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Codimension-2 bifurcations of the Kaldor model of business cycle
Chaos, Solitons & Fractals 44 (2011) 28–42
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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Codimension-2 bifurcations of the Kaldor model of business cycle Xiaoqin P. Wu Department of Mathematics, Computer & Information Sciences, Mississippi Valley State University, Itta Bena, MS 38941, USA
a r t i c l e
i n f o
Article history: Received 27 April 2010 Accepted 17 November 2010 Available online 15 December 2010
a b s t r a c t In this paper, complete analysis is presented to study codimension-2 bifurcations for the nonlinear Kaldor model of business cycle. Sufficient conditions are given for the model to demonstrate Bautin and Bogdanov–Takens (BT) bifurcations. By computing the first and second Lyapunov coefficients and performing nonlinear transformation, the normal forms are derived to obtain the bifurcation diagrams such as Hopf, homoclinic and double limit cycle bifurcations. Some examples are given to confirm the theoretical results. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction In this paper, we study the following Kaldor model of business cycle
(
dY dt dK dt
¼ a½IðY; KÞ SðY; KÞ; ¼ IðY; KÞ qK;
ð1:1Þ
where Y is gross product, K capital stock, a > 0 the adjustment coefficient in the goods market, q 2 (0, 1) the depreciation rate of capital stock, and I(Y, K) and S(Y, K) investment and saving functions, respectively. The first equation of Sys. (1.1) describes the dynamics of gross product and the second one the dynamics of capital stock. As in [10], also see [1,17], using the following saving and investment functions S and I, respectively,
SðY; KÞ ¼ cY;
IðY; KÞ ¼ IðYÞ bK;
where b > 0 and c 2 (0, 1) are constants, we obtain the following system
(
dY dt dK dt
¼ a½IðYÞ bK cY; ¼ IðYÞ ðb þ qÞK:
ð1:2Þ
The study of this model can be traced back to Kaldor [6] in 1940, who proposed nonlinear investment and saving functions so that the system may have cyclic behaviors or limit cycles caused by the instability at the equilibrium point. From the point of view of economics, limit cycles represent periodic oscillations of macro-economic variables of business systems. The model presented here is a little different from the original one that Kaldor studied since the depreciation of the capital stock is added to the second equation of Sys. (1.2). The earliest result in this line is that Chang and Smyth [4] in 1971 gave a critical value around which the system generates a limit cycle (for q = 0) by using a theorem from Poincare and Bendixson. Several authors also discussed similar models [5,19] and established the existence of limit cycles. However, Sys. (1.2) may exhibit more complicated dynamical behaviors if the characteristic equation of the linear part of the system at the equilibrium point has a pair of purely imaginary roots and a double-zero root under certain conditions. A pair of purely imaginary roots enables us to study Hopf bifurcation and a double-zero root Bogdanov–Takens (BT) bifurcation. In the case of Hopf bifurcation, the first Lyapunov coefficient may be zero E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2010.11.002
X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
29
under certain condition, which leads us to discuss degenerate Hopf or Bautin bifurcation and we have to compute the second Lyapunov coefficient. In the case of BT bifurcation, the system can display very interesting bifurcation diagrams such as Hopf, homoclinic, and double limit bifurcations. In this paper, the complete analysis is given for these two cases which are socalled codimension-2 bifurcations. To the author’s knowledge, the literature has not discussed these two cases and their corresponding bifurcation diagrams are unknown. Note that Kalecki [7,8] introduced the idea that there is a time delay for investment before a business decision. Krawiec and Szydlowski [10–12] incorporated Kalecki’s idea into the model of Kaldor by proposing the following Kaldor–Kalecki model of business cycles
( dYðtÞ
¼ a½IðYðtÞÞ bKðtÞ cYðtÞ;
dt dKðtÞ dt
ð1:3Þ
¼ IðYðt sÞÞ ðb þ qÞKðtÞ;
where s P 0 is a time lag representing delay for the investment due to the past investment decision. This model has been studied extensively by many authors, see [13–18]. Clearly Sys. (1.2) is a special case of Sys. (1.3) when s = 0. The rest of this manuscript is organized as follows. In Section 2, a detailed presentation is given for the distribution of eigenvalues of the linear part of Sys. (1.2) at an equilibrium point in the (k, a)-parameter space, where k is the slope of the investment function in the equilibrium point and a is the adjustment coefficient in the goods market. In Sections 3 and 4, the techniques in [9] are applied to compute the first and second Lyapunov coefficients to perform Bautin bifurcation analysis and to obtain the normal forms for Bautin and BT bifurcations, respectively. These normal forms are used to predict the bifurcation diagrams such as Hopf, homoclinic, and double limit cycle bifurcations for the original Sys. (1.2). Finally in Section 5, some numerical simulations are presented to confirm the theoretical results. 2. Eigenvalues Throughout the rest of this paper, we assume that
a; b > 0; q; c 2 ð0; 1Þ; and IðsÞ is a nonlinear C 6 function; and that (Y*, K*) is an equilibrium point of Sys. (1.2). Let I* = I(Y*), u1 = Y Y*, u2 = K K*, and i(s) = I(s + Y*) I*. Then Sys. (1.2) can be transformed as
( du
¼ a½iðu1 Þ bu2 cu1 ;
1
dt du2 dt
ð2:1Þ
¼ iðu1 Þ ðb þ qÞu2 :
Let the Taylor expansion of i at 0 be
iðuÞ ¼ ku þ i u2 þ i u3 þ i u4 þ i u5 þ Oðjuj6 Þ; ð2Þ
ð3Þ
ð4Þ
ð5Þ
where 0
k ¼ i ð0Þ ¼ I0 ðY Þ; ð4Þ
i
ð2Þ
i
¼
1 00 1 i ð0Þ ¼ I00 ðY Þ; 2 2
1 0000 1 0000 i ð0Þ ¼ I ðY Þ; 4 4!
¼
i
ð5Þ
¼
i
ð3Þ
¼
1 000 1 i ð0Þ ¼ I000 ðY Þ; 3! 3!
1 0000 0 1 0000 i ð0Þ ¼ I 0 ðY Þ: 5! 5!
The linear part of Sys. (2.1) at (0, 0) is
( du
1
dt du2 dt
Let
¼ a½ðk cÞu1 bu2 ;
ð2:2Þ
¼ ku1 ðb þ qÞu2 :
A¼
aðk cÞ
ab
k
ðb þ qÞ
:
Then the characteristic equation of A is
DðkÞ k2 þ ðq ak þ b þ acÞk þ a½kq þ ðq þ bÞc ¼ 0;
ð2:3Þ
which has two roots
k1;2 ¼
1 ½q þ ka b ac 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðq ka þ b þ acÞ2 4aðkq þ ðq þ bÞcÞ:
Define
k ¼
qþb
a
þ c;
k ¼
bc þ c; q
ð3Þ
a ¼
3i qðq þ bÞ ð2Þ 2
ð3Þ
2ði Þ q þ 3i bc
;
a ¼
qðq þ bÞ : bc
30
X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
Then we have the following result. Theorem 2.1. If k < min{k*, k**}, all roots of Eq. (2.3) have negative real parts, and hence (Y*, K*) is asymptotically stable. If k > min{k*, k**}, Eq. (2.3) has a positive root and a negative root, and hence (Y*, K*) is unstable. If k = k**, a = a**, Eq. (2.3) has a root 0 with multiplicity 2. In this paper, we make some assumptions. The first one is: Assumption (H1). a > a**. Then if (H1) holds, A has a pair of purely imaginary roots ±x0i where
x0 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2 qb þ abc:
on the half line
l ¼ fl ¼ ðk; aÞ : k ¼ k ; a > a g: Near l, D(k) = 0 has two complex roots k = r + ix and k where
1 2
rðk; aÞ ¼ ðq þ ka b acÞ;
xðk; aÞ ¼
1 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðq ka þ b þ acÞ2 þ 4aðkq þ ðq þ bÞcÞ:
Then r(k*, a) = 0 and x0 = x(k*, a). Define l = (k, a) and u = (u1, u2)T. Then Sys. (2.1) can be written as
u ¼ LðlÞu þ Fðu; lÞ; where
LðlÞ ¼
ð2:4Þ
aðk cÞ
ab
k
ðb þ qÞ
;
a 1 1 1 1 ð2Þ ð3Þ ð4Þ ð5Þ 6 6 Fðu; lÞ ¼ i u21 þ i u31 þ i u41 þ i u51 þ Oðjuj Þ ¼ Bðu; uÞ þ Cðu; u; uÞ þ Dðu; u; u; uÞ þ Eðu; u; u; u; uÞ þ Oðjuj Þ: 2 3! 4! 5! 1
