A Simple Economic Growth Model with a Time Lag―Reconsideration of Haavelmo’s 1954 Model―
Takumi Ohkuma∗
Graduate School of Economics, Chuo University

Abstract
It has been examined that the parameter eﬀected on the stability of the Haavelmo’s
economic growth model is diﬀerent whether it is modeled by continuous time or discrete
time. In this paper, we are attempt to bridge the gap by introducing the time lag into
the model. Also, we examine the relation between the ﬁxed time lag and the continuously
distribute time lag. This is shown by describing the partition curbe in the parameter space
for the case assuming the type of time lag. Moreover, we indicate the possibility that theclosed orbits occur.

1

Introduction

It has been recognized that there are time lags in the economy since early times and iswell known that Kalecki (1935) is the pioneering work of the study of macro economy withtime lags. Until recently, the study took cognizance of the existence of time lag has been
not many studyied, in spite of the issue was posed long ago. It is considered that tracting
the diﬀerential-diﬀerence equation is mathematical diﬃculty. But lately, the number of thestudy took cognizance of the existence of time lag has been increasing. For example, there are
Jarsulic (1993), Asada and Yoshida (2001), Cai (2005), De Casare and Sportelli (2005), Fanti
and Manfredi (2007), Yoshida and Asada (2007), Matsumoto (2008), De Casare and Sportelli(2012) and others. These study focused largely on the eﬀects of the government’s stabilization
policy with time lags. In this paper, we intoroduce a time lag into a simple economic growthmodel studied by Haavelmo (1954) and examin the eﬀect of a time lag by using the analyticalmethod be seen in the recent research.
As seen in section 2, the model studied by Haavelmo (1954) is the very simple. But theparameter eﬀected on the stability of the system is diﬀerent, according to whether it is modeledby the continuous time or by the discrete time. For example, Stutzer (1980) has fully characterized the discrete version macro model formulated by Haavelmo. He has shown that the chaotic
behavior occur at a certain value of the structural parameter. In this paper, we examine whichparameter eﬀect on the stability of the system with a time lag. This study may be attempt to∗ohkumatakumi@hotmail.co.jp

1bridge the gap that exists between the system modeled by the continuous time and the systemmodeled by the discrete time.
There are the issue when the time lag introduced to the model. It is the type of the timelag. In this paper, we consider that two types of the time lag―ﬁxed time lag and continuously
distributed time lag―. Those types are used by De Cesare and Sportelli (2005) and by Fanti
and Manfredi (2007) respectively. In this paper, we also consider a relation of those types.
The paper is organised as follows. Section 2 brieﬂy survey the Haavelmo’s 1954 model.
There, we concern that which parameter eﬀect on the stability of the molel formulated various
methodology. Section 3 contains a description of the model with a ﬁxed time lag and qualitative
study. On the other hand, in section 4, we study the model with continuously distributed timelag. Section 5 is devoted to the comparison of these lags. Section 6 concludes.

2

Haavelmo’s 1954 Model
The following simple economic growth model has been constructed by Haavelmo (1954):Ṅ (t)
N (t)
=α−β
N (t)
X(t)

(1)

X(t) = F [N (t)](2)

where α and β are positive constents and where X(t) is the total ﬂow of means of subsistenece
available to the population. N (t)/X(t) is the ‘population density’ at time t and denoted 1/x(t).
x(t) = X(t)/N (t) may be interpreted as subsistenece available to per capita the population.
If X(t) is constant, then the system is ‘logistic equation’. In that case, the high growth rateof population Ṅ (t)/N (t) is achieved by the low N (t) at least temporarily, then the low growthrate of population Ṅ (t)/N (t) as incresing N (t) due to pejoration of the economic situation. For
that reason, the function draw S curve. On the other hand, Haavelmo (1954) model has X(t)that depend on N (t). X(t) is result of ‘production’ and incresing N (t) product high X(t). In
this case, Ṅ (t)/N (t) increse due to incresing x(t) enrich people’s lives, but the stability of thesystem is not evident.
Also the stability conditions when the system with the production function is linear or
nonlinear has been examined by Haavelmo (1954). The ﬁrst is the following linear productfunctionX(t) = aN (t) + b(3)

where a and b are non-negative constants. The second is the folowing nonlinear prodct function
X(t) = A[N (t)]awhere A and a are positive cnstants and a is interprited as the elasticity of production.

