ON BAUTIN BIFURCATION IN A MONETARY DYNAMIC
MODEL
KATARÍNA KYSEL’OVÁ
Technical University in Zvolen, Faculty of Wood Sciences and Technology, Department of Mathematics and Descriptive Geometry,
T. G. Masaryka 24, Zvolen, Slovakia email: katarina.kyselova@tuzvo.sk
RUDOLF ZIMKA
Matej Bel University, Faculty of Economics,
Department of Quantitative Methods and Information Systems, Tajovského 10, Banská Bystrica, Slovakia
email: rudolf.zimka@umb.sk Abstract
In the paper a two-dimensional dynamic model of Asada (2011) describing the development of nominal rate of interest and expected rate of inflation is investigated. There are found conditions under which Bautin bifurcation (generalized Hopf bifurcation) arises. This kind of bifurcation enables the existence of so called corridor stability, when around the stable equilibrium two cycles emerge, the inner one which is unstable and the outer one which is stable. Numerical simulations of the model illustrating the achieved results are presented.
Key words: dynamic model, equilibrium, stability, limit cycles JEL Codes: E 30, E 32, E 52
DOI: 10.15611/amse.2017.20.21 1. Introduction
In the paper we are going to deal with the T. Asada’s model of central banking Y = Y (r − πe, G, τ),Yr−πe= ∂Y ∂ (r − πe)<0,YG= ∂Y ∂ G>0,Yτ= ∂Y ∂ τ<0 (1) M p = L(Y, r, π e), L Y = ∂ L ∂Y > 0, Lr = ∂ L ∂ r < 0, Lπ = ∂ L ∂ πe < 0 (2) π = ε (Y − ¯Y) + πe, Y¯ > 0, ε > 0 (3) ˙r = { α (π − ¯π ) + β (Y − ¯Y) if r> 0 max(0, α(π − ¯π ) + β (Y − ¯Y)) if r= 0 (4) ˙ πe = γ(θ ( ¯π − πe) + (1 − θ )(π − πe)), γ > 0, 0 ≤ θ ≤ 1 (5) σ = π + ηγ ˙ Y Y − ηr ˙r r− ηπe ˙ πe πe, σ = ˙ M M (6) where ˙r = dr(t)dt , ˙πe= dπ e(t) dt , ηγ = ∂ L ∂Y L Y > 0, ηr = − ∂ L ∂ r L r > 0, ηπe = − ∂ L ∂ π e L π e > 0 are elasticities of the real money demand with respect to changes of the real national income, nominal rate of interest and the expected rate of inflation, respectively. Variables in (1)-(6) have the following meanings:
Y - real national income (real output), ¯
Y - natural output level corresponding to the natural rate of unemployment (fixed),
G - real government expenditures (fixed), τ - marginal tax rate (fixed), 0 < τ < 1, M - nominal money supply,
L - real money demand, p - price level,
π = pp˙ - rate of inflation,
πe - expected rate of inflation, ¯
π - target rate of inflation, r - nominal rate of interest, r− πe - expected rate of interest,
σ - growth rate of nominal money supply M.
The model possesses five parameters α, β , γ, ε and θ .
Substituting equations (1) and (3) into equations (4), (5) and (6) we receive
˙r = { f1(r, π e, α, β , ε, G, τ) if r> 0 max(0, f1(r, πe, α, β , ε, G, τ)) if r= 0 (7) ˙ πe= f2(r, πe, γ, θ , ε, G, τ), (8) σ = ε (Y (r − πe, G, τ)−¯Y) + πe+ ηγ Yr−πe(˙r − ˙πe) Y(r − πe, G, τ)−ηr ˙r r− ηπe ˙ πe πe, (9) where f1(r, πe, α, β , ε, G, τ) = α(ε(Y (r − πe, G, τ) − ¯Y) + πe− ¯π ) +β (Y (r − πe, G, τ) − ¯Y), f2(r, πe, γ, θ , ε, G, τ) = γ(θ ( ¯π − πe)+(1 − θ )ε(Y (r − πe, G, τ)−¯Y)).