Here
Bðx; yÞ ¼
2 X @ 2 Fðu; lÞ ð2Þ a ju1 ¼u2 ¼0 xj yk ¼ 2i x1 y1 ; @uj @uk 1 j;k¼1
Cðx; y; zÞ ¼
2 X @ 3 Fðu; lÞ ð3Þ a x1 y 1 z 1 ; ju1 ¼u2 ¼0 xj yk zl ¼ 3i @uj @uk @ul 1 j;k;l¼1
Dðx; y; z; v Þ ¼
2 X j;k;l;r¼1
Eðx; y; z; v ; wÞ ¼
@ 4 Fðu; lÞ ð4Þ a x1 y1 z1 v 1 ; ju1 ¼u2 ¼0 xj yk zl v r ¼ 24i @uj @uk @ul @ur 1
2 X j;k;l;r;s¼1
@ 5 Fðu; lÞ ð5Þ a x1 y1 z1 v 1 w1 : ju1 ¼u2 ¼0 xj yk zl v r ws ¼ 120i @uj @uk @ul @ur @us 1
3. Bautin bifurcation In this section, we compute the first and second Lyapunov coefficients by using the formulas in [9]. Near k = k*, define
Q ðkÞ ¼ ððq þ ka þ b ac þ 2ixÞ=ð2kÞ; 1ÞT ; 1 PðkÞ ¼ Dððq b þ ðq ka þ b þ ac þ ixÞÞ=ðabÞ; 1Þ; 2
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X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
where 2
D¼
q2 þ k a2 qð2ka 2b þ 2ac 2ixÞ kað2b þ 2ac 2ixÞ þ ðb ac þ 2ixÞ 2
2ðq2 þ k a2 þ ðb acÞ2 þ 2qðka þ b acÞ 2kaðb þ acÞÞ
:
It is easy to check that
LT P ¼ kP;
LQ ¼ kQ;
hP; Q i ¼ P1 Q 1 þ P 2 Q 2 ¼ 1:
3.1. First Lyapunov coefficient and Hopf bifurcation Note that on l, k = k*, namely r = 0,x = x0. In order to calculate the first Lyapunov coefficient ‘1 on l, we have to evaluate g20, g11, g21 at k = k*. In fact, we have 2
ð2Þ
g 20 jk¼k ¼ hP; BðQ; QÞijk¼k ¼
i a2 ½2iq þ 2qðib þ x0 Þ þ bðiac þ x0 Þ ; x0 ðq þ b þ acÞ
g 11 jk¼k ¼ hP; BðQ; QÞijk¼k ¼
i a2 bðx0 iqÞ ; ðq þ b þ acÞx0
ð2Þ
ð3Þ
g 21 jk¼k ¼ hP; CðQ ; Q ; Q Þijk¼k ¼
3i a3 ðiq þ x0 Þ : ðq þ b þ acÞx0
Then the first Lyapunov coefficient ‘1 at k = k* can be computed as
‘1 jk¼k ¼
1
Re½ig 20 g 11 þ x0 g 21 jk¼k ¼ 2
2x0
a3 b½2ðið2Þ Þ2 qa þ 3ið3Þ ðq2 qb þ abÞ a3 b½2ðið2Þ Þ2 qa þ 3ið3Þ x20 ¼ : 2ðq þ b þ acÞx30 2ðq þ b þ acÞx30
Thus we obtain the following result regarding the stability of limit cycle bifurcating from Hopf bifurcation. Theorem 3.1. Suppose that a is fixed and satisfies (H1) and that k is sufficiently close to k*. Then Sys. (2.1) exhibits a Hopf bifurcation as k crosses k*. Moreover, if 2ðið2Þ Þ2 qa þ 3ið3Þ x20 < 0, then the Hopf bifurcation is supercritical and hence the limit cycle existing for k > k* is stalbe; and if 2ðið2Þ Þ2 qa þ 3ið3Þ x20 > 0, then the Hopf bifurcation is subcritical and hence the limit cycle existing for k > k* is unstable. 3.2. Second Lyapunov coefficient If ‘1 jk¼k ¼ 0, we have to consider Bautin bifurcation. Clearly, ‘1 jk¼k ¼ 0 if and only if a = a*. In order to guarantee that
a* > 0 and x0 > 0, we have to assume that Assumption (H2). i(2) – 0, and 2(i(2))2q + 3i(3)bc < 0. Under the assumption of (H2),
2ði Þ2 ð2Þ
k ¼
ð3Þ
3i
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ð2Þ bc qþb þ þ c; x0 ¼ 2ji jq ð2Þ 2 ; ð3Þ q 2ði Þ q þ 3i bc
and the second Lyapunov coefficient ‘2 at a = a* can be calculated by the following form
1 1 Re½g 32 þ 2 Im½g 20 g31 g 11 ð4g 31 þ 3g22 Þ g 02 ðg 40 þ g13 Þ g 30 g 12 3 x0 1 1 1 5 1 Re g 20 g11 ð3g 12 g30 Þ þ g 02 ðg12 g 30 Þ þ g02 g 03 þ g 11 g02 g30 þ 3g 12 þ g 02 g03 4g 11 g 30 þ 3 3 3 3 x0 2
1 þ3Im½g 20 g 11 Im½g 21 Þ þ 4 Im½g 11 g02 g20 3g20 g 11 4g 211 þ Im½g 20 g 11 3Re½g 20 g 11 2jg 02 j2 :
12‘2 ða Þ ¼
1
x0
x0
Let l* = (k*, a*). We first evaluate g32, g02, g30, g12, g03, g40, g31, g22, g13 at l = l* (for simplicity, for instance, we write g32 instead of g 32 jl¼l .):
g 32 ¼ hP; EðQ ; Q ; Q ; Q ; Q Þijl¼l ; g 03 ¼ hP; CðQ ; Q ; Q Þijl¼l ;
g 02 ¼ hP; BðQ; QÞijl¼l ;
g 40 ¼ hP; DðQ ; Q ; Q ; Q Þijl¼l ;
g 13 ¼ hP; DðQ; Q ; Q ; Q Þijl¼l :
g 30 ¼ hP; CðQ ; Q ; Q Þijl¼l ;
g 12 ¼ hP; CðQ ; Q ; Q Þil¼l ;
g 31 ¼ hP; DðQ ; Q ; Q ; Q Þijl¼l ;
g 22 ¼ hP; DðQ ; Q ; Q ; Q Þijl¼l ;
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X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
In fact, ð5Þ
g 32 ¼
60i a5 b2 ðx0 iqÞ ðq þ b þ acÞ2 x0 Þ
;
ð2Þ
g 02 ¼
i a2 ðx0 iacÞ½b2 þ abc þ qðb 2ix0 Þ 2ibx0 2x20 ðq þ b þ acÞ2 x20
;
ð3Þ
g 30 ¼
3i a3 ½x0 ðqb þ b2 þ abc 4x20 Þ iðqabc þ ab2 c 4qx2 3bx2 Þ ðq þ b þ acÞ2 x0
;
ð3Þ
g 12 ¼
3i a3 bðq þ b ix0 Þðx0 iacÞ ðq þ b þ acÞ2 x0 ð3Þ
g 03 ¼
3i a3 ðx0 iacÞ 3
ðq þ b þ acÞ x0
;
½b3 þ b2 ð2ab 3ix0 Þ bð5x0 þ 3iacÞx0 þ 4ix30 þ q b2 þ abc 3ibx0 x20 ;
h
ð4Þ g 40 ¼ 12i a4 x0 q b2 þ 4abc 8x20 þ x0 b b2 þ 5abc 8x20 i a2 b2 c2 4b2 x20 þ 8x40 þ abc b2 8x20 i
h = ðq þ b þ acÞ3 x0 ; þqb abc 4x20 ð4Þ
g 31 ¼
12i a4 b½ð2q þ bÞx0 þ ið2x20 abcÞ ðq þ b þ acÞ2 x0
;
ð4Þ
g 22 ¼
12i a4 b2 ðx0 iacÞ ðq þ b þ acÞ2 x0
;
ð4Þ
g 13 ¼
12i a4 bðx0 iacÞ½b2 þ abc þ qðb 2ix0 Þ 2ibx0 2x20 ðq þ b þ acÞ3 x0
:
Based on the above calculations, we have
‘2 ðl Þ ¼
a5 b ð2Þ ð2Þ ð3Þ ½2ði Þ4 qa3 bðq þ b þ acÞ3 ð3q2 2qb þ 2abcÞ þ ði Þ2 i a2 ð8q8 6ðq þ b þ acÞ5 x70 þ q7 ð3b 16acÞ þ q6 ð62b2 195abc þ 8a2 c2 Þ q5 bð118b2 þ 463abc þ 39a2 c2 Þ q3 b2 ð5b3 þ 105ab2 c 287a2 bc2 þ 3a3 c3 Þ þ a2 b3 c2 ð6b3 þ 9ab2 c24a2 bc2 þ 5a3 c3 Þ þ 2qab3 cð6b3 þ 3ab2 c 25a2 bc2 þ 10a3 c3 Þ q4 bð72b3 þ 391ab2 c 72a2 bc2 þ 49a3 c3 Þ þ q2 b2 ð6b4 10ab3 c þ 168a2 b2 c2 13a3 bc3 þ 36a4 c4 ÞÞ ð2Þ ð4Þ ð3Þ ð5Þ þ 52i i qabðq þ b þ acÞ3 x40 þ 3bðq þ b þ acÞ3 x40 ð3ði Þ2 qa þ 10i x20 Þ:
Now we impose the following assumption Assumption (H3). ‘2(l*) – 0. 3.3. Regularity To study the regularity of the map l = (k, a) ? (r, ‘1) near l* = (k*, a*), we have to calculate the determinant of the Jacobian Matrix of this map at l = l*, namely,
det
@r @k @‘1 @k
@r @a @‘1 @a
!
¼ l¼l
1 @‘ @‘ a 1 ðk cÞ 1 – 0: 2 @a @k l¼l
To do this, we have to calculate ‘1(k, a). We evaluate g20, g11, g02, g21 first: ð2Þ
g 20 ¼ hP; BðQ ; Q Þi ¼
i 2 ðq2 þ k a2 þ 2qðka þ b ac ixÞ þ ðb acÞðb ac 2ixÞ D1
þ 2kaðb þ ac þ ixÞÞðq þ ka b ac þ 2ixÞðq þ ka þ b ac þ 2ixÞ2 ;
X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
33
ð2Þ
g 02 ¼ hP; BðQ ; Q Þi ¼
i 2 ðq2 þ k a2 þ 2qðka þ b ac ixÞ þ ðb acÞðb ac 2ixÞ D1
2kaðb þ ac þ ixÞÞðq þ ka þ b ac 2ixÞ2 ðq þ ka b ac þ 2ixÞ; ð2Þ
i 2 ðq2 þ k a2 þ 2qðka þ b ac ixÞ þ ðb acÞðb ac 2ixÞ D1 2kaðb þ ac þ ixÞÞðq þ ka b ac 2ixÞðq þ ka þ b ac þ 2ixÞ;
g 11 ¼ hP; BðQ ; Q Þi ¼
g 21 ¼ hP; CðQ ; Q ; Q Þi ð3Þ
¼
3i 2 ðq2 þ k a2 þ 2qðka þ b ac ixÞ þ ðb acÞðb ac 2ixÞ 2kaðb þ ac þ ixÞÞðq þ ka b ac 2ixÞðq þ ka 2D1
þ b ac þ 2ixÞ2 ; where 2
2
D1 ¼ 8k bðq2 þ k a2 þ ðb acÞ2 þ 2qðka b acÞ 2kaðb þ acÞÞ: Now the first Lyapunov coefficient ‘1(k, a) can be expressed as the following
‘1 ðk; aÞ ¼
Re½c1
x
r
Im½c1
x2
ð3:1Þ
;
where
c1 ¼
g 21 jg 11 j2 jg 02 j2 g g ð2k þ kÞ þ 20 11 2 : þ þ 2 k 2ð2k kÞ 2jkj
Using Mathematica, we have
@ðr; ‘1 Þ ¼ det @ðk; aÞ l¼l
@r @k @‘1 @k
@r @a @‘1 @a
!
ð4Þ
¼ l¼l
81i bðq þ bÞx0 ð2Þ 4
16ði Þ ð2ði Þ2 q þ 3i ðq þ bÞcÞ ð2Þ
ð2Þ
:
Hence we assume that Assumption (H4). i(4) – 0. Lemma 3.1. Under the assumptions (H1)–(H4), the map (k, a) ? (l, ‘1) is regular near k = k*, a = a*. Now we assume that the conditions (H1)–(H4) hold. We give the normal form of Bautin bifurcation and its bifurcation diagram. Let k = k* + s1, a = a* + s2. Define
m1 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðk þ s1 ; a þ s2 Þ ; m2 ¼ j‘2 ða Þj‘1 ðk þ s1 ; a þ s2 Þ: xðk þ s1 ; a þ s2 Þ
Then after long calculation, we have ð3Þ
m1 ¼
3i qðq þ bÞ ð2Þ
ð3Þ
2ð2ði Þ2 þ 3i bcÞx0
s1 ;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m2 ¼ 27 j‘2 ða Þjbqðq þ bÞið2Þ ðið3Þ Þ3 ½2ðið3Þ Þ2 q2 ðq þ bÞððið2Þ Þ2 ð9q þ 19bÞ 15ið3Þ bcÞs1 þ ð2ðið2Þ Þ2 q þ 3i bcÞ3 s2 =½4x0 ði Þ3 ð2ði Þ2 q þ 3i ðq þ bÞcÞ: ð3Þ
ð2Þ
ð2Þ
ð3Þ
Then after performing nonlinear transforms, Sys. (2.4) is equivalent to the following truncated normal form
z_ ¼ ðm1 þ iÞz þ m2 zjzj2 þ szjzj4 ;
ð3:2Þ
where s = sign‘2(a*) = ±1. Theorem 3.2. Under the conditions (H1)–(H4), Sys. (1.3) exhibits Bautin bifurcation at k = k*, a = a*, around which Sys. (1.3) is equivalent to the normal form (3.2).