2(4)

The steady-state solution and the stability condition of the system (1) (3)Substituting (3) into (1), the system is(αa − β)N (t) + αbṄ (t)=.
N (t)aN (t) + b(5)

Setting Ṅ (t) = 0, the steady-state solution N̄ of the system is
N̄ =

αb
.
β − αa(6)

Total-diﬀerentiating at the steady-state solution,(αa − β)(aN̄ 2 + 2bN̄ + αb2 )dṄ (t)=.
dN (t) N (t)=N̄
(aN̄ + b)2(7)

Therefore, if the condition αa − β < 0 is satisﬁed, then the system (1) (3) is stable.

The steady-state solution and the stability condition of the system (1) (4)On the other hand, when we consider the steady-state solution and the stability condition of
the system (1) (4), the system (1) (4) is transformed the equation about N (t) into the equation
about x(t) for simplicity. Diﬀerentiating (4) with t and dividing by N (t), the system isẊ(t)= αax(t) − βa.
N (t)

(8)

SubtractingṄ (t) X(t)
X(t)
=α
−β
N (t) N (t)N (t)from (8), we obtainẋ(t) = α(a − 1)x(t) − β(a − 1).
Consequently, we can conﬁrm that the steady-state solution of the system is
x̄ =

β
α

(9)

and the general solution of the system is(β ) α(a−1)t βx(t) = x(0) −
e
+ .
α
α
Therefore, if the condition a < 1 is satisﬁed, then the system (1) (4) is stable.

3

(10)

The discrete time version of the system (1) (4)Stutzer (1980) has characterized the behavior of the discrete time system (1) (4) fully. As
contrasted with the continuous time Haavelmo’s model, it has been indicated that there is acomplex periodic and random aperiodic orbit structure dependent on the value of elasticity of
production a. The discrete time logistic model is
Nt+1 − Nt
Nt
=α−β
Nt
Xt

(11)

Xt = ANta(12)

where all symbols are deﬁned previously. Substituting (12) into (11) and letting Nt = [A(1 +1α)/β] (1−a) zt , we havezt+1 = (1 + α)zt [1 − zt1−a ].

(13)

Stutzer (1980) has indicated that (13) has chaotic behavior for certain value of a.
The parameter eﬀected on the stability of the system is diﬀerent by the shape of the productfunction as seen above. In this paper, we assume the nonlinear prodct function. That is, we
introduce a time lag into the system (1) (4) and examine the behavior of the system.

3

Fixed Time Lag

In Haavelmo’s 1954 model, there is no time lag. This meen that the growth rate of population
at time t is determined by ‘population density’ at time t . If X(t)/N (t) is interpreted as ‘average
income’, then we can guess that the growth rate of population react laggardly for it. Becauseof the people expect future income based on current income and then decide to bear. Thatis, the growth rate of population follow economic conditions. First, we examin a limit casethat the people expect future income at a speciﬁc point in past time. The time lag of thistype is plausible, if the lengths of the lags are not distributed in a random manner over all the
population.
To study the stability of the case noted above, we replace (1) by (14):Ṅ (t)N (t − θ)
=α−β
N (t)
X(t − θ)(14)

The equation (14) represents that the variation in growth rate of the population at time t
depend on subsistenece available to per capita population at time t − θ.
1There are two equiliblium points in the system, zero and (αA/β) 1−a . We assume that there
1is the equiliblium point N̄ = (αA/β) 1−a only, because of two populations are nessesary for thegrowth. Substituting (4) into (14) and approximeting at N̄ , we obtainṄ (t) − N̄ = −(1 − a)α(N (t − θ) − N̄ ).

4

(15)

Setting x(t) = N (t) − N̄ , (15) is represented byẋ(t) = −(1 − a)αx(t − θ).

(16)

Substituting the candidate solution x(t) = Ceλt for (16), we have the following characteristic
equation.
λ + (1 − a)αe−λθ = 0(17)

It is known that the necessary and suﬃcient condition for locally asymptotical stability is thatthe all real parts of the eigenvalues are negative. So, we will start by characterizating thecharacteristic root in the next subsection.

