameter β can be considered arbitrarily large compared with the values of the parameters α1 and α2.
Remark 2. Assumption 1 is analogous to the standard hypothesis of the Kaldorian model of business cycles [4]. This assumption automatically implies that F21>0 and
0 >
43
F Assumption 2 also agrees with economic theory.
Definition. A triple (α10,α20,β0) of parameters α1, α2 and β in (4) is called a critical triple of model (4) if the matrix A(α10,α20,β0) has eigenvalues λ1,2=±iω1,
2 3,4= ω
λ ±i , i= −1, λ5 <0.
The characteristic equation of A(α1,α2,β) is
0, = 5 4 2 3 4 2 4 1 5+aλ +a λ +aλ +a λ+a λ (6)
where: ), ( = trace = ) , , ( = 1 1 2 1 11 12 2 33 34 55 1 a α α β A αG G α G G F β a − − − − − − = ) , , ( = 2 1 2 2 a α α β
a sum of all the principal second-order minors of A,
34 11 1 33 31 13 11 2 1 12 21 12 11 1 2 0 0 = G G G G G G G F G G a α +αα +α 34 12 33 12 2 55 51 15 11 1 0 0 0 0 ) ( ) ( G G G G F F G G + + + α β β α ) ( ) ( ) ( 0 53 55 35 33 2 34 43 34 33 2 55 15 12 β β α α β F F G G G F G G F G G + + + , ) ( 0 55 35 34 β F G G + − = ) , , ( = 3 1 2 3 a α α β
a sum of all the principal third-order minors of A,
34 12 21 12 11 1 33 31 12 21 13 12 11 2 1 3 0 0 0 0 0 0 = G G F G G G G G F G G G a −αα −α 34 43 34 33 31 13 11 2 1 55 51 15 12 21 15 12 11 1 0 0 ) ( 0 ) ( F G G G G G G F F G G F G G G α α β β α − − ) ( 0 ) ( 0 0 ) ( ) ( ) ( 51 55 35 34 15 11 1 55 53 51 35 33 31 15 13 11 2 1 β β α β β β α α F F G G G G F F F G G G G G G − − ) ( ) ( 0 0 0 0 0 0 0 55 53 35 33 15 12 2 34 43 34 33 12 2 β β α α F F G G G G G F G G G − − ) ( 0 ) ( ) ( 0 0 0 0 55 53 35 34 43 35 34 33 2 55 35 34 15 12 β β α β F F G G F G G G F G G G G − − , = ) , , ( = 4 1 2 4 a α α β
) ( ) ( 0 ) ( 0 0 0 0 0 0 0 0 = 55 53 51 35 33 31 15 12 21 15 13 12 11 2 1 34 43 34 33 31 12 21 13 12 11 2 1 4 β β β α α α α F F F G G G G G F G G G G G F G G G G F G G G a + ) ( 0 ) ( ) ( 0 0 ) ( 0 0 ) ( 0 0 0 0 55 53 51 35 34 43 35 34 33 31 15 13 11 2 1 55 51 35 34 15 12 21 15 12 11 1 β β β α α β β α F F F G G F G G G G G G G F F G G G G F G G G + + , ) ( 0 ) ( 0 0 0 0 0 55 53 35 34 43 35 34 33 15 12 2 β β α F F G G F G G G G G + A a a5= 5(α1,α2,β)=−det ) ( 0 ) ( 0 ) ( 0 0 0 0 0 0 = 55 53 51 35 34 43 35 34 33 31 15 12 21 15 13 12 11 2 1 β β β α α F F F G G F G G G G G G F G G G G − ]. ) )( )[( ( =−α1α2G12G34F55 β G11−F21 G33−F43 −G13G31
On the basis of Assumption 2, we will arrange the coefficients of the characteristic equation (6) as polynomials with respect to the parameter β, expressing only their highest order terms explicitly. In these relations, we use the following notation:
0, > 1 = 2 2 1 1 1 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ − M r E M r d 0, > ) ( = 1 12 34 2 d G G d − + 0, > = 1 1 15 11 1 3 Y r G G d d ∂ ∂ + 0, > = 2 2 35 33 1 4 Y r G G d d ∂ ∂ −
