A Dynamic Model of a Small Open Economy Under Flexible Exchange Rates

A Dynamic Model of a Small Open Economy Under Flexible Exchange Rates

Petra Medveďová

Department of Quantitative Methods and Information Systems Faculty of Economics

Matej Bel University

Tajovského 10, 975 90 Banská Bystrica, Slovakia petra.medvedova@umb.sk

Abstract: In the paper a three dimensional dynamic model of a small open economy, describing the development of net real national income, real physical capital stock and the expected exchange rate of the near future, which was introduced by T. Asada in [1], is analysed under flexible exchange rates. We study the question of the existence of business cycles. Sufficient conditions for the existence of a pair of purely imaginary eigenvalues with the third one negative in the linear approximation matrix of the model are found. For the existence of business cycles and their properties the structure of the bifurcation equation of the model is very important. Formulae for the calculation of the bifurcation coefficients in the bifurcation equation of the model are derived. Theorem on the existence of business cycles in a small neighbourhood of the equilibrium point is presented.

Keywords: dynamical model; matrix of linear approximation; eigenvalues; bifurcation equation; business cycle

1 Introduction

Toichiro Asada formulated in [1] a Kaldorian business cycle model in a small open economy. He studied both the system of fixed exchange rates and that of flexible exchange rates with the possibility of capital mobility. In this article we investigate Asada’s model which was introduced in [1] under flexible exchange rates. In this case Asada’s model has the form

(

^{+ + + −}

)

^{, >0,}

=α C I G J Y α

Y , I

K = (1)

(

⁻

)

^{, >0,}

=γ π π γ

π^{e e}

where

(

−

)

+ 0,0< <1, 0 >0

,

=cY T C c C

C

, 0 , 1 0

,

₀

0

< < >

−

= Y T T

T τ τ

(

^{, ,}

)

^{, 0, 0, <0}

^,

∂

= ∂

∂ <

= ∂

∂ >

= ∂

=

r I Ir K

I IK Y

I IY r K Y I I

( ) , , 0 , < 0 ,

∂

= ∂

∂ >

= ∂

= r

L L Y L L r Y p L M

r

Y (2)

( ) ^{, , 0 , > 0 ,}

∂

= ∂

∂ <

= ∂

= π J

_π

π ^J

Y J J Y J

J

_Y

,

0 , >

− −

−

=

⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛

β

π π β π

e rf r Q

, Q J A = +

,

= 0 A

=

M

constant,

( ^{M = 0} ) ^.

The meanings of the symbols in (1) and (2) are as follows:

Y

- net real national income,

C

- real private consumption expenditure,

I

- net real private investment expenditure on physical capital,

G

- real government expenditure (fixed),

T

- real income tax,

K

- real physical capital stock,

M

- nominal money stock,

p

- price level,

r

- nominal rate of interest of domestic country, rf - nominal rate of interest of foreign country,

π

- exchange rate,

π

^e - expected exchange rate of near future,

J

- balance of current account (net export) in real terms,

Q

- balance of capital account in real terms,

A

- total balance of payments in real terms,

α

- adjustment speed in goods market,

β

- degree of capital mobility,

α , β , γ

- positive parameters, and the meanings of other symbols are as follows,

,

dt

Y = dY , dt

K = dK _,

dt d

^e

e

π

π = t

- time.

In the whole paper we assume as well as Asada in [1] a fixed-price economy, so that

p

is exogenously given and normalized to the value 1. Asada assumed that the equilibrium on the money market

M = L ( ) Y , r

is always preserved, which enables using the Implicit-function theorem to express interest rate

r

as the function of

Y

, so

( ) Y , r

r = > 0 .

∂

= ∂ Y r

_Y

r

Solving equation

( ) , ( ) ⎥ = 0

⎦

⎢ ⎤

⎣

⎡ −

−

− +

= π

π β π

π r Y r

_f ^e

Y

J

A

with respect

to

π

, we have

( ) Y , π

^e

, π

π = ,

π

2

βπ β π π

π e Y Y Y

J

r J

Y +

−

= −

∂

= ∂ > 0 .

