Directory UMM :Data Elmu:jurnal:J-a:Journal Of Business Research:Vol51.Issue3.2001:

Journal of Business Research 51 (2001) 179 ± 191

Non-linear dynamics and duopolistic competition: A R&D
model and simulation
Simon Whitbya,1, David Parkerb,*, Andrew Tobiasa,1
a

School of Manufacturing and Mechanical Engineering, University of Birmingham, Birmingham, B15 2TT, UK
b
Aston Business School, Aston University, Birmingham, B4 7ET, UK
Received 1 December 1998; accepted 1 December 1998

Abstract
In recent years, a number of studies have considered the application of chaos theory to economics; the primary focus, however, has been
the implications for stock prices, the foreign exchange market, and the macroeconomy. This paper describes a non-linear model of duopolistic
competition, which focuses on a firm's expenditure and the resulting quality or technological endowment of its product. Results from
computer iteration of the model are presented which indicate that chaotic outcomes are possible for a range of competing managerial policies;
the associated unpredictability is due solely to the dynamics of the interaction. The study also provides the results of some initial work on
how management adaptation may act to forestall chaotic outcomes. D 2000 Elsevier Science Inc. All rights reserved.
Keywords: Economics; Competition; R&D; Non-linear dynamics; Chaos

1. Introduction
In recent years, a number of studies have suggested that
non-linear dynamics offers a useful approach to economic
modeling (e.g. Baumol and Benhabib, 1989; Day, 1987;
Kelsey, 1988; Parker and Stacey, 1995; Radzicki, 1990 and
the papers in Anderson et al., 1988). The approach
requires a form of systems thinking which recognizes that
systems evolve and adapt, persistently existing far from
equilibrium. In particular, in a number of cases, systems of
simple non-linear modeling equations have been shown to
be capable of erratic and random-like behavior, which is in
fact deterministic; such behavior is described as deterministic chaos.
This article provides a non-linear model linking R&D
spending and product quality in a duopolistic industry.
Results from a computer iteration of the models are presented. These indicate that chaotic outcomes are possible for
a range of competing managerial policies with the associated
unpredictability being solely due to the dynamics of the

* Corresponding author. Tel.: +44-121-359-3611; fax: +44-121-3333474.
E-mail address: d.parker1@aston.ac.uk (D. Parker).

1
Tel.: +44-121-414-4263; fax: +44-121-414-7484.

interaction. The study also provides some initial results on
work into how management adaptation may act to forestall
chaotic outcomes. The article has relevance to anyone
researching or managing in the field of R&D and innovation.
The results demonstrate that, based on some simple behavioral rules, quite complex chaotic outcomes are possible.
They also show that, with an awareness of the possible
outcomes from certain adaptive adjustments, management
actions can prevent or remove chaotic outcomes. In particular, collaboration between the duopolists may dampen erratic
fluctuations and actually be welfare-enhancing in economic
terms. The paper begins with a summary of previous related
research into non-linear models and microeconomics.

2. Microeconomics and chaos
2.1. Chaotic systems
Chaos theory concerns systems where the relationships
between variables are such that unpredictable but deterministic outcomes can result. Evolving chaotically, system
variables are oscillatory with upper and lower bounds, the

fluctuations being erratic to the extent that standard statistical tests cannot distinguish them from purely random data.
The behavior, however, is deterministic and not a result of

0148-2963/01/$ ± see front matter D 2000 Elsevier Science Inc. All rights reserved.
PII: S 0 1 4 8 - 2 9 6 3 ( 9 9 ) 0 0 0 5 0 - 8

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S. Whitby et al. / Journal of Business Research 51 (2001) 179±191

outside (exogenous) factors. It is solely due to the mathematics of the relationship between variables endogenous to
the system. In addition, such systems can have periods of
regular behavior, such as equilibrium and periodicity, that
arise abruptly and disappear just as suddenly.
The state of an evolving system can be represented by a
number of time-dependent variables; the minimum number
of variables (n) needed to characterize a system is called its
dimension. As a system evolves, its state can be considered
to exist in a space of dimension n called phase space, the
components of which correspond to the system's n state

variables. A geometric space with axes given by the phase
space components (i.e. the system variables) provides a
setting for the graphic illustration of a system's evolution. In
particular, for a dissipative system (i.e. one possessing a
mechanism to dissipate perturbations from steady motion), a
steady-state behavior can take place in phase space on an
attractor. Crutchfield et al. (1986) define a system's attractors
as geometric forms that characterize the system's long-term
behavior in phase space. Chaotic evolutions of dissipative
systems in phase space occur on strange attractors, which are
complicated shapes of fractal structure, first observed by
Lorenz (1963). Many studies of non-linear dissipative dynamical systems use strange attractors to characterize chaos.
In order to predict future states of a time-dependent
system, a set of modeling equations can be developed. Such
a set of equations is called a dynamical system, which may
evolve continuously or discretely over time depending on
the real system modeled. A continuously evolving dynamical system is called a flow, while the discrete case is called a
mapping. Flows are represented by differential equations,
while difference equations describe the discrete case. The
dependencies governing the evolution over time, t, of a

deterministic flow can be expressed as a set of first-order
differential equations giving the rates of change of the n
system variables, xi, in terms of the same variables, i.e.:
d x1
ˆ f1 …x1 ; x2 ; . . . ; xn ; t†
dt
d x2
ˆ f2 …x1 ; x2 ; . . . ; xn ; t†
dt

