Journal of Economic Dynamics & Control 37 (2013) 2351–2370

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Journal of Economic Dynamics & Control
journal homepage: www.elsevier.com/locate/jedc

The bull and bear market model of Huang and Day:
Some extensions and new results
Fabio Tramontana a,n, Frank Westerhoff b, Laura Gardini c
a
b
c

Department of Economics and Management, University of Pavia, Via S.Felice 5, 27100 Pavia, Italy
Department of Economics, University of Bamberg, Feldkirchenstrasse 21, 96045 Bamberg, Germany
Department of Economics, Society, Politics, University of Urbino, Via Saffi 42, 61029 Urbino, Italy

a r t i c l e i n f o

abstract

Article history:
Received 14 November 2012
Received in revised form
8 May 2013
Accepted 10 June 2013
Available online 19 June 2013

We develop a financial market model with interacting chartists and fundamentalists that
embeds the famous bull and bear market model of Huang and Day as a special case. Their
model is given by a one-dimensional continuous piecewise-linear map. Our model, on the
other hand, is more flexible and is represented by a one-dimensional discontinuous
piecewise-linear map. Nevertheless, we are able to provide a more or less complete
analytical treatment of the model dynamics by characterizing its possible outcomes in
parameter space. In addition, we show that quite different scenarios can trigger realworld phenomena such as bull and bear market dynamics and excess volatility.
& 2013 Elsevier B.V. All rights reserved.

JEL classification:
C02
D84
G12
G14
Keywords:
Heterogeneous interacting agents
Bull and bear market dynamics
Piecewise-linear maps
Border collision bifurcations

1. Introduction
The development and analysis of financial market models with heterogeneous interacting agents began with the seminal
paper by Day and Huang (1990).1 Based on a few stylized institutional and behavioral facts, Day and Huang proposed a
simple financial market model with three types of agent: a market maker who adjusts prices with respect to ; excess
demand; chartists who believe in the persistence of bull and bear markets; and fundamentalists who bet on mean reversion.
In their model, market participants' transactions may cause apparently unpredictable price dynamics with randomly
alternating periods of generally rising or generally falling prices, so-called bull and bear market dynamics. Since then,
literally hundreds of follow-up papers have been presented, deepening on our understanding of how financial markets
function. For surveys of this burgeoning research field see, for instance, Chiarella et al. (2009), Hommes and Wagener
(2009), Lux (2009), and Westerhoff (2009).
The one-dimensional map studied in Day and Huang (1990) is nonlinear since fundamentalists become increasingly
aggressive as the price runs away from its fundamental value. Huang and Day (1993) reformulated their original model such
that it corresponds to a one-dimensional continuous piecewise-linear map by assuming that fundamentalists only start

n

Corresponding author. Tel.: +39 0382 986224.
E-mail addresses: fabio.tramontana@unipv.it (F. Tramontana), frank.westerhoff@uni-bamberg.de (F. Westerhoff), laura.gardini@uniurb.it (L. Gardini).
1
As usual, there are a number of predecessors, e.g. Zeeman (1974), Beja and Goldman (1980), and Frankel and Froot (1986).

0165-1889/$ - see front matter & 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.jedc.2013.06.005

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F. Tramontana et al. / Journal of Economic Dynamics & Control 37 (2013) 2351–2370

trading if the distance between the price and its fundamental value exceeds a critical threshold value. Otherwise,
speculators' trading strategies are linear and thus the model is represented by three connected linear branches. Of course,
simplifying assumptions must be made to derive such a piecewise-linear map. Due to the model's piecewise-linearity,
however, Huang and Day (1993) were able to study how certain model parameters, e.g. chartists' and fundamentalists'
reaction parameters, affect the distribution of prices. They found, for instance, that an increase in fundamentalists'
aggressiveness tends to fatten the tails of the distribution of prices. Obviously, such insights are quite important for our
understanding of financial markets.
In addition, the simplified model of Huang and Day (1993) initiated a substantial number of studies, some of which are
described below. For instance, Gu (1993) managed to estimate the model of Huang and Day (1993) on the basis of monthly
S&P 500 data, and discovered that the distribution of model-generated price changes does not differ statistically from its
empirical counterpart. In addition, the autocorrelation functions of simulated and actual returns are strikingly similar.
Gu concludes that although random elements play a role in the price formation process, the majority of the dynamics is due
to intrinsic market forces, as represented by the simple model of Huang and Day (1993), for instance.
Gu (1995) further extended the analysis of Huang and Day (1993) by deriving closed form density functions of prices. In
particular, he used this result to study how aggressively the market maker has to adjust prices with respect to excess
demand to make profits (density functions depend on market maker's price adjustment behavior and control her/his
profits). Gu showed that the market maker has to churn the market to make a living. Although it is undesirable for the
market maker to destabilize the market too considerably, she/he evidently has an incentive to keep prices moving. Gu, too,
admitted that the underlying piecewise-linear model is based on simplifying assumptions. Again, however, these
assumptions enabled rigorous results to be derived that have real practical relevance.
In reality, of course, there are more than two different types of speculator. Speculators may differ with respect to degree
of aggressiveness, their trading horizons, market entry levels, perceptions of the fundamental value, and so on. Day (1997)
thus considered the case where there are different types of fundamental and technical traders, and presented an example of
a one-dimensional continuous piecewise-linear map with 11 branches. As it turns out, this model is capable of generating
quite exciting dynamics. For instance, bull and bear markets may emerge at quite different levels and the price dynamics
may become even more intricate.
Tramontana et al. (2010a) developed a model in which chartists and fundamentalists react asymmetrically to bull and
bear market situations. For instance, market participants may trade more/less aggressively in overvalued markets than in
undervalued markets. As a result, they obtain a one-dimensional discontinuous piecewise-linear map (where the
discontinuity point is given by the market's fundamental value). Although the mathematical tools for studying such models
are still underdeveloped, they offer a detailed analytical treatment of the underlying map, which can produce all kinds of
dynamic behavior including fixed point dynamics, cycles, quasi-periodic motion, chaos, and exploding trajectories. This
model was further developed and investigated in Tramontana et al. (2010b, 2011a). Besides identifying economic
mechanisms that can produce endogenous bull and bear market dynamics, these papers enrich our knowledge about
how to deal with one-dimensional discontinuous piecewise-linear maps.2
Huang and Zheng (2012) also presented a model in the spirit of Huang and Day (1993) to explain different types of
financial crisis, in particular, the sudden crisis, the disturbing crisis, and the smooth crisis. In their model, chartists hold
regime-dependent beliefs about the fundamental value, motivated by psychologically grounded support and resistance
levels; the different types of crisis are explained by a constant switching between regimes. In addition, their model is also
able to explain a number of stylized facts concerning financial markets. The underlying map is essentially represented by a
one-dimensional discontinuous piecewise-linear map where all branches have a positive slope. Huang et al. (2010) modified
the latter model by allowing traders to switch between technical and fundamental trading strategies according to an
evolutionary fitness measure. On the basis of this model, Huang et al. (2012) demonstrated, amongst other things, that
major stock market movements and returns are asymmetric, i.e. that bubbles emerge gradually and crash suddenly and that
drawdowns tend to be more dramatic than drawups.
Another model inspired by Huang and Day (1993) is that by Venier (2008). Despite its deterministic nature, his model,
represented by a saw-tooth map, was able to reproduce some stylized facts concerning financial markets, such as fat tails
and volatility clustering. By buffeting their one-dimensional discontinuous piecewise-linear bull and bear market model
with dynamic noise, Tramontana and Westerhoff (2013) obtained a good match of the statistical properties of actual
financial market prices, also from a quantitative perspective. Westerhoff and Franke (2012) provided an even simpler but
nonetheless powerful stochastic bull and bear market approach. Further empirical evidence for regime-dependent trading
behavior of chartists and fundamentalists is provided by Chiarella et al. (2012), Manzan and Westerhoff (2007), and
Vigfusson (1997), amongst others.
To sum up, simple piecewise-linear models enable us to derive new analytical insights into how financial markets
function. Moreover, these models – despite their simplicity – are not at odds with reality. Firstly, their main building blocks
are based on stylized institutional and behavioral facts. For instance, the survey study conducted by Menkhoff and Taylor
(2007) revealed that market participants do indeed rely on technical and fundamental analysis to predict financial market

2
The formal analysis of one-dimensional discontinuous piecewise-linear models with one discontinuity can be traced back to Leonov (1959, 1962).
Recently, this research field has gained new momentum; see, for instance, Gardini et al. (2010), Avrutin et al. (2010), Gardini and Tramontana (2010).