+

∂ =

= ∂

π π βπ

β π

π π

π

π ^{e e}

J

e

Further we suppose that r_f is also given exogenously because of the assumption of a small open economy. Under these assumptions, taking into account (2) and the explicit expression for

r

, the model (1) takes the form

( ) ( ( ) ) ( ( ) )

[

^{c Y cT C G I Y K r Y J Y Y Y}

]

Y=

α

1−

τ

+ ₀+ ₀+ + , , + ,

π

,

π

^e −

( ) ( Y K r Y )

I

K = , ,

(3)

( )

[ ^,

^{e e}

] ^.

e

γ π Y π π

π = −

In the whole paper we suppose that:

(i) the model (3) has a unique equilibrium point

(

^∗

^,

^∗

^,

^∗

) ^,

∗

= Y K

^e

E π Y

^∗

> 0 , K

^∗

> 0 , π

^e∗

> 0 ,

to an arbitrary triple of positive parameters

( α , β , γ )

.

(ii)

0 < I

_Y

+ I

_r

r

_Y

< 1 − c ( 1 − τ ) − J

_Y at the equilibrium point.

(iii)

π

πe

− 1 < 0

at the equilibrium point.

(iv) The functions in the model (3) have the following properties: the function

I

is linear in the variable

K

and

r

. The function

π

is linear in the variable

π

^e. The function

J

is nonlinear in the variable

π

. In the variable

Y

the functions

I , J , r , π

are nonlinear, and have continuous partial derivatives with respect to

Y

up to the sixth order in a small neighbourhood of the equilibrium point.

In [1] Asada found sufficient conditions for local stability and instability of the equilibrium point. He studied how changes of the parameter

β

affect the dynamic characteristics of the model.

We analyse the question of the existence of business cycles analytically. Stable business cycles can arise only in the case when the linear approximation matrix of the model (3) has at the equilibrium point a pair of purely imaginary eigenvalues with the third one negative. In Section 2, Theorem 1 gives sufficient conditions for the existence of a pair of purely imaginary eigenvalues with the third one negative.

The bifurcation equation of the model (3) is very important for the existence of business cycles. In Section 3, Theorem 2 gives the formulae for the calculation of the bifurcation coefficients in the bifurcation equation. Theorem 3 speaks about the existence of business cycles in a small neighbourhood of the equilibrium point.

Such an analytical approach was applied to study similar models in [6], [7], [8], [9], [10].

2 The Analysis of the Model (3)

Consider an isolated equilibrium poin

^E

^∗

⁼ ( ^Y

^∗

^{, K}

^∗

^{, π}

^e∗

) ^{, Y}

^∗

^{> 0 , K}

^∗

^{> 0 ,}

,

> 0

e∗

π

of the model (3).

After the transformation

, ,

,

_{1 1}

1

−

∗

=

−

=

−

= Y Y

^∗

K K K

^{∗ e e e}

Y π π π

the equilibrium point

^E

^∗

⁼ ( ^Y

^∗

^{, K}

^∗

^{, π}

^e∗

)

goes into the origin

E

₁^∗

= ( 0 , 0 , 0 ) ,

and the model (3) becomes

( ) ( ) ( ( ) )

[

^{− + + + + + + +}

= ₁ ^∗ ₁ ^∗ ₁ ^∗ ₁ ^∗ ₀

1 c1 Y Y I Y Y ,K K ,rY Y cT

Y

α τ

^{+ C}

⁰

^{+ G + J} ( ^Y

¹

^{+ Y}

^∗

^{, π} ( ^Y

¹

^{+ Y}

^∗

^{, π}

^1e

^{+ π}

^e∗

) ) ^{− ( Y}

¹

^{+ Y}

^∗

⁾ ]

( )

( ⁺

^∗

⁺

^∗

⁺

^∗

)

= I Y Y K K r Y Y

K

_{1 1}

,

₁

,

₁ (4)

( ) ( )

[

1

^,

1 1

] ^.