…1†

#
d xn
ˆ fn …x1 ; x2 ; . . . ; xn ; t†
dt
For a discrete-time-mapping, the evolutionary laws can be
similarly expressed, but as difference equations giving the
values of the system variables in period t + 1(xi,t + 1) in
terms of their values in the previous period, t(xi,t), i.e.:

x1;t‡1
x2;t‡1

ˆ g1 …x1;t ; x2;t ; . . . ; xn;t ; t†
ˆ g2 …x1;t ; x2;t ; . . . ; xn;t ; t†
…2†

#
xn;t‡1

ˆ gn …x1;t ; x2;t ; . . . ; xn;t ; t†

The most fundamental requirement for chaos is nonlinearity. In the context of Eqs. (1) and (2), this essentially
means that at least one state variable will appear in the
functions fi or gi (where i denotes the index of a general
variable) with a non-linear relationship to the rate or the
next period value, respectively. Over and above this, precise
specification of the prerequisites for chaos of a general
system is difficult; there are presently no general criteria to
establish the necessary and sufficient conditions for a

system of differential or difference equations to be chaotic.
Requirements have been established, however, regarding
the minimum dimensionality of the system. Different conditions exist for flows and maps (Ott, 1993). For the case of
first-order, autonomous ordinary differential equations, i.e.
of the form of Eq. (1), the lowest dimension for which
chaos is possible (nmin) is 3. For a mapping, the minimum
required dimensionality differs depending on whether or not
the mapping is invertible. Using notation as in Eq. (2), an
invertible map is one whose equations, as well as giving a
unique value for each xi,t + 1 from the set of xi,t can also be
solved to give a unique value for each xi,t from the xi,t + 1.
For an invertible map, there cannot be chaos unless nmin 
2. For a non-invertible map, chaos is possible even for a
dimensionality of one. The logistic equation introduced by
May (1976) in a study of population, is a one-dimensional
map often used to illustrate chaotic behavior. It is a finite
difference equation with a quadratic non-linearity or hump
and takes the general form xt + 1 = l x t (1ÿxt). In population
studies, xt + 1 represents the population level and depends
on its immediate predecessor, xt. The parameter l is the

growth rate from one cycle to another and x is constrained
by a resource constraint modeled as 1ÿxt. Through feedback leading to period doubling (see later), the system's
evolution can become chaotic as the parameter value, l, is
increased, with chaos occurring when l is between around
3.6 and 4.
Whereas little can be said, a priori, concerning a system's
capability of producing chaos, for systems shown to be
chaotic some typical ``routes'' or ``scenarios'' have been
identified by which the onset of chaos may be approached.
In each case, the transition is effected by continually
changing (usually increasing) one or more ``control'' parameters, such as l in the logistics equation above. Normally,
only one parameter is varied, in which case the scenario is
known as a codimension Ð one route to chaos. Generally, at
distinct values of the changing parameter (l) bifurcations
occur. Bifurcations involve qualitative changes in the behavior of the system, for example, from equilibrium to
periodic or from periodic to chaotic (for a fuller explanation
see, e.g. Berge et al. 1984; Eckmann, 1981; Medio, 1992;
Ott, 1993). One scenario, known as intermittency, involves
interruption of otherwise regular periodic evolution with
bursts of irregular, chaos-like behavior as l is varied above/

below a critical value. The bursts are usually seen to become
more frequent as l is increased/decreased until the original,
regular behavior is lost and becomes chaotic.

S. Whitby et al. / Journal of Business Research 51 (2001) 179±191

Another route to chaos for an initial period system, is
period doubling. As l is increased, bifurcations occur at
particular values as additional periodicities complicate the
system's cycle. The frequency of bifurcation increases as l
is progressively increased until at a finite value of l the
number accumulates to infinity; increasing l beyond this
point results in the evolution becoming chaotic. Period
doubling will be described in further detail later, when
considering the results of our model.
In the light of what has been described above, we can
say that any possible candidate to represent a chaotic
system must satisfy the non-linearity and minimum dimensionality requirements. In addition, the model must be
oscillatory, or at least have the potential to be so, in order
not to preclude the possibility of its developing from

periodic to chaotic evolution.
2.2. Chaos and the firm; inventory control
If microeconomic institutions such as firms and markets
are governed by linear, or simple non-linear relationships, it
is possible to study them separately (adopt a reductionist
methodology) without losing crucial characteristics of their
behavior. Interactions with other firms or markets can be
relatively easily assimilated into the models as generally
well-behaved positive and negative feedback effects. Equilibrium outcomes result and uncertainty as to the model's
outcome is minimized. If economic systems are significantly
non-linear, however, then their behavior may be highly
unpredictable, and this limits the application of both traditional neoclassical market models and more recent games
theoretic modeling involving Bayesian techniques.
Many of the non-linear models, which have identified the
potential for chaos in microeconomics, involve discrete-time
queuing or consideration of inventory. A typical example is
the beer distribution model, an exercise in beer-stock
management involving non-linearity and feedback (Mosekilde and Larsen, 1988; Sterman, 1988; Van Ackere et al.,
1993). Mosekilde and Larsen presented an overview of this
model's behavior, producing a time series through iterations

of the model and identifying chaos for certain parameter
values. Sterman's approach was notably different in his use
of the model to provide a simulated environment in which
subjects could take part in a beer distribution game. To
describe the subjects' behaviors, a general heuristic was
constructed, which, on further iteration, suggested chaos for
a significant minority of values.
The common feature in these inventory control models
is a delay between the ordering of new stock and its
delivery. Furthermore, disruption in one part of the supply
chain leads to a sequence of changes in other parts;
expansion of output, for example, leads to excess capacity
developing when orders fall. Overall, this suggests a
tendency for oscillation in supply behavior, which, for
certain conditions, can become chaotic (Erramilli and
Forys, 1991; Levy, 1994).