The system of equations (7)-(9) determines the dynamics of three variables (r, πe, σ ). The expression max(0, f1(r, πe, α, β , ε, G, τ)) if r = 0 in (7) is so called nonnegative constraint. It
prevents the nominal rate of interest to obtain negative values. In this paper we do not take this nonnegative constraint into consideration.
The system (7)-(9) is a decomposable one in the sense that the dynamics of r and πe, which is determined by equations (7) and (8), does not depend on eq. (9). Equation (9) only plays the role to determine the growth rate of money supply σ . Therefore in the following consid-erations we deal only with the two-dimensional model represented by equations (7) and (8). Our aim is to find out if Bautin bifurcation can arise in the model. Bautin bifurcation which is also known as generalized Hopf bifurcation gives a complete phase portrait of solutions in a small neighborhood of the model equilibrium with respect to the values of the critical pair of parameters. This kind of bifurcation exhibits a special phenomenon that at some values of the critical pair of parameters two cycles emerge. The inner one which is unstable and the outer one which is stable. Solutions starting inside these two cycles cannot leave this domain, what means that this domain has the character of so called “corridor stability”. From economic point of view it is useful to study this phenomenon. The domain of “corridor stability” guarantees
that solutions starting inside it, though they are going out from the equilibrium, do not leave this domain. The existence of Bautin bifurcation was also studied in the paper of Wu (2011), where a nonlinear Kaldor model of business cycle is analyzed, in the paper of Bella (2013) who analyzed the Goodwin’s model of a class struggle, and also in the paper of Fan and Tang (2015) where a two-stage population model is studied.
2. The existence and stability of the equilibrium point
The question of the existence of an equilibrium point and its stability was rigorously solved in Asada (2016). As this part is necessary for our further considerations on the existence of Bautin bifurcation we present it also here in the whole scale.
The normal equilibrium point E = (r∗, πe∗) of the system (7)-(8) is determined by the rela-tions
˙r = 0, ˙πe= 0, Y = ¯Y. If we neglect the non-negative constraint of r, then we have
Y(r∗− πe∗, G, τ) = ¯Y, πe∗= ¯π . Denote
ρ∗= r∗− ¯π . (10)
Then ρ∗is determined by the equation
Y(ρ∗, G, τ) = ¯Y. Solving this equation with respect to ρ∗, we have
ρ∗= ρ∗(G, τ), and from (10) we get the equilibrium value of r
r∗= ρ∗+ ¯π .
Further we assume that r∗ > 0 and Y (r − πe, G, τ) ∈ C7 in a small neighbourhood of E = (r∗, πe∗).
Remark 1. As G, τ, ¯Y, ¯π are supposed to be fixed, then the value of the equilibrium E = (r∗, πe∗) does not depend on the values of parameters in the model.
First, transform the equilibrium E = (r∗, πe∗) into the origin by the shifting x1= r − r∗, x2= πe− πe∗.
After the shifting the model is ˙
x1 = α(ε(Y (x1− x2+ r∗− πe∗, G, τ) − ¯Y) + x2+ πe∗− ¯π )
+β (Y (x1− x2+ r∗− πe∗, G, τ) − ¯Y) ≡ F1(x1, x2; α, β , ε)
˙
x2 = γ(θ ( ¯π − x2− πe∗) + (1 − θ )ε(Y (x1− x2+ r∗− πe∗, G, τ) − ¯Y)
≡ F2(x1, x2; γ, ε, θ )
The Jacobian matrix of model (11) at the equilibrium E0= (x∗1= 0, x∗2= 0) has the form J(E0; α, β , γ, ε, θ ) = F11 F12 F21 F22 , (12) where F11 = ∂ F1 ∂ x1 = αεYr−π e+ βYr−πe= (αε + β )Yr−πe< 0, F12 = ∂ F1 ∂ x2 = −αεYr−π e+ α − βYr−πe= −(αε + β )Yr−πe+ α > 0, F21 = ∂ F2 ∂ x1 = γ(1 − θ )εYr−π e< 0, F22 = ∂ F2 ∂ x2 = γ(−θ − (1 − θ )εYr−π e).