0 > = 1 12 34 5 dG G d , ) ( = 3 34 1 12 21 11 6 d G dG F G d − − , , ) ( = 1 34 43 33 4 12 7 dG F G d G d − − 0, > ) ( = 5 21 11 8 d F G d − 0, > ) ( = 5 43 33 9 d F G d − 0, > ] ) )( [( = 1 12 34 21 11 43 33 13 31 10 dG G F G F G G G d − − − ] ) ( [ = 15 34 43 33 12 13 35 1 1 G G F G G G G Y r G − + ∂ ∂ − [ 12 35( 21 11) 15 31 34] 2 2 G G F G G G G Y r + − ∂ ∂ + )], ( ) ( ) ( [ 12 33 21 11 11 34 43 33 13 31 12 34 1 G G F G G G F G G G G G d − + − + + − 0, > ) 2 ( 1 = 3 9 4 8 3 4 8 9 2 (1) d d d d d d d d d G + + 0, > ) 2 ( 1 = 3 9 4 8 3 4 8 9 2 (2) d d d d d d d d d G + − ) ( ) ( = 13 35 15 33 1 1 31 13 33 11 1 G G G G Y r G G G G d H − ∂ ∂ − − ( 15 31 11 35). 2 2 G G G G Y r − ∂ ∂ +
The coefficients aj(α1,α2,β), j = 1, 2, 3, 4, 5, are given by the relations: ), , ( = ) , , ( 1 2 1 1 1 2 1 α α β dβ f α α a + ), , ( ) ( = ) , , ( 1 2 2 3 1 4 2 2 1 2 2 α α β d dα d α β f α α a − − + ), , ( ) ( = ) , , ( 1 2 5 6 1 7 2 1 2 3 1 2 3 α α β d dα d α Hαα β f α α a + − + + (7) ), , ( ) ( = ) , , ( 1 2 8 1 9 2 1 2 4 1 2 4 α α β dα dα Gαα β f α α a + − + , = ) , , ( 1 2 10 1 2 5 α α β d αα β a
where fj(α1,α2), j = 1, 2, 3, 4, 5, are polynomials with respect to the parameters α1, α2
of order not higher than two.
The following theorem gives sufficient conditions for the existence of a critical triple of model (4).
Theorem 1. If the inequalities , < 11 33 31 13G G G G 4 33 43 1 34 12 11 21 1 3 < < ( ) ) ( d G F d G G G F d d − − are satisfied and
) 2 ( 1 > 3 9 4 8 3 4 8 9 2 d d d d d d d d d G + + ,
then there exist two critical triples (α10,α20,β0) of model (4).
Before the proof of Theorem 1, several lemmas will be introduced.
Lemma 1. Characteristic equation (6) has two pairs of purely imaginary roots with the fifth one being negative if and only if the following relations are satisfied 0, > 0, > 0, > A.1 a1 a2 a4 0, 4 A.2 2 4 2 − a ≥ a (8) , = A.3 a1a2 a3 . = A.4 a1a4 a5 Proof. Denoting the roots of (6) as
0 < , 1 = , i = , = , = , = 1 2 1 3 2 4 2 5 1 ω λ ω λ ω λ ω λ λ i −i i − i −
and comparing (6) with its equivalent form
0, = ) )( )( )( )( (λ−iω1 λ+iω1 λ−iω2 λ+iω2 λ−λ5
we get the assertion of lemma. □
Lemma 2. The equations A.3 and A.4 from (8) can be expressed in the form 0 = ) , , ( ) ( = A.3 2 1 1 2 2 4 1 3 2 1 3 2 1a a d d d α d α β g α α β a − − − + (9) 0, = ) , , ( ) ( = A.4 2 2 1 2 2 1 2 9 1 8 1 5 4 1a a d dα dα Gαα β g α α β a − + − +
where for arbitrary fixed α1 and α2
1,2. = 0, = ) , , ( lim gj 1 22 j β β α α β→∞
Proof. If we set aj(α1,α2,β), j = 1, 2, 3, 4, 5, given by (7) into A.3 and A.4 in (8) and arrange the resulting equations as polynomials with respect to β, we obtain
asser-tion (9).
Consider the system of equations (10)–(11) 0, = 2 4 1 3 2 dα d α d − − (10) 0. = 2 1 2 9 1 8α dα Gαα d + − (11)
We are interested only in positive solutions α1, α2. Therefore, G must be positive.