1

∗

∗

− +

+ +

=

^{∗ e e e e}

e

γ π Y Y π π π π

π

Performing the Taylor expansion of the functions on the right-hand side of this system at the equilibrium point

E

₁^∗

= ( 0 , 0 , 0 )

the model (4) obtains the form

( )

[ − + + + + +

=

_{1 1 1 1 1}

1

c 1 Y I Y I K I r Y J Y

Y α τ

_{Y K r Y Y}

1 1 1

]

1

Y ~ Y J

Y

J

_Y

+

e ^e

− + +

_π

π

_π

π

_π

π

1 1 1

1 1

K~ Y r I K I Y I

K = _Y + _K + _{r Y} + (5)

[

1 1 1

] ^~

1

^,

1 e e e

Y e

Y π

e

π π π π

γ

π = +

_π

− +

where

( ) _,

Y E I

_Y

I

∂

= ∂

^∗

( ) _,

K E I

_K

I

∂

= ∂

^∗

( ) _,

r E I

_r

I

∂

= ∂

^∗

( ) _,

Y E r

_Y

r

∂

= ∂

^∗

( ) _,

Y E J

_Y

J

∂

= ∂

^∗

( ) _,

π

∂ π

= ∂ J E

^∗

J ( ) _,

Y E

Y

∂

= ∂ π

^∗

π ( ) _,

e

E

e

π

π

_π

π

∂

= ∂

^∗ and the functions Y~₁,K~₁,

π

~₁^e are nonlinear parts of the Taylor expansion.

The linear approximation matrix

A = A ( α , β , γ )

of the system (5) has the following form

( )

[ ]

( ¹ ) ^.

0

0 1

1

⎟ ⎟

⎟

⎠

⎞

⎜ ⎜

⎜

⎝

⎛

− +

− +

+ + +

−

=

e e

Y

K Y

r Y

K Y

Y Y r Y

I r

I I

J I

J J r I I c

π π π π

π γ γπ

π α α

π τ

α

A (6)

The characteristic equation of

A ( α , β , γ )

is given by ,

3 0 2 2

3 +b1λ +b λ+b =

λ (7) where

−

1

=

b

trace

A ( α , β , γ ) =

= − { α [ c ( 1 − τ ) + I

Y

+ I

r

r

Y

+ J

Y

+ J

_π

π

Y

− 1 ] + I

K

+ γ ( π

_πe

− 1 ) }

2

=

b

sum of all principal second – order minors of

A ( α , β , γ ) =

⁼ α I

K

[ c ( ^{1 −} τ ) ⁺ J

Y

⁺ J

_π

π

Y

^{− 1} ] ⁺ γ I

K

( π

_π^e

^{− 1} ) ⁺

+ αγ { [ c ( 1 − τ ) + I

Y

+ I

r

r

Y

+ J

Y

− 1 ] ( π

_πe

− 1 ) − J

_π

π

Y

}

−

3

=

b

det

A ( α , β , γ ) =

      

= − αγ I

K

{ ( π

_πe

− ¹ ) [ c ( ¹ − τ ) + J

Y

− ¹ ] − J

_π

π

Y

} ^.

 

As we are interested in the existence and stability of limit cycles we need to find such values of parameters

α , β , γ

at which the equation (7) has a pair of purely imaginary eigenvalues and the third one is negative. We will call such values of parameters

α , β , γ

the critical values of the model (3). We denote these critical values by

α

₀

, β

₀

, γ

₀

.

The mentioned types of eigenvalues are ensured by the Liu’s conditions [5].

. 0 ,

0 ,

0

_{3 1 2 3}

1

> b > b b − b =

b

(8) For an arbitrary

α

exists

γ

such that the first inequality is satisfied under the condition (iii). The second inequality is satisfied under the condition

( π

_πe

− 1 ) [ c ( 1 − τ ) + J

_Y

− 1 ] > J

_π

π

_Y

.

This condition is satisfied when

β

is sufficiently small.

The equation

b

₁

b

₂

− b

₃

= 0

gives

( )

[ ] ( )

{ α c ^{1 −} τ ⁺ I

Y

⁺ I

r

r

Y

⁺ J

Y

⁺ J

_π

π

Y

^{− 1 +} I

K

⁺ γ π

_π^e

^{− 1} } ^.

( )

{ − 1 + [ ( 1 − ) + + − 1 ] + . γ I

_K

π

_πe

α I

_K

c τ J

_Y

J

_π

π

_Y

( )

( ) ( )

[ ^{− + + + − − −} ] } ⁻

+ αγ c 1 τ I

_Y

I

_r

r

_Y

J

_Y

1 π

_πe

1 J

_π

π

_Y

( ) ^{[ ( ) ]}

{

^{−1 1− + −1 −}

}

^=0.