The eigenvalues of Jacobian (12) are the roots of the characteristic equation of (12) of the form
λ2− TrJλ + detJ = 0, which are given by the formula
λ1,2=
TrJ ±p(TrJ)2− 4detJ
2 , (13)
where
TrJ = (αε + β )Yr−πe+ γ(−θ − (1 − θ )εYr−πe) = (αε + β − γ(1 − θ )ε)Yr−πe− γθ ,
detJ = (αε + β )Yr−πeγ (−θ − (1 − θ )εYr−πe) − (−(αε + β )Yr−πe+ α)γ(1 − θ )εYr−πe
= −γ(θ β + αε)Yr−πe> 0
Conditions for pure imaginary eigenvalues of (12) are:
1. TrJ = (αε + β − γ(1 − θ )ε)Yr−πe− γθ = 0 (14)
2. detJ = −γ(θ β + αε)Yr−πe > 0 (15)
From (14) we have
Yr−πeβ = (1 − εYr−πe)γθ + (γ − α)εYr−πe.
From this we gain
β = 1 Yr−πe− ε γ θ + (γ − α )ε . (16)
Expression (16) is the equation of a line with respect to parameters θ and β . Its intersection ˜
θ with θ −axis is given by the equation 1 Yr−πe − ε γ ˜θ + (γ − α )ε = 0, and ˜ θ = −(γ − α)ε (Y 1 r−πe − ε)γ = γ − α γ −εYr−πe 1 − εYr−πe < 1. (17)
Theorem 1 The following statements hold: 1. If0 < α < γ and 0 < θ < ˜θ , ˜θ = γ −α
γ
−εYr−πe
1−εYr−πe, then the equilibrium E = (r
∗, πe∗) is
a) asymptotically stable for β >Y 1
r−πe − ε γ θ + (γ − α )ε , b) unstable for β < 1 Yr−πe − ε γ θ + (γ − α )ε .
2. If0 < α < γ and ˜θ ≤ θ < 1, then the equilibrium E = (r∗, πe∗) is asymptotically stable for all β and ε.
3. If γ ≤ α, then the equilibrium E = (r∗, πe∗) is asymptotically stable for all β , ε and 0 < θ < 1 .
3. Existence of the generalized Hopf bifurcation (Bautin bifurcation)
The question of the existence of the simple Hopf bifurcation was rigorously analyzed in Asada (2016).
At the points (θ , β ) lying on the segment β = 1 Yr−πe − ε γ θ + (γ − α )ε , 0 < θ < ˜θ , ˜ θ = γ − α γ −εYr−πe 1 − εYr−πe < 1, γ − α > 0 there is TrJ = (αε + β − γ(1 − θ )ε)Yr−πe− γθ = 0, detJ = −γ(θ β + αε)Yr−πe> 0, and λ1,2(θ ; α, γ, ε) = ±i √ detJ = ±iω0(θ ; α, γ, ε).
Consider an arbitrary credibility parameter θ = θ0, 0 < θ < ˜θ . Then at the pair (θ0, β0),
β0=
1 Yr−πe − ε
γ θ0+ (γ − α)ε there is λ1,2(θ0) = ±iω0(θ0). We shall call this pair the critical
pair of Jacobian (12). Let us fix parameters α, γ, ε in model (11). Further on we shall investigate the properties of the solutions of model (11) which respect to parameters θ and β from a small neighbourhood of the critical pair (θ0, β0). For this purpose it is suitable to transform θ0and β0
in model (11) to the origin by shifting
µ1= θ − θ0, µ2= β − β0.
We receive ˙
x1 = α(ε(Y (x1− x2+ r∗− πe∗, G, τ) − ¯Y) + x2+ πe∗− ¯π )
+(µ2+ β0)(Y (x1− x2+ r∗− πe∗, G, τ) − ¯Y) ≡ X1(x1, x2; µ2, α, ε) ˙ x2 = γ((µ1+ θ0)( ¯π − x2− πe∗) +(1 − µ1− θ0)ε(Y (x1− x2+ r∗− πe∗, G, τ) − ¯Y)) ≡ X2(x1, x2; µ1, γ, ε) (18)
In shorten form model (18) can be written as ˙ x= ˜X(x, µ), x= x1 x2 , µ = µ1 µ2 . (19)
Model (19) has the following properties:
1. ˜X(0, µ) = 0 for µ from a small neighbourhood of the critical value µ = 0.
