Nonlinear Duopoly Games with
Advertisement∗
Akio Matsumoto†Department of Economics
Chuo University
Hachioji, Tokyo, 192-0393, JAPAN
Ferenc Szidarovszky‡
Department of Systems and Industrial Engineering
University of Arizona
Tucson, Arizona, 85721-0020, USA
June 22, 2007

Abstract
This study introduces advertisement into the dynamic nonlinear
duopoly model. It demonstrates three major findings. The first isthat the model can be destabilized via either flip bifurcation or Hopf
bifurcation, the second is that a half-pitchfork bifurcation of output
occurs when the advertisement dynamics is periodic and the nonlinearity of the output dynamics becomes stronger, and the third is thatthe existence of attractor and the coexistence of attracting sets are
the main features of the model when it is locally unstable.
∗

We highly appreciate finanical supports from the Japan Ministry of Education, Culture, Sports, Science and Technology (Grant-in-Aid for Scientific Research (A) 16203019)
and the US Air Force Oﬃce of Scientific Research (MURI grant N14-03-1-150). We arealso grateful to J. Minagawa for his supporting computational studies.
†
742-1, Higashi-Nakano, Hachioji, Tokyo, 192-0393, Japan. akiom@tamacc.chuou.ac.jp; +81-426-74-3551 (tel); +81-426-74-3425 (fax).
‡
Tucson, AZ, 85721-0020, USA. szidar@sie.arizona.edu; +01-520-621-6557 (tel); +01520-621-6555 (fax).

1

1

Introduction

In the recent literature, several authors have demonstrated that nonlinear
oligopoly competition may be chaotic. Rand (1978) considers duopoly chaosfrom the mathematical point of view, Puu (1991) and Kopel (1996) examine duopoly chaos from the economic point of view, Puu and Sushko (2002)
and Bischi, Chiarella, Kopel and Szidarovszky (2008) give a comprehensive
summary of recent developments in nonlinear oligopoly theory. These works
indicate that dynamic nonlinear oligopoly models may explain various complex behavior observed in real economy. It has been well-known that advertisement is one of the most important activities in economy. Since thepublication of the seminal paper of Nerlove and Arrow (1962), many studies
have been devoted to determine optimal advertising expenditures over timein the dynamic optimal control framework in which the goodwill of the firm
is the stock variable and the advertisement expenditure is the flow variable.
See Sethi (1977) and Feichtinger, Hartl and Sethi (1994) for comprehensive
survey of the literature on advertisement policy. In the existing literature,
the authors have ignored the advertisement eﬀects on oligopoly competition.
The only exception is Ahmad, Agiza and Hassan (1999) (AAH, henceforth)
who have studied advertisement in Cournot duopoly models and consider
chaos control by applying OGY method. In spite of their eﬀorts, it has
not yet been fully known how advertisement aﬀects the nonlinear oligopoly
competition.
The main purpose of this paper is to provide a possible and positive
answer to this question. For this purpose, we use the dynamic model of
AAH and present a constructive method to derive stationary state outputs
in terms of model’s parameters. With this explicit form of the stationary
state, we can provide the explicit forms of local stability conditions and,with this condition, we can proceed to global dynamics in the case of local
unstable stationary points. Another interesting finding of this paper is thatthe dynamic system shows a half-pitchfork bifurcation on the way to chaos: atthe bifurcation points, only half of the periodic points gives birth of perioddoubling points.
This paper is organized as follows. Section 2 introduces the dynamic
model and determines its stationary state. Section 3 discusses the local
dynamics and derives the stability condition on which our main study isbased. Section 4 examines global dynamics under homogenous advertisement strategy in which the two firms adopt the same strategy and under theheterogeneous advertisement strategy in which the two firms adopt diﬀerent
strategies. Section 5 makes concluding remarks.

2

2

4D ModelWe reconsider the nonlinear dynamic duopoly model with advertisement introduced by AAH:µr¶
q2 (t)
q1 (t + 1) = (1 + x1 (t))− q2 (t) ,c1q2 (t + 1) = (1 + x2 (t))µr¶q1 (t)− q1 (t) ,c2(1)x1 (t + 1) = (1 − r1 )x1 (t) + a1 x1 (t)(1 − x1 (t)) − b1 x1 (t)x2 (t),x2 (t + 1) = (1 − r2 )x2 (t) + a2 x2 (t)(1 − x2 (t)) − b2 x1 (t)x2 (t).
Here ri represents a positive depreciation rate which is assumed to be lessthan unity, ai is a positive constant representing advertisement outlay, bi is apositive constant showing the eﬀect of the competition on the advertisementof both firms, ci is a positive constant giving the marginal cost of production,xi is the goodwill representing the eﬀects of current and past advertising outlays and qi is the output produced. Dynamics is considered in discrete timethat is denoted by t. The whole model is divided into two submodels. The
first two equations construct a dynamic model of output with advertisement
xi , which is a variant of the nonlinear duopoly model studies earlier by Puu(2003). The last two equations represent a dynamic model of the advertisement, which is based on the nonlinear models for advertisement introducedby Luhta and Virtanen (1996). It can be seen that the advertisement modelaﬀects output dynamics via oscillations of xi but not vise versa.

2.1

Advertisement Model

The advertisement equations incorporate three diﬀerent eﬀects: the depreciation eﬀect, denoted by ri , shows that the goodwill contributes less on demandas time goes on; the nonlinear second term indicates that the advertising outlays increase the goodwill when its level is relatively small and decrease it
when the level becomes larger than the threshold; and the interdependency
eﬀect is given by the third term. In this study, for the sake of simplicity, we
pay our attentions to the first two eﬀects on output dynamics and defer the
discussion on the third eﬀect by making the following assumption,1
Assumption 1 b1 = b2 = 0.
1

A more general model with positive bi will be considered in a subsequent paper.

3As a consequence of this assumption, the advertisement model becomesa system of two independent equations and thus its dynamics becomes uncoupled. Furthermore, to simplify the exposition, let us denote the righthand sides of the last two equations in (1) by ϕ1 (x1 ) and ϕ2 (x2 ). Then theadvertisement model can be rewritten as
x01 = ϕ1 (x1 ),(2)
x02 = ϕ2 (x2 ),where 0 denotes the unit-time advancement operator. Solving x0i = xi provides the fixed point, xei , of the advertisement model:
r1
r2
and xe2 = 1 − .
a1a2
To make these fixed points economically feasible, we assumexe1 = 1 −Assumption 2 ri < ai for i = 1, 2.
Under Assumption 1, the advertisement model becomes, the so called,
folded handkerchief map in which ϕi (xi ) is a unimodal map being topologically conjugated to the logistic map. Dynamic characteristic of this map
is now well-known. It generates rich dynamics ranging from periodic cycles
to chaotic behavior. In order to guarantee the feasibility (i.e., nonnegativity
and boundness) of the trajectories generated by the advertisement model,
we need to introduce a confinement condition. The advertisement model issubject to the non-negativity constraint if
0 ≤ xi ≤1 + ai − ri.
aiiSolving dϕ= 0 gives the maximizer,
dxi

xm
i =

1 + ai − ri,
2ai

and the maximum value of xi :
ϕ(xm
i ) =

(1 + ai − ri )2.
4ai

Given ai and ri , a trajectory xi stays within this feasible region for all t if
the maximum value is transformed into the non-negative region:(1 + ai − ri )2
1 + ai − ri
≤
,
4ai
ai
which implies the following confinement condition for xi :4

Assumption 3 ai − ri ≤ 3 for i = 1, 2.
Due to Assumptions 2 and 3, the feasible set of (a1 , a2 ) is defined as
A = {(a1 , a2 ) | ri < ai ≤ 3 + ri for i, j = 1, 2.} .