3.1

The characterization of characteristic root : The role of elasticity of
production and the number of characteristic roots with real partto be writen

The case of θ = 0
The characteristic equation is
λ = −(1 − a)α.

(18)

Therefore, if there is no lags, then stability of the system depend on the value of elasticity of
production a as is the case in Haavelmo’s 1954 model with nonliner production function;・If θ = 0 and a > 1, then λ > 0・If θ = 0 and a = 1, then λ = 0・If θ = 0 and 0 < a < 1, then λ < 0
The case of θ > 0
We rewrite the characteristic equation (17) by the form
Z1 (λ) ≡ e−λθ = −1
λ ≡ Z2 (λ).
(1 − a)α(19)(17) or (19) is transcendental equation, so we can not solve explicitly. It is known that there areinﬁnite number of roots in complex number renge. For this reason, we investigate the propertyof real part of roots by using Figure 1. Here, a1 < a2 < a3 .

Figre 1 is here

From Figure 1, the following proposition can be derived.
5Proposition 1. There are two roots with real part at most in the Haavelmo’s 1954 model with
a ﬁxed time lag. And the stability of the model depend on the value of elasticity of production
a. It is nessesary for the stable of the model that a exist in (0, 1):• If a > 1, then there is one positive real root.
• If a = 1, then there is no real root.
• If {a|0 < a < 1} is suﬃciently close to 1, then there are two negative real roots.
• If {a|0 < a < 1} is suﬃciently close to 0, then there is no real root.
• There are mltiple root at the intermediate values of {a|0 < a < 1}.

3.2

The partition curve that divide stability region from unstability region
in the parametar spase

Next, to examine the partition curve, substituting λ = iv into (17) and using Euler’s formula,
we obtain the following equation.
iv + (1 − a)α cos vθ − i(1 − a)α sin vθ = 0
Setting the real and imaginary parts is zero respectively, we have{
0 = cos vθ
v = (1 − a)α sin vθ.

(24)

(25)

Adding their squares, we obtainv2= sin2 vθ + cos2 vθ,[(1 − a)α]2
= 1,

(26)

v = (1 − a)α.

(27)that isSubstituting it into (25), we have{cos vθ = 0
sin vθ = 1.

(28)

Thus the stability switch may occur when following equalityvθf =π+ 2nπ,
2n = 0, 1, 2 · · · ,(29)or more speciﬁcally, eliminating v by substituting (27) into (29),
θf =

(π
)
1
+ 2nπ ,
(1 − a)α 2
6n = 0, 1, 2 · · · ,(30)where sbscript ‘f ’ indicates ‘ﬁxed time lag’. See Figure 2.

Figre 2 is here

And we can conﬁrm that this curve is the partition curves, because of[( dλ )−1 ] 1
Re= 2 > 0.
dθvThis calculation is given by Appendix A.
Next, we examine complete stability condition by using Heyes’ theorem1 . Useing this theorem, we can conﬁrm that the stable region is under the partition curve (30) when n = 0.
Therefor, we can derive the following proposition.
Proposition 2. Haavelmo’s 1954 model with a ﬁxed time lag is locally stable if a combinationof parameters is under the partition curb
θf =

1
π
(1 − a)α 2

in the parameter space (θ, (1 − a)α).
Proof. See Appendix B.

3.3

The existence of closed orbits in the case of the ﬁxed time lag

Proposition 2 represents when the time rag θ is suﬃciently small the system is locallystable and when the time rag θ is suﬃciently large the system is locally unstable for given
othere parameters. Therefor, the system has possibilities generating the closed orbits. In thissubsection, we conﬁrm that a closed orbit arise for a value of a and α by using the Hopf
bifurcation theorem2 .
Proposition 3. The system have the possibility that closed orbit occur at some velues of theelasticity of production a > 0.
Proof. See Appendix C.

1See, for example, Appendix B in this paper and Hayes (1950) and Gandlfo (1997) and Asada and Yoshida(2001).
2
See, for example, Appendix C in this paper and Kuang (1993) and Asada and Yoshida (2001).