The first equation implies
, = 4 1 3 2 2 d d d α α − (12)
which together with (11) means
. 0 = 4 1 3 2 1 4 1 3 2 9 1 8 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − + d d d G d d d d dα α α α
So, we obtain the following quadratic equation
. 0 = ) ( 4 8 3 9 2 1 2 9 2 1 3G d d d d d G d d d α + − − α + (13)
The roots are determined by the formula
, 2 4 = 2 (1,2) 1 a ac b b± − − α (14) where 0. > = , = 0, > =d3G b d4d8 d3d9 d2G c d2d9 a − − (15)
The roots of (13) are positive if and only if b<0 and b2− ac4 ≥0. From (15) we see that b<0 when
. > 2 9 3 8 4 d d d d d G − (16)
The inequality b2− ac4 ≥0 is satisfied if and only if
0. ) ( ) ( 2 2 3 9 4 8 3 9 4 8 2 2 2 2G − d d d +d d G+ d d −d d ≥ d
Since G>0, this inequality is satisfied for
, ) , (0, (2)〉∪〈 (1) ∞ ∈ G G G
where . 2 = 2 9 8 4 3 8 4 9 3 (1,2) d d d d d d d d d G + ±
Lemma 3. The system of equations (10)–(11)
a) has two positive solutions (α1(1,2),α2(1,2)) for G∈ G( (1),∞). b) has one positive solution (α1*,α2*) for G=G(1),
c) has no positive solutions for G∈(0,G(2)〉.
Proof. Let G∈(0,G(2)〉∪〈G(1),∞). Then (13) has real roots (2) 1 (1) 1 ,α
α with the
same sign. From (12) we obtain real values (1,2) 2 α . = = 4 (1) 1 3 2 (1) 2 4 (2) 1 3 2 (2) 2 d d d d d d α α α α − ≥ − (17) We have . ) ( = = 2 4 (2) 1 (1) 1 2 3 (2) 1 (1) 1 3 2 2 2 4 (2) 1 3 2 4 (1) 1 3 2 (2) 2 (1) 2 d d d d d d d d d d d α α α α α α α α − ⋅ − − + + (18) Since G d d d a c 3 9 2 (2) 1 (1) 1 α = = α and , = = 3 2 8 4 9 3 (2) 1 (1) 1 d G G d d d d d a b − + − +α α we have . 0 > = = 4 8 2 2 4 3 9 2 2 3 3 2 8 4 9 3 3 2 2 2 (2) 2 (1) 2 d G d d d G d d d d G d G d d d d d d d d − − + + α α (19)
Which means that both values have the same sign. So, consider
4 (2) 1 (1) 1 3 2 4 (2) 1 3 2 4 (1) 1 3 2 (2) 2 (1) 2 ) ( 2 = = d d d d d d d d d α α α α α α + − + − − + . = 2 = 4 2 9 3 8 4 4 3 2 8 4 9 3 3 2 G d G d d d d d d G d G d d d d d d d + − + − −
Both values (1) 2
α , (2)
2
α are positive if and only if . 0 > 2 9 3 8 4d d d d G d − + (20)
Conditions (16) and (20) are satisfied if and only if . | >| 4 8 3 9 2G d d d d d − (21) If , > 2 = 2 9 3 8 4 2 9 8 4 3 8 4 9 3 (1) d d d d d d d d d d d d d d G G≥ + + +
then (21) is satisfied and all four values (1,2) 1,2
α are positive. If 0<G≤G(2), then
. | | = 2 2 2 8 4 9 3 2 9 8 4 3 8 4 9 3 d d d d d d d d d d d d d d G≤ + − − Since , | | = | || | < | | 2 8 4 9 3 2 8 4 9 3 8 4 9 3 2 2 8 4 9 3 d d d d d d d d d d d d d d d d d d d − − + −
(21) is not satisfied and thus either (1) <0
1 α , (2)<0 1 α or (1) <0 2 α , (2) <0 2 α . □
Lemma 4. For any G∈ G( (1),∞) there exists >0
G
β such that for any β >βG
there are two positive solutions (α10(i),α20(i)), i=1,2, of system (9). Proof. Instead of β we will consider
β γ = 1 . Set ⎪⎩ ⎪ ⎨ ⎧ − − ≠ + − − . 0 = for ) ( 0 for ) 1 , , ( ) ( = ) , , ( 2 4 1 3 2 2 1 1 1 2 2 4 1 3 2 2 1 γ α α γ γ α α γ α α γ α α d d d g d d d d Φ ⎪⎩ ⎪ ⎨ ⎧ − + ≠ + − + . 0 = for ) ( 0 for ) 1 , , ( ) ( = ) , , ( 2 1 2 9 1 8 2 1 2 1 2 2 1 2 9 1 8 2 1 γ α α α α γ γ α α γ α α α α γ α α G d d g d G d d Ψ
The functions Φ and Ψ are polynomials in α1, α2 and γ. System (9) is equi-valent to the system
0 = ) , , (α1 α2 γ Φ . 0 = ) , , (α1 α2 γ Ψ (22)
For γ =0 we obtain the system (10)−(11), which has two positive solutions ) , ( () 2 ) ( 1i α i