181

The interplay between two departments functioning separately and in parallel, as opposed to in sequence as in the
stock management problem, was considered in a study by
Rasmussen and Mosekilde (1988). This study proposed a
model of a production company where resources are shared
between production and marketing in accordance with a
decision rule reflecting variations in inventory or the backlog of customer orders.
Other studies have suggested that the incidence of chaos
in business may be much wider. For example, it has been
suggested that there may be a tendency towards strongly
non-linear local interactions where there are nonconvexities
in production technology, due to technological indivisibilities for instance (Gordon, 1992). Gordon also used a
variation of the logistic equation to produce a plausible
model of the relationship between a company's advertising
budget and its resulting sales. Further economic relationships subject to feedbacks and timing, which have been
investigated for chaotic outcomes, are investment behavior
(Mosekilde et al., 1992), the planning of teacher ±student
ratios in education (Feichtinger and Novak, 1992), and
advertising outlays (Baumol and Benhabib, 1989).
The effect of research and development (R&D) activity
on a firm's time path has been modeled by Baumol and
Wolff (1983), Feichtinger and Kopel (1993), and Kopel
(1996). Baumol and Wolff focused on the relationship
between R&D and productivity growth, showing chaos to
be possible for certain conditions. The Feichtinger and
Kopel and Kopel models, on the other hand, described a
non-linear increase of sales for higher R&D allocations,
employing a treatment closer to that presented here. Notably, Feichtinger and Kopel placed particular emphasis on the
influence of the decision-maker (i.e. the manager) whose
input amounted to periodic adjustments to R&D funding in
the light of current sales. This was represented by a nonlinear decision rule that is well-founded in behavioral
decision theory, and which also features in the model used
in this study, namely the anchoring and adjustment heuristic, described in Tversky and Kahneman (1974). A variety
of temporal behavior was observed, including chaos.
In the above studies, the model parameters representing
human behavior were kept at a constant value throughout a
given evolution. This failure to allow parameter variability
within evolutions effectively ignores the fact that decisionmakers may adjust their responses over time; their actions
may evolve through some form of iterative process, for
example. There is a comment on this point in Sterman
(1988, p. 174).
2.3. Chaos and oligopoly
Oligopolistic industries demonstrate phases of co-operative behavior and phases of intense price and non-price
competition. Such phases result from oligopolists taking
into consideration the reactions of their competitors, as well
as consumers. Competitors may retaliate to price cuts, for

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S. Whitby et al. / Journal of Business Research 51 (2001) 179±191

example, by also cutting prices so as not to lose market
share. Developments in oligopoly theory have shown that
various other outcomes are possible depending, for example, upon the financial strength and production conditions
for each of the firms. In recent years, it has been recognized
that competition among a few firms can be usefully modeled
in a game theoretic framework that contains feedback, time,
and uncertainty. In game theoretic models of oligopoly,
rivalry is studied from an equilibrium (usually a Nash
equilibrium) perspective and uncertainty is dealt with by
an expected profit or payoff maximization. The outcome
then depends upon the assumptions made about actions and
reactions at the various stages of the game (e.g. see
Hargreaves Heap and Varoufakis, 1995; Phlips, 1995).
Game theory has introduced dynamism into the study of
the competitive process, opening up the possibility of
sufficiently non-linear reaction functions to give chaotic
outcomes. Rand (1978) considered the possible conditions
in which chaotic behavior might result, and more recently,
Byers and Peel (1994) flagged the possible contribution the
analysis of dynamic non-linear behavior might make to the
study of industrial economics. To date, however, there has
been limited analysis of the potential for chaos in oligopolistic markets. Noteworthy studies include that by Dana and
Montrucchio (1986), which showed how chaotic dynamical
paths might arise in duopoly where there are small rates of
discount, and that by Puu (1994), who considered an
adjustment process for a pair of Cournot duopolists. Puu
demonstrated mathematically that with an iso-elastic demand function and constant marginal costs, the duopoly
system could exhibit periodic, and even chaotic behavior.
In Section 3, a new, non-linear model of duopolistic
interaction is described in which the effect of managerial
policy plays a central role. The formulation is based on the
relationships specified in the R&D model of Feichtinger and
Kopel (1993). However, while their study included managerial judgements relative to an internally determined target
performance, the following extends the analysis by applying
the relationships to the interaction between two separate,
competing companies, the policy judgements of each being
made in light of its own performance relative to the other. In
addition, while Feichtinger and Kopel's treatment, along
with the vast majority of model simulation studies, makes
no allowance for policy adjustment over time, this study
makes some attempt to include the management's capacity
to adapt its strategy.