2.2

Output model

Assuming that the inverse demand function is isoelastic and the production
costs are linear, Puu (2003) constructs nonlinear duopoly and triopoly models and shows that these models can be chaotic if nonlinearities involved
become stronger. In (1), advertisement eﬀects are introduced into Puu’s
duopoly model in which two competitors, firm 1 and firm 2, produce nondiﬀerentiated good of quantities q1 and q2 with linear production costs, where
the marginal costs are c1 and c2 , respectively. Following the traditional spiritof the classical Nerlove-Arrow optimal advertisement model, we assume thateach firm has its own goodwill and can sell more product if it has better goodwill. One simple way to incorporate the eﬀects of the goodwill is to assume
the following form of the inverse demand function that firm i perceives,
pi =

1
qi
+ qj
1 + xiin which xi ≥ 0 denotes the advertisement eﬀect or the value of the goodwill.
xi = 0 implies no advertisement eﬀects and thus the two goods becomes
identical. When xi > 0, firm i can sell larger quantity due to the positive
advertisement eﬀects, taking the competitor’s output qj and market pricegiven. Thus the profit of firm i is given asπi =qi− ci qiqi+ qj
1 + xiwhich is maximized with respect to qi . Using the first-order condition, we
can solve for the reaction function,¶µr
qj− qj for i = 1, 2.
qi = (1 + xi )ci
These considerations provide a microeconomic interpretation of the reaction
functions assumed in (1).

5

To simplify the exposition, again, let us denote the right hand sides ofthe first two equations of (1) by φ1 (q2 , x1 ) and φ2 (q1 , x2 ). Then the output
model can be rewritten as
q10 = φ1 (q2 , x1 ),(3)
q20 = φ2 (q1 , x2 ).
For xi = 0, the output dynamic model is reduced to the nonlinear duopoly
model of Puu (2003) in which complex dynamics can be generated, depending on the production cost ratio. Indeed it has been shown that the stationary point of Puu’s model is asymptoticallystable if and only if the cost√
ratio belongs to the interval ( 3+21√2 , 3 + 2 2). It is unstable otherwise, butgoes through period-doubling bifurcation
to chaos if the ratio decreases from√1√4
to 25 or increases from 3 + 2 2 to 25. Furthermore, it is numerically4
3+2 2
established that the dynamics is symmetric with respect to unit value of themarginal cost ratio. Thus it is safe to assume that the ratio is greater thanunity:
Assumption 4 c2 ≥ c1 .
In order to obtain the fixed points of q1 and q2 in (3), we set qi0 = qi ,assume that the fixed point (xe1 , xe2 ) of the advertisement dynamics is given
and then rewrite the output dynamics asrq2eeq1 + (1 + x1 )q2 = (1 + x1 )
,
c1
(4)
r
q1
q2 + (1 + xe2 )q1 = (1 + xe2 ).
c2Squaring both sides of these equations, dividing the resulting first equationby the second one and introducing the new variables
z=

c2
q2
, c = , d1 = 1 + xe1 and d2 = 1 + xe2 ,q1
c1

we obtain the following relationcz =µd2 1 + d1 z
d1 d2 + z

¶2

.

(5)

This is a cubic equation for unknown z. Although it is possible to derive
an explicit form of the real root of (5), the form has a very complicated6expression which is not useful for the following analysis. Thus, instead ofsolving the equation, we look for the intersection of the right and left handsides of (5) which determines the fixed point of the output ratio.
Let us denote the left hand side by f (z) and the right hand side by g(z):f (z) = cz and g(z) =µd2 1 + d1 z
d1 d2 + z

¶2.

It can be seen that g(z) has upper and lower bounds and is positive-sloping,g(0) =

1< 1,(d1 )2

lim g(z) = (d2 )2 > 1,

z→∞

and

µ

d2
g (z) = 2d1
0¶2

(1 + d1 z)(d1 d2 − 1)
> 0.
(d2 + z)3

Clearly, f (z) is strictly increasing, linear and f (0) = 0. An examination of thesolution formula of the cubic equations shows that there is a unique positive
intersection α of f (z) and g(z) if 1 < di < 2. It corresponds to the real rootof the cubic equation and determines the ratio of outputs at the intersection:α = α(c, d1 , d2 ) =

q2e
,
q1e

where qie is the stationary output of firm i.
Figure 1 gives the graphical representation of the stationary output ratio,in which two graphs of f (z) with cA = 1.15 and cB = 1.5, two graphs ofg(z) with ( ar11 , ar22 ) = (0.6, 0.2) denoted by gI (x) and with ( ar11 , ar22 ) = (0.2, 0.6)
denoted by gII (x) are shown. There are four intersections, A, B, a, b of thesefunctions as illustrated. This figure reveals two issues: first, the ratio of
outputs can be larger than unity even under Assumption 4 and second, theeﬀect on the output ratio caused by a change in c is negative. To check thefirst issue, we consider g(z) at z = 1 which, after arranging terms, is given
by
⎧
r2
r1 ⎫2
⎪
⎨ (2 − )(3 − ) ⎪⎬a2a1g(1) =.
rr⎪⎩ (2 − 1 )(3 − 2 ) ⎪⎭a1
a2

It is obvious that the ratio α is less than unity if g(1) ≤ c because theintersection of f (z) and g(z) surely occurs in the interval (0, 1) as shown7

by points B, a and b, and that it is greater than unity if g(1) > c as given
by point A.2 It is also shown that g(1) ≤ 1 if ar11 ≤ ar22 and g(1) > 1 if
r1
> ar22 in which case g(1) > c could be possible. The second issue is clear
a1
as the intersection moves to B from A or to b from a, respectively, when themarginal cost ratio increases from cA to cB , These results are summarized in
the following theorem:
r2
r1
r2
r1
≤
or if
>
and c ≥ g(1), then α ≤ 1; (2) If
a1
a2
a1
a2
r1
r2
dα
>
and c < g(1), then α > 1, and (3)
< 0.
a1
a2
dc

Theorem 1 (1) If

cz
cBz

gIHzL

A

cAaA
B

cBaB

cAaa
cBab

cAz

gIIHzL

a
b

ab

aa aB

1 aA

z

Figure 1. Determinations of the output ratio
Substituting q1e = αq2e into the first and second equations of (4) andsolving each equation for qi provides the stationary values of the outputs interms of parameters ai , ci and ri ,
µ
¶2
µ
¶2
d1
d2
α
1
e
q1 =
=
,
c1 1 + αd1
c2 α + d2
(6)
µ
¶2
µ
¶2
αd1
d2
1
α
=
.
q2e =
c1 1 + αd1
c2 α + d2¡ ¢2
g(1) takes the maximum value, 43 , when ar11 = 1 and ar22 = 0. Therefore, α < 1
always if the marginal cost ratio c is larger than this maximum value.
28Since α is the real root of the cubic equation (5), it is diﬃcult and complicatedto analytically derive the eﬀects on the stationary outputs caused by changes
in parameter values but numerical investigations indicate that for firm 1,
∂q1e
∂q1e
∂q1e
<
0,
<
0,
< 0,
r2
∂( ar11 )
∂( a2
)
∂c
and for firm 2,

∂q2e
∂q2e
∂q2e
< 0.
r1 > 0,
r2 < 0,∂( a1 )∂( a2 )∂c

Negative derivatives of q1e , q2e and α with respect to c imply that the elasticity
of q2e with respect to c is larger in absolute value than the elasticity of q1e ,
¯ ¯
¯
¯
½
¾
¯ c ∂q2e ¯ ¯ c ∂q1e ¯∂αα c ∂q2ec ∂q1e¯<¯¯.
=− e< 0 ⇒ ¯¯ e
∂cc q2e ∂cq1 ∂c
q2 ∂c ¯ ¯ q1e ∂c ¯

As in the case of the advertisement dynamics, in order to avoid negative
output values, we need a confinement condition for the output. To this end,
we first derive the domain of the output generating function φi (qj , xi ) of firmi
1
0 ≤ qj ≤ ,ciand then look for the condition for which the output stays in this domain
for all t. It is seen that φi (qj , xi ) has a mound-shaped curve and takes its
maximum value
1 + xiqimax =
4ci
at qjm = 4c1i . The output trajectory stays within the feasible region if φ1 (q2max ) ≤1
and φ2 (q1max ) ≤ c11 . After arranging the terms, these conditions can bec2
rewritten as
4(1 + x1 )(1 + x2 )2(4 + (1 + x1 )(1 + x2 ))2≤c≤.
4(1 + x1 )2 (1 + x2 )(4 + (1 + x1 )(1 + x2 ))2For 0 ≤ x1 < 3 and 0 ≤ x2 < 3, the left hand side becomes strictly lessthan unity and the right hand side becomes strictly greater than unity. As
mentioned above, xi has its upper bound, 1+aaii−ri , below 3 if ai ≥ 0.5. The
confinement condition needs to be taken into account when the stationary
state is unstable. As it will be shown later, ai is larger than 2 + ri when thestationary state is unstable. This observation and Assumption 4 (i.e., c ≥ 1)
imply that the lower bound confinement condition on c is redundant. Thus
we rewrite the confinement condition of the output as follows:c ≤ ψ(x1 , x2 ),9(7)

where3(4 + (1 + x1 )(1 + x2 ))2
ψ(x1 , x2 ) =.
4(1 + x1 )2 (1 + x2 )(8)Local Stability