2. The eigenvalues of the Jacobian of (19) at x = 0 and µ from a small neighbourhood of µ = 0 are
λ1,2(µ) = δ (µ) ± iω(µ), δ (0) = 0, ω (0) = ω0.
Perform Taylor expansion of model (19) at the equilibrium E0= (x∗1, x∗2) = (0, 0). We shall
take into account that Y = Y (x1− x2+ r∗− πe∗, G, τ) and
∂Y ∂ x1 =∂Y (x1− x2+ r ∗− πe∗, G, τ) ∂ (r − πe∗) , ∂Y ∂ x2 = −∂Y (x1− x2+ r ∗− πe∗, G, τ) ∂ (r − πe∗) . After Taylor expansion model (19) takes the form
˙ x= J(E0; µ)x + X (x, µ), (20) where J(E0; µ) = (αε + β0+ µ2)Yr−πe −(αε + β0+ µ2)Yr−πe+ α γ (1 − µ1− θ0)εYr−πe γ (−µ1− θ0− (1 − µ1− θ0)εYr−πe) , X(x, µ) = X1(x1, x2; µ2, α, ε) X2(x1, x2; µ1, γ, ε) , X1(x1, x2; µ2, α, ε) = 1 2!(a (2,0)x2 1+ 2a(1,1)x1x2+ a(0,2)x22) +1 3!(a (3,0)x3 1+ 3a(2,1)x21x2+ 3a(1,2)x1x22+ a(0,3)x32) (21) +1 4!(a (4,0)x4 1+ 4a(3,1)x31x2+ 6a(2,2)x21x22+ 4a(1,3)x1x32+ a(0,4)x42) +1 5!(a (5,0)x5
1+ 5a(4,1)x41x2+ 10a(3,2)x31x22+ 10a(2,3)x12x32+ 5a(1,4)x1x24+ a(0,5)x52) +O(|x|6),
a(p,q)= (−1)q(αε + β0+ µ2) ∂kY(r∗− πe∗, G, τ) ∂ (r − πe∗)k , p+ q = k, and X2(x1, x2; µ1, γ, ε) = 1 2!(b (2,0)x2 1+ 2b(1,1)x1x2+ b(0,2)x22) +1 3!(b (3,0)x3 1+ 3b(2,1)x21x2+ 3b(1,2)x1x22+ b(0,3)x32) (22) +1 4!(b (4,0)x4 1+ 4b(3,1)x31x2+ 6b(2,2)x21x22+ 4b(1,3)x1x32+ b(0,4)x42)
+1 5!(b (5,0)x5 1+ 5b(4,1)x41x2+ 10b(3,2)x31x22+ 10b(2,3)x12x32+ 5b(1,4)x1x24+ b(0,5)x52) +O(|x|6), b(p,q)= (−1)qγ (1 − µ1− θ0)ε ∂kY(r∗− πe∗, G, τ) ∂ (r − πe∗)k , p+ q = k.
Denoting A = −(αε + β0+ µ2)Yr−πe, B = γ(1 − µ1− θ0)εYr−πe then Jacobian J(E0; µ) in
(20) can be written in the form
J(E0; µ) =
−A A+ α
B −B − γ(µ1+ θ0)
. (23)
In the next we shall express eigenvectors of Jacobian (23). An eigenvector u1=
u11
u21
is determined by the equation
(−A − λ1)u11+ (A + α)u21= 0, λ1= δ (µ) + iω(µ), µ =
µ1
µ2
. (24)
Taking u11 = A + α, we get from (24) that u21 = A + λ1. An eigenvector u2 =
u12 u22
is determined by the equation
(−A − λ2)u12+ (A + α)u22= 0, λ2= δ (µ) − iω(µ). (25)
Taking u12= A + α, we get from (25) that u22= A + λ2.