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X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
In the case of s = 1, the complete bifurcation diagrams of Sys. (3.2) can be found in [9]. Here, we just briefly list some results. Define
Hþ ¼ fðm1 ; m2 Þ : m1 ¼ 0; m2 > 0g; H ¼ fðm1 ; m2 Þ : m1 ¼ 0; m2 < 0g; Bþ ¼ fðm1 ; m2 Þ : m1 > 0; m2 ¼ 0g; T ¼ ðm1 ; m2 Þ : m22 þ 4m1 ¼ 0; m2 > 0g: For (m1, m2) small enough, (1) Between H+ and B+, there is a unique stable limit cycle bifurcating from (0, 0). (2) Between H+ and T, there are limit cycles of opposite stability which disappear and collide at the curve T. Applying the above results and using the expressions of m1, m2, we obtain
Hþ ¼ fðs1 ; s2 Þ : s1 ¼ 0; s1 s2 > 0g; H ¼ fðs1 ; s2 Þ : s1 ¼ 0; s1 s2 < 0g; Bþ ¼ fðs1 ; s2 Þ : s1 > 0; s2 ¼ s2 s1 g; T ¼ fðs1 ; s2 Þ : s2 ¼ s3 s1 s4
pffiffiffiffiffi
s1 þ Oðs12 Þ; s1 > 0g;
where
s1 ¼ 2ði Þ2 q þ 3i ðq þ bÞc; ð2Þ
ð3Þ
2ði Þ2 q2 ðq þ bÞð2ði Þ4 ð9q þ 19bÞ þ 15i Þbc ð3Þ
s2 ¼
ð2Þ
ð3Þ
ð2ði Þ2 q þ 3i bcÞ3 ð2Þ
ð3Þ
2ði Þ2 q2 bðq þ bÞðði Þ2 ð9q þ 19bÞ 15i bcÞ ð3Þ
s3 ¼
ð2Þ
ð3Þ
ð2ði Þ2 þ 3i bcÞ3 ð2Þ
;
ð3Þ
;
8qji j3 j2ði Þ2 þ 3i ðq þ bÞcj s4 ¼ pffiffiffi : ð2Þ ð3Þ ð3Þ ð3Þ 9 3bð2ði Þ2 q þ 3i bcÞðji j‘2 ða Þqx0 Þ1=2 ði Þ2 ð2Þ
ð2Þ
ð3Þ
Therefore the following theorem regarding the original Sys. (1.2) is attained. Theorem 3.3. Under the conditions (H1)–(H4), for sufficiently small s1, s2, (i) Between Hþ and Bþ , there is a unique stable limit cycle bifurcating from (0, 0). (ii) Between Hþ and T, there are two limit cycles of opposite stability which disappear and collide at the curve T undergoes a fold bifurcation in the curves.
The case of s = 1 can be treated similarly under the transformation (z, m, t) ? (z, m, t). 4. BT bifurcation In this section we give the normal form of BT bifurcation of Eq. (2.4) and its corresponding bifurcation diagrams. Set
l** = (k**, a**). From Section 2, we know that A has a double zero root when l = l**. Let s1 = k k**, s2 = a a**. Then Eq. (2.4) becomes
u_ ¼ L0 u þ Gðu; sÞ þ Oðkuk4 Þ þ Oðkskkuk2 Þ;
ð4:1Þ
where
0
qþb
qðqþbÞ c
q
q b
L0 ¼ @ ðqþbÞc 0
1
qbc Gðu; sÞ ¼ @
h
1 A;
ð2Þ
ð3Þ
ðq2 ðq þ bÞs1 þ b2 c2 s2 Þu1 þ i q2 ðq þ bÞu21 þ i q2 ðq þ bÞu31
s1 u1 þ ið2Þ u21 þ ið3Þ u31 :
i1 A:
X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
35
Let
aq c ; a
Q0 ¼
qðaþqþbÞ cðqþbÞ
Q1 ¼
1
!
cðbþqþbÞ qðqþbÞ
P0 ¼
;
! ;
1
P1 ¼
bqc
! ;
b;
where
a¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ 1 þ 4q þ 4b ; 2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1 þ 4q þ 4b : 2
b¼
Then we have
LT0 P1 ¼ 0;
L0 Q 0 ¼ 0;
L0 Q 1 ¼ Q 0 ;
LT0 P0 ¼ P1 ;
hP0 ; Q 0 i ¼ hP 1 ; Q 1 i ¼ 1:
Let u = Q0x1 + Q1x2; that is
u1 ¼
aq
c
x1 þ
qða þ q þ bÞ x ; cðq þ bÞ 2
u2 ¼ ax1 þ x2 :
Then Eq. (4.1) can be written as
x_ ¼ Jy þ qy þ f2 ðxÞ þ f3 ðxÞ þ h:o:t:
ð4:2Þ
where
J¼
0 1 0 0
0
;
q¼@
s1 þb2 ðbþqþbÞc2 s2 qbcðqþbÞ
2 ðbþqÞðqþbÞ
a½q
ðaþqþbÞðq
2 2 3 abðq s1qbþbc c s2 Þ
s1 þb2 ðbþqþbÞc2 s2 qbðqþbÞc
2 ðbþqÞðqþbÞ 3
s1 þb bðaþqþbÞðq qbcðqþbÞ
2 2 c
s2 Þ
1 A;
and
0 B f2 ðxÞ ¼ @ (2)
If i
i
2 1 þðaþqþbÞx2 bc2 ðqþbÞ2
ð2Þ 2 q ðbþbÞ½aðqþbÞx
ð2Þ 3 q ½aðqþbÞx
bi
1 þðaþqþbÞx2 2
2
1
0
C A;
B f 3 ðxÞ ¼ @
bc2 ðqþbÞ
i
3 1 þðaþqþbÞx2 bðqþbÞ3 c3
ð3Þ 3 q ðbþqÞ½aðqþbÞx
bi
ð3Þ 4 q ½aðqþbÞx
1 þðaþqþbÞx2 bðqþbÞ3 c3
3
1 C A:
– 0, using nonlinear transformation in [9], we transform Eq. (4.2) into the following system (still using x1, x2)
x_ 1 ¼ x2 ; x_ 2 ¼ h1 x1 þ h2 x2 þ a1 x21 þ b1 x1 x2 þ h:o:t:;
ð4:3Þ
where
h1 ¼
ðq þ bÞðq3 s1 þ b2 c2 s2 Þ ; qbc
a1 ¼
aq3 ðq þ bÞi bc2
ð2Þ
h2 ¼
q2 ðq þ bÞs1 þ b2 c2 s2 ; qbc ð2Þ
;
b1 ¼
2aq2 ðq þ bÞi bc2
:
It is easy to check
det
@h1 @ s1
@h1 @ s2
@h2 @ s1
@h2 @ s2
! ¼ bðq þ bÞ – 0
which means that the regularity condition holds. If i(2) = 0 and i(3) – 0, then similarly using nonlinear transformation in [9], we transform Eq. (4.2) into the following system (still using x1, x2)
x_ 1 ¼ x2 þ h:o:t:; x_ 2 ¼ h1 x1 þ h2 x2 þ a2 x31 þ b2 x21 x2 þ h:o:t:;
ð4:4Þ
where ð3Þ
a2 ¼
q4 ðq þ bÞi ða þ q þ bÞ ; bc3
ð3Þ
b2 ¼
3q3 ðq þ bÞi ða þ q þ bÞ : bc3
It is easy to see that a1b1 > 0 and a2b2 > 0. Theorem 4.1. Sys. (1.3) exhibits Bautin bifurcation at k = k**, a = a**. If i(2) – 0, then Sys. (1.3) is equivalent to the normal form (4.3); and if i(2) = 0 and i(3) – 0, then Sys. (1.3) is equivalent to the normal form (4.4).
36
X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
Now we use the truncated systems (4.3) and (4.4) to obtain bifurcation diagrams of Sys. (2.1). First, we consider the truncated system of (4.3):
x_ 1 ¼ x2 ; x_ 2 ¼ h1 x1 þ h2 x2 þ a1 x21 þ b1 x1 x2 :
ð4:5Þ
Noting that (a1, b1) ? (a1, b1) under the transformation (x1, x2) ? (x1, x2), we may assume that i(2) > 0. After the change of coordinates
x1 ¼
a1 2 b1
h1 ; n1 b1
x2 ! s
a21 3 b1
n2 ;
b1 t ! s; a1
we have (still using x1, x2 for simplicity)
x_ 1 ¼ x2 ; x_ 2 ¼ m1 þ m2 x1 þ x21 þ sx1 x2 ;
ð4:6Þ
where 2
m1 ¼ m2 ¼
b1 4½q3 ðq þ bÞs1 þ b2 ðq2 þ 2bÞc2 s2 ðb1 h1 a1 h2 Þ ¼ ; q4 bc a31
b1 4b2 cs2 ðb1 h1 2a1 h2 Þ ¼ 2 q3 a1
and s = signa1b1 = ± 1. Since a1b1 > 0, we have that s = 1. The complete bifurcation diagrams of Sys. (4.6) can be found in [9]. Here, we just briefly list some results. For (m1, m2) small enough, (i) Sys. (4.6) undergoes a fold bifurcation when (m1, m2) is on the parabola
T þ ¼ fðm1 ; m2 Þ : 4m1 m22 ¼ 0; m2 > 0g;
T ¼ fðm1 ; m2 Þ : 4m1 m22 ¼ 0; m2 < 0g:
(ii) Sys. (4.6) undergoes a Hopf bifurcation when (m1, m2) is on the curve
H ¼ fðm1 ; m2 Þ : m1 ¼ 0; m2 < 0g; and the Hopf bifurcation gives rise to a unstable limit cycle. (iii) Sys. (4.6) undergoes a saddle homoclinic bifurcation when (m1, m2) is on the curve
P¼
6 ðm1 ; m2 Þ : m1 ¼ m22 ; m2 < 0 ; 25
Moreover, when (m1, m2) is in the region between the curves H and P, Sys. (4.5) has a unique unstable periodic orbit. Applying the above results and using the expressions of m1, m2, we obtain the following result regarding the original Sys. (1.2). Theorem 4.2. Suppose that i(2) > 0. For sufficiently small s1, s2, (i) Sys. (1.2) undergoes a fold bifurcation in the curves
( ð s1 ; s2 Þ : s1
Tþ ¼
) b2 c2 2 3 2 ¼ 5 q ðq þ 2bÞs2 þ b cs2 ; s2 > 0 ; q ðq þ bÞ
( ð s1 ; s2 Þ : s1 ¼
T ¼
) b2 c2 2 3 2 ðq þ 2bÞ s þ b c s s < 0 : ; q 2 2 2 q5 ðq þ bÞ
(ii) Sys. (1.2) undergoes a Hopf bifurcation on the curve
( ð s1 ; s2 Þ : s1 ¼
H¼
) b2 c2 ðq þ 2bÞ s ; s > 0 : 2 2 q3 ðq þ bÞ
(iii) Sys. (1.2) undergoes a saddle homoclinic bifurcation on the curve
( P¼
ðs1 ; s2 Þ : s1 ¼
) b2 c2 3 2 2 ðq þ 2bÞ s þ 24b c s s < 0 : 25q ; 2 2 2 25q5 ðq þ bÞ
Moreover, if (s1, s2) is in the region between the curves H and P, Sys. (1.2) has a unique unstable periodic orbit.
X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
37
Next, we consider the truncated system of (4.4):
x_ 1 ¼ x2 ; x_ 2 ¼ q1 x1 þ q2 x2 þ a2 x31 þ b2 x21 x2 :
ð4:7Þ
The bifurcation diagrams of this system are more complicated and interesting. Since a2 and b2 have the same sign, we must consider two cases. Case 1: a2 > 0, b2 > 0. Sys. (4.7) can be transformed as
x_ 1 ¼ x2 ; x_ 2 ¼ e1 x1 þ e2 x2 þ x31 x21 x2 ;
ð4:8Þ
where
e1 ¼
2 b2 9ðq þ bÞðq3 s1 þ b2 c2 s2 Þ h1 ¼ ; q3 bc a2
e2 ¼
b2 3q2 ½q2 ðq þ bÞs1 þ b2 c2 s2 h2 ¼ ; q2 b c a2 2
b 2ffi under the transformation t ! ba22 t; x1 ! pbaffiffiffi x1 and x2 ! a2 p2affiffiffi2ffi x2 . The complete bifurcation diagrams of Sys. (4.8) can be 2 found, for example, in [2,3,9]. Here, we briefly list some results: for small e1, e2,
(i) When (e1, e2) is in the line
T 1 ¼ fðe1 ; e2 Þ : e1 ¼ 0; e2 2 Rg; Sys. (4.8) undergoes a pitchfork bifurcation; (ii) Sys. (4.8) undergoes a stable Hopf bifurcation for the trivial equilibrium point on the line
H1 ¼ fðe1 ; e2 Þ : e2 ¼ 0; e1 < 0g: (iii) On the curve
C¼
1 ðe1 ; e2 Þ : e2 ¼ e1 þ O je1 j3=2 ; e1 < 0 ; 5
Sys. (4.8) undergoes a heteroclinic bifurcation. Moreover, if (e1, e2) is in the region between the curves H1 and C Sys. (4.8) has a unique stable periodic orbit. Applying the above results and using the expressions of h1, h2, e1, e2, we obtain the following theorem regarding the original Sys. (1.2). Theorem 4.3. Suppose that i(2) = 0 and i(3) > 0. For sufficiently small s1, s2, (i) Sys. (1.2) undergoes a pitchfork bifurcation in the line
( ðs1 ; s2 Þ : s1 ¼
T1 ¼
) b2 c2 s ; s 2 R : 2 2 q3
(ii) Sys. (1.2) undergoes a stable Hopf bifurcation in the line
( ðs1 ; s2 Þ : s1 ¼
H1 ¼
) b2 c2 s ; s < 0 : 2 2 q2 ðq þ bÞ
(iii) Sys. (1.2) undergoes a branch of homoclinic bifurcation on the curve
( C¼
ðs1 ; s2 Þ : s1 ¼
) 3 b2 c2 ð2q þ 3bÞ 2 ;s < 0 s þ O j s j : 2 2 2 2q3 ðq þ bÞ
Moreover, if (s1, s2) is in the region between the curves H1 and C, Sys. (1.2) has a unique unstable periodic orbit.