3. A basic model of duopolistic competition with
non-linear dynamic interactions
The proposed model describes two manufacturers producing a similar technological product and competing
through product innovation. In both cases, managerial
decisions regarding the extent of an input or resource
allocation are made in the light of the organization's recent

standard of achievement in terms of the resulting product,
relative to that of the competition. In addition, this standard is, for both firms, a direct result of the level of inputs,
i.e. resources applied. In constructing the model, particular
emphasis was placed on the resources that each allocates
to R&D, and the resultant (measurable or subjective)
product standard/quality, such as the number of features
or durability. The latter are means by which discerning
consumers may differentiate in a market where product
standard is an important order winning criterion. Expenditure on R&D could also result in a change in the process
of production rather than a change in the product, having
implications for costs of supply; this impact on process is
not modeled here. In the model, an increase in a company's resource allocation to R&D is considered to lead to an
improved product standard (quality or technological endowment of the product), while having a product standard
superior to that of the competition is deemed to constitute
a competitive advantage leading ultimately to greater sales
and profits. Current sales, therefore, are considered to be a
direct result of the current, relative product standards of the
competitors and so do not need to feature explicitly in the
model. Each firm is assumed to be able to monitor readily
the product standard of its competitor.
The model is structured to represent two companies, A
and B, for which the resource allocations to R&D for period
t are denoted by RAt and RBt, resulting in product quality
standards denoted for A and B during period t by SAt and
SBt, respectively. Modeling R&D to impact on the product
standard in the same time period is admittedly a simplification Ð R&D might be expected to affect product standard in
t + n where n varies depending on the product and industry;
the simplification is introduced for ease of modeling. A
notable consequence of the simplification is that for conditions under which the system oscillates over time, the
fluctuations of the separate companies will be either in
phase (peaks coincide) or in anti-phase (one's peaks coincide with the other's troughs).
To facilitate description, X, RXt and SXt are used to
represent either company and its associated system variables for period t, in the general case. Arising from
managerial decisions made at the end of each time period,
changes to R&D allocation are modeled as dependent on
the current relative quality or technological standing of the
firm's products.
The interactions within the system are shown in Fig. 1,
where the arrows between variables represent deterministic
cause and effect Ð the ``tail'' variables affecting the ``heads,''
e.g. RAt affects SAt. The dotted arrows describe similar
effects but between tail variables and the head variables of
the following period, e.g. SAtÿSBt affects RAt + 1.
For both companies, the demand for the output in the
market is assumed to be a function of the product standard.
Other variables that may impact on demand, such as price,
are assumed to be unimportant or constants. While it is
recognized that the amount of resources dedicated to R&D

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S. Whitby et al. / Journal of Business Research 51 (2001) 179±191

Fig. 1. Model of duopoly comprising two similar manufacturers (A and B)
of technological products.

may have implications for costs and consequently affect
prices and demand, the introduction of this feedback would
complicate the analysis and is not, in any case, the relationship which we wished to study. Hence, in the analysis:
QAt =Yt ˆ f …SAt ÿ SBt †

…3†

QBt =Yt ˆ f …SBt ÿ SAt †

…4†

where Yt is the total market volume for period t and QAt and
QBt are sales of A and B, respectively for that period,
making QA/Y and QB/Y the respective market shares. As the
market is a duopoly,
QAt =Yt ‡ QBt =Yt ˆ 1

…5†

For both companies, the R&D expenditures, RAt+1 and
RBt+1, for a given period, t + 1, are assumed to arise from
management decisions at the end of the previous period, t,
(Eqs. (6) and (7)). The expenditure for a period t is
considered to determine directly the quality standard
achieved in production (SAt and SBt) during the same
period, t, (Eq. (8)).
RAt‡1 ˆ max‰0; RAt ‡ c…SAt ÿ SBt †Š

…6†

RBt‡1 ˆ max‰0; RBt ‡ d…SBt ÿ SAt †Š
m 
x
arctan…RXt † ÿ nx
SXt ˆ 2
p

…7†
…8†

The parameters mx and nx in Eq. (8), can be used to tune the
curvature of the function for company X.
If the SXt in Eqs. (6) and (7) are substituted by SXt =
f (RXt), from Eq. (8), then the decision rules take the form
R X t +1 = g(RXt); expressed in this way the representation of
the decision process is seen to be non-linear. Expressions (6)
and (7) are similar in form to that used by Feichtinger and
Kopel (1993), and are based on the anchoring and adjustment heuristic described in Tversky and Kahneman (1974).
The estimation of an unknown quantity by anchoring and
adjustment involves a preliminary estimate given by a
known reference point (the anchor) which is then adjusted
for the effects of other factors. Sterman (1989, p. 324)
reports that ``anchoring and adjustment has been shown to
apply to a wide variety of decision-making tasks'', and gives
appropriate references.

For the managerial decisions in this model, the allocation of the previous period, e.g. RAt for company A, is
identified as the anchor, while an adjustment is considered
to depend on the relative product standards of the two
companies. Eq. (6) for A, for example, shows RAt acting
as the anchor with an adjustment based on the advantage A
has over B in current product standard, i.e. SAtÿSBt. The
parameters c and d are used to define the two management
policies for allocation adjustment. For instance, again
considering A, a positive value of c is used to represent
a reinforcing policy of further increasing a company's
research funding when performance is good, i.e. SAt >
SBt, and reducing funds when performance is poor (SAt <
SBt). Conversely, a negative adjustment parameter reflects
a counteracting policy of reducing funds when performance is good, a poor performance resulting in their
increase. The two policies are summarized for company
A in Table 1. The modulus of the parameter is used to
model the extent to which the management alters funding
in light of the company performance, i.e. the sensitivity of
the decision-maker to performance outcomes. As the decision rules stand, only the two policies are possible; the
specification of a company's action when in a position of
advantage necessarily implies its action when disadvantaged and vice-versa. The formulation could be extended,
however, to allow a greater range of procedures where, for
example, a company could adopt a policy of reinforcement
when at an advantage but take counteracting action when at
a disadvantage.
For each company, the function describing the effect of
various R&D fundings on product standard should show a
general increase of the latter with the former, the relationship becoming markedly non-linear for higher R&D expenditures as diminishing returns set in. Furthermore, in
order to investigate the possibility of complicated behavior,
it is important to consider a description of SXt = f (RXt) for
which non-linearity is apparent over the range of values of
RXt displayed by the model in practice. A function is
required, therefore, whose position and extent of curvature
can be adjusted easily in the light of the range of variation

