IERCU
Institute of Economic Research, Chuo University
50th Anniversary Special Issues
Discussion Paper No.20６

Existence of a 2D Torus in a Continuous-Time Modelof a Liquidity Trap
Eiji Tsuzuki
Shunsuke Shinagawa
Tomohiro Inoue

August 2013I

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

Existence of a 2D Torus in a Continuous-Time Modelof a Liquidity Trap
Eiji Tsuzuki∗, Shunsuke Shinagawa†, and Tomohiro Inoue†

Abstract
The main purpose of this study is to show the existence of a 2D torus in amacrodynamic model of a liquidity trap. The model developed here has two steadystate points: a targeted steady state and a liquidity trap. When a 2D torus exists
around a targeted steady state, an economy will keep fluctuating on a torus andnever reach the state, unless the variables jump. We develop an analysis using thecontinuous-time New Keynesian dynamic general equilibrium model proposed by
Benhabib et al. (2003). Although their model takes the form of “money in the
production function,” we reconstruct it into the more typical “money in the utility
function.”

JEL Classification: E32; E52
Keywords: new Keynesian DGE model, backward-looking interest-rate rules, 2D torus,
homoclinic orbit, Hopf bifurcation

1

Introduction

Two types of New Keynesian dynamic general equilibrium (DGE) model exist: one attributes the cause of liquidity traps to a decrease in the natural rate of interest, while∗

Corresponding author. Institute of Economic Research, Chuo University, 742-1 Higashinakano,Hachioji, Tokyo 192-0393, Japan; Faculty of Economics, Daito Bunka University, 560 Iwadono, Higashimatsuyama, Saitama 355-8501, tel.: +81-493-31-1503 (ext. 6174); fax: +81-493-31-1512; e-mail:tsuzukie5@gmail.com.
†
Faculty of Political Science and Economics, Waseda University, 1-6-1 Nishiwaseda, Shinjuku-ku,
Tokyo 169-8050, Japan.

1the other to the emergence of multiple steady states. Eggertsson and Woodford (2003)
is representative of the former, while Benhabib et al. (2001) is that of the latter. This
study is based on the latter, in which the liquidity trap is regarded as the result of theemergence of multiple steady states.
Benhabib et al. (2001) demonstrate that a steady state other than that targeted by thecentral bank can emerge when a zero lower bound on nominal interest rates is considered.
If the central bank conducts an active interest rate policy that satisfies the Taylor principle
in the neighborhood of a targeted steady state, the economy will be trapped in a lowinflation (and sometimes deflationary) state.1 They call this low-inflation steady state aliquidity trap.
Benhabib et al.’s (2001) model compounds a two-dimensional system of differential
equations. They completely analyze the dynamics and demonstrate the existence of periodic solutions, a homoclinic orbit, and a path connecting the targeted steady state withthe liquidity trap (i.e., a saddle connection). Further, Benhabib et al. (2002a) propose amodel that compounds a system of difference equations to show the existence of chaos.
They suggest that fiscal policies that can prevent an economy from entering a liquidity
trap, considered by Benhabib et al. (2002b), are ineffective in avoiding chaos.
Moreover, Benhabib et al. (2003) propose a continuous-time model of liquidity traps
that incorporates a backward-looking interest-rate rule under which the central bank adjusts the nominal interest rate in response to a weighted mean of past inflation rates.
Their model compounds a three-dimensional system of differential equations. As in Benhabib et al. (2001), Benhabib et al. (2003) demonstrate that multiple steady states (i.e.,a targeted steady state and a liquidity trap) can emerge and that periodic solutions, homoclinic orbits, and saddle connections exist. However, they make little reference to a
torus.
This study examine the existence of a 2D torus under plausible parameter values ina continuous-time New Keynesian DGE model, which incorporates a backward-looking
interest-rate rule with zero lower bound on the nominal interest rate.
Applying Kopell and Howard’s (1975) theorem, Benhabib et al. (2003) demonstratethat a homoclinic orbit can exist around a targeted steady state. If a 2D torus does
not exist, a solution originating from inside the orbit will converge to the steady state,because it is a stable spiral point (indeterminate). However, if a 2D torus exists, a solution
1

For general arguments about the effects of interest-rate policies using the New Keynesian DGE model,
see Woodford (2003), Galı́ (2008), and Walsh (2010).