We next consider the local stability of the 4D dynamic system (1). The
Jacobian of the system at the fixed point reads⎛⎞
0 J12 J13 0⎜ J21 00 J41 ⎟⎟,J =⎜⎝ 0
0 J33 0 ⎠
0
0
0 J44
where

J12
J21

µ

¶
1
qe
√ e − 1 , J13 = 1 ,= d1
2 c1 q2d1µ¶
1qe√ e − 1 , J24 = 2 ,= d2
2 c2 q1
d2

J33 = 1 − (a1 − r1 ), and J44 = 1 − (a2 − r2 ).
Its characteristic equation has the factored form
(λ2 − J12 J21 )(λ − J33 )(λ − J44 ) = 0.
Denote by λ1,2 the roots of the first factor and by λ3 and λ4 the roots of thesecond and third factors, respectively. The four roots are
s
µ
¶µ
¶
1
1
√
√
λ1 = d1 d2−1−1 ,2 c1 q2e
2 c2 q1eλ2 = −λ1 ,λ3 = 1 − (a1 − r1 ) < 1,
λ4 = 1 − (a2 − r2 ) < 1.
The 4D dynamic system is locally asymptotically stable if all roots are insidethe unit circle of the complex plane.
Clearly λ3 and λ4 are the eigenvalues of the advertisement model andthey are less than unity due to Assumption 3, implying that the slope ofthe corresponding advertisement function evaluated at the fixed point is lessthan unity. Thus if the slope is greater than −1, then the advertisement
model is stable. It is obvious that |λi | < 1 if ai < 2 + ri and |λi | > 1 if
ai > 2 + ri . Consequently, we obtain the following result.
10

Lemma 1 The advertisement fixed point (xe1 , xe2 ) is locally asymptoticallystable if r1 < a1 < 2 + r1 and r2 < a2 < 2 + r2 .
The flip boundary is ai = 2 + ri on which λi is equal to −1 and thus
a loss of stability just occurs. As the parameter ai crosses this boundary
and increases further, a solution of the advertisement dynamic system goes
through a period doubling cascade to chaos, the most common route to chaos.
We now turn our attentions to the stability of the output dynamics. From(6), we have
¶2
µ
d2
e
c1 q2 =
α + d2
and
¶2
µ
αd1
e
,
c2 q1 =
1 + αd1
both of which are substituted into the right hand side of λ1 to obtain
1 p −1λ1 (= −λ2 ) =(α − d1 ) (α − d2 ).
2Denoting dm = min(d1 , d2 ) and noticing that 0 < di < 2, the expression
under the square root is negative, since(α−1 − d1 ) (α − d2 ) = d1 d2 − (αd1 + α−1 d2 )< d1 d2 − (α + α−1 )dm< d1 d2 − 2dm < 0.
Thus λ1 and λ2 are pure imaginary roots. (α−1 − d1 ) (α − d2 ) = −4 is the
Hopf boundary on which the absolute values of the eigenvalues become unity.
Hence, we obtain the following result.
Lemma 2 The fixed point (q1e , q2e ) of the output model is locally asymptotically stable if (α−1 − d1 ) (α − d2 ) > −4.
From Lemma 1 and 2, the stability conditions of the 4D dynamic system
are summarized in the following way.
Theorem 2 The 4D dynamic system (1) is locally asymptotically stable if
r1 < a1 < 2 + r1 , r2 < a2 < 2 + r2 and (α−1 − d1 ) (α − d2 ) > −4 where
di = 2 − arii and α is the real solution of the cubic equation,¶2µ
d2 1 + d1 α.
cα =d1 d2 + α114Global Behavior

If any one or more of the local stability conditions is violated, then a trajectory soon or later moves away from the fixed point. However, nonlinearities ofthe model prevent the trajectory from globally diverging. As a consequence,
the trajectory neither converges nor diverges but keeps oscillating around thefixed point. Indeed, if the trajectory crosses either the flip boundary or the
Hopf boundary, it exhibits various dynamics ranging from simple periodic
behavior to complex dynamics involving chaos. We will study such global
dynamics in this section. We will assume first that the firms adopt the homogeneous advertisement strategy in which ri = r and ai = a as a benchmark case in Section 4.1, and then we will proceed to the heterogeneousadvertisement case in Section 4.2 in which r1 6= r2 and/or a1 6= a2 .

4.1

Homogeneous Advertisement

We consider now the homogeneous case in which each firm has the samedepreciation rate and the same advertisement outlay, r1 = r2 = r and a1 =a2 = a. The advertisement equations become identical in the sense that time
evolutions of two firms’ advertisements behave identically over time if thesame initial conditions are chosen. By this simplifying assumption, we canwrite ϕ1 = ϕ2 = ϕ and call ϕ the advertisement map. The 4D dynamic
system is now written as
qi0 = φi (qj , xi ),(9)x0i = ϕ(xi ) for i, j = 1, 2 and i 6= j.
The analysis of the dynamic behavior of this system is further divided into
three parts. First, the basin structures of the advertisement model and theireﬀect on the output dynamics is studied. Second, we restrict (9) to the
diagonal of the (x1 , x2 ) space and characterize the output dynamics here,
and finally, we study the half-pitchfork bifurcation structure of the restricted
system, which means that at a bifurcation point, only a half of the periodic
points divides into two periodic points with period-doubling and the otherhalf of the periodic points does not .
4.1.1

Homogenous 4D Model

The stationary point of (9) in the homogenous case is the following:
µ
µ
¶2
¶2
d
αd
1
1
r
e
e
q1 =
, q2 =and xe = 1 − ,c2 α + d
c1 1 + αda12where xe = xe1 = xe2 , d = 1 + xe and α = α(c, d, d). The advertisement
dynamics is described by the double logistic map. It is well-known thateach logistic map ϕ(xi ) goes to chaos through period-doubling cascade; thestationary state is attracting for a < r + 2, the stationary state becomes√
repelling and an attracting period-2 cycle appears for r + 2 < a < r + 6,
the period-2
cycle becomes repelling and an attracting period-4 cycle appears√
for r + 6 < a < r + 2.54409, and so on. Cycles with√ odd period can also befound: the onset of the period-3 cycle is at a = r+2 2, the onset for period-5
is a = r + 2.73817, and so forth. Accordingly, one of the main features ofthe double logistic map is the birth of complicated dynamics and the otheris the coexistence of periodic cycles. We investigate the basin structures tocapture the latter feature.
In the period-2 region with a = 3 and r = 0.6, each ϕ(xi ) has two periodB
A
B
B
A
2 points xA
i and xi such that xi = ϕ(xi ) and xi = ϕ(xi ). Thus the double
logistic map has four points of the (x1 , x2 ) plane that form two period-2 cyA
B
B
A
B
cles; A = (xA
1 , x2 ) ↔ B = (x1 , x2 ) on the diagonal and a = (x1 , x2 ) ↔
BA
3
b = (x1 , x2 ) on the oﬀ-diagonal. To avoid the emergence of complicateddynamics, we consider the case in which the output model is stable if the
goodwill (that is, xi ) is constant. By doing so, we can see which kind of
dynamics is produced by the logistic map and how the advertisement dynamics aﬀects the output dynamics. Since it has two periodic cycles, thedouble logistic map has two basins of attractions as shown in Figure 2A inwhich the red regions form the basin of the diagonal attractor and the light
blue regions represent the basin of the oﬀ-diagonal attractor. The advertisement dynamics is path-dependent, so it depends on the choice of the initialpoint which cycle it eventually converges to. Since the output dynamics depends on the advertisement dynamics, it exhibits periodic oscillations withperiod-2 if the advertisement dynamics gives rise to a period-2 cycle. In thecase when the advertisement dynamics starts at either point A or B of theadvertisement space in Figure 2A, the output trajectory converges to theperiod-2 cycle moving between points A and B of the output space in Figure
2B. Similarly, if point a or b is selected as the initial point, then the output
trajectory converges to the other period-2 cycle oscillating between points aand b shown in Figure 2B. Note that these output cycles do not coexist while
two advertisement cycles do. It depends on the initial point selection of theadvertisement map which cycle the output trajectory is periodic to.
3
Since xi = 0 is also a fixed point of ϕi , we have two more period-2 cycles that periodically visit points on the horizontal or vertical axis of the (x1 , x2 ) plane as one of theperodic points is zero. We do not consider this kind of cycles in the following analysis.