−

αγ

I_K

π

_πe c

τ

J_Y J_π

π

_Y

Asada indicated in [1] that

b

₁

b

₂

− b

₃

> 0

for

π

_Y

< 0

and for

π

_Y

= 0 .

The equation b1b2 −b3 =0is satisfied under the condition

π

_Y

> 0 .

This inequality is satisfied if

.

Y Y

r

− J β <

The equation

b

₁

b

₂

− b

₃

= 0

can be expressed in the form

( , )

² ₂

( , )

₃

( , ) 0 ,

1

α β γ + f α β γ + f α β =

f

where

( ) J [ I

K

( J

Y

R ) r

Y

]

f

₁

α , β =

_π²

π

²

− α − − αβ

( ) = TJ r

_Y

+ J ( J

_Y

− R )( T − J r

_Y

) − f

₂

α , β α

²

β

² _π

π α

²

β

_π

π

_π

π

⁻

α

²J_π²

π

²

(

R⁻J_Y

)

²⁻2

αβ

I_KJ_π

π

T⁺2

α

I_KJ_π²

π

²

(

J_Y ⁻R

)

⁻

−

β

I_K²J_π

π

−I_K²J_π²

π

²

( ) = I

_K

TU + I

_K

J [ T ( P − J

_Y

) ( + U R − J

_Y

) ] + f

3

α , β α

²

β

²

α

²

β

_π

π

^{+ I}

K

^J ( ^{R − J}

Y

)( ^{P − J}

Y

) ^{+ I}

K

^{U +}

2 2 2

2

2

π αβ

α

_π

+ αβ I

K²

J

_π

π ( P − J

Y

+ U ) + α I

K²

J

_π²

π

²

( P − J

Y

)

where

( 1 − ) + − 1 < 0 ,

= c J

_Y

P τ

,

< 0 + +

= P I

_Y

I

_r

r

_Y

R

,

< 0

−

= R J r

_Y

T

_π

π

.

<0

−

=P J rY

U ππ

We see that for an arbitrary

γ

exists

α ˆ

such that

f

₁

( ) α ˆ , 0 = 0 , α ˆ > 0 .

Put

( ) ( ) ( ) ( ) 1 0 .

1 , , ,

,

, =

₁

+

₂

+

_{3 2}

=

β γ γ α

β α β

α γ

β

α f f f

F

Instead of

γ

introduce

ϑ = γ ¹

and consider an equation

Φ ( α , β , ϑ ) = 0

when

( ) ( )

( ) ( ) ( )

⎩ ⎨

⎧

≠ +

+

= =

Φ , , , , 0 .

0 ,

, ,

,

₂

3 2

1

1

ϑ ϑ β α ϑ β α β

α

ϑ β

ϑ α β

α f f f

f

We see that

Φ ( α , β , ϑ ) = 0

is equivalent to

F ( α , β , γ ) = 0 .

Analyze

( , , ) = 0 . Φ α β ϑ

It holds:

1.

Φ ( α ˆ , β = 0 , ϑ = 0 ) = 0

2.

( ^{ˆ , 0 , 0} ) ^{= −}

^{2 2}

( ⁻ ) ^{≠ 0 .}

∂

=

= Φ

∂ J π J

_Y

R

α ϑ β

α

π

By the implicit function theorem there exists a function

α = f ( β , ϑ )

in a small neighbourhood of

( β = 0 , ϑ = 0 )

such that

α ˆ = f ( ) 0 , 0

and

( )

( ^{, , ,} ) ^{= 0 .}

Φ f β ϑ β ϑ

We see that for a sufficiently large

γ

₀ of parameter

γ

and sufficiently small

β

₀

of parameter

β

there exists value

α

₀ of parameter

α

such that the triple is

( α

₀

, β

₀

, γ

₀

)

the critical triple of the model (3). The following theorem gives sufficient conditions for the existence of a critical triple of the model (3).

Theorem 1. Let the condition

π

πe

− 1 < 0

is satisfied. If parameter

Y Y

r

− J β <

is sufficiently small and parameter

γ

is sufficiently large, then there exists a critical triple

( α

₀

, β

₀

, γ

₀

)

of model (5).