F. Tramontana et al. / Journal of Economic Dynamics & Control 37 (2013) 2351–2370

2353

prices (this has also been found in laboratory experiments; see, e.g. Hommes, 2011). Secondly, these models are able to
explain some important stylized facts concerning financial markets such as bubbles and crashes, excess volatility, and longmemory effects. Given the repeated emergence of severe financial crises and the herewith associated risk to the real
economy, we believe that more work should be undertaken in this exciting research direction.
In this paper, we thus continue this line of research by proposing a financial market model that may be regarded as a
generalization of the model of Huang and Day (1993). Let us briefly recall why their one-dimensional piecewise-linear map
is continuous. Close to the fundamental value, only chartists are active in the market. Hence, the slope of the map's inner
branch is larger than one. If the price deviates too far from the fundamental value, additional fundamentalists enter the
market. Since their demand is zero when they enter the market, all three branches of the map are connected, and the slope
of the two outer branches is lower than the slope of the inner branch. Instead, we assume that a number of chartists and
fundamentalists are always active in the market, that additional chartists and additional fundamentalists may enter the
market when the distance between the price and its fundamental value exceeds a critical level, and that new traders'
demand may be non-zero at the market entry level. As a result, the dynamics of our model is due to a quite flexible onedimensional discontinuous piecewise linear map. In particular, the three branches are typically disconnected, and there are
no restrictions to the values their slopes may assume.
Nevertheless, we are able to provide a comprehensive analysis of the model dynamics. For instance, we are able to
determine the frontiers, in parameter space, that separate bounded dynamics from divergent dynamics. This analysis
demonstrates that both chartists and fundamentalists can contribute to or prevent market stability, making the regulation of
financial markets a delicate issue. Moreover, we find that quite different scenarios can lead to intricate bull and bear market
dynamics. As in Huang and Day (1993), we observe repeated price rallies and subsequent market crashes if the slope of the
two outer branches is negative, due to fundamentalists' aggressive trading behavior. However, we also observe such – and
even more intricate – boom-bust dynamics if the slope of the two outer branches is positive, due to chartists' aggressive
trading behavior. Overall, the emergence of endogenous bull and bear market dynamics may thus be regarded as a robust
and characteristic feature of speculative markets.
Our model results in a discontinuous map with two discontinuity points. Exploration of the so-called border collision
bifurcations that occur in this case is in its infancy. A few results can be found in Tramontana et al. (2012), in which, however,
only stable regimes are investigated (i.e. the slopes of all three branches are between 0 and 1). Clearly, such a scenario does
not occur in our model. Thus, from a mathematical point of view, our model reveals exciting dynamics and interesting
bifurcation structures, some of which have never yet been documented in the literature. As we will see, our analysis will also
highlight a number of open problems, presenting non-trivial (mathematical) challenges for future research. We also hope
that the detailed formal analysis of our model will help other researchers to explore their own discontinuous piecewiselinear maps. Unfortunately, as already noted, our knowledge about such maps is as yet rather limited. Variations of our
model (still with two discontinuity points) are studied in Tramontana et al. (2011b, in press). However, these models do not
embed the model of Huang and Day (1993) as a special case.
The remainder of our paper is organized as follows. In Section 2, we present our financial market model and show that it
corresponds to a one-dimensional discontinuous piecewise-linear map. In Section 3, we give an overview of the potential
model dynamics and illustrate its economic implications. Section 4 has a stronger mathematical orientation and explores
some interesting parameter regions in further detail. In Section 5, we conclude the paper and highlight avenues for further
research.

2. A simple financial market model
Our model is closely related to those of Day and Huang (1990) and Huang and Day (1993), and may even be interpreted
as a generalization of the latter. Following the original papers, we assume that a market maker adjusts prices with respect to
excess demand in the usual way. In our case, excess demand comprises the transactions of four different types of market
participant. So-called type 1 chartists and type 1 fundamentalists are always active in the market. Type 1 chartists bet on the
persistence of bull and bear markets while type 1 fundamentalists expect the price to return to its fundamental value.
So-called type 2 chartists and type 2 fundamentalists share the same general trading philosophy as their type 1
counterparts, but only enter the market if mispricing exceeds a critical threshold value. Huang and Day (1993) argue that
fundamentalists may perceive the chance of gains as small or zero near to the fundamental value and thus may opt against
the participation in the trading process. For chartists, it can be argued that they only learn about (or trust) bull and bear
markets if mispricing is sufficiently extreme. Of course, these are simplifying assumptions. Nonetheless, it appears natural to
us to assume that not all traders react immediately to their trading signals. We will now present the building blocks of our
model and show that they constitute a one-dimensional discontinuous piecewise-linear map.

2.1. The setup
Within our model, the price formation is due to a (standard) log-linear price adjustment rule according to which excess
buying drives prices up and excess selling drives them down. Such a rule may also be interpreted as the stylized price

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F. Tramontana et al. / Journal of Economic Dynamics & Control 37 (2013) 2351–2370

setting behavior of a risk-neutral market maker. Let P be the log of the price. Then we have
C;2
P tþ1 ¼ P t þ a ðDC;1
þ DF;1
þ DF;2
t
t þ Dt
t Þ:

ð1Þ

The four terms in brackets on the right-hand side of (1) capture the transactions conducted by type 1 chartists, type 1
fundamentalists, type 2 chartists, and type 2 fundamentalists, respectively. Parameter a is a positive price adjustment
parameter which we set, without loss of generality, equal to a ¼1.
Type 1 chartists optimistically buy (pessimistically sell) if prices are in the bull (bear) market, that is, if log price P is
above (below) its log fundamental value F. Their orders are formalized as
DC;1
¼ c1 ðP t −FÞ:
t

ð2Þ

1

Reaction parameter c is positive and indicates how aggressively type 1 chartists react to their perceived price signals. The
larger the c1 is, the more aggressive type 1 chartists are.
Type 1 fundamentalists always trade in the opposite direction of type 1 chartists. Orders placed by type 1
fundamentalists are written as
1

DF;1
t ¼ f ðF−P t Þ;

ð3Þ
1

where reaction parameter f is positive. In an overvalued market, therefore, type 1 fundamentalists sell; in an undervalued
market they buy.
Type 2 speculators are only active if prices are at least a certain distance from their fundamental value. The critical
distance for type 2 speculators is given by z, a positive constant.3 The orders of type 2 chartists are thus expressed as
8 2
3
for P t −F≥z
>
< c ðP t −FÞ þ c
C;2
for −z o P t −F oz
Dt ¼ 0
ð4Þ
>
: c2 ðP −FÞ−c3
for P −F ≤−z:
t

t

2

Again, reaction parameter c is positive, i.e. the trading intensity of type 2 chartists increases in line with the distance
between prices and fundamentals. However, their transactions can be adjusted using parameter c3 for which we assume
c3 ≥−c2 ðz−FÞ. Therefore, type 2 chartists' transactions are non-negative in the bull market and non-positive in the bear
market.
Orders placed by type 2 fundamentalists are based on the same principles, i.e. we have
8 2
3
for P t −F≥z
>
< f ðF−P t Þ−f
F;2
ð5Þ
Dt ¼ 0
for −z oP t −F o z
>
: 2
3
for P t −F ≤−z;
f ðP t −FÞ þ f
2

3

2

where restrictions f 4 0 and f ≥f ðF−zÞ hold.
2.2. The model's law of motion
To simplify the notation, let us define
1

S1 ¼ c1 −f ;

S2 ¼ c2 −f

2

and

3

M ¼ c3 −f :

Note first that there are no restrictions
yields
8
1
2
>
< P t þ ðS þ S ÞðP t −FÞ þ M
1
þ
S
ðP
−FÞ
P tþ1 ¼ P t
t
>
:
P t þ ðS1 þ S2 ÞðP t −FÞ−M

on these aggregate parameters (they can take any values). Combining (1)–(5) then
if P t −F≥z
if −z o P t −F oz

ð6Þ

if P t −F ≤−z:

Furthermore, it is convenient to express the model in terms of deviations from the fundamental value by defining P~t ¼ P t −F.
We then have
8
1
2 ~
~
>
< ð1 þ S þ S ÞP t þ M if P t ≥z
1 ~
~
if −z o P~ t oz
ð7Þ
P tþ1 ¼ ð1 þ S ÞP t
>
:
1
2 ~
if P~ t ≤−z;
ð1 þ S þ S ÞP t −M
i.e. prices are driven by a one-dimensional piecewise linear map that has three separate linear branches.
3
Two aspects deserve some more attention. First, many cases discussed in our paper are consistent with a model in which there are either only type 2
chartists or only type 2 fundamentalists. Second, different market entry levels for type 2 chartists and type 2 fundamentalists, say zC and zF (instead of z),
would result in a one-dimensional discontinuous piecewise-linear map with four discontinuity points and five linear branches. We leave this
mathematically intricate scenario for the future.