α for any G∈ G( (1),∞). Let
1
α and α2 satisfy (10)−(11). Consider the Ja-cobian 1 9 2 8 4 3 2 1 2 2 1 1 2 1 2 2 1 1 , , = ,0) , ( ,0) , ( ,0) , ( ,0) , ( α α α α α α α α α α α α α α G d G d d d Ψ Ψ Φ Φ − − − − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ). (2 = ) ( =d4d8−d3d9+G d3α1−d4α2 d4d8−d3d9+G d3α1−d2 The Jacobian is zero only for
. 2 = 2 = 3 2 8 4 9 3 1 a b G d Gd d d d d − + − α
The Jacobian is nonzero at both points ( , ())
2 ) ( 1i α i
α . System (22) has two solutions
for γ sufficiently close to 0. □
Remark 3. For G=G(1) we have one positive solution, because (2) 1 (1) 1 =α α and (2) 2 (1) 2 =α
α . However, the Jacobian is zero and hence the Implicit Function Theorem cannot be used.
Lemma 5. If the inequalities
4 33 43 1 34 12 11 21 1 3 < < ( ) ) ( d G F d G G G F d d − − , (23) 33 11 31 13G <G G G (24)
are satisfied, then a3(α1,α2,β) from (7) is positive for an arbitrary pair (α1,α2) of parameters α1, α2 and sufficiently large parameter .β
Proof. The coefficient a3(α1,α2,β) is given by the relation
), , ( ) ( = ) , , ( 1 2 5 6 1 7 2 1 2 3 1 2 3 α α β d dα d α Hαα β f α α a + − + + where:
), ( = 3 34 1 12 21 11 6 d G dG F G d − − , ) ( = 1 34 43 33 4 12 7 dG F G d G d − − ) ( ) ( = 13 35 15 33 1 1 31 13 33 11 1 Y G G G G r G G G G d H − ∂ ∂ − − ( 15 31 11 35). 2 2 G G G G Y r − ∂ ∂ +
According to (23) d6 is positive, d7 negative and according to (24) H is positive. Therefore, a3(α1,α2,β) is positive for an arbitrary pair of values (α1,α2) of parame-ters α1, α2 and sufficiently large value of parameter .β □ Proof of Theorem 1. Lemma 4 guarantees the existence of two triples
) , , ( () 2 ) ( 1 α β α i i of parameters 1
α , α2 and ,β which satisfy conditions A.3 and A.4 from Lemma 1. The condition ( , (), )>0
2 ) ( 1
1 α i α i β
a is satisfied for sufficiently large .β Lemma 5 guarantees the positiveness of ( , (), )
2 ) ( 1
3 α i α i β
a for sufficiently large β . As 0 > ) , , ( () 2 ) ( 1 1 α i α i β a , ( , (), )>0 2 ) ( 1 3 α i α i β a and ( , (), )>0 2 ) ( 1 5 α i α i β
a then from A.3 and
A.4 the positiveness of ( , (), )
2 ) ( 1 2 α i α i β a and ( , (), ) 2 ) ( 1 4 α i α i β
a follows for sufficiently large β . In this way, the condition A.1 in Lemma 1 is satisfied. It is clear that condi-tion A.2 is also satisfied for sufficiently large .β Therefore, there exist two critical triples (α10(i),α20(i),β0), i=1,2, of model (3). □ Now take an arbitrary critical triple (α10,α20,β0) of model (4), fix β0 and inves-tigate model (4) with respect to parameters α1 and α2, which are connected by
20 10 1
2 =κ(α α ) α
α − + , where κ is a real constant and α1∈(α10 −δ,α10 +δ), where 0
>
δ . After shifting the critical values α10,α20 to the origin by the translation
10 1 1=
~ α α
α − , α~2=α2−α20, denoting ε =α~1 and performing a linear mapping Tx
y = , which transforms the matrix A(α10,α20,β0) into its Jordan form, model (4) takes the form
), , , ( = Jx X x ε κ x& + (25) where x=(x1,x2,x3,x4,x5), x2 = x1, x4= x3, , = , = , = , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 2 1 4 3 5 4 3 2 1 5 4 3 2 1 X X X X X X X X X X J ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ λ λ λ λ λ
and the sign “ ” denotes the complex conjugate.