13

Figure 2. Two basins of attractors with a = 3 and corresponding period-2 cycles
In the period-4 region with a = 3.14 and r = 0.6, each ϕ(xi ) has four
period-4 points xki such that xki = ϕ4 (xki ) for i = 1, 2 and k = A, B, C, D thus
the double logistic map has the following coexisting four period-4 cycles:
A
B
B
C
C
D
D
A
A
(xA
1 , x2 ) ⇒ (x1 , x2 ) ⇒ (x1 , x2 ) ⇒ (x1 , x2 ) ⇒ (x1 , x2 ),
A
B
B
C
C
D
D
A
A
(x1 , x2 ) ⇒ (x1 , x2 ) ⇒ (x1 , x2 ) ⇒ (x1 , x2 ) ⇒ (x1 , xB
2 ),
A
C
B
D
C
A
D
B
A
(x1 , x2 ) ⇒ (x1 , x2 ) ⇒ (x1 , x2 ) ⇒ (x1 , x2 ) ⇒ (x1 , xC
2 ),
A
D
B
A
C
B
D
C
A
(x1 , x2 ) ⇒ (x1 , x2 ) ⇒ (x1 , x2 ) ⇒ (x1 , x2 ) ⇒ (x1 , xD
2 ),in which the first cycle moves in the diagonal and the other three in the oﬀdiagonal. The basin of attractor in the period-4 regions is shown in Figure
3A. For the sake of simplicity, we do not depict periodic points in Figure 3A
and illustrate only a period-4 cycle that corresponds to the diagonal dynamics
of advertisement in Figure 3B.

Figure 3. Four basins of attractors with a = 3.12 and a period-4 cycle of output
As a increases further, we can have more complicated and exotic basinsof attractions. It can be checked that m period-m cycles coexist if ϕ(x)14generates a period-m cycle. One cycle is in the diagonal and the remaining(m−1) cycles are in the oﬀ-diagonal. It can be supposed that the rectangularregions of the basin will be divided further. At the edge of chaos in which
a is at the accumulation point, Shin (2007) mathematically shows how thepatterns of those basin structures are composed.
4.1.2

Restricted Homogenous 4D Model

To characterize the dynamics, we restrict our attentions to the behavior in
the diagonal, since qualitative features of the oﬀ-diagonal dynamics is thesame as that of the diagonal dynamics. If the firms start with the symmetricinitial point, x1 (0) = x2 (0), then advertisements remain equal for all t ≥ 0,i.e., x1 (t) = x2 (t). The double logistic map may be interpreted as a simple
logistic map, since the evolution of the advertisement reflects the common
behavior of the two firms. In this case, we can write x1 = x2 = x and thenthe 4D dynamic system can be reduced to the 3D dynamic system,4
q10 = φ1 (q2 , x),q20 = φ2 (q1 , x),(10)x0 = ϕ(x).
In order to determine the confinement condition of the output dynamics, wewill distinguish between two cases depending on whether the advertisement
dynamics is stable or not, .
If the advertisement model is stable, then its trajectory eventually converges to the stationary point. Thus it is safe to assume that the advertisement evolution starts at its stationary value and stays there afterward.
Consequently, the confinement boundary of the output can be obtained bysubstituting xe into x1 and x2 of (7),2

(4 + (1 + xe )2 )c=.
4(1 + xe )3(11)

If the advertisement model is unstable, then the trajectory may oscillate
ranging from the maximum value xmax of ϕ(x) to the zero minimum value.
The most restrictive confinement constraint on the output dynamics is given
by substituting xmax into x1 and x2 of (7),2

(1 + a − r)2(4 + (1 + xmax )2 )
max
with
x
=
c=
.
4(1 + xmax )34a4

We denote the eigenvalue of ϕ by λ3 for convenience.

15

(12)

For the time being we call this relation the confinement condition of the output when the advertisement is unstable. It should be noted, however, thatthe condition may be violated if an unstable trajectory of the advertisement
map takes its maximum value less than xmax . This could happen if the trajectory is periodic and xmax is not a periodic point. In spite of this comment,
it can be used only as a rough estimate of the confinement condition.
The Hopf and flip boundaries divide the parameter region into stable andunstable regions while the confinement boundaries divide it into feasible andinfeasible regions. The 3D system contains three parameters, r, a and c.
Taking r given, we characterize the stability of the stationary point for thevalues of parameters c and a in the set,
D = {(c, a) | 1 ≤ c ≤25, r ≤ a ≤ 3 + r},
4

in which, as will be seen soon, 25
is the maximum upper bound of the mar4
ginal cost ratio for feasible solutions.
Depending on the parametric configuration of (c, a) in D, the following
cases should be considered:(1) a = rThis condition means the restriction of the 3D dynamic system to thelower boundary of D and implies that xe = 0 so no advertisement takes
place. The 3D system is further reduced to a 2D system, namely, the output
model without advertisement, which is equivalent to Puu’s duopoly model.
Therefore, the time evolution of the outputs coincides with that of Puu’s
model. In fact, it can be checked that α = 1c when a = r. Thus solving
the Hopf bifurcation
condition (c−1 − 1)(c − 1) = −4 for c gives the solution√
cu = 3 + 2 2 as c ≥√1 is assumed. The Hopf boundary crosses the horizontal
line a = r at (3 + 2 2, r). Therefore the stationary state is stable for c < cuand becomes unstable for c ≥ cu . Substituting xe = 0 into the confinement. The concondition gives the upper bound of the marginal cost ratio, c̄ = 25
425
finement boundary (11) crosses the horizontal line at ( 4 , r). As c increases
from cu , the stationary point loses its stability, bifurcates to periodic cycles
and then exhibits chaotic oscillations for suﬃciently higher value of c. Notethat c cannot be greater than the critical value c̄, beyond which the model
becomes infeasible. These results are the same as those already shown in
Puu (2003).
(2) r < a < 2 + r

16

This subset of D is further divided into two parts by the Hopf boundary,(α−1 − d)(α − d) = −4. In the left side of the boundary in which (α−1 −d)(α − d) > −4, |λi | < 1 for i = 1, 2, 3 so the output model as well as
the advertisement map are locally stable. In the right side in which (α−1 −d)(α − d) < −4, |λi | > 1 for i = 1, 2 and |λ3 | < 1. Thus the output model
becomes locally unstable but the advertisement map is still locally stable.
The unstable trajectories are bounded by the confinement boundary that isdefined by (11),2(4 + (1 + xe )2 )ee
ψ(x , x ) =.
4(1 + xe )3
For dynamic considerations, we assume that advertisement is at its stationary value since the advertisement map is stable. The output model withconstant advertisement is linearly conjugate to the output model with zeroadvertisement, which case is already discussed above.
Figure 4 presents two bifurcation diagrams with respect to c and a. The
3D system is iterated 5,000 times to eliminate transient behavior and thelast 50 iterated points are plotted. In Figure 4A, c ranges from 1 to 25
and4
a from r to 2 + r where r = 0.6 is fixed. Figure 4B is an enlargement of apart of Figure 4A, in which c ranges from 2.9 to 3.0 and a from 1.03 to 1.19.
Diﬀerent colors indicate diﬀerent periods of periodic cycles up to period 16.
Periodic points with a larger period than 16 or aperiodic points are colored ingrey. Trajectories are infeasible in the white region in which the confinement
condition is violated. Red regions in Figures 4A and 4B represent the region
where the 3D system is locally stable. The same color in the entire region
indicates that any trajectory converges to the stationary point, which, inturn, imply global stability of the 3D system. For (c, a) crossing the Hopf
boundary but not reaching the confinement boundary, the stationary point(q1e , q2e ) of the output model becomes unstable and generates essentially thesame time evolutions as the ones obtained in case (1), which indicates thatthe stationary point goes to chaos via Hopf bifurcation. In case (1), theemergence of complex√dynamics is associated with a highly asymmetric cost
structure (i.e., (3 + 2 2)c1 < c2 < 6.25c1 ). In contrast, the same bifurcation
scenario takes place in case (2) under more plausible cost structure if the17advertisement outlay is higher, as observed in Figure 4B.