2

originating within the orbit will never attain the steady state unless the variables jump,because solutions keep fluctuating on a torus. Therefore, alongside the liquidity trap,
there exists another possibility that an economy cannot reach a targeted steady state.
Moreover, as Behnabib et al. (2002a) suggest, these fluctuations cannot be avoided byfiscal policies that can prevent an economy from falling into liquidity traps.
This paper is organized as follows. Section 2 proposes a continuous-time new Keynesian DGE model incorporating a backward-looking interest-rate rule with zero lower
bound on the nominal interest rate. Section 3 presents a theorem needed to verify theexistence of a homoclinic orbit and applies it to the model proposed in Section 2. Section
4 set the parameters at plausible values and numerically examines the existence of a 2D
torus. Section 5 presents the conclusions.

2

The model

Although Benhabib et al.’s (2003) model takes the form of the “money in the production
function,” we reconstruct it into the more typical “money in the utility function.”
The model comprises three economic agents: firms, households, and the consolidated
government (central bank + government). We assume a perfectly competitive goods
market, a monopolistically competitive labor market, sticky nominal wages, and flexible
prices.
Firms
Firms aggregate a differentiated labor force li (i ∈ [0, 1]) supplied by households according
to the Dixit-Stiglitz function as follows:2
φZ 1 φ−1  φ−1l=li φ di,(1)0where l is the composite labor and φ (> 1) is the elasticity of labor supply.
Firms produce goods using composite labor l and a constant volume of input k thatthey possess (e.g., land). We specify the form of the production function as a constantelasticity of substitution:1y = [ζlδ + (1 − ζ)k δ ] δ ,2

See Dixit and Stiglitz (1977) and Blanchard and Kiyotaki (1987) for details.

3(2)where 0 < ζ < 1 and δ is the technical substitutability between l and k.3 For simplicity,
we employ the following assumption:
Assumption 1 l and k are perfectly substitutable (δ → 1).
Given the nominal wage rate of labor i Wi and the volume of composite labor l, firms
R1
determine the volume of li to minimize total cost 0 Wi li di. The first-order condition for
optimality is given by
li =



Wi
W

−φ

l,

(3)

where W is the nominal wage rate of composite labor defined as
W =

Z 1

Wi1−φdi

0

1
 1−φ

.

(4)

Because the goods market is perfectly competitive, firm profits are zero. Further, we
assume that the value created by input k is distributed to households in a lump sum.
Then ζp = W holds, where p is the price of goods.
Households
Households obtain utility from consumption c and real money balances m,4 and disutility
from labor li and wage negotiations ωi (reflected by the change rate in nominal wage).
We specify the form of the instantaneous utility function asη
li1+ψ
ln c + ln m −− (ωi − ω̄)2 ,1+ψ 2
where ψ (> 0) denotes the elasticity of marginal disutility of labor and η (> 0) denotes
the stickiness of nominal wages; if η > 0, nominal wages are sticky, and if η → 0, they are
flexible.5 ω̄ is a constant that reflects households’ subjective views about inflation. They
take wage adjustment costs when the rate of change in the nominal wages differs from ω̄.
3

In Benhabib et al.’s (2003) model, real money balances m enter Eq. (2) in place of k as an input.
For precision, we must write c as ci and m as mi ; however, because households are identical in
consumption and money holdings, we simply denote them by c and m, respectively.
5
We followed Rotemberg’s (1982) formulation of wage adjustment cost. η2 (ωi − ω̄)2 can be regarded
as psychological stresses caused by wage negotiations.
44Assuming that assets A are composed of money M and bonds B
A = M + B,
we obtainȦ = RB + Wi li + p(1 − ζ)k − pc − pτ,(5)where R is the nominal interest rate of bonds. Eq. (5) indicates that income p(1 − ζ)k as
well as capital and wage income increase assets and that consumption and tax pτ decrease
them. Rewriting Eq. (5) in terms of real variables, we obtainȧ = ra + wi li + (1 − ζ)k − c − τ − Rm,(6)where a = Ap is real asset balances, r = R − π is the real interest rate, and wi = Wpi is thereal wage rate of labor i.
Household i determines the volume of c, m, and ωi subject to the demand function
for labor i (3) and the budget constraint (6) to maximize the discounted present value ofthe stream of utility, given by
#
Z ∞"
li1+ψ
η
Ui =
ln c + ln m −− (ωi − ω̄)2 e−ρt dt,
1
+
ψ
2
0
where ρ (> 0) is the subjective discount rate.
First-order conditions for optimality are given by (see Appendix A.1)ċc+π+ρ=R = ,cm
φl1+ψ (1 − φ)ζlω̇ = ρ(ω − ω̄) −
−
.
η
ηc