On the basis of Lemma 1 in [4, pp. 22–23], we can formulate the following lemma.
Lemma 6. If p1ω1+ p2ω2 ≠0 for 0<|p1|+ p| 2|≤5, p1, p2∈Z, then there exists a transformation 5, 4, 3, 2, 1, = ), , , , , , ( =u h u1 u2 u3 u4 j xj j+ j ε κ (26) , = 4 2 } 1 , 0 { 2 4 4 3 3 2 2 1 1 ) , 4 , 3 , 2 , 1 ( 4 3 2 1
∑
− ∈ + + ≥ + + m m m m m m m m m m m m m m m m m j j c u u u u h εwhich transforms model (25) into the form
4 3 1 1 2 2 1 1 1 1 1 1 1= u ( )u u u uuu u& λ +ζ κ ε +η +χ ) , , , , , , ( ) , , , , , , ( * 1 2 3 4 5 1 5 4 3 2 1 0 1 u u u u u ε κ U u u u u u ε κ U + + , 4 3 2 2 2 2 1 2 2 2 2 2 2 = u ( )u uu u uu u& λ +ζ κ ε +η +χ ) , , , , , , ( ) , , , , , , ( * 1 2 3 4 5 2 5 4 3 2 1 0 2 u u u u u ε κ U u u u u u ε κ U + + , 2 1 3 3 4 2 3 3 3 3 3 3 3 = u ( )u u u uuu u& λ +ζ κ ε +η +χ (27) ) , , , , , , ( ) , , , , , , ( * 1 2 3 4 5 3 5 4 3 2 1 0 3 u u u u u ε κ U u u u u u ε κ U + + , 2 1 4 4 2 4 3 4 4 4 4 4 4= u ( )u u u u uu u& λ +ζ κ ε+η +χ ) , , , , , , ( ) , , , , , , ( * 1 2 3 4 5 4 5 4 3 2 1 0 4 u u u u u ε κ U u u u u u ε κ U + + , , ) , , , , , , ( ) , , , , , , ( = * 1 2 3 4 5 5 5 4 3 2 1 0 5 5 5 5 λu U u u u u u ε κ U u u u u u ε κ u& + +
where λj are the eigenvalues of J, u2 =u1, u4 = u3, U0(u1,u2,u3,u4,0,ε,κ)=0
j , = ) , , , , , , ( 1 2 3 4 5 * εu εu εu εu εu ε κ Uj ( ε)5U~j(u1,u2,u3,u4,u5,ε,κ), j = 1, 2, 3, 4, 5. In polar coordinates , = , = , = , = , = 2 5 2 4 2 2 3 1 1 2 1 1 1 e u e u e u e u v u ρ iϑ ρ −iϑ ρ iϑ ρ −iϑ
) , , , , , ( ) ( = 0 1 1 2 2 1 1 2 2 12 2 1 11 1 1 ρ ρ ρ ε ρ ϑ ρ ϑ ε ρ& a +a +q +R v *( 1, 1, 2, 2, , ), 1 ρ ϑ ρ ϑ v ε R + ) , , , , , ( ) ( = 0 1 1 2 2 2 2 2 2 22 2 1 21 2 2 ρ ρ ρ ε ρ ϑ ρ ϑ ε ρ& a +a +q +R v *( 1, 1, 2, 2, , ), 2 ρ ϑ ρ ϑ vε R + = 1 [ 10( 1, 1, 2, 2, , ) 1 1 2 2 12 2 1 11 1 1 ω ρ ρ ε ρ ρ ϑ ρ ϑ ε ϑ& +b +b +s + Θ v *( 1, 1, 2, 2, , )], 1 ρ ϑ ρ ϑ v ε Θ + (28) ) , , , , , ( [ 1 = 0 1 1 2 2 2 2 2 2 2 22 2 1 21 2 2 ω ρ ρ ε ρ ρ ϑ ρ ϑ ε ϑ& +b +b +s + Θ v *( 1, 1, 2, 2, , )], 2 ρ ϑ ρ ϑ v ε Θ + ) , , , , , ( =λ5v V0 ρ1ϑ1 ρ2 ϑ2 v ε v& + +V*(ρ1,ϑ1,ρ2,ϑ2,v,ε), where a11=Reη1, a12=Reχ1, a21=Reχ3, a22=Reη3, q1=Reζ1, q2 =Reζ3, 1 11=Imη b , b12=Imχ1, b21=Imχ3, b22=Imη3, s1=Imζ1, s2=Imζ3 and the
functions with superscripts “0” and “*” have analogous properties to the functions with such superscripts in model (27).