Figure 4. Bifurcation diagram in the (c, a) space(3) 2 + r ≤ a ≤ 3 + r
In this case |λ3 | > 1 always, so the advertisement map is unstable. The
output model is unstable if (α−1 − d)(α − d) > −4 (that is, |λi | > 1 for
i = 1, 2). Moreover, since the advertisement map plays the dominant role forcharacterizing the dynamics of the output, the output model becomes unstable even if (α−1 − d)(α − d) ≤ −4. Figure 5 depicts the two-dimensional
bifurcation diagram of (9) in the (c, a) plane in which c ranges from 1.8 to 2.3
and a from 2+r to 3+r. As in Figure 4, the regions of periodic and aperiodic
behavior are indicated with diﬀerent colors and the solutions are infeasible
in the white region. The negative sloping real line is the Hopf boundary. It
can be observed that nothing particular changes if the parameter configuration (c, a) crosses the Hopf boundary. The negative sloping dotted line is theconfinement boundary. Four interesting facts should be mentioned. First,
the confinement condition is satisfied for large values of a as chaos is bornfor higher values and its trapping interval includes xmax . Second, this is not
the case for smaller values of a, since only periodic cycles are born and xmax
is not a periodic point. Third, there are some higher values of a for whichwindows of chaos emerge with periodic points violating the confinement condition. Finally, there are no diﬀerences in dynamics whether the parameter
configuration (c, a) is located in the left or right side of the Hopf boundary,which shows sharp contrast to the result obtained in case (2).

18

Figure 5. Bifurcation diagram in (c, a)Parameters a (as well as r) and c can be thought as the sources of nonlinearities of the advertisement model and the output model, respectively.
The larger the parameter is, the stronger the nonlinearities of those models are. The eﬀect caused by a change in a on output dynamics is however
qualitatively diﬀerent from the eﬀect caused by a change in c. To see thediﬀerences, we first choose a as the bifurcation parameter and keep c fixed.
Since the advertisement map is qualitatively the same as the logistic map, it
exhibits the well-known period-doubling bifurcation after a crosses the flip
boundary (i.e., 2 + r) if we increase it to its upper bound, 3 + r. The time
evolution of the output is synchronized with that of the advertisement andthus exhibits the period-doubling sequence of bifurcations with respect to aas shown in Figure 6. Two simulations are performed with diﬀerent values ofc there. In the first simulation, a is increased along the vertical segment AB
of Figure 5 while c = 1.85 and r = 0.6 are fixed. The resultant bifurcation
diagram against a is depicted in Figure 6A. In the second simulation, only c
is changed to 1.5 (which value is outside the range of Figure 6 so that it is
not illustrated there) and the resultant bifurcation diagram is presented in
Figure 6B. The similarity between these two diagrams is clear. Bifurcation
occurs for exactly the same values of a, namely, period-2 cycle bifurcates toperiod-4 cycle for a = a4 , which, in turn, bifurcates to period-8 cycle fora = a8 , further, the window opens for a = a3 and so on. Diﬀerences can also
be observed: the range of q1 as well as the diameter of one of the period-2
cycles become much smaller as c decreases with all other parameter values
kept unchanged.

19

Figure 6. Period-doubling bifurcation diagrams against a

Alternatively, if we choose c as the bifurcation parameter and fix a = 3
and r = 0.6, then we have a diﬀerent type of bifurcation scenario as illustrated
in Figure 7 in which we increase c along the horizontal segment CD from 1.8
to 2.0935 as shown in Figure 5. Since a is unchanged along this segment, theadvertisement map keeps generating a two period cycle. As shown in Figure
7, a period-2 cycle of the output appears first, which is synchronized with theadvertisement evolution. For a larger value of c (i.e., larger than about 1.97),
a period-3 cycle is born because one of the two periodic points bifurcates toa period-2 cycle. As c increases further, the periodic cycle goes to chaos after
repeating many period doubling bifurcations. At the first glance, the wellknown bifurcations scenario seems to occur. However, careful observations
reveal, as shown below, that the whole model undergoes diﬀerent type of
period-adding bifurcations.

Figure 7. Period-adding bifurcation diagram against c
20

4.1.3Half-Pitchfork Bifurcation

We now consider the pitchfork bifurcation scenario of the restricted dynamic
system (10). To simplify the discussion, we specify the parameter values as
a = 3 and r = 0.6 and select the case in which the advertisement model ϕ(x)generates a period-2 cycle, the lowest-order cycle.5 Let x1 and x2 be the periodic points of the cycle, which are obtained by solving equation ϕ(ϕ(x)) = x:p
2 + a − r − (a − r)2 − 4
x1 =
,
2a
and

p
(a − r)2 − 4
x2 =
.
2a
Apparently these points are defined only for a > 2 + r. We have x1 = 0.51225
and x2 = 0.95442. Let x2 be an initial point (t = 0) of the advertisement
map. Then x(t) oscillates between these two points: x1 = ϕ(x(t)) if t is odd
and x2 = ϕ(x(t)) if t is even. It is thus convenient to re-define the output
generating functions as follows: when t is odd and x1 is fixed,¶µrq2− q2 ,
f1 (q2 , x1 ) = (1 + x1 )
c1
¶
µr
q1
f2 (q1 , x1 ) = (1 + x1 )− q1 ,c2
2+a−r+

and when t is even and x2 is fixed,¶
q2− q2 ,
g1 (q2 , x2 ) = (1 + x2 )
c1
¶
µr
q1
g2 (q1 , x2 ) = (1 + x2 )− q1 ,
c2µr

in which x1 and x2 are determined solely by a and r and thus they can betreated as positive constants when we consider output dynamics. To find theperiod-2 points of the output, we define the two-fold iterates as follows:2(q1 ) = g1 (f2 (q1 , x1 ), x2 ) and G212 (q2 ) = g2 (f1 (q2 , x1 ), x2 ),F125

We obtain qualitatively the same result even in cases when ϕ(x) generates higher ordercycles and another values of the parameters are specified.

21

and
2
F21
(q1 ) = f1 (g2 (q1 , x2 ), x1 ) and G221 (q2 ) = f2 (g1 (q2 , x2 ), x1 ),where the subscripts ”ij” mean that the time evolution of the advertisement
moves from xi to xj , and the superscript ”2” means that functions are twice
folded.
Let us select cA = 1.95 for which a period-2 cycle of output appears
as shown in Figure 7. The vertical dotted line at c = cA crosses the two
downward sloping curves. Two intersections are the q2 -periodic points of the2
period-2 cycle. Solving each of the four equations, F12(q1 ) = q1 , G212 (q2 ) = q2 ,2
F21
(q1 ) = q1 and G221 (q2 ) = q2 , yields two periodic points of the output
dynamic system,
QA = (q1A , q2A ) and QB = (q1B , q2B ),where
q1A = 0.35439, q2A = 0.08119, q1B = 0.39821 and q2B = 0.14056.
Diﬀerentiating each of the two-folded functions gives the following derivativevalues at the corresponding periodic points:
20 A
B
20
A
20 B
F21
(q1 ) = G20
12 (q2 ) = −0.39301 and G21 (q2 ) = F12 (q1 ) = −0.96651.

All derivatives are less than unity in absolute value so this period-2 cycle isstable and thus, any trajectory starting at any point of the feasible region is
asymptotically periodic to the period-2 cycle.
Next, fixing the values of a and r, we increase the value of c to cB =1.97725. Since a and r remain unchanged, the advertisement map still generates the period-2 cycle, x1 and x2 , while the change in c induces qualitativechanges in the output dynamics. The output model now seems to give rise toa period-3 cycle according to the bifurcation diagram of Figure 7 in which thevertical dotted line intersects three times with the curves depicted. However,in fact, it generates a period-4 cycle as will be seen shortly. To show the
emergence of the period-4 cycle, it is convenient to introduce the following
four-fold iterates:
4
2
2
F12
(q1 ) = F12(F12(q1 )) and G412 (q2 ) = G212 (G212 (q2 )),

and
4
2
2
F21
(q1 ) = F21(F21(q1 )) and G421 (q2 ) = G221 (G221 (q1 )).