Table 1
Reinforcing and counteracting policies for company A

Allocation
adjustment
policy

Corresponding
range of
adjustment
parameter c

Reinforcing

c>0

Counteracting c < 0

Action taken
if in a position
of advantage
(SAt > SBt)

Action taken
if in a position
of disadvantage
(SAt < SBt)

Attempt to reinforce
advantage, i.e.
increase in allocation
to research
Relax because of
advantage, i.e.
decrease allocation
to research

Retreat, i.e. decrease
research allocation

Attempt to improve
position, i.e. increase
research allocation

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S. Whitby et al. / Journal of Business Research 51 (2001) 179±191

Fig. 2. The relationship between R&D expenditure (RXt) and product
standard (SXt). Graph origin at (0,0).

observed on initial iterations of the model. Suitable control
is obtained with the arctan form in Eq. (8) above, the shape
of which can be tuned by changing mx and nx. Together,
these parameters determine the upper and lower bound of
SXt in the model. In particular, varying nx changes that
value of product standard (SXt) corresponding to a research
allocation (RXt) of zero; SXt = 0 and RXt = 0 only coincide
for the parameter value nx = 0. It is reasonable to expect SXt
= 0 to correspond to some positive value of RXt, reflecting
some element of fixed costs that do not impact on product
standard, and this is represented in Eq. (8) by a positive
value of nx. The general shape of Eq. (8) is shown in Fig. 2
for which the origin is (0,0) and the negative intercept on the
SXt axis is equal to the value of nx used.

4. Experimentation and results
In general, non-linear dynamical systems, such as the
model presented here, describe temporal evolution with
equations for which there is no simple solution. In such
cases, investigation takes the form of computer iteration, i.e.
computer simulation, which, rather than attempting to solve
the equations, repeatedly implements the model and so
produces actual temporal behavior for analysis. Accordingly, computer simulation was used to investigate the
evolution of our model. Along with quantification of parameters for Eqs. (6) ± (8), each simulation of a given system
arrangement required specification of initial R&D allocations for A and B, i.e. RA0 and RB0. To facilitate the study,
it was assumed that the relationship between funding for
R&D and product standard, Eq. (8), was identical for the
two companies. As a consequence, if one firm spends more
than the other on R&D, this will imply an advantage in
product standard for the same company for that period, e.g.
RAt > RBt () SAt > SBt.
Often, the time paths of an evolving system are considered in phase space, a mathematical space whose n coordinates correspond to the system's n state variables (n being
the minimum number of variables needed to characterize the
system, i.e. its dimension). If n  3, or a subset of phase
space with less than three components is considered, then

this space provides a setting for the graphic illustration of a
system's evolution (see Crutchfield et al., 1986 for an
introduction to phase space, attractors, and the chaotic
strange attractors). However, as will be seen, for the
interesting oscillatory behavior of this system, the two
companies evolve either in phase or in anti-phase (i.e. with
a phase difference of p). It follows that if an appropriate
two-dimensional phase space ((RAt, RBt) or (SAt, SBt)) is
used to analyze system behavior, any oscillatory motion
(including chaos) will produce an unilluminating, straightline phase trajectory. For this model, therefore, phase space
offers a less effective illustration of system behavior than
that given by time series, and accordingly, the latter are used
here to present the results.
The evolutionary properties of the system were found to
depend on the coupling of company strategies. Particular
couplings were specified by the parameter pairings (c,d),
referred to here as the system arrangement. Arrangements
were then classified into four groups, each comprising a
different permutation of the paired signs of c and d, i.e.
(+,+), (+,ÿ), (ÿ,+), and (ÿ,ÿ), and each group containing a
continuous range of system arrangements corresponding to
different moduli of c and d.
4.1. Arrangement group (+,+)
For this class of arrangement, both companies implement
reinforcing policies and incorporate a positive feedback
loop, i.e. RAt!RAt + 1 and RBt!RBt + 1, the two interacting through the comparison of SAt and SBt (see Fig. 1).
When a firm is ahead in product standard, it invests more in
the next period to maintain or increase its competitive
advantage; when it is behind, it retreats and invests less,
perhaps with a view to eventual withdrawal from the
market. It was found that for this group of policy arrangements, the system was not prone to oscillation, but instead,
RAt and RBt repeatedly diverged, one to zero, the other
continuously increasing. As would be expected, SAt and SBt
also diverged, approaching the upper and lower limits
bound by the function (8). The company gaining the
advantage, therefore, spends an increasing amount on development for marginal increases in product quality/sales.
The tendency for divergence rather than oscillation can
be understood in terms of the feedback structure of the
system for these arrangements. Here, positive feedback in a
company reflects the fact that the policy of its management
will tend to reinforce any initial difference between SAt and
SBt. It follows that for (+,+) arrangements, any initial
difference in product standard of A and B will be amplified
by managerial actions in both cases and so oscillation will
not occur. Rather, SAt and SBt will repeatedly diverge to the
upper and lower bounds of the model, as observed.
Coupled with non-linearity, system oscillation is an
essential condition for the onset of chaotic behavior, and
the lack of oscillation for (+,+) system arrangements implies
an inability to produce chaotic behavior.