(7)
(8)

Consolidated government
We formulate the central bank’s policy rule as follows:Dp

R = R∗ e R∗ (π −π ) ,∗(9)which represents a backward-looking interest-rate rule with zero lower bound on the nominal interest rate.6 D (> 0) denotes the elasticity of the interest rate with respect to the6

The original Taylor rule by Taylor (1993) assumes that the nominal interest rate responds both tothe inflation rate and the output (GDP gap). However, there is a positive correlation between the twovariables; hence, there are no qualitative differences in the results of two cases where only inflation isconsidered and where both inflation and output are considered. For the sake of simplicity, we consider aspecial case where the nominal interest rate responds only to the inflation rate.

5

inflation rate. If D > 1, the central bank conducts an active policy around the targetedsteady state (π ∗ ); however, it conducts a passive policy if D < 1. π p is a weighted meanof past inflation rates, defined as
Z tp
π (t) = bπ(s)e−b(t−s) ds,(10)−∞

where b (> 0) denotes the degree of backward looking of the central bank. As Benhabib etal. (2003) show, if the central bank conducts an active policy around the targeted steadystate, another steady state with low inflation (i.e., a liquidity trap) emerges.
Differentiating Eq. (10) with respect to time t, we obtainπ̇ p = b(π − π p ).

(11)

Government expenditures are assumed to be zero. Then the budget consitarint of theconsolidated government is given byḂ + Ṁ = RB − pτ.
Rewriting this equation in terms of real variables, we obtainȧ = ra − τ − Rm.
The present discounted value of total government liabilities must converge to zero:Rt

lim a(t)e− 0 r(s)ds = 0.

t→∞This type of fiscal-monetary regime is refered to the Ricardian policy (see Benhabib etal. 2002b).
System of differential equations
Using the goods market equilibrium condition y = c, our model can be summarized in
the following system of differential equations:Dpċ = [R∗ e R∗ (π −π ) − π − ρ]c,π̇ = ρ(π − π̄) −∗φ( ζc − 1−ζ
k)1+ψ
ζ
η

π̇ p = b(π − π p ),where π̄ = ω̄ because ζp = W .
6(1 − φ)[1 − (1 − ζ) kc ]
−
,
η

(12)

Steady states
The steady-state value of π and π p in the targeted steady state is given by π ∗ = (π p )∗ =R∗ − ρ.
Define that
1+ψ



k
c 1−ζ
−
k
− (1 − φ) 1 − (1 − ζ) ,h(c) ≡ −φζζcwe then obtainπ̇ = ρ(π − π̄) + h(c)/η.
Hence, the long-run Phillips curve is given by π − π̄ = −h(c)/(ρη). Given the value of π,
the steady-state value of c is determined to satisfy this equation. Because the volume of
employment l = 1ζ (y − (1 − ζ)k) = 1ζ (c − (1 − ζ)k) must be non-negative, c must satisfy
the condition that c ≥ c ≡ (1 − ζ)k.
Function h(c) can be rewritten as follows:


φ
1
ψ
h(c) = − 1+ψ (c − c) − (1 − φ) (c − c) ≡ ĥ(c)(c − c).
ζ
c
∗

−π̄)
is the steadyTherefore, the value of c that satisfies the equation ĥ(c) = g(c) ≡ −ρη(πc−cstate value.