The same principle is used for defining the subscripts and superscripts ofthese functions as before. The fixed points of these four-folded functions are22composed of the periodic points of the period-4 cycle, which are denoted byA, B, C and D in Figure 6. The fixed points are calculated as given below,
where the superscript indicates an element of the corresponding periodic
points:4
F21(q1 ) = q1 =⇒ q1B = q1D = 0.350469
G421 (q2 ) = q2 =⇒ q2B = 0.06613 and q2D = 0.09524,
FI4 (q1 ) = q1 =⇒ q1A = 0.41266 and q1C = 0.37336,
andG412 (q2 ) = q2 =⇒ q2A = q2C = 0.13647.
Derivatives of these functions at the fixed points are also calculated as
40
40
40
= G40
F12
21 = 0.99266 and F21 = G12 = 0.17544.

All of these values are less than unity in absolute value so the period-4
cycle is stable and any trajectory is asymptotically periodic to it. Fixing
the parameter values of the advertisement map and increasing the marginal
cost ratio of the output model result in a bifurcation of the period-2 cycle toperiod-4 cycle. The question that we naturally raise is how such a bifurcation
developes.
To understand the birth of the period-4 cycle, consider, first, the twofolded functions Fij2 and G2ij for i, j = 1, 2 and i 6= j with c = cB . Following
the same procedure we have done above, we can obtain a period-2 cycle whose
periodic points are depicted as two red points, a and b, in the first quadrantof Figure 8. Two red points in the second and fourth quadrants correspondto periodic points in the output-advertisement plane. However this period-2
cycle is unstable. To show this fact, we solve Fij2 (q1 ) = q1 and G2ij (q2 ) = q2 to
obtain the periodic points,
q1a = 0.39462, q2a = 0.13647 and q1b = 0.35227, q2b = 0.07881,
and then diﬀerentiate the functions at the fixed points to obtain the derivatives
0

20 a
2
a
F12
(q1 ) = G221 (q2b ) = −1.00183 and F21(q1b ) = G20
12 (q2 ) = −0.41886.

These results lead to the following conclusion. At point a, q1a is unstable
while q2a is stable. On the other hand, at point b, q1b is stable while q2b isunstable. The period-2 cycle with c = cB , therefore, becomes one half stable
and the other half unstable.
2
It is evident that these unstable fixed points of F12(q1 ) and G221 (q2 ) are4
40 a
also the fixed points of F12(q1 ) and G221 (q2 ). Moreover since F12(q1 ) =
23

¡ 20 b ¢2
20 a 2b(F12(q1 )) > 1 and G40> 1, q1a and q2b are repelling. As
21 (q2 ) = G21 (q2 )44done above, equations F12 (q1 ) = q1 and G21 (q2 ) = q2 provide two additional
stable fixed points, which are illustrated in Figure 8 as the blue points. In
the fourth quadrant, which corresponds to the (q1 , x) plane, there are two
blue dots and one red dot for x1 ; q1A and q1C are shown by the blue dots and
q1a is shown by the red dot. This picture indicates that in the case when2
the eigenvalue of F12(q1 ) equals −1, the stability of the period-2 cycle is in4
this case just lost and at the same time, two new fixed points of F12(q1 ) areborn. This phenomenon, as well-known, is called the period doubling bifur4
cation. However, the fixed point of F21(q1 ) does not bifurcate as far as theadvertisement map gives rise to the period-2 cycle. In the second quadrant,
which corresponds to the (x, q2 ) plane, there are also two blue dots and one
red dot. The same phenomenon is occurred in G221 and G421 so at c, for which
b
2
4
G20
21 (q2 ) = −1, G21 becomes unstable and G21 undertakes a period-doubling2
bifurcation and two new fixed points of G21 are created.
We can summarize the above results as follows. The period-2 cycle isattracting for c = cA but becomes repelling for c = cB . FI4 (q1 ) does not havea period-4 cycle for c = cA but it has an attracting period-4 cycle of output
for c = cB . There is a critical value of c between these two values, at which
the eigenvalues of the two-fold iterate are equal to −1, and those of the fouriterates equal 1.6 The bifurcation from period-2 cycle to period-4 cycle takes
place just at this critical value.

6

We encountered diﬃculties when we tried to derive a general expression of this criticalvalue without specifiying the values of a and r, since, we needed to solve a cubic equation
whose solution may have very complicated expression in general.

24

q2

A,C,a

q2A

D

q2 C

D

b

q2 b

b

B

q2 B

B

x2

q1B

x1
x1

x2

C

a

A

q1C
C

q1 a
a

q1A
A

q1

B,D,b

Figure 8. Birth of period-4 cycle from period-2 cycle.

As c increases further, additional bifurcation occurs and new fixed points
are created repeatedly in the same way as just described. A sequence of thistype of bifurcation leads to complicated dynamics. Notice however that even
if aperiodic cycle of the output emerges, the advertisement trajectory still
oscillates between two points x1 and x2 since it’s dynamics is independent of
the value of c. In Figure 9, two diﬀerent attractors are illustrated: on the left
side, c = 2.093 and (a, r) = (3, 0.6) for which the advertisement map gives
rise to the period-2 cycle and on the right side, c = 1.9 and (a, r) = (3.5, 0.6)
for which the advertisement map gives rise to chaotic behavior, therefore theoutput also behaves chaotically.

25

Figure 9. Two attractors with diﬀerent values of a

4.2

Heterogeneous Advertisement

In this section we remove the simplifying assumption of homogeneity andreturn to the original 4D dynamic model (1) that is composed of the output
dynamic model, qi0 = φi (qj , xi ),and the advertisement dynamic model, x0i =
ϕi (xi ) for i, j = 1, 2. As shown in Lemmas 1 and 2, there are three ways toviolate the local stability conditions: a set of parameters crosses one of the flip
boundaries a1 = 2+r1 , a2 = 2+r2 or the Hopf boundary (α−1 −d1 )(α−d2 ) =−4. Since the time-evolution of the stationary point depends on the values ofri , ai and c, we consider dynamic characterization in the parametric region A
defined above by changing the value of c with fixed values of r1 and r2 at 0.6.7
Each of the two flip boundaries and the Hopf boundary divides A into two
subsets; the stable region in which the absolute value of the corresponding
eigenvalue is less than unity and the unstable region in which |λi | > 1.
Thus, set A can be divided into 23 parts because there are essentially threeeigenvalues and each of them can be greater or less than unity in absolutevalue.8 The actual number of the division depends on the value of c, theparameter controlling the nonlinearity of the output model. In the first case,the division of A by the two flip boundaries is examined:
ASS = {(a1 , a2 ) ∈ A | r1 ≤ a1 < 2 + r1 and r2 ≤ a2 < 2 + r2 },ASU = {(a1 , a2 ) ∈ A | 2 + r1 ≤ a1 ≤ 3 + r1 and r2 ≤ a2 < 2 + r2 },AU S = {(a1 , a2 ) ∈ A | r1 ≤ a1 < 2 + r1 and 2 + r2 ≤ a2 ≤ 3 + r2 },AU U = {(a1 , a2 ) ∈ A | 2 + r1 ≤ a1 ≤ 3 + r1 and 2 + r2 ≤ a2 ≤ 3 + r2 },7

Needless to say, a choice of the particular values of ri does not aﬀect the qualitative
aspects of the results obtained below.
8
Although there are actually four eigenvalues, it is suﬃcient to consider the division of
A by only three of them, since λ1 = −λ2 with idential absolute values.