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4.2. Arrangement group (+,ÿ)/(ÿ,+)
The groups of opposing managerial policies (+,ÿ) and
(ÿ,+) are identical due to the structural symmetry of the
model and while the following refers to (+,ÿ), it applies
equally to (ÿ,+). For these arrangements, on occasions
when the firm with the reinforcing policy is ahead in terms
of product standard, this firm invests more in R&D. In
such cases, the trailing competitor (with a counteracting
policy) does the same in the hope of making good its
present loss of competitive advantage. Equally for this
arrangement group, circumstances occur for which the
counteracting firm has the advantage and, true to its policy,
relaxes and invests less in R&D, perhaps expecting to
maintain its competitive advantage without additional R&D
effort. In this case, the trailing, reinforcing competitor also
invests less, representing a firm which, when trailing, sees
no point in trying to compete through R&D to improve its
market share.
Investigation of the model showed the potential for
oscillation to depend not only on the modulus and sign
of the parameters, but also on the initial relative standing
of the two companies. This led to the consideration of four
general scenarios.
4.2.1. Scenario 1
Arrangements were investigated for which one company
(say B) had a sensitive reinforcing policy (d = a large
positive number), while the other (A) had a less sensitive
counteracting policy (c = a smaller negative number), with
initial conditions such that the reinforcing company started
with the greater allocation (RB0 > RA0). For these conditions, as for (+,+) arrangements, the model showed no
potential for oscillation or chaos. In this case, the RAt and
RBt (and so SAt and SBt) variables both increased repeatedly over time. As before, this tendency for monotonic
variation rather than oscillation can be understood from the
system's feedback structure, as determined by the policies.
For these arrangements and initial conditions, the larger
initial funding of B translates to a larger initial product
standard SB0. As the model is iterated and the system
evolves, B, with its sensitive reinforcing policy, will
further increase funding and so amplify its advantage. At
the same time, A, with its counteracting policy, attempts to
remedy its disadvantageous initial position by increasing
its funding. Due to the lower sensitivity of its policy,
however, A repeatedly redresses B's advantage less than
the extent to which B further increases it. Overall, therefore, SBt and SAt will diverge to the upper bound of the
model, but with SBt > SAt at all times before the upper
bound is reached.
4.2.2. Scenario 2
Further (+,ÿ) arrangements were investigated for policies with relative sensitivities as above, but with initial
conditions such that the reinforcing company started with

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the smaller allocation (RA0 > RB0). The system again
showed monotonic behavior rather than oscillation or chaos.
This behavior can be explained by the fact that here, B, with
its sensitive reinforcing policy, increases its initial disadvantage and so directs SBt towards zero. Meanwhile, the
smaller counteracting policy of A means this company acts
so as to relax on its advantage. The overall result of these
tendencies is that both SBt and SAt fall to zero, with the
former having the more sensitive policy, thus reducing the
more rapidly.
4.2.3. Scenario 3
Investigations were also carried out on a third type of
(+,ÿ) arrangement, somewhat different from the two
considered above. In this case, the one company (B) had
a sensitive counteracting policy (d = a large negative
number), while the other (A) had a less sensitive reinforcing policy. Further, the initial conditions were such that
the counteracting company started with a greater allocation
(RB0 > RA0), which translated to SB0 > SA0. For these
conditions, for certain parameter values, oscillatory behavior was observed, shown for variables RAt and RBt in
Fig. 3(a) and (b), respectively. It is notable that while the
two variables took values over different ranges, their
oscillations were in phase.
The potential for chaos in non-linear systems is often
investigated by varying one system parameter while the
others are kept constant. The occurrence of increasingly
complicated periodic behavior leading to chaos as a parameter is varied and is well-recognized in chaotic systems.
Where one parameter is varied, the scenario is known as a

Fig. 3. (a) Period 1 oscillations of RAt for system arrangement (c, d) =
(+0.1, ÿ3.8), initial conditions (RA0, RB0) = (10.0, 15.0). (b) Period 1
oscillations of RBt, with system arrangement and initial conditions as in (a).

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codimension-one route to chaos. Generally, at distinct
values of the changing parameter, bifurcations occur. Bifurcations involve qualitative changes in the behavior of the
system, for example, from equilibrium to periodic (for a
fuller explanation see e.g. Berge et al., 1984; Eckmann,
1981; Medio, 1992; Ott, 1993). Such a process was observed for these system conditions, as shown in Fig. 4(a),
(b), and (c) for which the system's behavior was studied for
increasingly negative settings of d. These changes correspond to a company (B in this case) becoming increasingly
sensitive in its counteracting policy, while sensitivity at the
reinforcing competitor remains unchanged.
The plots in Fig. 4 indicate classic period doubling. In
Fig. 4(a), for which policy parameter d = ÿ4, the evolution
of RBt was found to undergo two oscillations, with two
different maxima and minima, before a full cycle back to the
initial value was completed. Behavior of this type is known
as a period 2 evolution, while the most basic oscillations of
Fig. 3 are termed period 1, the system bifurcating from the
latter to the former as the control parameter (d) is varied. In
Fig. 4(b), d has become more negative, d = ÿ4.18, and
another bifurcation has occurred so that the periodic behavior has become even more complicated; each complete
period now comprises four sub-oscillations: a period 4
evolution. In theory, this period doubling as a system
parameter is increased/decreased will continue for even
higher periods. For increasingly higher period numbers,
however, the bifurcations occur for progressively smaller
changes in the increasing/decreasing parameter, and therefore, become progressively more difficult to observe.