Case 2: a2 < 0, b2 < 0. After rescaling, Sys. (4.7) can be transformed as
x_ 1 ¼ x2 ; x_ 2 ¼ e1 x1 þ e2 x2 x31 x21 x2 ;
ð4:9Þ
38
X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
where
e1 ¼
2 b2 9ðq þ bÞðq3 s1 þ b2 c2 s2 Þ h1 ¼ ; q3 bc a2
e2 ¼
b2 3½q2 ðq þ bÞs1 þ b2 c2 s2 h2 ¼ : q2 bc a2
This system has one trivial equilibrium point E0 and other two nontrivial equilibrium points E1,2. The complete bifurcation diagrams of this system can be found, for example, in [9]. Here, we briefly list some results. For small e1, e2, (i) When (e1, e2) is in the line
T 2 ¼ fðe1 ; e2 Þ : e1 ¼ 0; e2 2 Rg; Sys. (4.9) undergoes a pitchfork bifurcation. (ii) When (e1, e2) is in the half line
H2 ¼ fðe1 ; e2 Þ : e2 ¼ e1 ; e1 > 0g; Sys. (4.9) undergoes a stable Hopf bifurcation at E1,2 and the bifurcation is subcritical. (iii) When (e1, e2) is the curve
C0 ¼
4 ðe1 ; e2 Þ : e2 ¼ e1 þ O je1 j3=2 ; e1 > 0 ; 5
Sys. (4.9) has a unique homoclinic orbit connecting E1 and E2 and two homoclinic orbits simultaneously at E0. Moreover, if (e1, e2) is in the region between the curves H2 and C0 , Sys. (4.9) has three limit periodic orbits: a ‘‘large’’ one and two ‘‘small’’ ones. (iv) When (e1, e2) is the curve
C d ¼ fðe1 ; e2 Þ : e2 ¼ ce1 þ Oðje1 j3=2 Þ; e1 > 0g; where c 0.752, Sys. (4.9) undergoes a double limit cycle bifurcation. Moreover, if (e1, e2) is in the region between the curves C0 and Cd, Sys. (4.9) has two large limit cycles: the outer one which is stable and the inner one which is unstable, and these two cycles collide on Cd. Applying the above results and using the expressions of h1, h2, e1, e2, we obtain the following theorem regarding the original Sys. (1.2). Theorem 4.4. Suppose that i(2) = 0 and i(3) < 0. For sufficiently small s1, s2, (i) Sys. (1.2) undergoes a pitchfork bifurcation in the line
( ð s1 ; s2 Þ : s1
T1 ¼
) b2 c2 ¼ 3 s2 ; s2 2 R : q
(ii) Sys. (1.2) undergoes a branch of stable Hopf bifurcation on the curve
( ðs1 ; s2 Þ : s1
H2 ¼
) b2 c2 ð2q þ 3bÞ ¼ s2 ; s2 < 0 : 2q3 ðq þ bÞ
(iii) Sys. (1.2) has two small homoclinic orbits simultaneously at (Y*, K*) and a large homoclinic orbit on the curve
( C0 ¼
ðs1 ; s2 Þ : s1 ¼
) 3 b2 c2 ð7q þ 12bÞ 2 ;s < 0 s þ O j s j : 2 2 2 7q3 ðq þ bÞ
Moreover, if (s1, s2) is in the region between the curves H2 and C 0 , Sys. (1.2) has three limit periodic orbits: a ‘‘large’’ one and two ‘‘small’’ ones. (iv) Sys. (1.2) undergoes a branch of a double limit cycle bifurcation on the curve
( Cd ¼
ðs1 ; s2 Þ : s1
) 3 b2 c2 ½ð1 þ 3cÞq þ 3cb 2 ¼ s2 þ O js2 j ; s2 < 0 ð1 þ 3cÞq3 ðq þ bÞ
where the constant c 0.752. Moreover, if (s1, s2) is in the region between the curves C 0 and C d , Sys. (1.2) has two large different limit cycles. The outer one is stable, the inner one is unstable, and these two cycles collide on C d .
39
X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
5. Numerical Simulations In this section, we give some examples to verify the theoretical results obtained Sections 3 and 4. For simplicity, we assume that (0, 0) is one of the equilibrium points. Example 1. This example demonstrates the result of Theorem 3.1. Let a = 1, b = 0.8, c = 0.5625, q = 0.3. Then k* = 1.6625 and the condition (H1) holds. Take k = k* + 0.01 and
IðsÞ ¼ ks þ 0:5s2 0:4s3 : Then (0, 0) is the trivial equilibrium point. Simple calculation shows that the first Lyapunov coefficient ‘1(k*) = 0.199685 < 0 and hence, by Theorem 3.1, there is a stable limit cycle generated by Hopf bifurcation. Fig. 1 verifies this result. Example 2. This example supports the result of Theorem 3.3. Let b = 0.8, c = 0.5625, q = 0.3 and
IðsÞ ¼ ks þ 0:5s2 0:4s3 þ 0:1s4 0:2s5 : Then k* = 1.6458333333333335, a* = 1.0153846153846153. Easy calculation shows that the first Lyapunov coefficient ‘1(k*) = 0 and the second Lyapunov coefficient ‘2(a*) = 0.788942 < 0. Also
Hþ ¼ fðs1 ; s2 Þ : s1 ¼ 0; s2 < 0g;
H ¼ fðs1 ; s2 Þ : s1 ¼ 0; s2 > 0g;
pffiffiffiffiffiffiffiffiffi T ¼ fðs1 ; s2 Þ : s2 ¼ 3:83189s1 1:24375 s1 ; s1 < 0g; Bþ ¼ fðs1 ; s2 Þ : s2 ¼ 1:27267s1 ; s1 > 0g; K 2.0
1.5
1.0
0.5
0.0
Y
0.5 0.5
0.0
0.5
1.0
1.5
Fig. 1. A stable limit cycle bifurcating from (0, 0) for k > k*.
Fig. 2. Bautin bifurcation—left: a stable limit cycle when (s1, s2) is the region between B+ and H+, right: two limit cycles when (s1, s2) is the region between T and H+.
40
X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
Fig. 3. An unstable limit cycle bifurcates from (0, 0) for k = k** + 0.0001 and a = a** + 0.01.
Fig. 4. Three limit (two small and one large) cycles are generated when (s1, s2) is in the region between H2 and C 0 .
Thus the system exhibits a Bautin bifurcation at k = k*, a = a*. If taking s1 = 0.00001, s2 = 0.00002, then we have that (s1, s2) is the region between B+ and H+ and hence there is an unstable limit cycle (see Fig. 2(left)). If taking s1 = 0.000035, s2 = 0.010688, then we have that (s1, s2) is the region between T and H+ and hence there are two limit cycles (the big one is stable and the small one is unstable) (see Fig. 2(right)). Example 3. This example verifies the result of Theorem 4.2. Let b = 0.5, c = 0.5625, q = 0.1. Then the condition (H6) holds and k** = 3.375099999999996, a** = 0.21333333333333332. Easy calculation shows that
H ¼ fs1 ¼ 145:01953124999997s2 ; s2 > 0g; P ¼ fs1 ¼ 145:01953125s2 þ 889:8925781249998s22 ; s2 < 0g: If we take s1 = 0.000147 and s2 = 0.000001, then (s1, s2) is the region between H and P. Let k = k** + s1 = 3.3751469999999997, a = a** + s2 = 0.21333233333333332, and
IðsÞ ¼ ks þ 0:005s2 : According to (iii) of Theorem 4.2, there is a unique unstable limit cycle bifurcating from (0, 0). Fig. 3 verifies this result.
X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
41
Fig. 5. Two ‘‘big’’ limit cycles are generated when (s1, s2) is in the region between C 0 and C d : the outer one is stable while the inner one is unstable.
Example 4. This example verifies (iii) and (iv) of Theorem 4.4. Let b = 0.5, c = 0.5625, q = 0.8. Then k** = 0.914063, a** = 3.69778. Easy calculation shows that
H2 ¼ fs1 ¼ 0:184206s2 ; s2 < 0g; C 0 ¼ fs1 ¼ 0:196939s2 ; s2 < 0g; C d ¼ fs1 ¼ 0:201805s2 ; s2 < 0g: If we choose s1 = 1.8446108 and s2 = 0.0000001, then (s1, s2) is the region between H2 and C 0 . Let k = k** + s1, a = a** + s2. Choose
IðsÞ ¼ ks 0:00001s3 : According to (iii) of Theorem 4.4, there are three limit cycles bifurcating from (0, 0) (Fig. 4). However if we choose s1 = 0.000000201756 and s2 = 0.000001, then (s1, s2) is the region between C 0 and C d . Let k = k** + s1, a = a** + s2. Choose
IðsÞ ¼ ks 0:0001s3 : We have two ‘‘big’’ limit cycles (Fig. 5). According to (iv) of Theorem 4.4, the outer one is stable while the inner one is unstable.
6. Conclusion In this paper, we have investigated dynamical behaviors of the Kaldor model of business cycle through studying both Bautin (or degenerate Hopf) and BT singularities using (k, a) as the bifurcation parameter. By applying the normal form theory, we are able to predict their corresponding bifurcation diagrams such as Hopf, homoclinic and double limit cycle bifurcations. In economics, limit cycles are very important because they stand for cyclic oscillations of variables, which represent coexistence and future behaviors of business systems. Acknowledgement The authors would like to express their gratitude for valuable comments on this manuscript from the referees and the editor. References [1] [2] [3] [4] [5]
Bisichi GI, Dieci R, Rodano G, Saltari E. Multiple attractors and global bifurcations in a Kaldor-type business cycle model. J Evol Econ 2001;11:527–44. Carr J. Applications of center manifold theory. New York: Springer-Verlag; 1981. S.N. Chow, C. Li and D. Wang, Normal forms and bifurcation of planar vector fields, Cambridge, 1994. Chang WW, Smyth DJ. The existence and persistence of cycles in a nonlinear model: Kaldor’s 1940 model re-examined. Rev Econ Study 1971;38:37–44. Grasman J, Wentzel JJ. Co-existence of a limit cycle and an equilibrium in Kaldor’s business cycle model and its consequences. J Econ Behav Organ 1994;24:369–77.