3 Existence of Limit Cycles and their Stability

According to the assumption (iv) the model (5) can be itemized in the form

( )

( )

[ ]

{ ^{+ − − − − + + +} } ⁺

= I

_Y

I

_r

r

_Y

c J

_Y

J

_Y

Y I

_K

K J

e ^e

Y

₁

α 1 1 τ

_π

π

_{1 1 π}

π

_π

π

₁

⁺ (

^{( )}

⁺

^{( )}

) ⁺

^{( )}

( ) ⁺ (

^{( )}

⁺ )

1³

⁺

3 2 3

1 2 2

1 2 2

6 1 2

1 2

1 α I

_Y

J

_Y

Y α J

_πe

π

^e

α I

_Y

J

_Y

Y

^{( )}

( ) (

Y^{( )} Y^{( )}

)

^{( )}

( )

^e

(

^e

)

e

I J Y J O Y

J

e e 5 1 1

4 1 4 4

1 4 3 4

1

3

,

24 1 24

1 6

1 α

_π

π + α + + α

_π

π + π

+

( ⁺ ) ^{+ +}

^{( )}

⁺

^{( )}

⁺

^{( )}

⁺

=

_{1 1} ² ₁^{2 3} ₁^{3 4} ₁⁴

1

24

1 6

1 2

1 I Y I Y I Y

K I Y r I I

K

_{Y r Y K Y Y Y}

+ O

₅

( ) Y

₁ (9)

( ⁻ ) ⁺

^{( )}

⁺

^{( )}

⁺

^{( )}

⁺

+

=

_{1 1} ² ₁^{2 3} ₁^{3 4} ₁⁴

1

24

1 6

1 2

1 1 Y Y Y

Y

^e _{Y Y Y}

Y e

e

π γπ γπ γπ

π γ γπ

π

_π

+ O

₅

( ) Y

₁

,

where

( )

( ) _,

2 2 2

Y E I

_Y

I

∂

= ∂

^{∗ ( )}

( ) _,

3 3 3

Y E I

_Y

I

∂

= ∂

^{∗ ( )}

( ) _,

4 4 4

Y E I

_Y

I

∂

= ∂

^{∗ ( )}

( ) _,

2 2 2

Y E J

_Y

J

∂

= ∂

^∗

( )

( ) _,

3 3 3

Y E J

_Y

J

∂

= ∂

^{∗ ( )}

( ) _,

4 4 4

Y E J

_Y

J

∂

= ∂

^{∗ ( )}

( )

( )

²

^,

2 2

e

E J

e

J

π

∂ π

= ∂

^{∗ ( )}

( ) ( )

³

^,

3 3

e

E J

e

J

π

∂ π

= ∂

^∗

( )

( )

( )

⁴

^,

4 4

e

E J

e

J

π

∂ π

= ∂

^{∗ ( )}

( ) _,

2 2 2

Y E

Y

∂

= ∂ π

^∗

π

^{( )}

( ) _,

3 3 3

Y E

Y

∂

= ∂ π

^∗

π

^{( )}

( ) _.

4 4 4

Y E

Y

∂

= ∂ π

^∗

π

Consider a critical triple

( α

₀

, β

₀

, γ

₀

)

of the model (3). Let us investigate the behavior of

Y

₁

, K

₁ and

π

₁^e around the equilibrium

E

₁^∗

= ( 0 , 0 , 0 )

with respect to the parameter

α , α ∈ ( α

₀

− ε , α

₀

+ ε ) , ε > 0 ,

and the fixed parameters

0

, β

β = γ = γ

₀

.

After the shifting

α

₀ into the origin by relation

α

₁

= α − α

₀

,

the model (9) becomes

( )

( )

[ ^{+ − − − − +} ] ^{+ +}

=

_{0 1 0 1}

1

I I r 1 c 1 J J Y I K

Y

α

_{Y r Y}

τ

_{Y π}

π

_Y

α

_K

⁺ α

0

J

_π

π

_πe

π

1^e

⁺ [ I

_Y

⁺ I

_r

r

_Y

⁻ ( 1 ⁻ c ( 1 ⁻ τ ) ⁻ J

_Y

) ⁺ J

_π

π

_Y

] Y

1

α

1

⁺

^{+ + +} (

^{( )}

⁺

^{( )}

) ⁺

^{( )}

( )