F. Tramontana et al. / Journal of Economic Dynamics & Control 37 (2013) 2351–2370

2355

Note that this map embeds the famous models of Day and Huang (1990) and, in particular, Huang and Day (1993) as
1
special cases. We obtain their model if we assume that f ¼ 0, i.e. there are no type 1 fundamentalists; c2 ¼ c3 ¼ 0, i.e. there
3
2
are no type 2 chartists; and f ¼ f ðF−zÞ, i.e. type 2 fundamentalists' demand is zero at the market entry level. In this case,
the two outer branches are connected with the inner branch. The inner branch has a slope that is always larger than one.
However, interesting bull and bear market dynamics only occurs in their model if the slope of the two outer branches is less
than minus one. Note that the latter condition requires that fundamentalists trade quite aggressively. Our model is more
flexible by imposing less restrictive assumptions on traders' behavior. In general, the slopes of its outer and inner branches
can be positive or negative, and the map is discontinuous. As we will show in the next section, this leads to quite interesting
bifurcation properties and new bull and bear market scenarios.
3. Dynamics of the model: a first overview
In this section, we present an overview of the model dynamics. In Section 4, we will proceed to study certain aspects of
the model dynamics in further detail. Following Huang and Day (1993), we assume from here on that ð1 þ S1 Þ 4 1.
Economically, this implies that type 1 chartists react more aggressively to a given price signal than type 1 fundamentalists
1
(i.e. c1 4 f and thus S1 4 0).4 Apart from this assumption, both M and ð1 þ S1 þ S2 Þ can take any positive or negative value,
and z 40.
Note first that market entry level z can be considered as a scale variable. By using the change of variable x ¼ P~ =z and by
setting m ¼M/z, we obtain
8
1
2
>
< F R ðxÞ ¼ ð1 þ S þ S Þx þ m if x 41
F:

x′ ¼

>
:

F C ðxÞ ¼ ð1 þ S1 Þx

if −1 ox o 1

F L ðxÞ ¼ ð1 þ S1 þ S2 Þx−m

if x o−1;

ð8Þ

where “′” denotes the unit time advancement operator.
One useful property of the model is that its dynamics is symmetric with respect to the origin x ¼0. Using the definition
y ¼ −x results in
8
1
2
>
< ð1 þ S þ S Þy þ m if y 41
y′ ¼

>
:

ð1 þ S1 Þy
ð1 þ S1 þ S2 Þy−m

if −1 o yo 1
if y o−1;

ð9Þ

which is identical to F in (8), meaning that the following property holds:5
Property 1 (Symmetry). Any invariant set of the map F in (8) is either symmetric with respect to x ¼0 or the symmetric set also
exists.
A direct consequence of Property 1 is that cycles of odd periods are never unique since the symmetric cycle also exists.
That is, if a cycle with periodic points ðx1 ; …; xk Þ exists where k is odd (and whatever is the sign of xi, positive or negative),
then the cycle ð−x1 ; …; −xk Þ also necessarily exists. As we shall see, this property will lead to a peculiar dynamic behavior
described in Section 4.
In general, map F is discontinuous, with two discontinuity points in x ¼ −1 and x ¼1. In fact, map F is continuous only if:
F C ð1Þ ¼ F R ð1Þ:

ð10Þ

With respect to the model parameters, this condition can be rewritten as
ðCÞ :

S2 ¼ −m;

ð11Þ

and always holds in the framework of Huang and Day (1993).
Let us next consider the two-dimensional bifurcation diagram in Fig. 1. Here we fix S1 ¼ 0:5 and vary S2 and m as
indicated on the axes.6 The parameter conditions for which our model is continuous are represented by the straight line ðCÞ.
Again, all possible parameter combinations of the (continuous) model of Huang and Day (1993) are located on this line.
Since Huang and Day (1993) only consider a market entry of additional fundamentalists, S2 is in their model, by definition,
negative. Hence, only part of the parameter combinations given by line (C) corresponds to their model. As will become
clearer below, the model of Huang and Day (1993) may produce two symmetric stable fixed points (line (C), yellow region in
Fig. 1), chaotic dynamics (line (C), white region in Fig. 1), which can be with two disjoint chaotic intervals (see Fig. 2a for an
example) or in a unique chaotic interval (see Fig. 2b for an example), and divergent dynamics (line (C), gray region in Fig. 1).
4
As a result, type 1 chartists' transactions drive the price monotonically away from the fundamental value. Of course, for ð1 þ S1 Þo −1 the inner regime
also becomes unstable. Then, however, type 1 fundamentalists' transactions cause explosive improper oscillations. A detailed mathematical analysis of this
case is quite involved and will necessitate a separate paper.
5
Recall that an invariant set is a set A that satisfies FðAÞ ¼ A.
6
In all of our simulations, we set S1 ¼ 0:5. However, we obtain similar results for any other value of S1.

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F. Tramontana et al. / Journal of Economic Dynamics & Control 37 (2013) 2351–2370

Fig. 1. Two-dimensional bifurcation diagram in the parameter plane ðm; S2 Þ at S1 ¼ 0:5 fixed. (For interpretation of the references to color in this figure
caption, the reader is referred to the web version of this article.)

Fig. 2. In (a) m¼ 3, S2 ¼ −3; in (b) m ¼5, S2 ¼ −5.

Naturally, our model includes a much larger parameter space and, as already visible from Fig. 1, many different dynamic
peculiarities.
Since the central branch FC(x) of map (8) is a straight line running through the origin with a slope always greater than 1,
the dynamic properties of our model strongly depend on external branches FL(x) and FR(x), that is on parameter m and
ð1 þ S1 þ S2 Þ. As we will see in further detail in Section 4, the parameter regions
ðR1Þ : ð1 þ S1 þ S2 Þ 4 0 and m 40;
ðR2Þ : ð1 þ S1 þ S2 Þ o 0 and m 40;
ðR3Þ : ð1 þ S1 þ S2 Þ 4 0 and m o0;
ðR4Þ : ð1 þ S1 þ S2 Þ o 0 and m o0;

ð12Þ
2

1

2

produce quite different dynamics. In Fig. 1, we thus draw the lines m ¼0 and S ¼ −1:5 (which results from ð1 þ S þ S ¼ 0Þ
and S1 ¼ 0:5) to mark the four regions.
Economically, ð1 þ S1 þ S2 Þ o 0 implies that the joint price-dependent trading impact of type 1 and type 2 fundamentalists strongly overcompensates the joint price-dependent trading impact of type 1 and type 2 chartists. Similarly, it may
be argued that for 0 oð1 þ S1 þ S2 Þ o1 the joint price-dependent trading impact of type 1 and type 2 fundamentalists
weakly overcompensates the joint price-dependent trading impact of type 1 and type 2 chartists while for ð1 þ S1 þ S2 Þ 4 1
the joint price-dependent trading impact of type 1 and type 2 fundamentalists is overcompensated by the joint pricedependent trading impact of type 1 and type 2 chartists. Moreover, m 40 (m o 0) can be interpreted in the sense that the

F. Tramontana et al. / Journal of Economic Dynamics & Control 37 (2013) 2351–2370

2357

price-independent trading impact of type 2 chartists is stronger (weaker) than the price-independent trading impact of type
2 fundamentalists.7
Since our map is piecewise-linear and since the slope of the central branch is higher than 1, it follows that stable cycles
can only exist in the strip
−1 oð1 þ S1 þ S2 Þ o1;

ð13Þ

that is
−2−S1 oS2 o−S1 :

ð14Þ
1

2

This can also be seen in Fig. 1 (where S ¼ 0:5 turns (14) into −1:5 o S o −0:5).
The fixed points of map (8), apart from x ¼0 which is always unstable, are associated with the symmetric external
branches. Solving F L ðxÞ ¼ x and F R ðxÞ ¼ x leads to
xnL ¼

m
S1 þ S2

;

xnR ¼ −

m
S1 þ S2

:

ð15Þ

Of course, the fixed points only exist if xnL o −1 and xnR 41. For instance, for xnR we thus have −m=ðS1 þ S2 Þ 41, which occurs
for (a) S2 o −m−S1 if m o0 and S1 þ S2 40, or (b) S2 4 −m−S1 if m 4 0 and S1 þ S2 o 0. Thus in Fig. 1 we also show the
straight line
ðf Þ :