26where the first (respectively, the second) subscript of A denotes S or U
according to whether the stationary point of firm 1’s (respectively, firm 2’s)advertisement is stable or unstable. The first question we are concerned is
how the value of c further divides A.
To answer this question, let us begin our analysis by defining three criticalvalues of c for which the Hopf boundary just passes through the three points,(3 + r1 , 3 + r2 ), (2 + r1 , 2 + r2 ), and (r1 , r2 ), respectively. Let ai be equal tok + ri for k = 0, 2, 3. Considering z as an unknown, we first solve the Hopf
boundary condition, −4 = (z −1 − d1 )(z − d2 ) for z, where di = 2 − arii andai = k + ri . Then we substitute the solution into equation (5) and then solve
it for c to obtain the critical value in terms of ri and k. Taking ri = 0.6 for
i = 1, 2 as given, we obtain the following critical values:√
ck=3 = 2.05865, ck=2 = 2.15917 and ck=0 = 5.828...(= 3 + 2 2).
√
The value 3 + 2 2 is the marginal cost
√ ratio for which Puu’s duopoly model
loses stability. Therefore ck=0 = 3+2 2 is intuitively clear because with ai =ri , the model is reduced to Puu’s model as seen in case (1) of Section 4.1.2.
According to the three critical value of c, four cases have to be considered.
(1) c ≤ ck=3
If the marginal cost ratio is small enough, so strict inequality holds, thenthere is no intercept of the Hopf boundary curve with any part of A. As aresult, (α−1 − d1 )(α − d2 ) > −4 for all (a1 , a2 ) ∈ A, so the output modelwith constant advertisement is stable. Notice that the properties of theadvertisement dynamics depend on the choice of (a1 , a2 ) from A, and the time
evolution of the output is aﬀected by the dynamics of the adjustment model.
Therefore, the properties of the output dynamics also depend on the choice of(a1 , a2 ) from A. In particular, for (a1 , a2 ) ∈ ASS , the advertisement dynamics
is stable so is the output dynamics. On the other hand, for (a1 , a2 ) ∈ A\ASS ,the complementary set of ASS , one or both of the advertisement maps isunstable and so is the output model.
Let us examine the unstable cases in more detail. If (a1 , a2 ) is chosenfrom either ASU or AUS , one of the advertisement maps is unstable and theother is stable. Thus we can limit our discussion to dynamics generated by
the output model and the unstable advertisement map, because the stable
advertisement map does not aﬀect asymptotic behavior of the other variables.
To simplify the discussion, the 4D dynamic system can be reduced to a 3D
dynamic system whose dynamics is essentially the same as the one generatedby (10) where ϕ(x) is replaced with the unstable advertisement map ϕi (xi ).
27

For (a1 , a2 ) ∈ AU U , both advertisement maps are unstable and produce
complex dynamics as ai is approaching its upper bound. It is easily expectedthat the output model generates complex dynamics aﬀected by the complex
behavior of the advertisement. It is repeatedly stated that the emergence ofcomplex dynamics is one feature of the output dynamics. It also follows fromthe discussions of Section 4.1.1 that another feature of the output dynamics
is multistability, namely coexistence of stable periodic cycles. Here we turn
our attention to this issue and address the following important question: if
ϕ1 (x) and ϕ2 (x2 ) have a stable period-m cycle and a stable period-n cycle,
respectively, then how many cycles does the advertisement model generate.
We start with the simple case in which ϕ1 (x) gives rise to a stable
B
period-2 cycle and ϕ2 (x2 ) a stable period-3 cycle. Let X1 = {xA
1 , x1 }0B
be a set of two periodic points with the eigenvalue λ1 = ϕ01 (xA
1 )ϕ1 (x1 )B
Cand X2 = {xA
2 , x2 , x2 } a set of three periodic points with the eigenvalue
0
A
0
B
0
λ2 = ϕ2 (x2 )ϕ2 (x2 )ϕ2 (xC
2 ). In considering the iteration of the advertisement
maps, we first redefine the advertisement dynamics as a two-dimensional map
Φ : R2 → R2 given byΦ(x1 , x2 ) = (ϕ1 (x1 ), ϕ2 (x2 )),and then focus on the 2 × 3 points of the (x1 , x2 ) space obtained by the
Cartesian product X1 × X2 . It can be checked that a period-6 cycle is formedby these six points because the least common multiple period of period-2 andperiod-3 cycles is six,A{Φt (xA
1 , x2 ), t = 1, ..., 6}.

The stability of the period-6 cycle can be shown as follows. Denote the0iteration of the advertisement map by ϕti (xi ) = ϕi (ϕt−1i (xi )) and ϕi (xi ) = xi .
2
A3
Since xA
1 is the fixed point of ϕ1 (x1 ) and x2 is the fixed point of is ϕ2 (x2 ),diﬀerentiating the two-dimensional map at the fixed points
A
6 A
A
2×3 A
3×2 A
(xA
1 , x1 ) = Φ (x1 , x2 ) = (ϕ1 (x1 ), ϕ2 (x2 ))yields the Jacobi matrix
!
Ã ¡
¢
0
A
0
A 3
0
ϕ1 (x1 )ϕ1 (x1 )A¡ 0 A 0 B 0 C ¢2 .
J(xA
1 , x2 ) =0
ϕ2 (x2 )ϕ2 (x2 )ϕ2 (x2 )A
32The eigenvalues of the period-6 cycle are λA
1 = (λ1 ) and λ2 = (λ2 ) so theperiod-6 cycle is stable if the original period-2 and the period-3 cycles arestable, that is |λi | < 1 for i = 1, 2.

28

We proceed to the next case in which ϕ1 (x) gives rise to a period-2 cyB
cle with periodic points X1 = {xA
1 , x1 } and ϕ2 (y) a period-4 cycle withB
C
D
periodic points X2 = {xA
2 , x2 , x2 , x2 }. Eigenvalues of the two cycles are
0
A
0
B
0
A
0
C
0
B
λ1 = ϕ1 (x1 )ϕ1 (x1 ) and λ2 = ϕ2 (x2 )ϕ02 (xB
2 )ϕ2 (x2 )ϕ2 (x2 ), respectively. Set
X1 × X2 has eight points, and since the least common multiple period of
period-2 and period-4 cycles is 4, there are 8/4 = 2 period-4 cycles:A{Φt (xA
1 , x2 ), t = 1, ..., 4},andB{Φt (xA
1 , x2 ), t = 1, ..., 4}.

A
Since the first period-4 cycle visits its periodic points in the order of (xA
1 , x2 ) →
B
B
A
C
B
D
(x1 , x2 ) → (x1 , x2 ) → (x1 , x2 ), the Jacobi matrix isµ ¡¶¢2
ϕ01 (xA)ϕ01 (xB
)
0
A
A
1
1
J(x1 , x2 ) =
,
0
B
0
C
0
D
0
ϕ02 (xA
2 )ϕ2 (x2 )ϕ2 (x2 )ϕ2 (x2 )BBCand by the same way, since the second cycle is formed as (xA
1 , x2 ) → (x1 , x2 ) →ADBA(x1 , x2 ) → (x1 , x2 ), its Jacobi matrix is¶µ ¡¢2)ϕ01 (xB)
0ϕ01 (xA
A
B
1
1.
J(x1 , x2 ) =
0
C
0
D
0
A
0
ϕ02 (xB
2 )ϕ2 (x2 )ϕ2 (x2 )ϕ2 (x2 )The two period-4 cycles have the same eigenvalues, λ1 = (λ1 )2 and λ2 = (λ2 )1
so each period-4 cycle is stable if the original period-2 cycle and the period-4
cycle are stable.
Now we move to the general case in which ϕ1 (x1 ) gives rise to a periodm cycle with periodic points X1 = {x11 , x21 , ..., xm
1 } and ϕ2 (x2 ) a period-n12ncycle with periodic points X2 = {x2 , x2 , ..., x2 }. The least common periodof period-m and period-n cycles is the least common multiple, q, of m andn, and there are mn points in X1 × X2 . Let k = mn/q, then this formula
implies that the mn points obtained by the Cartesian product X1 × X2 canbe divided into k subsets, each of which consists of q points:{Φt (x11 , xα2 ) = (ϕt1 (x11 ), ϕt2 (xα2 )), t = 1, ..., q} for α = 1, ..., k,so we have k period-q cycles. The stability of the period-q cycle is determinedby the eigenvalues of ϕti (xi ), λ1 = Πqi=1 ϕ01 (xi1 ) and λ2 = Πqi=1 ϕ02 (xi2 ). Thenany period-q cycle is stable if the original periodic cycles are stable. We
summarize these results as follows:29

Theorem 3 Suppose that ϕi (xi ) has a period-m cycle, ϕj (xj ) has a periodn cycle and q is the least common multiple of m and n. Let k = mn/q.
Then the advertisement model generates k coexisting period-q cycles
and each period-q cycle is stable if both of the period-m cycle and theperiod-n cycle are stable.
(2) ck=3 < c ≤ ck=2
If c > ck=3 , then the Hopf boundary crosses some part of A so that A
is further divided. From an economic point of view, the output must be
non-negative and thus must satisfy the confinement conditions c ≤ ψ(x1 , x2 )where ψ is defined in (8). In Figure 10A in which c = 2.075, three downward
sloping curves are depicted in AU U ∪ ASU . The right most curve is the Hopf
boundary, the curve depicted in AUU is the confinement boundary to be
defined in the case where both advertisement stationary points, xe1 and xe2 ,are unstable and the curve illustrated in ASU is the confinement boundary to
be defined in the case where xe1 is unstable but xe2 is stable. The confinementmaxboundary depicted in AUU is defined by c = ψ(xmax
1 , x2 ) where
xmax
=
i