1 ²

⁺

2 0 2 1 2 2 0 1 1 1

1

2

1 2

1

e

Y Y e

K

K J

^e

I J Y J

^e

I α

_π

π

_π

π α α α

_π

π

⁺ (

^{( )}

⁺

^{( )}

) ⁺

^{( )}

( ) ⁺ (

^{( )}3

⁺

^{( )}3

)

1³

⁺

0

1 2 1 2 1

2 1 2 2

6 1 2

1 2

1 I

_Y

J

_Y

Y α J

_πe

π

^e

α α I

_Y

J

_Y

Y

⁺

^{( )}

( ) ⁺ (

^{( )}

⁺

^{( )}

) ⁺

^{( )}

( )

1

⁺

3 1 3 1

3 1 3 3 3

1 3

0

6

1 6

1 6

1 α J

_πe

π

^e

I

_Y

J

_Y

Y α J

_πe

π

^e

α

(10)

(

^{( ) ( )}

)

^{( )}

( )

5

(

1 1 1

)

4 1 4 0 4

1 4 4

0

, ,

24 1 24

1 α I

_Y

+ J

_Y

Y + α J

_πe

π

^e

+ O Y π

^e

α +

( )

_{1 1} ^{( )2} ₁^{2 ( )3} ₁^{3 ( )4} ₁⁴ ₅

( )

₁

1

24

1 6

1 2

1 I Y I Y I Y O Y

K I Y r I I

K =

_Y

+

_{r Y}

+

_K

+

_Y

+

_Y

+

_Y

+

( ⁻ ) ⁺

^{( )}

⁺

^{( )}

⁺

^{( )}

⁺

+

=

_{0 1 0 1 0} ² ₁² ₀ ³ ₁³ ₀ ⁴ ₁⁴

1

24

1 6

1 2

1 1 Y Y Y

Y

^e _{Y Y Y}

Y e

e

π γ π γ π γ π

π γ π γ

π

_π

+ O

₅

( ) Y

₁

.

Denote the eigenvalues of (6) as

( , , ) ( , , ) ,

₂

( , , ) ( , , ) ,

1

ξ α β γ ω α β γ λ ξ α β γ ω α β γ

λ = + i = − i

( α β γ )

λ λ

₃

=

₃

, ,

and let

( , , ) , ( , , ) .

,

,

_{2 0 0 0 0 0 30 3 0 0 0}

0

1

ω λ ω ω ω α β γ λ λ α β γ

λ = i = − i = =

Express the model (10) in the form

( ) ~ ( , ) ,

,

,

_{0 0 1}

0

β γ α

α x Y x

A

x = +

where

,

1 1 1

⎟ ⎟

⎟

⎠

⎞

⎜ ⎜

⎜

⎝

⎛

=

e

K Y π

x

.

~

~

~

~

3 2 1

⎟⎟

⎟ ⎟

⎠

⎞

⎜⎜

⎜ ⎜

⎝

⎛

= Y Y Y Y

Consider a matrix M

= ( ) m

_ij

, i , j = 1 , 2 , 3 ,

which transfers the matrix

( α

₀

, β

₀

, γ

₀

)

A

into its Jordan form J

,

and its inverse matrix

M

⁻¹

⁼ ( ) m

ij⁻¹

^.

By the transformation x

=

My

,

⎟ ⎟

⎟

⎠

⎞

⎜ ⎜

⎜

⎝

⎛

=

e

K Y

2 2 2

π

y we obtain

( y , α

₁

) , F Jy

y = +

(11) Where

, 0

0

0 0

0 0

30 0 0

⎟ ⎟

⎟

⎠

⎞

⎜ ⎜

⎜

⎝

⎛

−

=

λ ω ω

i i

J

( ) ( )

( )

( ) ^{~ ~ ~ ,}

~

~

~

~

~

~

, , , ,

3 1 33 2 1 32 1 1 31

3 1 23 2 1 22 1 1 21

3 1 13 2 1 12 1 1 11

1 3

1 2

1 1

1

⎟⎟

⎟ ⎟

⎠

⎞

⎜⎜

⎜ ⎜

⎝

⎛

+ +

+ +

+ +

⎟ =

⎟ ⎟

⎠

⎞

⎜ ⎜

⎜

⎝

⎛

=

−

−

−

−

−

−

−

−

−

Y m Y m Y m

Y m Y m Y m

Y m Y m Y m

F F F

α α α α

y y y y

F

2

,

2

Y

K = F

₂

= F

_1, and

F

₃ is real function (the symbol

"− "

means complex conjugate expression).