S2 ¼ −m−S1 :

ð16Þ
1

2

1

8

Hence, the symmetric fixed points are stable only for m 4 0 in the strip −2−S o S o −S (the yellow area in Fig. 1). As we
will study in further detail below, in a portion of this area the fixed points coexist with a stable period 2-cycle. For m o 0, the
fixed points, if they exist, are always unstable.
From an economic perspective, a fixed point implies that the market maker adjusts the price no further. The reason for
this is, obviously, that excess demand is zero. For x¼ 0, excess demand is zero since the price is equal to its fundamental
value, and thus speculators do not perceive any trading signals. For xnL and xnR , however, excess demand is zero since
speculators' buying and selling orders cancel each other out.
Outside the strip −2−S1 o S2 o −S1 , the dynamics is either chaotic (as all the slopes are in modulus larger than 1) or the
generic trajectory explodes (marked by white and gray in Fig. 1, respectively). As it turns out, it is possible to determine the
frontiers separating bounded dynamics from the regions of divergent trajectories analytically. Consider the case of unstable
fixed points for m o 0 (the qualitative shape of map F is as shown in Fig. 3a).
As long as the dynamics is bounded, the unstable fixed points are on the boundary of the basin of attraction:
BðAc Þ ¼ xnL ; xnR ½

ð17Þ

and a condition leading to a so-called final bifurcation is the homoclinic bifurcation of the unstable fixed points, which
occurs when the offsets of the central branch reach the fixed points, i.e. when
F C ð1Þ ¼ xnR

ð18Þ

and clearly also at the same time F C ð−1Þ ¼ xnL , which leads to the condition
ðd1Þ :

S2 ¼ −

m
1 þ S1

−S1

ð19Þ

(straight line (d1) in region m o0 in Fig. 1a).
However, a homoclinic bifurcation of the fixed points may also occur due to a change in the slopes and offsets of the
external branches, when the condition
F L ð−1Þ ¼ xnR

ð20Þ

occurs, which leads to the following condition to be satisfied:
ðS2 Þ2 þ S2 ð1 þ 2S1 þ mÞ þ ðS1 ð1 þ S1 Þ þ mS1 −mÞ ¼ 0;
the solution of which is expressed by the two branches of equation
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðd2Þ : S2 ¼ 12−ð1 þ 2S1 þ mÞ 7 ð1 þ 2S1 þ mÞ2 −4ðS1 ð1 þ S1 Þ þ mS1 −mÞ;

ð21Þ

ð22Þ

shown as (d2) in Fig. 1a.
7
Note that for S2 4 −1:5 (S2 o −1:5) we have that right (left) of the line m ¼0 even small parameter changes may lead to many different cycles. As
pointed out by one of the referees, this aspect may have interesting economic implications. Suppose, for instance, that the model parameters are changing
in this area from time to time. Then, the model dynamics alternates between different cycles, and the overall dynamics may become quite intricate. This
parameter space may be a good candidate if one seeks to bring the model closer to the data.
8
Note that fixed point dynamics, yellow area in Fig. 1, may instantaneously turn into chaotic motion, white area in Fig. 1. Hence, even a tiny change in a
model parameter, respectively a tiny change in the trading behavior of the market participants, may result in a completely different dynamic outcome. In
contrast, nonlinear smooth maps usually produce more gradual “routes to chaos”.

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F. Tramontana et al. / Journal of Economic Dynamics & Control 37 (2013) 2351–2370

Fig. 3. In (a) m ¼ −3, S2 ¼ 0:8; in (b) m¼5, S2 ¼ −4.

Fig. 4. In (a) m¼2.5, S2 ¼ −3:3; in (b) versus time trajectory. Points belonging to the grey strip on the bull region are mapped in one iteration in the bear
region. A symmetric strip exists in the bear region, mapping points in the bull region in one iteration.

We have to reason differently in the case m 40. If the fixed points do not exist (an example is shown in Fig. 4), then an
unstable 2-cycle exists, with periodic points xR and xL ¼ −xR in the external branches of F (i.e. with symbolic sequence RL),
where xR satisfies F L ○F R ðxR Þ ¼ xR . This leads to
xR ¼

−m
2 þ S1 þ S2

:

ð23Þ

This unstable 2-cycle bounds the basin of attraction of the existing chaotic set:
BðAc Þ ¼ xL ; xR ½

ð24Þ

and the bifurcation leading to divergent trajectories occurs at the homoclinic bifurcation of this 2-cycle, which occurs when
F C ð1Þ ¼ xR :

ð25Þ

This leads to the condition
ðd3Þ :

S2 ¼

−m
1 þ S1

−2−S1 ;

ð26Þ

which is the straight line (d3) in Fig. 1a.
A second way to obtain the homoclinic bifurcation of the unstable 2-cycle on the basin boundary is due to the external
branches (see the qualitative shape of the map in Fig. 3b, and the unstable fixed points are included in the chaotic intervals).
This occurs when the condition
F R ð1Þ ¼ xR

ð27Þ

F. Tramontana et al. / Journal of Economic Dynamics & Control 37 (2013) 2351–2370

2359

Fig. 5. In (a) m ¼ −3, S2 ¼ 0:8; in (b) versus time trajectory. Points belonging to the gray strip in the bull region are mapped in one iteration in the bear
region. A symmetric stripe exists in the bear region, and points in this reason are mapped in one iteration in the bull region.

is satisfied, which leads to
ðS2 Þ2 þ S2 ð2ð1 þ S1 Þ þ 1 þ mÞ þ ð1 þ S1 Þ2 þ ð1 þ S1 Þð1 þ mÞ þ 2m ¼ 0:
The solution of this is expressed by the two branches of equation
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðd4Þ : S2 ¼ 12−ð2ð1 þ S1 Þ þ 1 þ mÞpm ð2ð1 þ S1 Þ þ 1 þ mÞ2 −4ðð1 þ S1 Þ2 þ ð1 þ S1 Þð1 þ mÞ þ 2mÞ;

ð28Þ

ð29Þ

shown as (d4) in Fig. 1a. Thus we have proved the following
Property 2 (divergence). For any value of m, outside the region satisfying −2 o S1 þ S2 o 0 either chaotic dynamics exist or the
generic trajectory is divergent. The region with divergent trajectories is bounded by the surfaces given in (19) and (22) for m o0,
and in (26) and (29) for m 40. Chaos occurs in robust chaotic intervals of map F.
Let us take a closer look at Fig. 1 and the region bounded by (d1)–(d4). Naturally, one important goal of a central
authority would be to prevent a market from collapsing, i.e. to prevent the price trajectory from settling on an explosive
course. However, as is clear from Fig. 1, the regulation of financial markets is a non-trivial, challenging task. Note that either
an increase or decrease in parameter S2 and/or parameter m, and thus an increase or decrease in the aggressiveness of type 2
chartists and type 2 fundamentalists, may benefit or harm market stability. On the other hand, we learn from Fig. 1 that
markets at least do not explode as long as the condition −2−S1 oS2 o−S1 is met. Given our model, a central authority may
thus seek to manipulate the slopes of the outer branches of the underlying map such that they fall into this corridor, for
instance by following appropriate intervention strategies (e.g. Wieland and Westerhoff, 2005 studied the impact of chaos
control-inspired intervention strategies within the original model of Day and Huang, 1990).
A number of examples of chaotic dynamics outside the strip −2 oS1 þ S2 o0 are shown in Fig. 4 for m 40 and decreasing
external branches, in Fig. 5 for m o 0 and increasing branches. As can be seen, both parameter combinations, despite being quite
different, can produce complex bull and bear market dynamics, which underlines the robustness of this finding. In the following,
we sketch how these price fluctuations come about. However, let us first return to Fig. 2, which illustrates the dynamics of the
original model of Huang and Day (1993). Suppose, for a start, that the price is slightly above the fundamental value. Since only
chartists are active in that market area, prices are driven upwards, period for period, until fundamentalists enter the market. Since
fundamentalists trade quite aggressively, prices are pushed downwards. Fig. 2 reveals that as long as the price drop is not too
extreme, i.e. as long as the market remains in its bull state, prices will rise again, due to chartists' transactions. This pattern repeats
itself in a complex manner, enabling us to observe irregular bull market dynamics. However, when prices reach a very high level,
fundamentalists' transactions may become so forceful that the resulting price drop turns the market into a bear market. From
then on, the chartists' previously optimistic mood turns pessimistic, and they expect a further price decrease. And indeed, prices
decrease – until fundamentalists re-enter the market again and push prices upwards. We may now observe a longer bear market
episode or, alternatively, a faster comeback of a bull market. It is virtually impossible to predict the duration of bull and bear
markets, which is one of the aspects that make the model of Huang and Day so interesting.
Let us now take a look at Fig. 4. Essentially, the dynamics is similar to that just discussed. In our generalized model, we
also see more or less randomly alternating periods of complex bull and bear market dynamics, despite the map's two
discontinuities. Depending on the size of parameter m 4 0, however, we may observe larger price drops more/less
frequently. Recall that in our model, chartists and fundamentalists are always jointly active. In the inner regime, the trading
activity of type 1 chartists is stronger than that of type 1 fundamentalists. For this reason, the slope of the map's inner
branch is higher than one, and prices are driven away from the fundamental value. At some point, however, additional