(1 + ai − ri )2
for i = 1, 2.
4ai

For (a1 , a2 ) ∈ AU U , ϕi (xi ) may be chaotic so it may take its maximum
value, denoted by xmax
along the chaotic trajectory. To prevent output fromi
becoming negative, we require the most strict confinement constraint whenϕ(xi ) is unstable. The confinement boundary depicted in AU S is defined byeeec = ψ(xmax
1 , x2 ), since x1 is unstable and x2 is stable. In the same way,c = ψ(xe1 , xmax2 ) can be defined as the confinement boundary in ASU in whichex1 is stable but xe2 is unstable. It is not shown in Figure 10A, since thisconfinement curve does not intersect with ASU so it is ineﬀective. One orboth confinement conditions are violated in the white region. The output
model is stable in the left side of the Hopf boundary and unstable in theright side. The whole system is stable in ASS . Dynamics in ASU and in thefeasible (i.e., colored) region of AU S is described by the 3D system obtained byeliminating the stable advertisement map from the 4D system. The dynamics
in the feasible region of AU U is qualitatively the same as the one consideredin case (1) just above.
(3) ck=2 < c ≤ ck=0
We set c = 2.35 in Figure 10(B) where four downward sloping curves are
depicted. The most left curve is the Hopf boundary. Three other downward30

sloping curves give the confinement boundaries, c = ψ(xe1 , xmax2 ) in ASU ,ec = ψ(xe1 , xe2 ) in ASS and c = ψ(xmax
,
x
)
in
A
,
respectively.
The whiteSU1
2
region that includes the entire AUU is infeasible. It can be seen that anincrease in the value of c shifts the Hopf boundary and the confinementboundaries leftward. Two further observations emerge by comparing Figure10(A) with Figure 10(B): (1) the eﬀect on the Hopf boundary is so strongthat it shifts the boundary from the right most line to the left most line and(2) the c = ψ(xe1 , xmax2 ) curve divides AU S and thus the output trajectories
become infeasible (i.e., negative) for some configurations of (a1 , a2 ). Dynamics
in the feasible regions is governed by a 3D or 2D system according to which
equation is stable or unstable.

Figure 10. 2-parameter bifurcation diagrams with diﬀerent values of c

We now turn our attentions to the dynamics in ASS in order to consider
multistability due to nonlinearities of the output model. In ASS between the
Hopf boundary and the confinement boundary c = ψ(xe1 , xe2 ), the advertisement model is stable while the output model is unstable but may be feasible.
Thus in the following, we limit our analysis to the case in which the advertisement adjustment is so rapid that the asymptotic output dynamics can beconsidered to be governed by the iteration of the two-dimensional map
0q1 = φ1 (q2 , xe1 ),(3’)
0q2 = φ2 (q1 , xe2 ).
This is the special case of the output model (3) in which the advertisementtakes its stationary value.9 It is the topological conjugate of Puu’s nonlinear
9

The very special model with xe1 = xe2 has already been considered in case (2) of Section
4.1.2.

31

duopoly model. Puu (2003) has already demonstrated that this model withnaive and adaptive expectations can give rise to not only rich dynamics ranging from periodic behavior to chaotic dynamics but also to the coexistenceof attracting sets by applying the results of Bischi, Mammana and Gardini(2001) and Agliar, Bischi and Gardini (2002). Therefore multistability canbe a characteristic property of the output model (3’). It should be notedthat output trajectories could be negative on the way to the attractor if theconvergence of the stable advertisement dynamics is supposed to be sluggish.
As will be suggestged later, we need to take the nonnegativity constraint into
account in the best response behavior in this case.
(4) ck=0 < c√
In Figure 12, we set c = 3 + 2 2 for which the Hopf boundary curve
passes through the point (r1 , r2 ). We have |λi | > 1 in A for i = 1, 2, but
the solutions for the output are feasible only inside of the triangular region
surrounded by the confinement boundary, c = ψ(xe1 , xe2 ). It can be seen thatthe periodic cycles double their periods as the pair (a1 , a2 ) moves toward
the upper bound. In Puu’s model, the marginal cost ratio c is the source of
nonlinear dynamics. Introducing the advertisement into Puu’ s model showsthat advertisement outlays also can be the source.

√
Figure 11 2-parameter bifurcation diagaram with c = 3 + 2 2.

5

Conclusion

The present study considers local as well as global dynamics in nonlinear
duopoly models with advertisement, which consist of two submodels, the32

output dynamic model and the advertisement dynamic model. It first determines the stationary state of the model which is the basis of the stability
analysis. We show that the model can be destabilized in two ways: one isthrough destabilizing the advertisement model when the parameter configurations cross over the flip boundary where one of the eigenvalues is −1.
The other is through destabilizing the output model when the parameter
configurations cross over the Hopf boundary on which the eigenvalues are
pure imaginary and their common absolute value is unity. It demonstrates
that a half-pitchfork bifurcation of the output occurs when the advertisement
model gives rise to periodic cycle and the nonlinearity of the output model becomes stronger. The explicit treatment of the confinement conditions reveals
the appropriate parameter configurations which prevent unstable trajectories from becoming negative. We also demonstrated that the multistability coexistence of attracting sets - is the distinguished feature of the model.
The present model should be extended at least in two diﬀerent directions.
One direction is to take away Assumption 1, the simplifying assumption on
the advertisement. Under the general condition, bi > 0, the advertisment
model also has the multistability and thus enhances output dynamics. The
other direction is to take the nonnegativity condtion into account. The feasibility of the output trajectories can also be enforced by modifying the best
response mapping of the firm as follows:
⎧
1
⎪
⎪
0
if qj ≥ ,
⎪
⎪
ci
⎨
qi0 =
¶
µr
⎪
⎪
qj
⎪
⎪
otherwise.
− qj⎩ (1 + xi )ci
In this case the best responses are defined as the profit maximizing outputs
under the nonnegativity condition. This new method results in diﬀerent
dynamic properties than those discussed in this paper. We will return tothis modifed model with positive bi in a future paper.

33

References[1] Agliari, A., G-I., Bischi and L. Gardini, "Some Methods for the Global
Analysis of Dynamic Games Represented by Iterated Noninvertible
Maps," in Puu and Sushko (2002), 31-83, 2002, Springer.
[2] Ahmed, E., H. N. Agiza and S. Z. Hassan, "On Modeling Advertisementin Cournot Duopoly," Chaos, Solitions and Fractals, vol. 10, 11791184, 1994.
[3] Bischi, G-I., C. Chiarella, M. Kopel and F. Szidarovszky, Nonlinear
Oligopolies: Stability and Bifurcations, to be published in 2008.
[4] Bischi, G-I., C. Mammana, and L. Gardini, "Multistability and cyclic
attractors in duopoly games," Chaos, Solitions and Fractals, vol. 11,
543-564, 2000.
[5] Feichtinger, G., R. F. Hartl and S. P. Sethi, "Dynamic Optimal Control
Models in Advertising: Recent Developments," Management Science, vol. 40, 195-226, 1994.
[6] Kopel, M., "Simple and Complex Adjustment Dynamics in Cournot
Duopoly Models," Chaos, Solitons and Fractals, vol.7, 20312048,1996.
[7] Luhta, I., and I. Virtanen, "Non-linear Advertisement Capital Modelwith Time Delayed Feedback between Advertising and Stock of
Goodwill," Chaso, Solitions and Fractals, vol. 7, 2083-2014, 1996.
[8] Nerlove, M., and K. Arrow, "Optimal Advertising Policy under Dynamic
Conditions," Econometrica, vol. 29, 129-142, 1962.
[9] Puu, T., "Chaos in Duopoly Priceing," Chaos, Solitions and Fractals,
vol.1, 573-581, 1991.
[10] Puu, T., and I. Sushko, Oligopoly Dynamics, Springer, 2002.
[11] Rand, D., "Exotic Phenomena in Games and Duopoly Models," Journalof Mathematical Economics, vol. 5, 173-184. 1978.
[12] Shin, G-S, "Interwoven Basin Structure of Double Logistic Map at theEdge of Chaos," mimeo, 2007.
[13] Sethi, S. P., "Dynamic Optimal Control Models in Adverting: A Survey," SIAM Review, vol. 19, 685-725, 1977.

34