• ĥ(c) is monotonically decreasing for c > c:φψĥ′ (c) = − 1+ψ (c − c)ψ−1 − (φ − 1)c−2 < 0.
ζ• ĥ(c) = (φ − 1) 1c > 0 and limc→∞ ĥ(c) = −∞.
In addition, if π ∗ < π̄, function g(c) is drawn as shown in Fig. 1; if π ∗ > π̄, it isdrawn as shown in Fig. 2. Whenever π ∗ < π̄ and the value of π̄ is sufficiently close to π ∗ ,ĥ(c) = g(c) has two real roots (c∗ and ĉ1 ) in the range of c > c (see Fig. 1). On the other
hand, if π ∗ > π̄, ĥ(c) = g(c) has a unique real root (ĉ2 ) in the range of c > c independentof parameter values.7
If π ∗ = π̄, c becomes a steady-state value. The steady-state value c represents an equilibrium atwhich employment is zero. In our model, k and l are perfectly substitutable; hence, an economy can
produce and continue to consume goods using k only (without labor).
7

7

r

r

0

c c

ĉ1

∗

g(c)
c
ĥ(c)

Figure 1: Case where π ∗ < π̄ (two real roots exist)

0

c

ĉ2

c
g(c)

r

ĥ(c)
Figure 2: Case where π ∗ > π̄3

Existence of a homoclinic orbit

We present Kopell and Howard’s (1975) theorem that provides conditions for assuring theexistence of a homoclinic orbit.
Theorem 1 (Kopell and Howard, 1975; Theorem 7.1 and Corollary 7.1) : Let
8

Ẋ = Fµ,ν (X) be a two-parameter family of ordinary differential equations on ℜn , F smoothin all its arguments, such that Fµ,ν (0) = 0. Using a Taylor expansion, Ẋ = Fµ,ν (X) canbe writtenẊ = (A + µA1 + νA2 )X + Q(X, X) + R1 (X, µ, ν),
where A, A1 , A2 are n × n matrices, the vector Q(X, X) contains terms quadratic in xi
and is independent of (µ, ν), and R1 (X, µ, ν) = o(µxi , νxi , xi xj ). Also assume
1. dF0,0 (0) has a double-zero eigenvalue corresponding to a single eigenvector e.
2. The mapping (µ, ν) → (det dFµ,ν (0), σ(dFµ,ν (0))) has a nonzero Jacobian at (µ, ν) =(0, 0), where σ(dFµ,ν (0)) is the sum of the principal minors of dFµ,ν (0).
3. [dF0,0 (0)Q(e, e)] has rank n.
Then there is a curve f (µ, ν) = 0 such that if f (µ0 , ν0 ) = 0, Ẋ = Fµ0 ,ν0 (X) has ahomoclinic orbit. This one-parameter family of homoclinic orbits (in X, µ, ν space) is on
the boundary of a two-parameter family of periodic solutions. For all sufficiently smallvalues of |ν|, |µ|, if Ẋ = Fµ,ν (X) has neither a homoclinic orbit nor a periodic solution,
there is a unique trajectory joining the critical (fixed) points (i.e., a saddle connection).
Applying Theorem 1
Consider the three-dimensional system of differential equations (12). We transform variables so as to make the origin a fixed point:x1 = − ln(c/c∗ ),x2 = π − π ∗ ,x3 = π p − π ∗Then System (12) can be rewritten asDẋ1 = −R∗ e R∗ x3 + (x2 + π ∗ ) + ρ,
1+ψ
(1 − φ)[1 − (1 − ζ) ck∗ ex1 ]φ c∗ −x1 1 − ζ∗ẋ2 = ρ(x2 + π − π̄) −
e
−
k
−
,
η ζ
ζ
η

(13)

ẋ3 = b(x2 − x3 ).

Further, we assume that (µ, ν) = (b + Aρ21 , 1 − D),8 where A21 ≡ φ(1+ψ)( cζ − 1−ζk)ψ cζ +ηζ∗(1−φ)(1−ζ) ck∗.
η8Changing the value of µ produces a corresponding change in the value of b.

9∗

Denote that X = [x1 , x2 , x3 ]′ , then Eq. (13) can be written asẊ = Fµ,ν (X) = (F 1(X, µ, ν), F 2(X, µ, ν), F 3(X, µ, ν)).
Because F is smooth and Fµ,ν (0) = 0, the premises of Theorem 1 are satisfied. We canassert the following proposition.
Proposition 1 If A21 6= 0, there is a homoclinic orbit around the targeted steady state
for a certain set of parameter values.
Proof: See Appendix A.2.