Theorem 2. There exists a polynomial transformation

(

_{3 3 1}

)

1 3

2

Y h Y , K , α

Y = +

(

_{3 3 1}

)

2 3

2

K h Y , K , α

K = +

(12)

(

_{3 3 1}

)

3 3

2

π , , α

π

^e

=

^e

+ h Y K

,

where

h

_j

( Y

₃

, K

₃

, α

₁

) , j = 1 , 2 , 3 ,

are nonlinear polynomials with constant coefficients of the kind

( )

^{( )}

{ }

,

,

,

³

3 3 2 1

3 2 1 3 2 1

2 4

1 , 0 , 2

1 3 3 , , 1

3

3

∑

⁻

∈

≥ + +

=

^m

m m m m

m m m m

m m j

j

Y K h Y K

h α α j = 1 , 2 , 3 , h

₂

= h

₁

,

which transforms the model

(

2 2 2 1

)

1 2 0

2

i ω Y F Y , K , π

^e

, α Y = +

(

2 2 2 1

)

2 2 0

2

i ω K F Y , K , π

^e

, α

K = − +

(11)

(

2 2 2 1

)

3 2 30

2

λ π , , π , α

π

^e

=

^e

+ F Y K

^e

into its partial normal form on a center manifold

(

3 3 3 1

) (

3 3 3 1

)

3 2 3 2 1 3 1 3 0

3

i ω Y δ Y α δ Y K U Y , K , π

^e

, α U Y , K , π

^e

, α

Y = + + +

^D

+

^∗

( ) ⁺

+ +

+

−

=

_{0 3 1 3 1 2 3 3}² _{3 3 3 1}

3

i ω K δ K α δ Y K U Y , K , π

^e

, α

K

^D

+ U

^∗

( Y

3

, K

3

, π

3^e

, α

1

)

(13)

(

3

^,

3

^,

3

^,

1

) (

3

^,

3

^,

3

^,

1

) ^,

3 30

3

λ π π α π α

π

^e

=

^e

+ V

^D

Y K

^e

+ V

^∗

Y K

^e where

U

^D

( Y

₃

, K

₃

, 0 , α

₁

) = V

^D

( Y

₃

, K

₃

, 0 , α

₁

) = 0 ,

( ^{, , ,} ) ( ) ^~ (

3

^,

3

^,

3

^,

1

) ^,

5 1 1

3 1 3 1 3

1

α α π α α π α

α Y K

^e

U Y K

^e

U

^∗

=

( ^{, , ,} ) ( ) ^~ (

3

^,

3

^,

3

^,

1

) ^,

5 1 1

3 1 3 1 3

1

α α π α α π α

α Y K

^e

V Y K

^e

V

^∗

=

and

U ~ , V ~

are continuous functions.

The resonant coefficients

δ

₁ and

δ

₂ in the model (13) are determined by the formulae

( )

[ ]

{

11 21 31

}

1 11

1

m I

Y

I

r

r

Y

c 1 τ 1 J

Y

J

π

π

Y

m I

k

m J

π

π

πe

m

δ =

⁻

+ + − − + + + +

( ) ( )

( )

^{( )}

⎢⎣ ⎡ + + +

=

₁₁⁻¹ ₀ ^{3 3} ₁₁² _{12 0} ³ ₃₁² ₃₂

2

2

1 2

1 I J m m J m m

m α

Y Y

α

πe

δ

⁺ ( ^I

Y^{( )}

^{+ J}

Y^{( )}

) ^{A + J}

^{( )e}

^B ] ⁺

2 0 2

2

0

α

π

α

^{( ) ( ) ( ) ( )}

,

2 1 2

1

₂

0 12 2 11 3 0 1 13 2

12 2 11 3 1

12

⎥⎦ ⎤

⎢⎣ ⎡ +

⎥⎦ +

⎢⎣ ⎤

⎡ +

+ m

⁻

I

_Y

m m I

_Y

A m

⁻

γ π

_Y

m m γ π

_Y

A

where