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F. Tramontana et al. / Journal of Economic Dynamics & Control 37 (2013) 2351–2370

speculators enter the market. Depending on the relative strength of type 2 chartists and type 2 fundamentalists, the slope
and location of the two outer branches change. The specification in Fig. 4 is such that the aggregate transactions of all
speculators produce more rapid price corrections than observed in Huang and Day's original model.
The situation differs in Fig. 5, where a new price pattern also emerges. What do the parameters now imply? Note that
m o0 can be interpreted in the sense that the price-independent trading behavior of type 2 fundamentalists is quite
pronounced, while S2 40 can be interpreted in the sense that type 2 chartists trade rather aggressively on their price
signals. The new feature of Fig. 5 is as follows: observe, for instance, that a transition from a bull market to a bear market
now emerges for more intermediate price levels rather than for very high price levels. Here we have a situation in which
fundamentalists' selling orders exceed chartists' buying orders to such an extent that the consequent price drop triggers a
bear market. If the price had been somewhat higher, however, we would still have observed a price drop. Due to the
increased number of buying orders placed by chartists and the higher price level, however, the bull market would have
survived. Hence, our model is able to generate a different regime-shifting pattern, and it may be argued that the price
dynamics becomes even more erratic. Other examples of complex bull and/or bear market dynamics are depicted in
Figs. 7–9, 11 and 12, further supporting the robustness of bull and bear market dynamics.
Apart from the regions with chaotic dynamics outside the strip −2 oS1 þ S2 o0, another quite interesting region is given
exactly by this strip. As we will see, attracting cycles of any period can exist in this particular strip, and the boundaries can
also be analytically determined. In the enlargement in Fig. 1b we illustrate the periodicity regions associated with stable
cycles using different colors. We can also see from the structure of the regions that we have different dynamic properties in
regions m o 0 and m 4 0 and also, depending on the sign of the slope ð1 þ S1 þ S2 Þ, of the external branches, i.e. in the four
regions defined in (12), as described in the next section. First of all, however, the separating case occurring at value m ¼0
deserves special mention. Recall that a very important property of piecewise-linear maps is whether or not they are
invertible. In fact, when a piecewise-linear map is invertible inside an invariant interval (i.e. with a unique inverse in the
invariant interval), then only stable dynamics can exist, i.e. no unstable cycle or chaotic dynamics can occur, as proved in
Keener (1980). However, when the map is non-invertible, stable and unstable cycles can exist, as well as chaos. In our case,
considering the discontinuity point on the positive side, where the map is increasing/decreasing (i.e. all slopes are positive),
the invertibility condition, given by
F C ○F R ð1Þ 4 F R ○F C ð1Þ

ð30Þ

reads as
ð1 þ S1 Þm 4 m;

ð31Þ

which is always satisfied in region (R1). Moreover, the condition
F C ○F R ð1Þ ¼ F R ○F C ð1Þ

ð32Þ

is satisfied for m ¼0, which implies that for m ¼0 map F is conjugated with a circle map and, in particular, with a linear
rotation, whose dynamics are well known. That is, either all points of the absorbing interval are periodic (of the same
period) or any trajectory is quasiperiodic and dense in the absorbing interval (see also Gardini et al., 2010; Gardini and
Tramontana, 2010). This depends on the rotation number, rational or irrational, respectively, and can also be observed
numerically: the points associated with rational rotation numbers are the points on m ¼0; the periodicity regions in Fig. 1b
emanate from there. Thus we can state the following:
Property 3 (Linear rotation). For m¼ 0 and −2 oS1 þ S2 o 0 map F is conjugated with a linear rotation.

4. Dynamics of the model in regions (R1)–(R4)
In this section, we characterize the model dynamics in parameter regions (R1)–(R4). The dynamics in regions (R1) and
(R2) can easily be related to well-known results of discontinuous maps with only one discontinuity point (see Gardini et al.,
2010; Avrutin et al., 2010; Gardini and Tramontana, 2010). However, new dynamic behaviors and new bifurcation structures
may occur when all three branches of the map start to interact (although a few results exist for maps with two discontinuity
points by Tramontana et al., 2011b, 2012, in press, they cannot be applied in our case).
To make our mathematical analysis as stark as possible, we decided to refrain from giving a deeper economic discussion
of our results in Section 4. In doing so, we hope that our analysis will be easier to follow and may encourage the study of
other economic models and further discontinuous piecewise-linear maps. So far, our knowledge about such systems is,
unfortunately, quite limited. In any case, the economic meaning of parameters S1, S2 and m is well defined and thus the
economic implications of our study should be straightforward to deduct.
Due to the symmetry of our map, and the useful notation in terms of symbolic sequences to represent existing cycles, we
shall use R and L to describe the right-most and left-most branches of our map F, while the central branch is separated by
the origin on the right and the left, denoted by C+ and C−, respectively.

F. Tramontana et al. / Journal of Economic Dynamics & Control 37 (2013) 2351–2370

2361

4.1. Region (R1): adding structure and disjoint symmetric attractors
All the branches of map F are increasing in region (R1). It is clear that for F R ð1Þ 41, i.e. for S2 4−m−S1 (as we have seen in
(16) and (15)), a stable fixed point exists on the right, and similarly on the left. It follows that xnR attracts all initial conditions
x0 4 0, while xnL attracts all negative ones.
Regarding the remaining portion, in (R1) and m o−ðS1 þ S2 Þ, the map is uniquely invertible in two disjoint (symmetric)
absorbing intervals, one on the positive side and one on the negative side, and without fixed points. Any initial condition
x0 4 0 converges to the unique attractor existing in
I R ¼ ½F R ð1Þ; F C ð1Þ ¼ ½1 þ S1 þ S2 þ m; 1 þ S1 ;

ð33Þ

and similarly in the negative one in the absorbing interval
I L ¼ ½F C ð−1Þ; F L ð−1Þ ¼ ½−ð1 þ S1 Þ; −ð1 þ S1 þ S2 þ mÞ:

ð34Þ

As already noted, in this range we can take advantage of the properties of a map with one discontinuity point. Hence we can
state that all dynamics is regular. Stable cycles of any period exist. Applying the formulas reported in several papers (see, for
example, Gardini et al., 2010), we can also write the explicit analytic equations of BCB curves (really surfaces in our case,
since the parameter space is three-dimensional), leading to a stable cycle in any period. Such cycles are organized in
complexity levels. As we can see from Fig. 1b, the so-called regions of cycles of the first complexity level (or maximal cycles) of
period (n+1) for any n≥1 have symbolic sequences C þ Rn above the region of the attracting 2-cycle, and symbolic sequences
RC nþ below the region of the 2-cycle. Then, between any two consecutive regions of cycles of the first complexity level, for
example between C þ Rk and C þ Rkþ1 , we can detect two infinite families of periodicity regions of cycles of the second
complexity level with symbolic sequence ðC þ Rk Þj C þ Rkþ1 and C þ Rk ðC þ Rkþ1 Þj for any j≥1, and so on: between any two
consecutive periodicity regions of cycles of complexity level r we can find two infinite families of cycles of complexity level
r+1 by using the same adding rule in the symbolic sequences. That is, the adding mechanism leads to the structure of the
attracting cycles and their periodicity regions, as evident in Fig. 1b.
A one-dimensional bifurcation diagram showing the dynamic behavior of our variable x is illustrated in Fig. 6a for m 4 0
at fixed parameters for S1 and S2 (along the horizontal path (A) shown in Fig. 1b). The attracting set on the positive side is
shown in black. The symmetric dynamics clearly also exists on the negative side (shown in red in Fig. 6a). The symbolic
sequence of the orbits is obtained from those on the right, substituting C+ and R with C− and L, respectively. The adding
mechanism in the structure of the attracting cycles is clearly evident in Fig. 6a. We also recall that for 0 o m o−ðS1 þ S2 Þ the
intervals associated with stable cycles are dense in the interval ð0; −ðS1 þ S2 ÞÞ. The parameter points that do not belong to
the closure of a periodicity region of a certain cycle are associated with stable non-periodic dynamics: the trajectory
ultimately converges to a non-chaotic Cantor set attractor, on which the trajectories are quasiperiodic and dense.
4.2. Region (R2): increment structure and chaotic intervals
The external branches of F are decreasing in the region (R2). For F R ð1Þ 41, i.e. for m 4−ðS1 þ S2 Þ (as we have seen in (16)
and (15)), a stable fixed point may exist on the right, and similarly on the left. However, the stable fixed point may now no
longer be the unique attractor. In fact, when the fixed points exist, the shape of the map is increasing/decreasing in the two
disjoint absorbing intervals on the two opposite sides. For such kinds of maps, with one discontinuity only in each invariant