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X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42 Kaldor N. A model of the trade cycle. Econ J 1940;40:78–92. Kalecki N. A macrodynamic theory of business cycles. Econometrica 1935;3:327–44. Kalecki N. A theory of the business cycle. Rev Stud 1937;4:77–97. Kuznetsov Yu. Elements of applied bifurcation theory. 3rd ed. Springer; 2004. Krawiec A, Szydlowski M. The Kaldor–Kalecki business cycle model. Ann Oper Res 1999;89:89–100. Krawiec A, Szydlowski M. The Kaldor–Kalecki model of business cycle. J Nonlinear Math Phys Supp Proc NEED 99 2001;8. Krawiec A, Szydlowski M. The Hopf bifurcation in the Kaldor–Kalecki model. In: Holly S, Greenblatt S, editors. Computation in economics, finance and engineering: economics systems. Elsevier; 2000. p. 391–6. Krawiec A, Szydlowski M. On nonlinear mechanics of business cycle model. Regular Chaotic Dyn 2001;6:101–18. Krawiec A, Szydlowski M. The stability problem in the Kaldor–Kalcki business cycle model. Chaos, Solitons Fractals 2005;25:299–305. Szydlowski M, Krawiec A, Tobola J. Nonlinear oscillations in business cycle model with time lags. Chaos, Solitons Fractals 2001;12:505–17. Takeucki Y, Yamamura T. The stability analysis of the Kaldor with time delays: monetary policy and government budget constraint. Nonlinear Anal (RWA) 2004;5:277–308. Wu XP, Wang L. Multi-parameter bifurcations of the Kaldor–Kalecki model of business cycle with delay. Nonlinear Anal (RWA) 2009;11:869–87. Wang L, Wu XP. Bifurcation analysis of a Kaldor–Kalecki model of business cycle with delay. Electron J Qual Theor Differ Equat. Spec. Ed. I 2009. Varian HR. Catastrophe theory and the business cycle. Econ Inquiry 1979;17:14–28.
Contents lists available at ScienceDirect
Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Codimension-2 bifurcations of the Kaldor model of business cycle Xiaoqin P. Wu Department of Mathematics, Computer & Information Sciences, Mississippi Valley State University, Itta Bena, MS 38941, USA
a r t i c l e
i n f o
Article history: Received 27 April 2010 Accepted 17 November 2010 Available online 15 December 2010
a b s t r a c t In this paper, complete analysis is presented to study codimension-2 bifurcations for the nonlinear Kaldor model of business cycle. Sufficient conditions are given for the model to demonstrate Bautin and Bogdanov–Takens (BT) bifurcations. By computing the first and second Lyapunov coefficients and performing nonlinear transformation, the normal forms are derived to obtain the bifurcation diagrams such as Hopf, homoclinic and double limit cycle bifurcations. Some examples are given to confirm the theoretical results. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction In this paper, we study the following Kaldor model of business cycle
(
dY dt dK dt
¼ a½IðY; KÞ SðY; KÞ; ¼ IðY; KÞ qK;
ð1:1Þ
where Y is gross product, K capital stock, a > 0 the adjustment coefficient in the goods market, q 2 (0, 1) the depreciation rate of capital stock, and I(Y, K) and S(Y, K) investment and saving functions, respectively. The first equation of Sys. (1.1) describes the dynamics of gross product and the second one the dynamics of capital stock. As in [10], also see [1,17], using the following saving and investment functions S and I, respectively,
SðY; KÞ ¼ cY;
IðY; KÞ ¼ IðYÞ bK;
where b > 0 and c 2 (0, 1) are constants, we obtain the following system
(
dY dt dK dt
¼ a½IðYÞ bK cY; ¼ IðYÞ ðb þ qÞK:
ð1:2Þ
The study of this model can be traced back to Kaldor [6] in 1940, who proposed nonlinear investment and saving functions so that the system may have cyclic behaviors or limit cycles caused by the instability at the equilibrium point. From the point of view of economics, limit cycles represent periodic oscillations of macro-economic variables of business systems. The model presented here is a little different from the original one that Kaldor studied since the depreciation of the capital stock is added to the second equation of Sys. (1.2). The earliest result in this line is that Chang and Smyth [4] in 1971 gave a critical value around which the system generates a limit cycle (for q = 0) by using a theorem from Poincare and Bendixson. Several authors also discussed similar models [5,19] and established the existence of limit cycles. However, Sys. (1.2) may exhibit more complicated dynamical behaviors if the characteristic equation of the linear part of the system at the equilibrium point has a pair of purely imaginary roots and a double-zero root under certain conditions. A pair of purely imaginary roots enables us to study Hopf bifurcation and a double-zero root Bogdanov–Takens (BT) bifurcation. In the case of Hopf bifurcation, the first Lyapunov coefficient may be zero E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2010.11.002
X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
29
under certain condition, which leads us to discuss degenerate Hopf or Bautin bifurcation and we have to compute the second Lyapunov coefficient. In the case of BT bifurcation, the system can display very interesting bifurcation diagrams such as Hopf, homoclinic, and double limit bifurcations. In this paper, the complete analysis is given for these two cases which are socalled codimension-2 bifurcations. To the author’s knowledge, the literature has not discussed these two cases and their corresponding bifurcation diagrams are unknown. Note that Kalecki [7,8] introduced the idea that there is a time delay for investment before a business decision. Krawiec and Szydlowski [10–12] incorporated Kalecki’s idea into the model of Kaldor by proposing the following Kaldor–Kalecki model of business cycles
( dYðtÞ
¼ a½IðYðtÞÞ bKðtÞ cYðtÞ;
dt dKðtÞ dt
ð1:3Þ
¼ IðYðt sÞÞ ðb þ qÞKðtÞ;
where s P 0 is a time lag representing delay for the investment due to the past investment decision. This model has been studied extensively by many authors, see [13–18]. Clearly Sys. (1.2) is a special case of Sys. (1.3) when s = 0. The rest of this manuscript is organized as follows. In Section 2, a detailed presentation is given for the distribution of eigenvalues of the linear part of Sys. (1.2) at an equilibrium point in the (k, a)-parameter space, where k is the slope of the investment function in the equilibrium point and a is the adjustment coefficient in the goods market. In Sections 3 and 4, the techniques in [9] are applied to compute the first and second Lyapunov coefficients to perform Bautin bifurcation analysis and to obtain the normal forms for Bautin and BT bifurcations, respectively. These normal forms are used to predict the bifurcation diagrams such as Hopf, homoclinic, and double limit cycle bifurcations for the original Sys. (1.2). Finally in Section 5, some numerical simulations are presented to confirm the theoretical results. 2. Eigenvalues Throughout the rest of this paper, we assume that
a; b > 0; q; c 2 ð0; 1Þ; and IðsÞ is a nonlinear C 6 function; and that (Y*, K*) is an equilibrium point of Sys. (1.2). Let I* = I(Y*), u1 = Y Y*, u2 = K K*, and i(s) = I(s + Y*) I*. Then Sys. (1.2) can be transformed as
( du
¼ a½iðu1 Þ bu2 cu1 ;
1
dt du2 dt
ð2:1Þ
¼ iðu1 Þ ðb þ qÞu2 :
Let the Taylor expansion of i at 0 be
iðuÞ ¼ ku þ i u2 þ i u3 þ i u4 þ i u5 þ Oðjuj6 Þ; ð2Þ
ð3Þ
ð4Þ
ð5Þ
where 0
k ¼ i ð0Þ ¼ I0 ðY Þ; ð4Þ
i
ð2Þ
i
¼
1 00 1 i ð0Þ ¼ I00 ðY Þ; 2 2
1 0000 1 0000 i ð0Þ ¼ I ðY Þ; 4 4!
¼
i
ð5Þ
¼
i
ð3Þ
¼
1 000 1 i ð0Þ ¼ I000 ðY Þ; 3! 3!
1 0000 0 1 0000 i ð0Þ ¼ I 0 ðY Þ: 5! 5!
The linear part of Sys. (2.1) at (0, 0) is
( du
1
dt du2 dt
Let
¼ a½ðk cÞu1 bu2 ;
ð2:2Þ
¼ ku1 ðb þ qÞu2 :
A¼
aðk cÞ
ab
k
ðb þ qÞ
:
Then the characteristic equation of A is
DðkÞ k2 þ ðq ak þ b þ acÞk þ a½kq þ ðq þ bÞc ¼ 0;
ð2:3Þ
which has two roots
k1;2 ¼
1 ½q þ ka b ac 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðq ka þ b þ acÞ2 4aðkq þ ðq þ bÞcÞ:
Define
k ¼
qþb
a
þ c;
k ¼
bc þ c; q
ð3Þ
a ¼
3i qðq þ bÞ ð2Þ 2
ð3Þ
2ði Þ q þ 3i bc
;
a ¼
qðq þ bÞ : bc
30
X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
Then we have the following result. Theorem 2.1. If k < min{k*, k**}, all roots of Eq. (2.3) have negative real parts, and hence (Y*, K*) is asymptotically stable. If k > min{k*, k**}, Eq. (2.3) has a positive root and a negative root, and hence (Y*, K*) is unstable. If k = k**, a = a**, Eq. (2.3) has a root 0 with multiplicity 2. In this paper, we make some assumptions. The first one is: Assumption (H1). a > a**. Then if (H1) holds, A has a pair of purely imaginary roots ±x0i where