4

Existence of a 2D torus

This section set the parameters at plausible values and demonstrates that a 2D torus cansurround the targeted steady state in the system of differential equations (13).
Following Benhabib et al. (2003), we set parameter values as shown in Table 1.9 In
the case where π̄ = 0.08 (π̄ > π ∗ ), we obtain c∗ = 0.2012 and ĉ1 = 0.8840 under Table
1.10 On the other hand, if π̄ = 0.001, we obtain ĉ2 = 0.8875.
π∗

D

φ

0.01 1.5 21

ρψ

0.005 1

η

ζ

k

350

0.8 1

Table 1: Parameter values

Steady state c∗
When the steady-state value of c is given by c∗ = 0.2012, we obtain A21 = −0.0568 6= 0.
Therefore, System (13) satisfies the conditions of Theorem 1 and has a homoclinic orbit.
Taking b as a bifurcation parameter, the dynamics of the system around the targetedsteady state c∗ vary as shown in Table 2, depending on the value of b.
9

We set ζ = 0.8 and k = 1 here, but the main results in this study are not dependent on these values
and hold for a relatively wide range of values for ζ and k.
10 ′h (0.2012) > 0 and h′ (0.8840) < 0. The long-run Phillips curve is given by π − π̄ = −h(c)/(ρη).
Hence, the relation is negative between c and π when h′ (c) > 0, whereas the relation is positive whenh′ (c) < 0. Thus, c = 0.8840 is viewed as the “usual” steady state.

10

Value of bb ∈ (0, bh )b = bh ≃ 0.0033
b ∈ (bh , bsc )b = bsc ≃ 0.0054
bsc < b

Unstable spiral (determinate).
Hopf bifurcation.
Stable spiral (indeterminate) & unstable cycles.
Homoclinic bifurcation.
Saddle connection.

Table 2: Dynamics around the targeted steady state c∗bh is the value of b that satisfies the conditions for a Hopf bifurcation, provided inAppendix A.3. In the setting of Table 1, the Hopf bifurcation is subcritical, and therefore
an unstable cycle exists around the stable spiral point. Moreover, we calculate the homoclinic bifurcation value bsc as a value where cycles vanish.11 Fig. 3 shows the numericalresult when b = bsc . A homoclinic orbit is evident.12
In addition, after plotting D on the abscissa and b on the ordinate, we can see thechanges in bh and bsc when D increases (Fig. 4).

x 10

-3

2
0
-2

x

3

-4
-6
-8
-10
-12
0.4
1

0.2
0.5
0

0
-0.5

-0.2
-1
x

-0.4

-1.5
x

2

1

Figure 3: Homoclinic orbit

11

Technically, the homoclinic bifurcation value can be obtained using the Melnikov method. See Guckenheimer and Holmes (1983, p. 371) for details.
12
Initial values of (x1 , x2 , x3 ) are (x1 , x2 , x3 ) = (0.811, 0.02, 0.001).

11

x 10

-3

7

6.5

6

Homoclinic bifurcation

b

5.5

5

4.5

Hopf bifurcation

4

3.5

3
1.1

1.2

1.3

1.4

1.5

1.6

1.7

D

Figure 4: Changes in bh and bsc
Fig. 5 shows the dynamics of a solution starting from a point inside the homoclinic
orbit. The two of three Lyapunov exponents converge to zero (Fig. 613 ). Hence, a 2D
torus exists inside the homoclinic orbit.
Such dynamics of a solution can also be seen for the values of b in the neighborhoodof bsc .
Steady states ĉ1 and ĉ2
When steady-state values of c are given by ĉ1 = 0.8840 (where π̄ > π ∗ ) and ĉ2 = 0.8875(where π̄ < π ∗ ), the values of A21 are given by A21 = 0.1004 6= 0 and A21 = 0.1015 6= 0,
respectively. Therefore, in these cases, System (13) also satisfies the conditions of Theorem
1 and has a homoclinic orbit.
We also confirmed that the two of three Lyapunov exponents converge to zero. Hence,a 2D torus exists.

13

We construct this figure based on a Matlab code presented by Wolf et al. (1985).