For some non-linear systems, if the control parameter is
progressed past a critical value, the motion becomes aperiodic, producing the phenomenon known as deterministic
chaos. Fig. 4(c) shows the model exhibiting such chaotic
behavior; there is no longer a discernible repeating pattern to
the series, which has become aperiodic. A comparison of
Fig. 4(c) and (d) illustrates sensitive dependence on initial
conditions (sdic), known to be a characteristic of chaotic
evolutions. Both series emanate from identical system
arrangements, i.e. the company policies are the same in
each case, but the initial values of the evolutions, as
specified by (RA0, RB0), differ slightly: initial conditions
were (10.0, 15.0) and (10.1, 15.0) for Fig. 4(c) and (d),
respectively. The marked difference in the two (chaotic)
evolutions, as a result of sdic, suggests serious, fundamental
limitations to forecasting, control and planning where such
system arrangements exist.
A useful illustrative tool in the analysis of systems,
which pursue the period doubling route to chaos, is a
bifurcation diagram. A bifurcation diagram indicates how
the post-transient, limiting behavior of a system alters as one
parameter is varied. One of its great strengths is that it
provides a more general, global view of the system's
behavior for a variety of parameter settings; in this case,
company policies. More specifically, for this discretely
evolving model, the bifurcation diagram shown in Fig. 5
identifies the values of a particular system variable (say
RAt) present in the post-transient evolution, plots these
values against the corresponding value of the parameter,
and repeats this for a range of parameter values. The

Fig. 4. (a) Period 2 oscillations of RBt for system arrangement (c, d) = (+0.1, ÿ4.0), initial conditions (RA0, RB0) = (10.0, 15.0). (b) Period 4 oscillations
of RBt for system arrangement (c, d) = (+0.1, ÿ4.18), initial conditions (RA0, RB0) = (10.0, 15.0). (c) Chaotic behavior of RBt for system arrangement
(c, d) = (+0.1, ÿ4.5), initial conditions (RA0, RB0) = (10.0, 15.0). (d) Chaotic behavior of RBt for system arrangement as in (c), initial conditions (RA0,
RB0) = (10.1, 15.0).

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Fig. 5. Bifurcation diagram for (+,ÿ) system arrangements; policy
parameter d varying.

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4.2.4. Scenario 4
Oscillations, period doubling, and chaotic behavior similar to that described above were again obtained for certain
parameter values for a fourth type of (+,ÿ) arrangement.
The relative sensitivities of the policies were similar to those
for the conditions producing the oscillations and chaos
described above, but here, the initial conditions were such
that the counteracting company started with the smaller
allocation (RA0 > RB0). The tendency for oscillation could
be explained by the fact that for these conditions, the
sensitive counteracting B acts to remedy its initial disadvantage, while the less sensitive reinforcing A tends to
increase its advantage. As before, B's effect is the stronger,
so that the system evolves to a state where RBt > RAt, and
the policies now engender opposite effects. This change in
direction of the effects due to policy occurs repeatedly, and
it follows that for these conditions, there is a tendency for
the system again to oscillate.
4.3. Arrangement group (ÿ,ÿ)

evolutions have the same starting values (RA0 and RB0) in
each case.
A comparison between a time series and its representation in a bifurcation diagram helps clarify the structure of
the latter. The post-transient periodic motion of Fig. 3(a), for
example, comprises two values of RAt, which together
constitute the oscillations. Looking at the system's bifurcation diagram in Fig. 5, it can be seen that for policy
parameter d = ÿ3.8 (with c = 0.1) two values of RAt are
plotted, shown at f in the diagram, thereby indicating a
period 1 oscillation. It follows that the single value of RAt
plotted for policy d = ÿ3 indicates one value of RAt in the
corresponding time series, i.e. equilibrium (see g in the
diagram). In a similar way, period 2 and period 4 behaviors,
such as those shown by RBt in Fig. 4(a) and (b), respectively, correspond to instances where evolutions comprise
four and eight variable-values in turn, as shown on the
diagram at h and i, respectively. The parameter region of the
bifurcation diagram, for which a seemingly continuous but
bounded range of RAt are present, represents chaos.
As before, analysis of the system's feedback structure
sheds light on the tendency of (+,ÿ) arrangements to
oscillate for these conditions. On iteration, company B, with
its sensitive counteracting policy, tends to relax given its
initial advantage and spends less on R&D in the next period.
Simultaneously, the smaller reinforcing policy of A acts to
reinforce its initial disadvantage. Due to its more sensitive
counteracting policy, however, it is B, which has the
stronger initial reduction, and accordingly, the system soon
evolves so that RBt < RAt (and by evolution SBt < SAt). In
this state, B's counteracting policy then acts to increase RBt
while A's reinforcement has a similar but weaker selfincreasing effect. Together, these effects cause evolution to
a state where SBt > SAt. This vacillation of the position of
advantage continues throughout the evolution, i.e. in these
conditions the system is prone to oscillation.