7

4Continuously Distributed Time Lags

In this section, we introduce uncertain delay into the model. As Fanti and Manfredi (2007)
remarks, the aggregate ﬁxed delay is attained as a limit case of the continuouse-time distributed
lag. When the diﬀerent agents have the diﬀerent length of the lag, the length of the lags are
distributed over all the population. For this reason, it may be regarded that using continuously
distributed time lags is appropriate in this case.
We will start by replacing (14) by (31).
Ṅ (t)β= α − [N e (t)]1−a
N (t)
A

(31)
1

1−a ,
where N e (t) is the expected population at time t. Linearizing (31) at N̄ = ( αAβ )ẋ(t) = −(1 − a)αxe (t)(32)

here x(t) = N (t) − N̄ , xe (t) = N e (t) − N̄ . Substituting∫ tex (t) =

w(t − s, θ, m)x(s)ds

(33)

0

into (32), we obtainẋ(t) = −(1 − a)α∫ tw(t − s, θ, m)x(s)ds.

(34)

0

The weighting function w(t − s, θ, m) is deﬁned as the following3 .
{1 − t−sθifTew(t − s, θ, m) =m(t−s)
1 m m+1(t − s)m e− θifm! ( θ )

m=0
m≥1

(35)

Figure 3 is here

Here m is an non-negative integer and θ is a positive real parameter. Substituting a candidate
solution x(t) = x0 eλt into (34), we ﬁnd
λ = −(1 − a)α∫ tw(t − s, θ, m)e−λ(t−s) ds

(36)

0

Letting t − s = u, (36) is
λ = −(1 − a)α

∫ t
0

3

See, for example, Bischi et al. (2010).

8w(u, θ, m)e−λu du(37)

When t → ∞

∫ ∞
0

(
λθ )−(m+1)
w(u, θ, m)e−λu du = 1 +
q

with

{
q=

1 if
m if

m=0
m ≥ 1,

so that we ﬁnally obtain the following form of the characteristic equation:(
λθ )m+1
λ 1++(1 − a)α = 0.
q

4.1

(38)

(39)

The partition curve in the case of the continuously distribute time lag

The case of θ = 0 and m ≥ 0
If θ = 0 and m ≥ 0, then (39) is equal to (18). Therefore, the stability of the system (34)with θ = 0 and m ≥ 0 depends on the value of the elasticity of production a only.
The case of θ > 0 and m = 0
If θ > 0 and m = 0, then (39) is1(1 − a)αλ2 + λ += 0.
θθSetting a0 = 1, a1 = 1/θ, a2 = (1 − a)α/θ, then

(40)

1
> 0,
θ
(1 − a)α∆2 = a1 a2 =>0(0 < a < 1).
θ2
Under a ∈ (0, 1), the system (31) with with θ > 0 and m = 0 is locally stable due to RouthHurwitz criterion4 .
∆1 = a1 =Proposition 4. If the weight function is exponential-type, the system (31) with continuously
distributed time lags is locally stable for any length of the time lag.
The case of θ > 0 and m = 1
If θ > 0 and m = 1, then (39) is2
1(1 − a)αλ3 + λ 2 + 2 λ +
= 0.
θ
θ
T2
Setting b0 = 1, b1 = 2/θ, b2 = 1/θ2 , b3 = (1 − a)α/θ2 , then
∆ 1 = b1 =

2
> 0,
θ

}
1 {2
∆ 2 = b1 b 2 − b0 b3 = 2− (1 − a)α ,θ θ(1 − a)α∆3 = b3 (b1 b2 − b0 b3 ) =
∆2 .
θ2
4

See, for example, Gandlfo (1997) for Routh-Hurwitz criterion.

9

(41)

We ﬁnd that ∆2 and ∆3 have the same sign under a ∈ (0, 1). Thus, using Routh-Hurwitz
criterion, the system (31) with θ > 0 and m = 1 is locally stable when the combination ofparameter meet θ < 2/{(1 − a)α}. Thus, in this case, the partition curve is depicted in Figure
4.