x0 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2 qb þ abc:
on the half line
l ¼ fl ¼ ðk; aÞ : k ¼ k ; a > a g: Near l, D(k) = 0 has two complex roots k = r + ix and k where
1 2
rðk; aÞ ¼ ðq þ ka b acÞ;
xðk; aÞ ¼
1 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðq ka þ b þ acÞ2 þ 4aðkq þ ðq þ bÞcÞ:
Then r(k*, a) = 0 and x0 = x(k*, a). Define l = (k, a) and u = (u1, u2)T. Then Sys. (2.1) can be written as
u ¼ LðlÞu þ Fðu; lÞ; where
LðlÞ ¼
ð2:4Þ
aðk cÞ
ab
k
ðb þ qÞ
;
a 1 1 1 1 ð2Þ ð3Þ ð4Þ ð5Þ 6 6 Fðu; lÞ ¼ i u21 þ i u31 þ i u41 þ i u51 þ Oðjuj Þ ¼ Bðu; uÞ þ Cðu; u; uÞ þ Dðu; u; u; uÞ þ Eðu; u; u; u; uÞ þ Oðjuj Þ: 2 3! 4! 5! 1
Here
Bðx; yÞ ¼
2 X @ 2 Fðu; lÞ ð2Þ a ju1 ¼u2 ¼0 xj yk ¼ 2i x1 y1 ; @uj @uk 1 j;k¼1
Cðx; y; zÞ ¼
2 X @ 3 Fðu; lÞ ð3Þ a x1 y 1 z 1 ; ju1 ¼u2 ¼0 xj yk zl ¼ 3i @uj @uk @ul 1 j;k;l¼1
Dðx; y; z; v Þ ¼
2 X j;k;l;r¼1
Eðx; y; z; v ; wÞ ¼
@ 4 Fðu; lÞ ð4Þ a x1 y1 z1 v 1 ; ju1 ¼u2 ¼0 xj yk zl v r ¼ 24i @uj @uk @ul @ur 1
2 X j;k;l;r;s¼1
@ 5 Fðu; lÞ ð5Þ a x1 y1 z1 v 1 w1 : ju1 ¼u2 ¼0 xj yk zl v r ws ¼ 120i @uj @uk @ul @ur @us 1
3. Bautin bifurcation In this section, we compute the first and second Lyapunov coefficients by using the formulas in [9]. Near k = k*, define
Q ðkÞ ¼ ððq þ ka þ b ac þ 2ixÞ=ð2kÞ; 1ÞT ; 1 PðkÞ ¼ Dððq b þ ðq ka þ b þ ac þ ixÞÞ=ðabÞ; 1Þ; 2
31
X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
where 2
D¼
q2 þ k a2 qð2ka 2b þ 2ac 2ixÞ kað2b þ 2ac 2ixÞ þ ðb ac þ 2ixÞ 2
2ðq2 þ k a2 þ ðb acÞ2 þ 2qðka þ b acÞ 2kaðb þ acÞÞ
:
It is easy to check that
LT P ¼ kP;
LQ ¼ kQ;
hP; Q i ¼ P1 Q 1 þ P 2 Q 2 ¼ 1:
3.1. First Lyapunov coefficient and Hopf bifurcation Note that on l, k = k*, namely r = 0,x = x0. In order to calculate the first Lyapunov coefficient ‘1 on l, we have to evaluate g20, g11, g21 at k = k*. In fact, we have 2
ð2Þ
g 20 jk¼k ¼ hP; BðQ; QÞijk¼k ¼
i a2 ½2iq þ 2qðib þ x0 Þ þ bðiac þ x0 Þ ; x0 ðq þ b þ acÞ
g 11 jk¼k ¼ hP; BðQ; QÞijk¼k ¼
i a2 bðx0 iqÞ ; ðq þ b þ acÞx0
ð2Þ
ð3Þ
g 21 jk¼k ¼ hP; CðQ ; Q ; Q Þijk¼k ¼
3i a3 ðiq þ x0 Þ : ðq þ b þ acÞx0
Then the first Lyapunov coefficient ‘1 at k = k* can be computed as
‘1 jk¼k ¼
1
Re½ig 20 g 11 þ x0 g 21 jk¼k ¼ 2
2x0
a3 b½2ðið2Þ Þ2 qa þ 3ið3Þ ðq2 qb þ abÞ a3 b½2ðið2Þ Þ2 qa þ 3ið3Þ x20 ¼ : 2ðq þ b þ acÞx30 2ðq þ b þ acÞx30
Thus we obtain the following result regarding the stability of limit cycle bifurcating from Hopf bifurcation. Theorem 3.1. Suppose that a is fixed and satisfies (H1) and that k is sufficiently close to k*. Then Sys. (2.1) exhibits a Hopf bifurcation as k crosses k*. Moreover, if 2ðið2Þ Þ2 qa þ 3ið3Þ x20 < 0, then the Hopf bifurcation is supercritical and hence the limit cycle existing for k > k* is stalbe; and if 2ðið2Þ Þ2 qa þ 3ið3Þ x20 > 0, then the Hopf bifurcation is subcritical and hence the limit cycle existing for k > k* is unstable. 3.2. Second Lyapunov coefficient If ‘1 jk¼k ¼ 0, we have to consider Bautin bifurcation. Clearly, ‘1 jk¼k ¼ 0 if and only if a = a*. In order to guarantee that
a* > 0 and x0 > 0, we have to assume that Assumption (H2). i(2) – 0, and 2(i(2))2q + 3i(3)bc < 0. Under the assumption of (H2),
2ði Þ2 ð2Þ
k ¼
ð3Þ
3i
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ð2Þ bc qþb þ þ c; x0 ¼ 2ji jq ð2Þ 2 ; ð3Þ q 2ði Þ q þ 3i bc
and the second Lyapunov coefficient ‘2 at a = a* can be calculated by the following form
1 1 Re½g 32 þ 2 Im½g 20 g31 g 11 ð4g 31 þ 3g22 Þ g 02 ðg 40 þ g13 Þ g 30 g 12 3 x0 1 1 1 5 1 Re g 20 g11 ð3g 12 g30 Þ þ g 02 ðg12 g 30 Þ þ g02 g 03 þ g 11 g02 g30 þ 3g 12 þ g 02 g03 4g 11 g 30 þ 3 3 3 3 x0 2
1 þ3Im½g 20 g 11 Im½g 21 Þ þ 4 Im½g 11 g02 g20 3g20 g 11 4g 211 þ Im½g 20 g 11 3Re½g 20 g 11 2jg 02 j2 :
12‘2 ða Þ ¼
1
x0
x0
Let l* = (k*, a*). We first evaluate g32, g02, g30, g12, g03, g40, g31, g22, g13 at l = l* (for simplicity, for instance, we write g32 instead of g 32 jl¼l .):
g 32 ¼ hP; EðQ ; Q ; Q ; Q ; Q Þijl¼l ; g 03 ¼ hP; CðQ ; Q ; Q Þijl¼l ;
g 02 ¼ hP; BðQ; QÞijl¼l ;
g 40 ¼ hP; DðQ ; Q ; Q ; Q Þijl¼l ;
g 13 ¼ hP; DðQ; Q ; Q ; Q Þijl¼l :
g 30 ¼ hP; CðQ ; Q ; Q Þijl¼l ;
g 12 ¼ hP; CðQ ; Q ; Q Þil¼l ;
g 31 ¼ hP; DðQ ; Q ; Q ; Q Þijl¼l ;
g 22 ¼ hP; DðQ ; Q ; Q ; Q Þijl¼l ;
32
X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
In fact, ð5Þ
g 32 ¼
60i a5 b2 ðx0 iqÞ ðq þ b þ acÞ2 x0 Þ
;
ð2Þ
g 02 ¼
i a2 ðx0 iacÞ½b2 þ abc þ qðb 2ix0 Þ 2ibx0 2x20 ðq þ b þ acÞ2 x20
;
ð3Þ
g 30 ¼
3i a3 ½x0 ðqb þ b2 þ abc 4x20 Þ iðqabc þ ab2 c 4qx2 3bx2 Þ ðq þ b þ acÞ2 x0
;
ð3Þ
g 12 ¼
3i a3 bðq þ b ix0 Þðx0 iacÞ ðq þ b þ acÞ2 x0 ð3Þ
g 03 ¼
3i a3 ðx0 iacÞ 3
ðq þ b þ acÞ x0
;
½b3 þ b2 ð2ab 3ix0 Þ bð5x0 þ 3iacÞx0 þ 4ix30 þ q b2 þ abc 3ibx0 x20 ;
h
ð4Þ g 40 ¼ 12i a4 x0 q b2 þ 4abc 8x20 þ x0 b b2 þ 5abc 8x20 i a2 b2 c2 4b2 x20 þ 8x40 þ abc b2 8x20 i
h = ðq þ b þ acÞ3 x0 ; þqb abc 4x20 ð4Þ
g 31 ¼
12i a4 b½ð2q þ bÞx0 þ ið2x20 abcÞ ðq þ b þ acÞ2 x0
;
ð4Þ
g 22 ¼
12i a4 b2 ðx0 iacÞ ðq þ b þ acÞ2 x0
;
ð4Þ
g 13 ¼
12i a4 bðx0 iacÞ½b2 þ abc þ qðb 2ix0 Þ 2ibx0 2x20 ðq þ b þ acÞ3 x0
:
Based on the above calculations, we have
‘2 ðl Þ ¼
a5 b ð2Þ ð2Þ ð3Þ ½2ði Þ4 qa3 bðq þ b þ acÞ3 ð3q2 2qb þ 2abcÞ þ ði Þ2 i a2 ð8q8 6ðq þ b þ acÞ5 x70 þ q7 ð3b 16acÞ þ q6 ð62b2 195abc þ 8a2 c2 Þ q5 bð118b2 þ 463abc þ 39a2 c2 Þ q3 b2 ð5b3 þ 105ab2 c 287a2 bc2 þ 3a3 c3 Þ þ a2 b3 c2 ð6b3 þ 9ab2 c24a2 bc2 þ 5a3 c3 Þ þ 2qab3 cð6b3 þ 3ab2 c 25a2 bc2 þ 10a3 c3 Þ q4 bð72b3 þ 391ab2 c 72a2 bc2 þ 49a3 c3 Þ þ q2 b2 ð6b4 10ab3 c þ 168a2 b2 c2 13a3 bc3 þ 36a4 c4 ÞÞ ð2Þ ð4Þ ð3Þ ð5Þ þ 52i i qabðq þ b þ acÞ3 x40 þ 3bðq þ b þ acÞ3 x40 ð3ði Þ2 qa þ 10i x20 Þ:
Now we impose the following assumption Assumption (H3). ‘2(l*) – 0. 3.3. Regularity To study the regularity of the map l = (k, a) ? (r, ‘1) near l* = (k*, a*), we have to calculate the determinant of the Jacobian Matrix of this map at l = l*, namely,
det
@r @k @‘1 @k
@r @a @‘1 @a
!
¼ l¼l
1 @‘ @‘ a 1 ðk cÞ 1 – 0: 2 @a @k l¼l
To do this, we have to calculate ‘1(k, a). We evaluate g20, g11, g02, g21 first: ð2Þ
g 20 ¼ hP; BðQ ; Q Þi ¼
i 2 ðq2 þ k a2 þ 2qðka þ b ac ixÞ þ ðb acÞðb ac 2ixÞ D1
þ 2kaðb þ ac þ ixÞÞðq þ ka b ac þ 2ixÞðq þ ka þ b ac þ 2ixÞ2 ;
X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
33
ð2Þ
g 02 ¼ hP; BðQ ; Q Þi ¼
i 2 ðq2 þ k a2 þ 2qðka þ b ac ixÞ þ ðb acÞðb ac 2ixÞ D1
2kaðb þ ac þ ixÞÞðq þ ka þ b ac 2ixÞ2 ðq þ ka b ac þ 2ixÞ; ð2Þ
i 2 ðq2 þ k a2 þ 2qðka þ b ac ixÞ þ ðb acÞðb ac 2ixÞ D1 2kaðb þ ac þ ixÞÞðq þ ka b ac 2ixÞðq þ ka þ b ac þ 2ixÞ;
g 11 ¼ hP; BðQ ; Q Þi ¼
g 21 ¼ hP; CðQ ; Q ; Q Þi ð3Þ
¼
3i 2 ðq2 þ k a2 þ 2qðka þ b ac ixÞ þ ðb acÞðb ac 2ixÞ 2kaðb þ ac þ ixÞÞðq þ ka b ac 2ixÞðq þ ka 2D1
þ b ac þ 2ixÞ2 ; where 2
2
D1 ¼ 8k bðq2 þ k a2 þ ðb acÞ2 þ 2qðka b acÞ 2kaðb þ acÞÞ: Now the first Lyapunov coefficient ‘1(k, a) can be expressed as the following
‘1 ðk; aÞ ¼
Re½c1
x
r
Im½c1
x2
ð3:1Þ
;
where
c1 ¼
g 21 jg 11 j2 jg 02 j2 g g ð2k þ kÞ þ 20 11 2 : þ þ 2 k 2ð2k kÞ 2jkj
Using Mathematica, we have
@ðr; ‘1 Þ ¼ det @ðk; aÞ l¼l
@r @k @‘1 @k
@r @a @‘1 @a
!
ð4Þ
¼ l¼l
81i bðq þ bÞx0 ð2Þ 4
16ði Þ ð2ði Þ2 q þ 3i ðq þ bÞcÞ ð2Þ
ð2Þ
:
Hence we assume that Assumption (H4). i(4) – 0. Lemma 3.1. Under the assumptions (H1)–(H4), the map (k, a) ? (l, ‘1) is regular near k = k*, a = a*. Now we assume that the conditions (H1)–(H4) hold. We give the normal form of Bautin bifurcation and its bifurcation diagram. Let k = k* + s1, a = a* + s2. Define
m1 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðk þ s1 ; a þ s2 Þ ; m2 ¼ j‘2 ða Þj‘1 ðk þ s1 ; a þ s2 Þ: xðk þ s1 ; a þ s2 Þ
Then after long calculation, we have ð3Þ
m1 ¼
3i qðq þ bÞ ð2Þ
ð3Þ
2ð2ði Þ2 þ 3i bcÞx0
s1 ;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m2 ¼ 27 j‘2 ða Þjbqðq þ bÞið2Þ ðið3Þ Þ3 ½2ðið3Þ Þ2 q2 ðq þ bÞððið2Þ Þ2 ð9q þ 19bÞ 15ið3Þ bcÞs1 þ ð2ðið2Þ Þ2 q þ 3i bcÞ3 s2 =½4x0 ði Þ3 ð2ði Þ2 q þ 3i ðq þ bÞcÞ: ð3Þ
ð2Þ
ð2Þ
ð3Þ
Then after performing nonlinear transforms, Sys. (2.4) is equivalent to the following truncated normal form
z_ ¼ ðm1 þ iÞz þ m2 zjzj2 þ szjzj4 ;
ð3:2Þ
where s = sign‘2(a*) = ±1. Theorem 3.2. Under the conditions (H1)–(H4), Sys. (1.3) exhibits Bautin bifurcation at k = k*, a = a*, around which Sys. (1.3) is equivalent to the normal form (3.2).