12

x 10

-3

2

0

-2

-4

-6

-8

-10

-12
0.4
0.2
0
-0.2
-0.4

1

0.5

0

-0.5

-1

-1.5

Figure 5: Dynamics of a solution starting from a point inside the homoclinic orbit

0.02

0.015

Lyapunov Exponents

0.01

0.005

λ1 = 0.00023508

0λ2 = -0.00010045-0.005λ3 = -0.00061044-0.01-0.015

-0.02
0

0.5

1

1.5
Time

2

2.5

3
x 10

4

Figure 6: Dynamics of Lyapunov exponents

135Conclusion

This study has analytically demonstrated the existence of a homoclinic orbit by applying
Kopell and Howard’s (1975) theorem to a variant of a continuous-time New Keynesian
DGE model of a liquidity trap (MIUF version of Benhabib et al’s 2003 model).
Under a plausible set of parameter values, we confirmed the existence of a 2D torus
inside a homoclinic orbit. Therefore, it is possible that a solution originating from a point
inside a homoclinic orbit keeps fluctuating on a torus and can never attain the targetedsteady state.

A

Appendix

A.1

Household optimization

The household optimization problem is described as
Z "∞Maximizec,m,ωi

Ui =

0

#
li1+ψ
η
2
ln c + ln m −− (ωi − ω̄) e−ρt dt,
1+ψ 2

subject toȧ = ra + wi li + (1 − ζ)k − c − τ − Rm,Ẇi = ωi Wi ,
 −φ
Wi
li =
l.
W
The Hamiltonian function of this problem can be written asi −φ 1+ψ
) l]
[( W
η
W
H = ln c + ln m −− (ωi − ω̄)2
1+ψ
2
"
#
 −φ
Wi Wi
+ µ1 ra +l − c − Rm + µ2 ωi Wi ,p
Wwhere µ1 and µ2 are the costate variables of a and Wi , respectively. We obtain the14

first-order conditions for optimality:∂H1= − µ1 = 0,
∂c
c
1
∂H
=
− µ1 R = 0,∂mm
∂H
= −η(ωi − ω ∗ ) + µ2 Wi = 0,∂ ω̄∂Hµ̇1 = ρµ1 −= (ρ − r)µ1,∂aφli1+ψ
li∂Hµ̇2 = ρµ2 −= ρµ2 − [+ µ1 (1 − φ) + µ2 ωi ].
∂Wi
Wi
p

(A.1)
(A.2)
(A.3)
(A.4)
(A.5)

The transversality conditions arelim a(t)e−ρt = 0,t→∞lim Wi (t)e−ρt = 0.

t→∞

The Hessian of H(c, m, ωi ), which we denote as |H|, is given by
− c12
0
0
1
|H| = 0 − m2 0 .
00−ηThe principal minors of |H|, which we denote |H1 |, |H2 |, and |H3 |, are given by
1
− c12
0
1
1
|H1 | = − 2 , |H2 | == 2 2 , |H3 | = |H| = − 2 2 .
1
c
cm
c mη
0 − m2
Because |H1 | < 0, |H2 | > 0, and |H3| < 0, the second-order conditions for maximizing H
are satisfied.
Because households are symmetric, we can drop subscript i from li , Wi , and ωi ; li = l
Wi = W and ωi = ω. Considering these expressions, Eqs. (A.1)–(A.5) are combined toobtainċc+π+ρ=R = ,cm
φl1+ψ (1 − φ)ζlω̇ = ρ(ω − ω̄) −
−
.
η
ηc

15

(A.6)
(A.7)

A.2

Applying Theorem 1 to the system of differential equations(13)

Condition 1 The Jacobian matrix of F is given byD
0 1 −De R∗ x3

dFµ,ν (X) = J = Z21 ρ
0
,
0 b
−b
where Z21 ≡ φ(1+ψ)( cζ e−x1 − 1−ζk)ψ cζ e−x1 +ηζEvaluating J at X = 0, we obtain∗∗(1−φ)(1−ζ) ck∗ ex1
.
η




0 1 −DdFµ,ν (0) = J ∗ = A21 ρ 0  .
0 b −b

The characteristic equation of J is written as∆(λ) = det(J ∗ − λI)


−λ
1
−D


= det A21 ρ − λ0 0b−b − λ= λ3 − (ρ − b)λ2 − (ρb + A21 )λ − A21 b(1 − D).