Figure 4 is here

Proposition 5. If the length of the time lag satisfy the inequality
θ < 2/(1 − a)α,then the system (33) with θ > 0 and m = 1 is local stable.
The case of θ > 0 and m = 2
If θ > 0 and m = 2, then (39) is12
6
88(1 − a)αλ 4 + λ3 + 2 λ2 + 3 λ +
=0
θ
θ
θ
θ3

(42)

Setting c0 = 1, c1 = 6/θ, c2 = 12/θ2 , c3 = 8/θ3 , c4 = 8(1 − a)α/θ3 ,∆1 = c1 = 6/θ > 0∆2 = c1 c2 − c0 c3 =

64
>0
θ3

}
1 { 512
−
288(1
−
a)α
θ5 θ
4608(1 − a)α∆4 = c1 c2 c3 c4 =>0
θ9∆3 = c1 c2 c3 − c21 c4 − c23 =Thus, if the combination of parameter meet θ < 16/9(1 − a)α then ∆3 > 0, so that the system(31) with θ > 0 and m = 2 is locally stable. Thus, in this case, the partition curve is depicted
in Figure 5.

Figure 5 is here

Proposition 6. If the length of the time lag satisfy the inequality
θ < 16/9(1 − a)α,then the system (34) with θ > 0 and m = 2 is locally stable.
10

4.2

The existence of closed orbits in the case of continuously distributed
time lagBy the analysis in the previous subsection, we will expect that the system with continuously
distributed time lag has closed orbits. So that, we can conﬁrm the possibility that the system(33) with θ > 0 and m = 1, 2 have closed orbits by using Hopf bifurcation theorem5 .
Proposition 7. There exist some nonconstant periodic solutions in the system with θ > 0 and
m = 1, 2 at some combinations of the parameter values θ > 0 and a.
Proof. See Appendix D about the case of the system with θ > 0 and m = 1. And see Appendix
E about the case of the system with θ > 0 and m = 2.

5Comparison of ﬁxed time lag and continuously distributed
time lagBy overlapping Figure 2 and Figure 4 and Figure 5, we can intuitively know the relation
between ﬁxed time lag and continuously distributed time lag. See Figure 6.

Figure 6 is here

Increasing the velue of the parameter m, the stability regioin narrow. In other word, the more
the degree of uncertainty of expectation is low, the more the system is going to unstable. But
we calculated to m = 2 due to the diﬃculty, we can expect that the partition curve with m = ∞coincide with the case of the ﬁxed time lag.

6

Conclusions

Unlike the nomal logistic equation, the stability is not certain in the Haavelmo’s 1954 modelthat the total ﬂow of means of subsistence available to the population X depqnded on the
population N . The parameter eﬀected on the stability of the system is diﬀerent whether tomodel by the continuous time or by the discrete time. And so, we examine which parametereﬀect on the stability of the system introduced the time lag―diﬀerential-diﬀerence equation―.
In this realistic case, All of the parameters exept for the parameter β eﬀect on the stability ofthe system. When the value of the time lag is large, the system often result in unstable. But
even so, the system often result in stable when the value of (1 − a)α is small. The stable regionof the system with the ﬁxed time lag is narrow then it with the continuously distributed timelag exceptin the case of m = ∞. It can be expected that if we make the parameter m larger
without limit than the system coinside with the system with the ﬁxed time lag. Moreover, thesystem have the potential of the occurring the closed orbits for two types of the time lag.
5

See Appendix D in this paper about Hopf bifurcation theorem.

11

Appendix A
To conﬁrm the dependence of λ on θ, we derive stability swichi condition from (17) by usinge−λθ = −λ/(1 − a)α,dλ = λ(1 − a)αe−λθ dθ + θ(1 − a)αe−λθ dλ,λ(1 − a)αe−λθdλ=dθ
1 − θ(1 − a)αe−λθ−1λ(1 − a)α (1−a)αλ=−λθ
1 − θ(1 − a)αe
−λ2=
1 + θλ−(iv)2=
1 + ivθ
v2=
1 + ivθ( dλ )−11ivθ
= 2+ 2.
dθv
v
That is[( dλ )−1 ] 1
Re= 2 > 0.
dθv

Appendix B
Theorem 1 (Heyes (1950)). All the roots of pes + q − ses = 0 have negative real parts if, andonly if,(1) p < 1,
1(2) p < −q < (p21 + p2 ) 2 ,where p1 is the root of the equation x = p tan x such that 0 < x < π. If p = 0, we take p1 = π/2.
Proof of Proposition 2.
Setting λθ = s and transforming the characteristic equation (17), we obtain−(1 − a)αθ − ses = 0.