Now, consider the matrix
M = (u1, u2) = A+ α A+ α A+ λ1 A+ λ2 . Its inverse matrixM−1is
M−1= 1 detM A+ λ2 −A − α −A − λ1 A+ α , detM = (A + α)(A + λ2) − (A + α)(a + λ1)
= A(λ2− λ1) + α(λ2− λ1) = −2i(A + α)ω(µ). Therefore M−1= 1 −2i(A + α)ω(µ) A+ λ2 −A − α −A − λ1 A+ α . Now, perform in (20) the transformation
x=M y, y = y1 y2 . We receive M ˙y = J(0; µ)M y + X(M y, µ), and ˙ y=M−1J(0; µ)M y + M−1X(M y, µ), (26)
where M = A+ α A+ α A+ λ1 A+ λ2 , M−1= 1 −2i(A + α)ω(µ) A+ λ2 −A − α −A − λ1 A+ α , J(0; µ) = −A A+ α B −B − γ(µ1+ θ0) , A= −(αε + β0+ µ2)Yr−πe, B= γ(1 − µ1− θ0)εYr−πe.
The relation between variables x1, x2 and the new variables y1, y2 is given by the formula
x=M y, what gives x1 x2 = A + α A+ α A+ λ1 A+ λ2 y1 y2 = (A + α)(y1+ y2) (A + λ1)y1+ (A + λ2)y2 . From this we have
x1= (A + α)(y1+ y2), x2= (A + λ1)y1+ (A + λ2)y2.
Putting these expressions into (21) and (22) instead of x1and x2we receive
X(M y, µ) = H(y, µ) = H 1(y 1, y2; µ2, α, ε) H2(y1, y2; µ1, γ, ε) . System (26) gives ˙y1 ˙ y2 = δ (µ ) + iω (µ ) 0 0 δ (µ ) − iω (µ ) y1 y2 + 1 2Aαω (µ ) iA2H1(y, µ) − iAαH2(y, µ)
−iA1H1(y, µ) + iAαH2(y, µ)
and ˙y1 ˙ y2 = δ (µ ) + iω (µ ) 0 0 δ (µ ) − iω (µ ) y1 y2 + 1 2Aαω (µ ) H1(y, µ) H2(y, µ) , (27) where y2= ¯y1, Aα = A + α, A1= A + λ1, A2= A + λ2, A = −(αε + β0+ µ2)Yr−πe, λ1,2(µ) = δ (µ) ± iω(µ), δ (0) = 0, ω(0) = ω0,
H1(y, µ) = iA2H1(y, µ) − iAαH2(y, µ), H2(y, µ) = −iA1H1(y, µ) + iAαH2(y, µ).
The symbol ¯(·) means here and also in the following part of the paper the complex conjugate expression to an expression (·) .
As the second equation in system (27) is complex conjugate to the first one, further we will work only with the first equation of (27) in the form
˙
y= λ (µ) y + 1 2Aαω (µ )
H1(y, ¯y, µ), (28)
y= y1, ¯y= y2, λ (µ) = δ (µ) + iω(µ), δ (0) = 0, ω(0) = ω0> 0.
In the following considerations we utilize the procedure which is described in Kuznetsov (2004).
Let us express equation (28) in the form ˙ y= λ (µ) y +
∑
2≤k+l≤5 1 k!l!gkl(µ)y ky¯l+O(|y|6). (29)Equation (29) can be transformed by an invertible parameter-dependent change of complex coordinate, smoothly depending on the parameters:
y= v +
∑
2≤k+l≤5 1 k!l! hkl(µ)v k ¯ vlfor all sufficiently small ||µ|| into the equation ˙ v= λ (µ) v + c1(µ) v2v¯+ c2(µ) v3v¯2+O(|v|6), (30) c1(µ) = g21 2 + g20g11(2λ + ¯λ ) 2|λ |2 + |g11|2 λ + |g02|2 2(2λ − ¯λ ), where λ = λ (µ ), gkl= gkl(µ), g21 = 2ω(µ)i (A21A2(a12− b03) + A2Aα2(a30− b21) − A1Aα(A1b12+ 2Aαb21) +2A1A2Aα(a21− b12) + 2A1A22a12+ A22Aαa21− A3αb30) + iA21A22a03 2ω(µ)Aα, g20 = 2ω(µ)i (2A1A2a11+ A2Aαa20− A21b02− 2A1Aαb11− A2αb20) + iA21A2a02 2ω(µ)Aα, g11 = 2ω(µ)i (A1A2(a11− b02) + A2Aα(a20− b11) − A1Aαb11+ A22a11 −A2 αb20) + iA1A22a02 2ω(µ)Aα, g02 = 2ω(µ)i (A2Aα(a20− 2b11) + A22(2a11− b02) − A2αb20) + iA32a02 2ω(µ)Aα.