The equation , 0 = 2 ε q ρ + B (29) where ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 2 1 2 2 2 1 2 22 21 12 11 , = , = = q q q a a a a ρ ρ ρ B
is the bifurcation equation of model (28). Suppose that detB≠0. Denote the solution of (29) as ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 2 1 2 = w w ε ρ
and consider the matrix
. | | | | = 2 22 2 1 21 2 1 12 1 11 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ w a w w a w w a w a P
From Theorem 1 in [4, p. 25], we can formulate the following Theorem.
Theorem 2. Suppose that the coordinates w and 1 w have the same sign. If the2 eigenvalues of matrix P are not purely imaginary, then model (28) has an invariant torus for every ε∈(0,ε0), – if w1>0, and for every ε∈(−ε0,0), if w1<0, where ε0 is sufficiently small. The torus is given by the equations
) , , ( | | = 3 1 1 2 1 1 ε ε ϑ ϑ ε ρ w + f , ) , , ( | | = 2 1 2 3 2 2 ε ε ϑ ϑ ε ρ w + f , ) , , ( | | = ε5 g ϑ1 ϑ2 ε v
where f , 1 f and g are continuous functions with respect to 2 ϑ1, ϑ2, and ε for ar-bitrary ϑ1, ϑ2 and ε∈(0,ε0), resp. ε∈(−ε0,0), and 2π-periodic in ϑ1, ϑ2.
References
[1] ASADA T., A Two-regional Model of Business Cycles with Fixed Exchange Rates: A Kaldorian
Ap-proach, Discussion Paper Series No. 44, Chuo University, Tokyo, Japan 2003.
[2] ASADA T., INABA T., MISAWA T., An Interregional Dynamic Model: The Case of Fixed Exchange
Rates, Studies in Regional Science, 2001, 31–2, 29−41.
[3] BIBIKOV Yu.N., Multi-frequency non-linear oscillations and their biffurcations, The Publishing House of The Saint Petersburg University, Saint Petersburg 1991 (in Russian).
[4] GANDOLFO G., Economic Dynamics, Springer-Verlag, Berlin 1997.
Czy torusy mogą się pojawiać w modelu dwuregionalnym przy ustalonej stopie wymiany?
W artykule omówiono dwuregionalny model wprowadzony przez T. Asada [1], opisujący dynamicz-ne wzajemdynamicz-ne oddziaływanie dwóch regionów połączonych poprzez międzyregionalny handel i przepływ kapitału. Model pokazuje rozwój dochodu, akcji kapitałowych i akcji pieniężnych w rozważanych regio-nach. T. Asada [1] dokonał analizy istnienia punktu równowagi dla modelu, znalazł warunki wystarczają-ce dla jego lokalnej stabilności, a także przeanalizował problem istnienia cykli biznesowych wokół punktu równowagi. Ponieważ badanym modelem jest pięciowymiarowy dynamiczny system, problem istnienia torusa wokół punktu równowagi jest uzasadniona. Artykuł daje odpowiedź na to pytanie. Torusy mogą pojawiać się tylko w przypadku, kiedy macierz aproksymacji liniowej modelu w punkcie równowa-gi ma dwie pary czysto urojonych wartości własnych. Twierdzenie 1 daje wystarczające warunki istnienia dwóch par czysto urojonych wartości własnych z pozostałą jedną wartością ujemną. Twierdzenie 2 sta-nowi komentarz istnienia torusów w bliskim sąsiedztwie punktu równowagi. Model rozważany w arty-kule może być zastosowany do analizy dynamicznego wzajemnego oddziaływania dwóch krajów w ob-szarze strefy Euro.
Słowa kluczowe: model dynamiczny, równowaga, macierz przybliżenia liniowego, wartości własne macierzy,