(A-1)

(A-1) is equivalence to the characteristic equation in Heyes’ theorem when p = 0 and q =−(1 − a)αθ. Thus, these conditions in this theorem are replaced by the following inequalities.
0 < (1 − a)αθπ
(1 − a)αθ <
2

(A-2)
(A-3)

Rewriting (A-3),
θ<

π
1
.
(1 − a)α 2(A-4)

Therefor, we can conﬁrm that the stabe region is under the partition curve (30) when n = 0
and a exist in (0, 1).
12Appendix C
Theorem 2 (Hopf bifurcation theorem―diﬀerential-diﬀerence equation version―). Let ẋ(t) =F (x(t), x(t−θ); ε), x ∈ R, ε ∈ R be a miced diﬀerence and diﬀerential equation with a parameter
ε. Suppose that the following properties are satisﬁed.
1. This equation has smooth curve of equilibria F (x∗ , x∗ ; ε) = 0.
2. The characteristic equation Γ(ρ) ≡ ρ − a − be−θρ = 0 has a pair of pure imaginary
roots ρ(ε0 ), ρ̄(ε0 ) and no other root with zero real part, where a ≡ (∂F/∂x(t))∗ andb ≡ (∂F/∂x(t − θ))∗ are partial derivatives of F which are evaluated at (x∗ (ε0 ), ε0 ) withthe parameter ε0 .
3. d{Reρ(ε)}/dε 6= 0 at ε = ε0 , where Reρ(ε) is the real part of ρ(ε).
Then, there exists a continuous function ε(γ) with ε(0) = ε0 , and for all suﬃcientry smallvalues of γ 6= 0 there exists a continuous family of nonconstant periodic solutions x(t, γ) for theabobe dynamical equation, which collapses to the equilibrium point x∗ (ε0 ) as γ → 0. Theperiodof the cycle is close to 2π/Imρ(ε0 ), where Imρ(ε0 ) is the imaginary part of ρ(ε0 ).
Proof of Proposition 3.
We will start by getting Lemma 1 before proof Proposition 3. we can describe the curvethat divide the characteristic solution with two from the characteristic solution with no realpart. Setting s = λθ, (19) is
Z1 (s) ≡ e−s = −1
s ≡ Z2 (s).
(1 − a)αθ(B-1)

In the tangent point, the slope of Z1 and Z2 is equivalent. Therefor, diﬀerentiating both sidesof (B-1) with s, we ﬁnd
Z10 (s) ≡ −e−s = −1≡ Z20 (s).
(1 − a)αθ(B-2)

That iss = log[(1 − a)αθ].

(B-3)

Substituting (B-3) into (B-1),
e− log[(1−a)αθ] = −log[(1 − a)αθ],(1 − a)αθlog[(1 − a)αθ] = −1.

(B-4)

or
θ=

1
.
e(1 − a)α(B-5)

Lemma 1. In the parameter space (θ, (1 − a)α), if the characteristic equation (17) has no realroot, then the following inequality is satisﬁed.
− log[(1 − a)αθ] < 1
13

Next, we will proof Proposition 3 by reference to Asada and Yoshida (2001). Substituting√
λ = u ± vi (i ≡ −1) into (17), we obtainu ± vi + (1 − a)αe−θ(u±vi) = 0.

(B-6)

u + (1 − a)αe−θu cos(θv) + {±v ∓ (1 − a)αe−θu sin(θv)}i = 0.

(B-7)

Rewriting (B-6),

Setting real part equal zero and imaginary part equal zero respectively, we have{u = −(1 − a)αe−θu cos(θv)
v = (1 − a)αe−θu sin(θv).

(B-8)

Rewriting the second equation of (B-8), it isθ(1 − a)α sin(θv)
θv

(B-9)

log[θ(1 − a)α] + log[ sin(θv)
θv ]
.
θ

(B-10)

eθu =
or
u=

Substituting the second equation of (B-8) and (B-10) into the ﬁrst equation of (B-8), we obtainlog[θ(1 − a)α] + log[ sin(θv)θv ]= −(1 − a)αe−θu cos(θv),
θ
log

log

[ sin(θv) ]θv+θ(1 − a)αe−θu cos(θv) = − log[θ(1 − a)α],[ sin(θv) ] θ(1 − a)αe−θu sin(θv)= − log[θ(1 − a)α],+θvtan(θv)

A(θv) ≡ log[ sin(θv) ]
θv

+

θv
= − log[θ(1 − a)α] ≡ B(θ, a).
tan(θv)(B-11)

Let ﬁx θ and a to θ0 and a0 respectively. Here a0 is suﬃcently close to 0. In this case, there is
no real root due to Proposition 1. Due to Lemma 1, the solution of v is found by Figure B-1.