Remark 2. We do not present here the formula for the coefficient c2(µ) as it is rather long
expression.
The coefficients c1(µ), c2(µ) are complex. They can be transformed into real expressions
l1(µ), l2(µ) by the following considerations.
Perform a time reparametrization ˜
which gives du d ˜τ = (ν(µ) + i) v + d1(µ) v|v| 2+ d 2(µ) v|v|4+O(|v|6) (31) with ν (µ ) = δ (µ ) ω (µ ), d1(µ) = c1(µ) ω (µ ), d2(µ) = c2(µ) ω (µ ). Next, we introduce a new time κ such that
d ˜τ = (1 + e1(µ) |v|2+ e2(µ) |v|4) dκ,
where the real functions e1,2have yet to be defined. System (31) can be written as
dv dκ = (ν + i) v + ((ν + i) e1+ d1) v|v| 2+ ((ν + i) e 2+ e1d1+ d2) v|v|4+O(|v|6). Therefore, setting e1(µ) = −Im[d1(µ)] e2(µ) = −Im[d2(µ)] + (Im[d1(µ)])2 yields dv dκ = (ν(µ) + i) v + l1(µ) v|v| 2+ l 2(µ) v|v|4+O(|v|6), (32) where l1(µ) = Re[d1(µ)] − ν(µ) Im[d1(µ)] = Re[c1(µ)] ω (µ ) − δ (µ) Im[c1(µ)] ω2(µ) , l2(µ) = Re[d2(µ)] − Re[d1(µ)] Im[d1(µ)] + ν(µ) ((Im[d1(µ)])2− Im[d2(µ)]).
The functions ν(µ), l1(µ), l2(µ) are smooth and real-valued. The real function l1(µ) is
called the first Lyapunov coefficient. The real function l2(µ) is called the second Lyapunov
coefficient.
At point µ = 0 there is
l1(0) = Re[cω (0)1(0)] = 2ω10(Re[g21] −ω10Im[g20g11]),
l2(0) = Re[c2(0)] ω (0) = 1 12( 1 ω0Re[g32] + 1 ω02Im[g20g¯31− g11(4g31+ 3 ¯g22) − 1 3g02(g40+ ¯g13) − g30g12] + 1 ω03(Re[g20( ¯g11(3g12− ¯g30) + g02( ¯g12− 1 3g30) + 1 3g¯02g03) +g11( ¯g02(53g¯30+ 3g12) +13g02g¯03− 4g11g30)] + 3Im[g20g11]Im[g21]) + 1 ω04(Im[g11g¯02( ¯g 2 20− 3 ¯g20g11− 4g211)] + Im[g20g11](3Re[g20g11] − 2|g02|2))), where gp,q= i 2 Aαω0 d1(p,q)(0), p, q = 0, 1, 2, 3, 4, ¯ gp,q= −i 2 Aαω0 d2(p,q)(0), p, q = 0, 1, 2, 3, dk(p,q)(0) =∂ p+qH k(y = 0, µ = 0) ∂ ykp+q , k = 1, 2. The point µ = 0 is called the Bautin bifurcation point if l1(0) = 0.
4. Numerical simulations
Consider model (7)-(8) consisting of the following functions
Y = C + I + G, C = c(Y − T ) +C0, T = τ Y − T0, I = ˜ κ er−πe, which gives Y(r − πe, G, τ) = G+C0+ c T0 1 − c(1 − τ) + 1 1 − c(1 − τ) ˜ κ er−πe. Take the following values in this model:
α = 0.1, ε = 1, θ0= 0.4, γ = 2, c = 0.8, τ = 0.4, G = 40, C0= 247 25 , T0= 2, ¯π = 0.02, ¯Y = 100, ˜κ = 13 25e 0.03.