34
X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
In the case of s = 1, the complete bifurcation diagrams of Sys. (3.2) can be found in [9]. Here, we just briefly list some results. Define
Hþ ¼ fðm1 ; m2 Þ : m1 ¼ 0; m2 > 0g; H ¼ fðm1 ; m2 Þ : m1 ¼ 0; m2 < 0g; Bþ ¼ fðm1 ; m2 Þ : m1 > 0; m2 ¼ 0g; T ¼ ðm1 ; m2 Þ : m22 þ 4m1 ¼ 0; m2 > 0g: For (m1, m2) small enough, (1) Between H+ and B+, there is a unique stable limit cycle bifurcating from (0, 0). (2) Between H+ and T, there are limit cycles of opposite stability which disappear and collide at the curve T. Applying the above results and using the expressions of m1, m2, we obtain
Hþ ¼ fðs1 ; s2 Þ : s1 ¼ 0; s1 s2 > 0g; H ¼ fðs1 ; s2 Þ : s1 ¼ 0; s1 s2 < 0g; Bþ ¼ fðs1 ; s2 Þ : s1 > 0; s2 ¼ s2 s1 g; T ¼ fðs1 ; s2 Þ : s2 ¼ s3 s1 s4
pffiffiffiffiffi
s1 þ Oðs12 Þ; s1 > 0g;
where
s1 ¼ 2ði Þ2 q þ 3i ðq þ bÞc; ð2Þ
ð3Þ
2ði Þ2 q2 ðq þ bÞð2ði Þ4 ð9q þ 19bÞ þ 15i Þbc ð3Þ
s2 ¼
ð2Þ
ð3Þ
ð2ði Þ2 q þ 3i bcÞ3 ð2Þ
ð3Þ
2ði Þ2 q2 bðq þ bÞðði Þ2 ð9q þ 19bÞ 15i bcÞ ð3Þ
s3 ¼
ð2Þ
ð3Þ
ð2ði Þ2 þ 3i bcÞ3 ð2Þ
;
ð3Þ
;
8qji j3 j2ði Þ2 þ 3i ðq þ bÞcj s4 ¼ pffiffiffi : ð2Þ ð3Þ ð3Þ ð3Þ 9 3bð2ði Þ2 q þ 3i bcÞðji j‘2 ða Þqx0 Þ1=2 ði Þ2 ð2Þ
ð2Þ
ð3Þ
Therefore the following theorem regarding the original Sys. (1.2) is attained. Theorem 3.3. Under the conditions (H1)–(H4), for sufficiently small s1, s2, (i) Between Hþ and Bþ , there is a unique stable limit cycle bifurcating from (0, 0). (ii) Between Hþ and T, there are two limit cycles of opposite stability which disappear and collide at the curve T undergoes a fold bifurcation in the curves.
The case of s = 1 can be treated similarly under the transformation (z, m, t) ? (z, m, t). 4. BT bifurcation In this section we give the normal form of BT bifurcation of Eq. (2.4) and its corresponding bifurcation diagrams. Set
l** = (k**, a**). From Section 2, we know that A has a double zero root when l = l**. Let s1 = k k**, s2 = a a**. Then Eq. (2.4) becomes
u_ ¼ L0 u þ Gðu; sÞ þ Oðkuk4 Þ þ Oðkskkuk2 Þ;
ð4:1Þ
where
0
qþb
qðqþbÞ c
q
q b
L0 ¼ @ ðqþbÞc 0
1
qbc Gðu; sÞ ¼ @
h
1 A;
ð2Þ
ð3Þ
ðq2 ðq þ bÞs1 þ b2 c2 s2 Þu1 þ i q2 ðq þ bÞu21 þ i q2 ðq þ bÞu31
s1 u1 þ ið2Þ u21 þ ið3Þ u31 :
i1 A:
X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
35
Let
aq c ; a
Q0 ¼
qðaþqþbÞ cðqþbÞ
Q1 ¼
1
!
cðbþqþbÞ qðqþbÞ
P0 ¼
;
! ;
1
P1 ¼
bqc
! ;
b;
where
a¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ 1 þ 4q þ 4b ; 2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1 þ 4q þ 4b : 2
b¼
Then we have
LT0 P1 ¼ 0;
L0 Q 0 ¼ 0;
L0 Q 1 ¼ Q 0 ;
LT0 P0 ¼ P1 ;
hP0 ; Q 0 i ¼ hP 1 ; Q 1 i ¼ 1:
Let u = Q0x1 + Q1x2; that is
u1 ¼
aq
c
x1 þ
qða þ q þ bÞ x ; cðq þ bÞ 2
u2 ¼ ax1 þ x2 :
Then Eq. (4.1) can be written as
x_ ¼ Jy þ qy þ f2 ðxÞ þ f3 ðxÞ þ h:o:t:
ð4:2Þ
where
J¼
0 1 0 0
0
;
q¼@
s1 þb2 ðbþqþbÞc2 s2 qbcðqþbÞ
2 ðbþqÞðqþbÞ
a½q
ðaþqþbÞðq
2 2 3 abðq s1qbþbc c s2 Þ
s1 þb2 ðbþqþbÞc2 s2 qbðqþbÞc
2 ðbþqÞðqþbÞ 3
s1 þb bðaþqþbÞðq qbcðqþbÞ
2 2 c
s2 Þ
1 A;
and
0 B f2 ðxÞ ¼ @ (2)
If i
i
2 1 þðaþqþbÞx2 bc2 ðqþbÞ2
ð2Þ 2 q ðbþbÞ½aðqþbÞx
ð2Þ 3 q ½aðqþbÞx
bi
1 þðaþqþbÞx2 2
2
1
0
C A;
B f 3 ðxÞ ¼ @
bc2 ðqþbÞ
i
3 1 þðaþqþbÞx2 bðqþbÞ3 c3
ð3Þ 3 q ðbþqÞ½aðqþbÞx
bi
ð3Þ 4 q ½aðqþbÞx
1 þðaþqþbÞx2 bðqþbÞ3 c3
3
1 C A:
– 0, using nonlinear transformation in [9], we transform Eq. (4.2) into the following system (still using x1, x2)
x_ 1 ¼ x2 ; x_ 2 ¼ h1 x1 þ h2 x2 þ a1 x21 þ b1 x1 x2 þ h:o:t:;
ð4:3Þ
where
h1 ¼
ðq þ bÞðq3 s1 þ b2 c2 s2 Þ ; qbc
a1 ¼
aq3 ðq þ bÞi bc2
ð2Þ
h2 ¼
q2 ðq þ bÞs1 þ b2 c2 s2 ; qbc ð2Þ
;
b1 ¼
2aq2 ðq þ bÞi bc2
:
It is easy to check
det
@h1 @ s1
@h1 @ s2
@h2 @ s1
@h2 @ s2
! ¼ bðq þ bÞ – 0
which means that the regularity condition holds. If i(2) = 0 and i(3) – 0, then similarly using nonlinear transformation in [9], we transform Eq. (4.2) into the following system (still using x1, x2)
x_ 1 ¼ x2 þ h:o:t:; x_ 2 ¼ h1 x1 þ h2 x2 þ a2 x31 þ b2 x21 x2 þ h:o:t:;
ð4:4Þ
where ð3Þ
a2 ¼
q4 ðq þ bÞi ða þ q þ bÞ ; bc3
ð3Þ
b2 ¼
3q3 ðq þ bÞi ða þ q þ bÞ : bc3
It is easy to see that a1b1 > 0 and a2b2 > 0. Theorem 4.1. Sys. (1.3) exhibits Bautin bifurcation at k = k**, a = a**. If i(2) – 0, then Sys. (1.3) is equivalent to the normal form (4.3); and if i(2) = 0 and i(3) – 0, then Sys. (1.3) is equivalent to the normal form (4.4).
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X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
Now we use the truncated systems (4.3) and (4.4) to obtain bifurcation diagrams of Sys. (2.1). First, we consider the truncated system of (4.3):
x_ 1 ¼ x2 ; x_ 2 ¼ h1 x1 þ h2 x2 þ a1 x21 þ b1 x1 x2 :
ð4:5Þ
Noting that (a1, b1) ? (a1, b1) under the transformation (x1, x2) ? (x1, x2), we may assume that i(2) > 0. After the change of coordinates
x1 ¼
a1 2 b1
h1 ; n1 b1
x2 ! s
a21 3 b1
n2 ;
b1 t ! s; a1
we have (still using x1, x2 for simplicity)
x_ 1 ¼ x2 ; x_ 2 ¼ m1 þ m2 x1 þ x21 þ sx1 x2 ;
ð4:6Þ
where 2
m1 ¼ m2 ¼
b1 4½q3 ðq þ bÞs1 þ b2 ðq2 þ 2bÞc2 s2 ðb1 h1 a1 h2 Þ ¼ ; q4 bc a31
b1 4b2 cs2 ðb1 h1 2a1 h2 Þ ¼ 2 q3 a1
and s = signa1b1 = ± 1. Since a1b1 > 0, we have that s = 1. The complete bifurcation diagrams of Sys. (4.6) can be found in [9]. Here, we just briefly list some results. For (m1, m2) small enough, (i) Sys. (4.6) undergoes a fold bifurcation when (m1, m2) is on the parabola
T þ ¼ fðm1 ; m2 Þ : 4m1 m22 ¼ 0; m2 > 0g;
T ¼ fðm1 ; m2 Þ : 4m1 m22 ¼ 0; m2 < 0g:
(ii) Sys. (4.6) undergoes a Hopf bifurcation when (m1, m2) is on the curve
H ¼ fðm1 ; m2 Þ : m1 ¼ 0; m2 < 0g; and the Hopf bifurcation gives rise to a unstable limit cycle. (iii) Sys. (4.6) undergoes a saddle homoclinic bifurcation when (m1, m2) is on the curve
P¼
6 ðm1 ; m2 Þ : m1 ¼ m22 ; m2 < 0 ; 25
Moreover, when (m1, m2) is in the region between the curves H and P, Sys. (4.5) has a unique unstable periodic orbit. Applying the above results and using the expressions of m1, m2, we obtain the following result regarding the original Sys. (1.2). Theorem 4.2. Suppose that i(2) > 0. For sufficiently small s1, s2, (i) Sys. (1.2) undergoes a fold bifurcation in the curves
( ð s1 ; s2 Þ : s1
Tþ ¼
) b2 c2 2 3 2 ¼ 5 q ðq þ 2bÞs2 þ b cs2 ; s2 > 0 ; q ðq þ bÞ
( ð s1 ; s2 Þ : s1 ¼
T ¼
) b2 c2 2 3 2 ðq þ 2bÞ s þ b c s s < 0 : ; q 2 2 2 q5 ðq þ bÞ
(ii) Sys. (1.2) undergoes a Hopf bifurcation on the curve
( ð s1 ; s2 Þ : s1 ¼
H¼
) b2 c2 ðq þ 2bÞ s ; s > 0 : 2 2 q3 ðq þ bÞ
(iii) Sys. (1.2) undergoes a saddle homoclinic bifurcation on the curve
( P¼
ðs1 ; s2 Þ : s1 ¼
) b2 c2 3 2 2 ðq þ 2bÞ s þ 24b c s s < 0 : 25q ; 2 2 2 25q5 ðq þ bÞ
Moreover, if (s1, s2) is in the region between the curves H and P, Sys. (1.2) has a unique unstable periodic orbit.