Figure B-1 is here

This show that there exist inﬁnite numbere of solutions vn for all n ∈ {0, 1, 2, 3, · · · }. Substituting this solution vn into (B-10), we obtain the solutions for u ;un =0 vn )log[θ0 (1 − a)α] + log[ sin(θθ0 vn ]

θ0
14,n ∈ {0, 1, 2, 3, · · · }.

(B-12)

Because ofduh< 0,d(θ0 vn )(B-13)

u 0 > u 1 > u 2 > u3 > · · · .

(B-14)For a value of a, the stability of the sysmate switch, that is u0 = 0. Moreover, considering
du0 /da0 6= 0, the point a = a0 is the Hopf bifurcation point.

Appendix D
Theorem 3 (Hopf bifurcation theorem―existence part―).
ẏ = ϕ(y, α)Consider this general system and suppose that for each α in the relevant interval it has an
isolated equilibrium point ye = ye (α). Assume that the jacobian matrix of ϕwith respect to y,evaluated at (ye (α), α) has the following properties:(H1) it possesses a pair of simple complex conjugate eigenvalues θ(α) ± iw(α) that become
pure imaginary at the critical value α0 of the parameter―i.e., θ(α0 ) = 0, while ω(α0 ) 6= 0―andno other eigenvalues with zero real part exist at (ye (α0 ), α0 );(H2) dθ(α)
dα α=α0 6= 0;
THEN the system has a family of periodic solutions.
Proof of Proposition 5.
The following theorems is used to conﬁrm that whether our system satisfy the condition(H1) and (H2) in Hopf bifurcation theorem6 .
Theorem 4 (Asada and Semmler(1995)). The characteristic equation, ∆(λ) ≡ λ3 + b1 λ2 +√b2 λ + b3 = 0, has a pair of pure imaginary roots ±hi (i ≡ −1, h 6= 0) if and only if b2 > 0 andb1 b2 − b3 = 0.
It is also indicated by Asada and Semmler(1995) that the following equation adaptate thecondition (H2).
∂∆(ρ)
6= 0
∂ρ ρ=ρ0
Here ρ0 is bifurcation value. We can get the next proposition by using both Theorem 4 andthis condition. If θ < 2{(1 − a)α} is satysfy then ∆3 > 0, so that the system is locally stable
by Routh-Hurwitz criterion. we can consider two case that θ is bifurcation parameter or a isbifurcation parameter. If T is suﬃcient small or a is suﬃcient close to 1, then ∆3 > 0. On thecontrary, if θ is suﬃcient large or a is suﬃcient close to 0, then ∆3 < 0. This meens that there
exist at least one value of θ̄ ∈ (0, ∞) and ā ∈ (0, 1) at which ∆3 = 0 by continuity. In addition,
6

See, for example, Asada and Semmler(1995) and Gandlfo(2010).

15

b2 is always positive by assumption, so that the condition (H1) in Hopf bifurcation theorem issatisyﬁed by theorem 3.
It is seen that the condition (H2) in Hopf bifurcation theorem is also stisﬁed by the following
inequality.
d∆2 (T ; ā)2{3}= 3 (1 − ā)α −
dθ
θ0
θ0
θ=θ0
{
3 }
2
(1
−
ā)α
−
=
2
2
[ (1−ā)α]3(1−ā)α=−[(1 − ā)α]4<08d∆2 (a; θ̄)α= 2 >0daθ̄a=a0
For that reason, under θ = θ0 or a = a0 , all condition in Hopf bifurcation theorem is satisﬁed.

Appendix E
The following theorems are also used to conﬁrm that whether our system satisfy the condition(H1) and (H2) in Hopf bifurcation theorem.
Theorem 5 (Asada and Yoshida(2003)).