Because b = b̃ ≡ −A21 /ρ and D = 1 hold at (µ, ν) = (0, 0), we obtain
0 1 −1

∗
dF0,0 (0) = J0,0
= A21 ρ 0  .
0 b̃ −b̃.

The characteristic equation of this matrix is given by∆0,0 (λ) = λ3 − (ρ − b̃)λ2 = λ2 (λ − (ρ − b̃)),which has double-zero eigenvalues and one nonzero eigenvalue of ρ − b̃. Hence, Condition
1 is satisfied.

16

The eigenvector associated with zero eigenvalues e = [e1 , e2 , e3 ]′ satisfies
dF0,0 (0)e = 0e,  e2 − e30  A21 e1 + ρe2  = 0 ,b̃e2 − b̃e30and therefore

where q is an arbitrary constant.

 
1
 
e = q  b̃  ,b̃Condition 2
det dFµ,ν (0) = A21 b(1 − D)


A21
ν.
= A21 µ −
ρHence, the Jacobian matrix of the map (µ, ν) → (det dFµ,ν (0), σ(dFµ,ν (0))) is given by#"
A21 (1 − D) A21 b.
σµσν
Evaluating this matrix at (µ, ν) = (0, 0), we obtain"#
0
A21 b̃
.
σµ (dF0,0 (0)) σν (dF0,0 (0))
This matrix cannot become zero as long as b̃ 6= 0 (i.e., A21 6= 0). Therefore, if b̃ 6= 0,
Condition 2 is satisfied. There is no need to calculate σ because it is only necessary toconfirm that the above matrix is nonzero.
Condition 3 The Hessian matrix of F 1 is given by


0 0
0
2
d F1 
H1 == 0 0
0
,
dX 2
D
2
0 0 −De R∗ x 3R∗
17the Hessian matrix of F 2 is given by−x1 (1+ψ)
−A
(1
+
ψ)e
0
0
21
d2 F 2 

H2 =
=
0
0 0 ,
2dX
0
0 0
and the Hessian matrix of F 3 is given by



0
0
0
d2 F 3 H3 == 0 0 0 .
2
dX
0 0 0

Evaluating these matrices at (X, µ, ν) = (0, 0, 0), we obtain−A21 (1 + ψ) 0 0
0 0
0




H 1 = 0 0
00 0 ,
0  , H2 = 0
0 0
0 0 − R1∗

Using these expressions, we obtain

H 3 = 0.

x2
1 ′X H1 X− 21 R3∗2 
Q(X, X) =  12 X ′ H2 X  = − 12 A21 (1 + ψ)x21  .
1 ′X H3 X
0
2







Substitute the eigenvector e = [1, b̃, b̃]′ into X to obtain2− 12 Rb̃ ∗Q(e, e) = − 12 A21 (1 + ψ) .
0

Therefore,20 1 −1− 21 Rb̃ ∗[dF0,0 (0)Q(e, e)] = −b̃ρ ρ 0 − 21 A21 (1 + ψ) .
0 b̃ −b̃0As long as b̃ 6= 0 (A21 6= 0), the rank of this matrix is 3. Hence, if b̃ 6= 0, Condition 3 issatisfied.

18

A.3

Hopf bifurcation theorem

The Hopf bifurcation theorem for a three-dimensional system of differential equations
represented by the coefficients of a characteristic equation is proposed by Asada (1995).
Theorem 2 (Asada, 1995) : There are periodic solutions that bifurcate from the fixed
point X ∗ if and only if the characteristic equation ∆(λ; µ) = λ3 + a1 λ2 + a2 λ + a3 = 0
satisfies the following conditions at µ = µ1 :(i) a1 6= 0, a2 > 0, ∆1 ≡ a1 a2 − a3 = 0,1(ii) d∆6= 0.
dµAcknowledgments
We appreciate comments and suggestions from Professors Hiroyuki Yoshida (Nihon University), Toichiro Asada (Chuo University), Akio Matsumoto (Chuo University), and theseminar participants at Chuo University.

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