X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
37
Next, we consider the truncated system of (4.4):
x_ 1 ¼ x2 ; x_ 2 ¼ q1 x1 þ q2 x2 þ a2 x31 þ b2 x21 x2 :
ð4:7Þ
The bifurcation diagrams of this system are more complicated and interesting. Since a2 and b2 have the same sign, we must consider two cases. Case 1: a2 > 0, b2 > 0. Sys. (4.7) can be transformed as
x_ 1 ¼ x2 ; x_ 2 ¼ e1 x1 þ e2 x2 þ x31 x21 x2 ;
ð4:8Þ
where
e1 ¼
2 b2 9ðq þ bÞðq3 s1 þ b2 c2 s2 Þ h1 ¼ ; q3 bc a2
e2 ¼
b2 3q2 ½q2 ðq þ bÞs1 þ b2 c2 s2 h2 ¼ ; q2 b c a2 2
b 2ffi under the transformation t ! ba22 t; x1 ! pbaffiffiffi x1 and x2 ! a2 p2affiffiffi2ffi x2 . The complete bifurcation diagrams of Sys. (4.8) can be 2 found, for example, in [2,3,9]. Here, we briefly list some results: for small e1, e2,
(i) When (e1, e2) is in the line
T 1 ¼ fðe1 ; e2 Þ : e1 ¼ 0; e2 2 Rg; Sys. (4.8) undergoes a pitchfork bifurcation; (ii) Sys. (4.8) undergoes a stable Hopf bifurcation for the trivial equilibrium point on the line
H1 ¼ fðe1 ; e2 Þ : e2 ¼ 0; e1 < 0g: (iii) On the curve
C¼
1 ðe1 ; e2 Þ : e2 ¼ e1 þ O je1 j3=2 ; e1 < 0 ; 5
Sys. (4.8) undergoes a heteroclinic bifurcation. Moreover, if (e1, e2) is in the region between the curves H1 and C Sys. (4.8) has a unique stable periodic orbit. Applying the above results and using the expressions of h1, h2, e1, e2, we obtain the following theorem regarding the original Sys. (1.2). Theorem 4.3. Suppose that i(2) = 0 and i(3) > 0. For sufficiently small s1, s2, (i) Sys. (1.2) undergoes a pitchfork bifurcation in the line
( ðs1 ; s2 Þ : s1 ¼
T1 ¼
) b2 c2 s ; s 2 R : 2 2 q3
(ii) Sys. (1.2) undergoes a stable Hopf bifurcation in the line
( ðs1 ; s2 Þ : s1 ¼
H1 ¼
) b2 c2 s ; s < 0 : 2 2 q2 ðq þ bÞ
(iii) Sys. (1.2) undergoes a branch of homoclinic bifurcation on the curve
( C¼
ðs1 ; s2 Þ : s1 ¼
) 3 b2 c2 ð2q þ 3bÞ 2 ;s < 0 s þ O j s j : 2 2 2 2q3 ðq þ bÞ
Moreover, if (s1, s2) is in the region between the curves H1 and C, Sys. (1.2) has a unique unstable periodic orbit.
Case 2: a2 < 0, b2 < 0. After rescaling, Sys. (4.7) can be transformed as
x_ 1 ¼ x2 ; x_ 2 ¼ e1 x1 þ e2 x2 x31 x21 x2 ;
ð4:9Þ
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X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
where
e1 ¼
2 b2 9ðq þ bÞðq3 s1 þ b2 c2 s2 Þ h1 ¼ ; q3 bc a2
e2 ¼
b2 3½q2 ðq þ bÞs1 þ b2 c2 s2 h2 ¼ : q2 bc a2
This system has one trivial equilibrium point E0 and other two nontrivial equilibrium points E1,2. The complete bifurcation diagrams of this system can be found, for example, in [9]. Here, we briefly list some results. For small e1, e2, (i) When (e1, e2) is in the line
T 2 ¼ fðe1 ; e2 Þ : e1 ¼ 0; e2 2 Rg; Sys. (4.9) undergoes a pitchfork bifurcation. (ii) When (e1, e2) is in the half line
H2 ¼ fðe1 ; e2 Þ : e2 ¼ e1 ; e1 > 0g; Sys. (4.9) undergoes a stable Hopf bifurcation at E1,2 and the bifurcation is subcritical. (iii) When (e1, e2) is the curve
C0 ¼
4 ðe1 ; e2 Þ : e2 ¼ e1 þ O je1 j3=2 ; e1 > 0 ; 5
Sys. (4.9) has a unique homoclinic orbit connecting E1 and E2 and two homoclinic orbits simultaneously at E0. Moreover, if (e1, e2) is in the region between the curves H2 and C0 , Sys. (4.9) has three limit periodic orbits: a ‘‘large’’ one and two ‘‘small’’ ones. (iv) When (e1, e2) is the curve
C d ¼ fðe1 ; e2 Þ : e2 ¼ ce1 þ Oðje1 j3=2 Þ; e1 > 0g; where c 0.752, Sys. (4.9) undergoes a double limit cycle bifurcation. Moreover, if (e1, e2) is in the region between the curves C0 and Cd, Sys. (4.9) has two large limit cycles: the outer one which is stable and the inner one which is unstable, and these two cycles collide on Cd. Applying the above results and using the expressions of h1, h2, e1, e2, we obtain the following theorem regarding the original Sys. (1.2). Theorem 4.4. Suppose that i(2) = 0 and i(3) < 0. For sufficiently small s1, s2, (i) Sys. (1.2) undergoes a pitchfork bifurcation in the line
( ð s1 ; s2 Þ : s1
T1 ¼
) b2 c2 ¼ 3 s2 ; s2 2 R : q
(ii) Sys. (1.2) undergoes a branch of stable Hopf bifurcation on the curve
( ðs1 ; s2 Þ : s1
H2 ¼
) b2 c2 ð2q þ 3bÞ ¼ s2 ; s2 < 0 : 2q3 ðq þ bÞ
(iii) Sys. (1.2) has two small homoclinic orbits simultaneously at (Y*, K*) and a large homoclinic orbit on the curve
( C0 ¼
ðs1 ; s2 Þ : s1 ¼
) 3 b2 c2 ð7q þ 12bÞ 2 ;s < 0 s þ O j s j : 2 2 2 7q3 ðq þ bÞ
Moreover, if (s1, s2) is in the region between the curves H2 and C 0 , Sys. (1.2) has three limit periodic orbits: a ‘‘large’’ one and two ‘‘small’’ ones. (iv) Sys. (1.2) undergoes a branch of a double limit cycle bifurcation on the curve
( Cd ¼
ðs1 ; s2 Þ : s1
) 3 b2 c2 ½ð1 þ 3cÞq þ 3cb 2 ¼ s2 þ O js2 j ; s2 < 0 ð1 þ 3cÞq3 ðq þ bÞ
where the constant c 0.752. Moreover, if (s1, s2) is in the region between the curves C 0 and C d , Sys. (1.2) has two large different limit cycles. The outer one is stable, the inner one is unstable, and these two cycles collide on C d .
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X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
5. Numerical Simulations In this section, we give some examples to verify the theoretical results obtained Sections 3 and 4. For simplicity, we assume that (0, 0) is one of the equilibrium points. Example 1. This example demonstrates the result of Theorem 3.1. Let a = 1, b = 0.8, c = 0.5625, q = 0.3. Then k* = 1.6625 and the condition (H1) holds. Take k = k* + 0.01 and
IðsÞ ¼ ks þ 0:5s2 0:4s3 : Then (0, 0) is the trivial equilibrium point. Simple calculation shows that the first Lyapunov coefficient ‘1(k*) = 0.199685 < 0 and hence, by Theorem 3.1, there is a stable limit cycle generated by Hopf bifurcation. Fig. 1 verifies this result. Example 2. This example supports the result of Theorem 3.3. Let b = 0.8, c = 0.5625, q = 0.3 and
IðsÞ ¼ ks þ 0:5s2 0:4s3 þ 0:1s4 0:2s5 : Then k* = 1.6458333333333335, a* = 1.0153846153846153. Easy calculation shows that the first Lyapunov coefficient ‘1(k*) = 0 and the second Lyapunov coefficient ‘2(a*) = 0.788942 < 0. Also
Hþ ¼ fðs1 ; s2 Þ : s1 ¼ 0; s2 < 0g;
H ¼ fðs1 ; s2 Þ : s1 ¼ 0; s2 > 0g;
pffiffiffiffiffiffiffiffiffi T ¼ fðs1 ; s2 Þ : s2 ¼ 3:83189s1 1:24375 s1 ; s1 < 0g; Bþ ¼ fðs1 ; s2 Þ : s2 ¼ 1:27267s1 ; s1 > 0g; K 2.0
1.5
1.0
0.5
0.0
Y
0.5 0.5
0.0
0.5
1.0
1.5
Fig. 1. A stable limit cycle bifurcating from (0, 0) for k > k*.
Fig. 2. Bautin bifurcation—left: a stable limit cycle when (s1, s2) is the region between B+ and H+, right: two limit cycles when (s1, s2) is the region between T and H+.
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X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
Fig. 3. An unstable limit cycle bifurcates from (0, 0) for k = k** + 0.0001 and a = a** + 0.01.
Fig. 4. Three limit (two small and one large) cycles are generated when (s1, s2) is in the region between H2 and C 0 .
Thus the system exhibits a Bautin bifurcation at k = k*, a = a*. If taking s1 = 0.00001, s2 = 0.00002, then we have that (s1, s2) is the region between B+ and H+ and hence there is an unstable limit cycle (see Fig. 2(left)). If taking s1 = 0.000035, s2 = 0.010688, then we have that (s1, s2) is the region between T and H+ and hence there are two limit cycles (the big one is stable and the small one is unstable) (see Fig. 2(right)). Example 3. This example verifies the result of Theorem 4.2. Let b = 0.5, c = 0.5625, q = 0.1. Then the condition (H6) holds and k** = 3.375099999999996, a** = 0.21333333333333332. Easy calculation shows that
H ¼ fs1 ¼ 145:01953124999997s2 ; s2 > 0g; P ¼ fs1 ¼ 145:01953125s2 þ 889:8925781249998s22 ; s2 < 0g: If we take s1 = 0.000147 and s2 = 0.000001, then (s1, s2) is the region between H and P. Let k = k** + s1 = 3.3751469999999997, a = a** + s2 = 0.21333233333333332, and
IðsÞ ¼ ks þ 0:005s2 : According to (iii) of Theorem 4.2, there is a unique unstable limit cycle bifurcating from (0, 0). Fig. 3 verifies this result.
X.P. Wu / Chaos, Solitons & Fractals 44 (2011) 28–42
41
Fig. 5. Two ‘‘big’’ limit cycles are generated when (s1, s2) is in the region between C 0 and C d : the outer one is stable while the inner one is unstable.
Example 4. This example verifies (iii) and (iv) of Theorem 4.4. Let b = 0.5, c = 0.5625, q = 0.8. Then k** = 0.914063, a** = 3.69778. Easy calculation shows that
H2 ¼ fs1 ¼ 0:184206s2 ; s2 < 0g; C 0 ¼ fs1 ¼ 0:196939s2 ; s2 < 0g; C d ¼ fs1 ¼ 0:201805s2 ; s2 < 0g: If we choose s1 = 1.8446108 and s2 = 0.0000001, then (s1, s2) is the region between H2 and C 0 . Let k = k** + s1, a = a** + s2. Choose
IðsÞ ¼ ks 0:00001s3 : According to (iii) of Theorem 4.4, there are three limit cycles bifurcating from (0, 0) (Fig. 4). However if we choose s1 = 0.000000201756 and s2 = 0.000001, then (s1, s2) is the region between C 0 and C d . Let k = k** + s1, a = a** + s2. Choose
IðsÞ ¼ ks 0:0001s3 : We have two ‘‘big’’ limit cycles (Fig. 5). According to (iv) of Theorem 4.4, the outer one is stable while the inner one is unstable.
6. Conclusion In this paper, we have investigated dynamical behaviors of the Kaldor model of business cycle through studying both Bautin (or degenerate Hopf) and BT singularities using (k, a) as the bifurcation parameter. By applying the normal form theory, we are able to predict their corresponding bifurcation diagrams such as Hopf, homoclinic and double limit cycle bifurcations. In economics, limit cycles are very important because they stand for cyclic oscillations of variables, which represent coexistence and future behaviors of business systems. Acknowledgement The authors would like to express their gratitude for valuable comments on this manuscript from the referees and the editor. References [1] [2] [3] [4] [5]
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