Fig. 6. One-dimensional bifurcation diagram as a function of m. In (a) along path (A) of Fig. 1b, at S2 ¼ −1:2; in (b) along path (B) of Fig. 1b, at S2 ¼ −1:8.
(For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

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F. Tramontana et al. / Journal of Economic Dynamics & Control 37 (2013) 2351–2370

interval, we know that there exists a regime of bistability, and a so-called increment structure for existing cycles (see Gardini
and Tramontana, 2010). Thus we have to separate the cases in which two disjoint absorbing intervals exist, or only a unique
one. For the particular value ð1 þ S1 þ S2 Þ ¼ 0 the external branches are flat, which corresponds to the transition between
regions (R1) and (R2). This case is also well known: the interval ð0; −ðS1 þ S2 ÞÞ ¼ ð0; −1Þ consists of consecutive segments
associated with one unique cycle in each absorbing interval; on the right they have the symbolic sequence RC nþ (on the left LC n− Þ.
The boundary points between one period and the other correspond to so-called big bang bifurcation points in the parameter
space (see Fig. 1b). They correspond to the intersection of two different border collision bifurcation curves. Infinitely many BCB
curves are issued on one side from those points (in our case for ð1 þ S1 þ S2 Þ 40Þ, associated with the adding mechanism. On the
other side (in our case for ð1 þ S1 þ S2 Þ o0Þ, we have the overlapping of two periodicity regions of cycles with symbolic sequence
RC kþ and RC kþ1
on the right, and cycles with symbolic sequence LC k− and LC kþ1
on the left. A few regions can be seen in Fig. 6b for
þ
−
m 40 (those on the positive side are black those on the opposite side are red).
Thus as long as we have two absorbing intervals, on the right the existing attracting cycle is the unique attractor or two
attracting cycles coexist (in which case, the two basins of attraction within the absorbing interval are separated by
discontinuity point x ¼1 and its preimages). The eigenvalue of a cycle with symbolic sequence RC nþ is given by
λðRC nþ Þ ¼ ð1 þ S1 þ S2 Þð1 þ S1 Þn :

ð35Þ

Note that this holds for any n≥0, as for n ¼0 we get fixed point xnR . It follows that a flip bifurcation9 of these cycles occurs
when λðRC nþ Þ ¼ −1 at
S2 ¼ −

1
ð1 þ S1 Þn

−ð1 þ S1 Þ

ð36Þ

(see the horizontal segments that form the lower bounds of the colored periodicity regions in Fig. 1b for m 40).
In the piecewise linear case we have chaotic dynamics in invariant intervals for discontinuous maps after the flip
bifurcation of the maximal cycles. Note that the chaotic intervals may be or not cyclically invariant (as remarked in Avrutin
et al., in press) or may not be made up by disjoint absorbing intervals. Now (contrary to case (R1)) we have
I R ¼ ½F R ○F C ð1Þ; F C ð1Þ ¼ ½ð1 þ S1 Þð1 þ S1 þ S2 Þ þ m; 1 þ S1 

ð37Þ

(due to the negative slope of the external branches), and
I L ¼ ½F C ð−1Þ; F L ○F C ð−1Þ ¼ ½−ð1 þ S1 Þ; −ð1 þ S1 Þð1 þ S1 þ S2 Þ−m;

ð38Þ

so that when
m ¼ −ð1 þ S1 Þð1 þ S1 þ S2 Þ

ð39Þ

occurs, the two intervals merge into one unique interval:
I ¼ ½F C ð−1Þ; F C ð1Þ ¼ ½−ð1 þ S1 Þ; 1 þ S1 :

ð40Þ

One example of the dynamics in two disjoint intervals (i.e. before the occurrence of the bifurcation described above, and
given in (39)) is shown in Fig. 7, where the dynamics is either in the bull market or in the bear market, depending on the
initial condition. The dynamic behavior after the bifurcation is shown in Fig. 8: a unique absorbing interval I exists, and the
dynamics are oscillating (in a chaotic regime) between the two markets.10
A one-dimensional bifurcation diagram showing the dynamic behavior of our variable x is illustrated in Fig. 6b for m 4 0 along
horizontal path (B) shown in Fig. 1b. In Fig. 6b, the point corresponding to the bifurcation described above, given in (39), is
marked m1 . In the same figure, we also see that the dynamics is only chaotic throughout interval I ¼ ½F C ð−1Þ; F C ð1Þ ¼ ½−ð1 þ
S1 Þ; 1 þ S1  in the range m2 o m o m1 , while for 0 o m o m2 , even if the dynamics involve all three branches of map F, the
asymptotic behavior is confined in two symmetric intervals
½F C ð−1Þ; F R ð1Þ∪½F L ð−1Þ; F C ð1Þ⊂I ¼ ½F C ð−1Þ; F C ð1Þ

ð41Þ

and the chaotic intervals are bounded by the iterates of the offsets at the discontinuity points. The second bifurcation of the
chaotic intervals occurs when the signs of the offsets of the external branches change, that is when
F R ð1Þ ¼ 0;

ð42Þ

which occurs at
m ¼ −ð1 þ S1 þ S2 Þ:

ð43Þ

In Fig. 6b, the point corresponding to the bifurcation given in (43) is marked m2 .
9

To be precise, a degenerate flip bifurcation, see Sushko and Gardini (2010).
Note that the dynamics depicted in Fig. 8 resembles the dynamics depicted in Fig. 2 of the asset-pricing model of Brock and Hommes (1998). Indeed,
if the intensity of choice parameter goes to plus infinity and if the prediction rules of the traders are linear, the dynamics of their model is driven by a
discontinuous piecewise-linear map. A similar remark also holds for other discrete-choice models, e.g. the cobweb approach of Brock and Hommes (1997).
We hope that the mathematical tools developed in this and related papers make it possible to study the dynamics of such models in more detail.
10

F. Tramontana et al. / Journal of Economic Dynamics & Control 37 (2013) 2351–2370

2363

Fig. 7. In (a) m¼ 0.5, S2 ¼ −1:8; in (b) versus time trajectory.

Fig. 8. In (a) m¼ 0.5, S2 ¼ −1:9; in (b) versus time trajectory.

Thus we have proved the following:
Property 4. Let the parameters belong to the chaotic region in (R2), then

 for m 4−ð1 þ S1 Þð1 þ S1 þ S2 Þ two disjoint invariant intervals exist, on opposite sides with respect to the origin (disjoint bull
and bear regimes);

 for −ð1 þ S1 þ S2 Þ om o−ð1 þ S1 Þð1 þ S1 þ S2 Þ the dynamics is chaotic in a unique invariant interval I ¼ ½−ð1 þ S1 Þ; 1 þ S1 
including the origin (bull and bear intermingled regimes);

 for 0 o m o −ð1 þ S1 þ S2 Þ the dynamics is chaotic in an invariant set that does not include the origin (but still involves bull
and bear intermingled regimes).
One example of the chaotic dynamics occurring in the third case, after the bifurcation given in (43) for 0 o m o m2 is
illustrated in Fig. 9.
4.3. Region (R3): even increment structure and chaotic intervals
In region (R3) the external branches of F are increasing ð1 þ S1 þ S2 Þ 4 0. For values of this slope close to 0, taking m o 0
such that F R ð1Þ ¼ ð1 þ S1 þ S2 Þ þ m o 0, it is possible to have stable cycles of any even period, involving all three branches
of F except for the 2-cycle. In fact, the stable 2-cycle existing for m o 0 is that with symbolic sequence RL (one point of which
has been already determined in (23), xR ¼ −m=ð2 þ S1 þ S2 Þ and xL ¼ −xR Þ, so that its region of existence is bounded by the
BCB curve of equation
S2 ¼ −m−S1 −2

ð44Þ

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F. Tramontana et al. / Journal of Economic Dynamics & Control 37 (2013) 2351–2370

Fig. 9. In (a) m¼ −0.3, S2 ¼ −2:2; in (b) versus time trajectory.