In this group, both companies adopt a counteracting
policy. At all times, the leading company invests less in
the next period while the trailing company moves to invest
more. For these arrangements, the managers of both firms
act so as to decrease any extant difference in their performance. Investigations were carried out on arrangements
where one company, say B, had a decidedly more sensitive
policy than its competitor and initial conditions such that
RA0 6ˆ RB0 ()SA0 6ˆ SB0). For these conditions, oscillation occurred for certain parameter values, as shown for
variables RAt and RBt in Fig. 6(a) and (b). It is interesting
that while for the (ÿ,+) group, RAt and RBt oscillated in
phase, as shown earlier, the (ÿ,ÿ) results show oscillations
in anti-phase, i.e. one's maxima coincide with the other's
minima. Period doubling bifurcations to chaos were observed as the modulus of the more strongly negative parameter was increased (representing an increase in the
sensitivity of that company's counteracting policy) as illustrated in Fig. 6(c) ± (e). The corresponding bifurcation
diagram is shown for variable RAt in Fig. 7, while a
comparison between Fig. 6(e) and (f) shows that the antiphase relationship between the evolutions of RAt and RBt
(and so SAt and SBt) persists into the chaotic regime.
It was found that for (ÿ,ÿ) arrangements where the
competitors adopted policies of similar sensitivity, the
tendency for oscillation was reduced and RAt and RBt
(and so with SAt and SBt) pursued a monotonic approach
to an equilibrium between RA0 and RB0 (and between SA0
and SB0).
Again, the system's oscillatory behavior can be understood from a consideration of its feedback structure. For
the case when B has the notably more sensitive policy, its
corrections (i.e. reductions to RBt when in a position of
advantage, and increases when disadvantaged) will be
larger than those of its competitor. Reasoning as before

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S. Whitby et al. / Journal of Business Research 51 (2001) 179±191

Fig. 6. (a) Period 1 oscillations of RAt for system arrangement (c, d) = (ÿ0.1, ÿ3.0), initial conditions (RA0, RB0) = (10, 5.0). (b) Period 1 oscillations of RBt
for system arrangement and initial conditions as for (a). (c) Period 2 oscillations of RBt for system arrangement (c, d) = (ÿ0.1, ÿ3.8), initial conditions (RA0,
RB0) = (10, 5.0). (d) Period 4 oscillations of RBt for system arrangement (c, d) = (ÿ0.1, ÿ3.99), initial conditions (RA0, RB0) = (10, 5.0). (e) Chaotic behavior
of RAt for system arrangement (c, d) = (ÿ0.1, ÿ4.15), initial conditions (RA0, RB0) = (10, 5.0). (f) Chaotic behavior of RBt, system arrangement and initial
conditions as for (e).

suggests that for these conditions, the system is prone to
oscillation where the scale of variation of the more
sensitive company is substantially greater than that of
the competition.

5. An adaptive model: the learning organization
The basic model outlined above describes managerial
adjustments to resource allocations, but makes no allowance
for any change over time in the policy concerning these
adjustments. This constitutes a serious limitation of the
model as, in reality, it is to be expected that the strategy
adopted will change, reflecting the decision-maker's perception of recent-policy efficacy. For an intelligent management
(or the ``learning organization''), there will be attempts to
adapt to previous shortcomings. The basic model was
therefore extended to include a mechanism for this adapta-

tion process by permitting variation over time of the policy
parameters c and d.
5.1. Adaptation mechanisms
Two different representations of adaptation were included separately in the adaptive model, the management
in both cases contemplating a policy change every f
periods (e.g. quarterly or annually). In deciding whether
to make a change, the manager (of, say, A) was considered
to examine the company's performance relative to that of
the competitor over each of the f periods since the last
deliberation. The sum of these relative performances (M)
was used
P as an overall measure, where, at period t,
M ˆ ttÿf SAi ÿ SBi . If M > 0, the current policy was
considered satisfactory, while M < 0 was taken to indicate
a need for change. This is admittedly a simple criterion for
assessment, which makes no allowance for trends. In

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Fig. 7. Bifurcation diagram for (ÿ,ÿ) system arrangements; policy
parameter d varying.

particular, a negative value of M would be returned for a
trailing company that is steadily reducing its deficit over
the f periods. In such cases, the policy would, to some
extent, be working satisfactorily and yet the negative value
of M would precipitate its modification. Nevertheless, the
criterion does provide a basis for introducing managerial
assessment of performance into the model; the effect of
more complicated adaptation must await future research.
It was in the adjustment rule that the two adaptation
mechanisms differed. In one, termed proportional policy
adaptation, a change amounted to a proportional reduction
in the extremity of a policy while maintaining its nature

Fig. 8. (a) Evolution of RBt with proportional adaptation mechanism. (b)
Further evolution of RBt with proportional adaptation mechanism.

189

(counteracting or reinforcing, as represented by the sign of
the parameter). Specifically, this reduction was executed in
the model by multiplying the appropriate company's policy
parameter by a reducing factor, ac or ad (0 < ac, ad < 1),
so that ck = acckÿ1 or dk = ad dkÿ1, where the index k
describes a particular stretch of f time periods. The same
values of ac and ad were used throughout a given evolution, i.e. for each change, policy extremi