1. The polynomial equation∆(λ) ≡ λ4 + c1 λ3 + c2 λ2 + c3 λ + c4 = 0
has a pair of pure imaginary roots and two roots with non-zero real parts if and only ifeither of the following set of conditions (A) or (B) is satisﬁed:(A) c1 c3 > 0, c4 6= 0, and ∆3 ≡ c1 c2 c3 − c21 c4 − c23 = 0.
(B) c1 = 0, c3 = 0, and c4 < 0.
2. The polynomial equation ∆(λ) has a pair of pure imaginary roots and two roots withnegative real parts if and only if the following set of conditions (C) is satisﬁed:(C) c1 > 0, c3 > 0, c4 > 0, and ∆3 ≡ c1 c2 c3 − c21 c4 − c23 = 0.
Theorem 6 (Asada and Yoshida(2003)).
1. Suppose that the conditions c1 (ε0 )c3 (ε0 ) > 0,c4 (ε0 ) 6= 0, and ∆3 (ε0 ) ≡ c1 (ε0 )c2 (ε0 )c3 (ε0 ) − c21 (ε0 )c4 − (ε0 )c3 (ε0 )2 = 0 are satisﬁed in
the following characteristic equation.
∆(λ; ε0 ) ≡ λ4 + c1 (ε0 )λ3 + c2 (ε0 )λ2 + c3 (ε0 )λ + c4 (ε0 ) = 0
Then, the condition (H2) in Hopf bifurcation theorem is equivalent to the following condition (D):(D) d∆3 (ε)/dε ε=ε0 6= 0
2. Suppose that the conditions c1 (ε0 ) = 0, c3 (ε0 ) = 0, c4 (ε0 ) < 0, and ∆3 (ε0 ) ≡ c1 (ε0 )c2 (ε0 )c3 (ε0 )−c21 (ε0 )c4 − (ε0 )c3 (ε0 )2 = 0 are satisﬁed in the characteristic equation. Then, the condition(H2)[in the Hopf bifurcation theorem
is equivalent to the following condition (E):]√02(E) c2 (ε0 ) + c2 (ε0 ) − 4c4 (ε0 ) c1 (ε0 ) − 2c03 (ε0 ) 6= 0
16

Proof of Proposition 6.
We can get the next proposition 6 by using both Theorem 5-ii and Theorem 6-i. If θ is
suﬃcient small or a is suﬃcient close to 1, then ∆3 > 0. On the contrary, if θ is suﬃcientlarge then ∆3 < 0. On the other hand, when a is suﬃcient close to 0, the following equation issatisﬁed.
)
1 ( 512lim ∆3 = 5− 288αa→0θθThus if 16/(9α) < θ, then ∆3 < 0. It is possible that taking a as bifurcation parameter, when
θ is suﬃcient large only. Here, we wont to ﬁnd the possible of existence of periodic behavior, sothat we assume θ satisfying16/(9α) < θ. Under this assumption, if a is close to 0, then ∆3 < 0.
These meens that there exist at least one value of θ̄ ∈ (0, ∞) and ā ∈ (0, 1) at which ∆3 = 0
by continuity. In addition, for any θ, c1 > 0 and c3 > 0 are satisﬁed always and c4 > 0 under
Assumption 1, so that the condition (H1) in Hopf bifurcation theorem is satisyﬁed by ii -(A) intheorem 5.
It is conﬁrmed by using Theorem 6 that the condition (H2) in Hopf bifurcation theorem isalso stisﬁed . ∆3 at θ = θ0 is∆3 = c1 c2 c3 − c21 c4 − c23}
1 { 512= 5− 288(1 − a)α
θ0 θ0
= 0.
Thus
T0 =

16
.
9(1 − a)αBy using T0 ,[]d∆3 (θ; ᾱ)3072 1440(1 − ā)α= sign − 7 +
dT
θ0
θ06
θ=θ0
= sign[−3072 + 1440(1 − ā)αθ0 ][16= sign − 3072 + 1440(1 − ā)α9(1 − ā)α]= sign[−512]< 0.
On the other hand, ∆3 at a = a0 isd∆3 (α; θ̄)
288α
=
>0
dα
θ̄
a=a0
here a0 = 1 − 16/(9θα). For that reason, under θ = θ0 or a = a0 , θ satisfying16/(9α) < θ, allcondition in Hopf bifurcation theorem is satisﬁed.

17

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