After these specifications model (7)-(8) obtains the form ˙r = f1(r, πe, β ) ˙ πe = f2(r, πe, θ ), where f1 = 1 10 e0.03 er−πe + π e−51 50 + β e 0.03 er−πe − 1 , f2 = 2 θ 1 50− π e +(1 − θ ) e 0.03 er−πe− 1 .
The equilibrium point of the model is E = (r∗= 0.05, πe∗= 0.02). The intersection ˜θ of the line β =Y 1
r−πe − ε
γ θ + (γ − α )ε with θ -axis is ˜θ = 0.475. Critical pair of parameters (θ , β ) has values (θ0= 0.4, β0= 0.3).
At the value µ = 0 there is
l1(0) = 0, l2(0)= −0.0499549..
This means that µ = 0 is the Bautin bifurcation point.
In the next part there are depicted three solutions of the model with different initial values and the same values of the critical parameters. In Fig.1 we can see all three solutions together which indicate two limit cycles - the stable one and an unstable cycle, which lies inside the first one. In Fig.2 and 3 there are depicted separated solutions. In Fig.2 there is a stable limit cycle between two solutions, in Fig.3 there is an unstable limit cycle.
Figure 1: Values of the critical parameters are θ = 0.401 and β = 0.299. Initial val-ues (red points): Inner solution starts at the point (0.050000095, 0.02), middle solution at (0.0500003, 0.02) and outer solution at (0.0500028, 0.02).
0.049997 0.049998 0.049999 0.050001 0.050002 0.050003 r 0.019996 0.019998 0.020002 0.020004 Πã
Figure 2: Values of the critical parameters are θ = 0.401 and β = 0.299. Initial val-ues (red points): Inner solution starts at the point (0.0500003, 0.02) and outer solution at (0.0500028, 0.02). 0.049997 0.049999 0.050001 0.050003 r 0.019996 0.019998 0.020002 0.020004 Πã
Source: The authors’ work.
Figure 3: Values of the critical parameters are θ = 0.401 and β = 0.299. Initial values (red points): Inner solution starts at the point (0.050000095, 0.02) and outer solution at (0.0500003, 0.02). 0.0499985 0.0499995 0.0500005 0.0500015 r 0.019998 0.019999 0.020001 0.020002 Πã
5. Conclusion
In the first part of the paper a two-dimensional dynamic model describing the development of nominal rate of interest and expected rate of inflation is introduced. The second part presents the results concerning the existence of an equilibrium of the model and its stability that were reached in Asada (2014). In the third part the question of the existence of Bautin bifurcation is studied. There are found conditions under which Bautin bifurcation can arise. Numerical example presented in the fourth section illustrates the reached results.
The model that is analyzed in this paper expresses monetary policy of a central bank. Its managers should control the development of nominal rate of interest and expected rate of infla-tion in the way to keep them in a stabilized domain. But Minsky (1982, 1986) showed that in an economy with developed financial system crises are inevitable. Managers of monetary and fiscal policy can only reduced them. But for doing this it is necessary to model the development of basic macroeconomic variables and to study their mutual relations. Such a knowledge can be very useful at controlling macroeconomic processes. The results that are reached in this paper contributes to this domain.
Asada in his paper Asada (2014) expanded the analyzed two-dimensional model to the four-dimensional one by adding to nominal rate of interest an expected rate of inflation also the developments of private firms debt and output, and to the six-dimensional model, adding to the four-dimensional model the developments of government bond and government expenditures. So far these models have been analyzed only with respect to the existence of equilibria and their stability, and the possible existence of cycles was only indicated. The question of the existence of cycles has not been rigorously analyzed yet.
Acknowledgements
The paper was financially supported by the grant scheme VEGA 1/0859/16 Dynamics of non-linear economic processes of the Ministry of Education, Science, Research and Sport of the Slovak Republic.
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