Fig. 10. In (a) m¼ −1, S2 ¼ −1:4; in (b) m¼−0.35, S2 ¼ −1:4.

(see the straight line in Fig. 1b for m o 0 at the boundary of the region of the 2-cycle). All other stable cycles that may be
observed in Fig. 1b in an increment structure with even periods have the symbolic sequence RC n− LC nþ for any n≥0 (n ¼0
leading to the 2-cycle RL).
One example of the 4-cycle is shown in Fig. 10a; a 10-cycle is given in Fig. 10b. Moreover, as in an usual period increment
nþ1
structure, there is a region of bistability between any two consecutive pairs RC n− LC nþ and RC nþ1
for any n≥0. This is a
− LC þ
totally new increment scenario, associated with the existence of two discontinuity points, that leads to cycles with periodic
points in all partitions. The BCB curves leading to their appearance and disappearance is due to the collision with the
discontinuity points as follows. For any n≥1 one boundary (that on the left side in Fig. 1b for m o0) is determined via the
condition
F Cn−1 ○F L ○F nC ○F R ○F C ð1Þ ¼ 1
leading to the equation, for any n≥1:
"
#
.
1
1
2 2
1 nþ1
−ð1
þ
S
þ
S
Þ
ð1
þ
S
Þ
½ð1 þ S1 þ S2 Þð1 þ S1 Þn −1
m¼
1 n−1
ð1 þ S Þ

ð45Þ

ð46Þ

while the other boundary (that on the right side in Fig. 1b for m o 0) is determined via the condition
F nC ○F L ○F nC ○F R ð1Þ ¼ 1

ð47Þ

leading to the equation, for any n≥1:
m ¼ ½1−ð1 þ S1 þ S2 Þ2 ð1 þ S1 Þ2n =½ð1 þ S1 þ S2 Þð1 þ S1 Þ2n −ð1 þ S1 Þn :

ð48Þ

F. Tramontana et al. / Journal of Economic Dynamics & Control 37 (2013) 2351–2370

2365

Fig. 11. In (a) m¼−2, S2 ¼ 0:319; in (b) versus time trajectory.

The boundaries of the 4-cycle obtained for n ¼1 are shown in Fig. 1b. This region overlaps with the region of the 2-cycle RL,
leading to the related region of bistability.
The eigenvalue associated with these cycles of symbolic sequence RC n− LC nþ is
λðRC n− LC nþ Þ ¼ ð1 þ S1 þ S2 Þ2 ð1 þ S1 Þ2n
so that they are stable as long as
S2 4

1
ð1 þ S1 Þn

λðRC n− LC nþ Þ o 1

ð49Þ
after which, for

−ð1 þ S1 Þ;

ð50Þ

the cycle of period ð2n þ 2Þ becomes unstable and we detect 2ð2n þ 2Þ chaotic intervals. Thus we have proved the following:
Property 5. Let the parameters belong to region (R3), then

 the existence region of a2-cycle with symbolic sequence RL, xR ¼ −m=ð2 þ S1 þ S2 Þ and xL ¼ −xR , is bounded by the border
collision bifurcation curve given in (44);

 the existence region of a cycle with symbolic sequence RC n− LC nþ for any n≥1 is bounded by two border collision bifurcation
curves given in (46) and in (48);

 for S2 o 1=ð1 þ S1 Þn −ð1 þ S1 Þ cycle RC n− LC nþ for any n≥0 is stable (n¼0, corresponding to the2-cycle RL);
nþ1
 overlapping regions leading to bistability occur for each pair RC n− LC nþ and RC nþ1
for any n≥0.
− LC þ

One example of the dynamics as a function of m along path (A) of Fig. 1b is shown in Fig. 6a: a few periodicity regions
(of cycles of period 2, 4, 6 and overlapping by pair) are crossed before chaotic dynamics emerges. In the white region in
Fig. 1b we have attracting chaotic intervals, and we can observe that, as m increases, the dynamics first involves all three
branches of the map, then two disjoint invariant absorbing intervals appear. That is, following arguments similar to those
used in the previous region, it is easy to see that, as long as we have F R ○F C ð1Þ o0, then the dynamics are invariant in two
intervals that do not include the origin, that is, inside the set
½F C ð−1Þ; F R ○F C ð1Þ∪½F L ○F C ð1Þ; F C ð1Þ
and a bifurcation occurs when F R ○F C ð1Þ ¼ 0 at
m ¼ −ð1 þ S1 Þð1 þ S1 þ S2 Þ

ð51Þ

since then, as long as F R ○F C ð1Þ 40 and F R ð1Þ o 0, the dynamics is chaotic in the interval
I ¼ ½F C ð−1Þ; F C ð1Þ ¼ ½−ð1 þ S1 Þ; ð1 þ S1 Þ:
In both cases, the chaotic dynamics involves all branches of F, so that bull and bear dynamics are intermingled, an example
of which is shown in Fig. 11.
This occurs as long as bifurcation F R ð1Þ ¼ 0 takes place, when
m ¼ −ð1 þ S1 þ S2 Þ;

ð52Þ

leading to two disjoint absorbing intervals, still with chaotic dynamics, but one in the bull region and the other in the bear
region. One example of the disjoint chaotic dynamics is shown in Fig. 12.

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Fig. 12. In (a) m¼−1.5, S2 ¼ 0:1; in (b) versus time trajectory.

Fig. 13. Two-dimensional bifurcation diagram. In (a) enlarged portion of Fig. 1b in region (R4). In (b) enlarged portion of the rectangle in (a).

Thus we have proved the following:
Property 6. Let the parameters belong to the chaotic region in (R3), then

 for m o −ð1 þ S1 Þð1 þ S1 þ S2 Þ the dynamics is chaotic in an invariant set not including the origin (but bull and bear
intermingled regimes);

 for −ð1 þ S1 Þð1 þ S1 þ S2 Þ om o−ð1 þ S1 þ S2 Þ the dynamics is chaotic in a unique invariant interval I ¼ ½−ð1 þ S1 Þ; 1 þ S1 
including the origin (bull and bear intermingled regimes);

 for −ð1 þ S1 þ S2 Þ o m o 0 the dynamics is chaotic in two disjoint invariant intervals, on opposite sides with respect to the
origin (disjoint bull and bear regimes).

4.4. Region (R4): even adding structure and bistability
In this region (R4) with m o0 we are interested in the dynamics inside the strip −2 oS1 þ S2 o−1,
i.e. −1 o ð1 þ S1 þ S2 Þ o 0. After all, we know that below, for ð1 þ S1 þ S2 Þ o−1, only divergent dynamics occur, as described
in Section 3, and as is visible in Fig. 1. Here, as for region (R3), in the stability strip we have a new dynamic behavior of the
map, that has never been investigated before. The branches of the map are increasing/decreasing in the positive
discontinuity point and decreasing/increasing in the negative discontinuity; all three branches are involved in the periodic
points of the existing cycles.
In Fig. 13a we have enlarged the portion of the parameters' plane associated with a new adding mechanism. The
overlapping regions associated with stable cycles determined in (R3) and described in the previous subsection intersect at
points of line 1 þ S1 þ S2 ¼ 0, creating infinitely many big bang bifurcation points. In fact, we can see that between the

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2367

Fig. 14. In (a) shape of the map in P : ðm ¼ −0:5; S2 ¼ −1:94Þ with a stable 14-cycle; in (b) enlarged part with the first return map Tr, as defined in (54).
kþ1
principal tongues (or main cycles) of symbolic sequence RC k− LC kþ and RC kþ1
for any k≥0 (whose existence has been
− LC þ
determined in region (R3)) are now, in the region with (1 þ S1 þ S2 Þ o 0, disjoint from one another, and we have an adding
bifurcation structure. However, the adding mechanism is totally new (since it involves three branches).
The main property that characterizes the map in the stability region considered here (the strip defined above) is that for
the invariant absorbing interval which includes the asymptotic behaviors of the map, intervals which are bounded by the
critical points (iteration of the offsets in the discontinuity points), the map has a unique inverse. Thus, the map cannot have
complex behaviors, and the invariant sets may only be stable cycles (when the dynamics are associated with a rational
rotation number) or Cantor set attractors (when the rotation number is irrational). It is worth noticing that having two
discontinuity points, the concept of “rotation number” is not yet very well defined although it has already been used for this
kind of map, for example in Tramontana et al. (2012). However, the property of “invertibility”, which characterizes this
regime, can be used to show that, although we have branches with negative slopes, all cycles can only have a positive
eigenvalue. This is due to the fact that we can properly investigate the dynamics of the map making use of its first return
map at interval J ¼ ½1; F C ð1Þ, say Tr:

T r ðxÞ : J-J;

J ¼ ½1; F C ð1Þ

ð53Þ

as any trajectory of F must necessarily visit this interval J, and the first return map Tr completely determines the dynamics of F.
It is clear that the problem is not very simplified, as also the first return map in this interval J may be a discontinuous map
with two discontinuity points: point d− , which is the preimage in J of the discontinuity x ¼ −1 of F, and point dþ , which is
the preimage in J of the discontinuity point x ¼ þ 1. However, the advantage is that in the first return map we must have all
positive slopes. This is because in order for a trajectory to return in interval J, the branch with a negative slope must necessarily
be applied an even number of times.
As an example, consider parameter point P in Fig. 13b P : ðm ¼ −0:5; S2 ¼ −1:94Þ, which is associated with a unique cycle
of period 14. Fig. 14a shows the map and the points of the cycle of period 14; the enlarged part shows the first return map at
interval J, and the three branches are defined by the symbolic sequence evidenced there, that is Tr(x) is defined as follows:
8
>
< F L ○F C− ○F R ðxÞ if 1 o x od−
ð54Þ
T r ðxÞ : x′ ¼ F Cþ ○F L ○F R ðxÞ if d− o x odþ
>
: F ○F ðxÞ
if dþ ox o F C ð1Þ;
L
R
where
d− ¼ F −1
R ð−1Þ;

−1
dþ ¼ F −1
R ○F L ð1Þ:

We can see that the symbolic sequence of the cycle is RC − LRC − LRLC þ RLC þ RLRL. However, in the adding mechanism we
can also see 14 periodic points belonging to two coexisting symmetric cycles of period 7, which occurs when the parameters
are at point Q of Fig. 13b, Q : ðm ¼ −0:5; S2 ¼ −1:993Þ, as illustrated in Fig. 15. The definition of the first return map (53) is the
same as before in (54). We can see that two symmetric cycles exist, with symbolic sequence RC − LRLRL and the symmetric
cycle, whose symbolic sequence is obtained by changing R, L and C− into L, R and C+, respectively.
We recall that for a map with only one discontinuity point, in the summation rule of two cycles, the new one has the
symbolic sequence obtained by concatenation of the symbols of the two starting cycles, so that the number of periodic
points on the two sides of the discontinuity (say L and R) are given by the sum of letters L and R of the two starting cycles,
and the period is the sum of the periods of the two starting cycles. In our case, we can reason similarly. However, we add
cycles with four symbols (L, C− and R, C+), and in the summation rule of two cycles we have points in the four partitions.
kþ1
Regarding the result of the summation rule between two cycles RC k− LC kþ and RC kþ1
− LC þ , we have the symbolic sequences of

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F. Tramontana et al. / Journal of Economic Dynamics & Control 37 (2013) 2351–2370

Fig. 15. In (a) the shape of the map in Q : ðm ¼ −0:5; S2 ¼ −1:993Þ with two symmetric stable 7-cycles; in (b) an enlarged part with the first return map Tr,
as defined in (54).
kþ1
k
k
kþ1
kþ1 n
type ðRC k− LC kþ Þn ðRC kþ1
− LC þ Þ and ðRC − LC þ ÞðRC − LC þ Þ and we can either have a unique cycle or a pair of symmetric cycles
(with odd period), which is a feature of the symmetry property and the two discontinuity points. What is relevant (in order
to have one case or the other) is the number of periodic points existing in the external branches (L and R) which, due to the
symmetry of the map, are the same in number and symmetric with respect to x¼ 0. Hence, if the summation of two cycles
leads to an odd number of periodic points on the L side (and, equivalently, on the R side), then a unique cycle exists (as in the
example of the 14-cycle in Fig. 14a, where we have 5 periodic points in branches L and R). If the summation of the two cycles
leads to an even number of periodic points on the L side (and, equivalently, on the R side), then two symmetric cycles appear,
with an odd period (as in the example in Fig. 15a, the summation rule leads to 6 periodic points in branches L and R, so that a
pair of symmetric cycles exist).
Although we are unable to give rigorous proof of this empirical result (which has been left for further research work), it
has been numerically verified in our simulations and bifurcation diagrams (see Fig. 13b). As the principal cycles are those of
symbolic sequence RC k− LC kþ having only one point in the external branches, we have that the adding of two consecutive
kþ1
sequences, say RC k− LC kþ and RC kþ1
− LC þ , leads to 2 points in the external branches, which is even. Hence two symmetric
cycles exist (with odd period). However, this is only one step of the adding mechanism. For example, for k ¼0, between the
periodicity regions of 2-cycle RL and of 4-cycle RC − LC þ , we have families ðRLÞn ðRC − LC þ Þ and ðRLÞðRC − LC þ Þn for any n≥1 and at
n ¼1 we have the pair of 3-cycles, as commented above. Then in family ðRLÞn ðRC − LC þ Þ as n increases, we add only one point
in the external branches, so that we have alternating single cycles or pairs of cycles of odd periods (see the enlargement in
Fig. 13b).

5. Conclusions
In our paper we aimed to generalize the famous bull and bear market models of Day and Huang (1990) and, in particular,
Huang and Day (1993) and to improve our understanding of the dynamics of discontinuous piecewise-linear maps. Indeed,
by relaxing some restrictive behavioral assumptions about speculators' trading activities, we succeeded in extending the
model of Huang and Day such that it changes from a continuous piecewise-linear map to a discontinuous one. Since the map
has two discontinuity points, little is known about its dynamic properties. Nevertheless, we have managed to provide a
more or less complete analytical and numerical characterization of our model.
From a mathematical point of view, we describe a number of dynamics and bifurcation structures that have never been
investigated before. While standard results can be used to explain the dynamic behaviors in regions (R1) and (R2), the
dynamics occurring in region (R3) is due to a new kind of phenomenon. In Section 4.3 we introduced a totally new period
increment scenario, with cycles that have periodic points in all three partitions of the map. The symbolic sequences and
bifurcation structure depend on the existence of two discontinuity points. We analytically determined the equations of the
border collision bifurcation curves associated with their existence, the region of bistability between any two consecutive
nþ1
pair of cycles RC n− LC nþ and RC nþ1
for any n≥0, as well as their degenerate flip bifurcation. The conditions under which
− LC þ
chaotic regimes are intermingled in the bull and bear regions were also determined. Moreover, in the stability strip observed
in region (R4) we described a new period adding structure. The peculiarity, due to the symmetry property and to the two
discontinuity points, is that in the adding mechanism we can have either a unique cycle or a pair of symmetric cycles (with
odd period). Bistability cannot occur in maps with one discontinuity only and increasing branches, and we showed that our
map can be studied making use of the first return map in a suitable interval. The increasing branches and invertibility of the

F. Tramontana et al. / Journal of Economic Dynamics & Control 37 (2013) 2351–2370

2369

first return map explain the existence of the adding structure. We also empirically understood the peculiarity of the
bistability, as described in Section 4.4, although rigorous proof has been left for further studies.
From an economic point of view, we conclude that the alternating switching between irregular bull and bear markets, as
first reported by Day and Huang, is a rather robust finding for such types of model. Moreover, we detect a number of boombust cycles that are even more intricate than those observed in Day and Huang's original models. For instance, while in their
model a bull market only turns into a bear market if prices have reached certain extremely high price levels, our model may
also produce such market crashes for intermediate price levels. In addition, we determined, in parameter space, the border
that separates bounded dynamics from divergent dynamics. Our analysis reveals that both chartists and fundamentalists
may contribute to market stability/instability—which makes the regulation of financial markets such a delicate issue.
Our model can be extended in many ways. For instance, one could assume that the market entry level of type 2 chartists
differs from that of type 2 fundamentalists, where the result would be a map with five branches. It could also be assumed
that speculators react differently to overvalued and undervalued market situations, leading to an asymmetric map. Note,
however, that the current map has not even been fully explored yet. For instance, the case where the inner regime is
unstable due to aggressive transactions by fundamentalists has not yet been addressed (as far as we know, the same is true
for a pure mathematical analysis).
Instead of considering a deterministic model, as in our paper, a stochastic model specification could also be explored.
Preliminary work we have performed reveals that our setup, if polluted with random noise, is quite capable of mimicking
some important stylized facts of financial markets. For instance, extreme price changes may be found close to the market
entry levels of type 2 speculators, even when prices reach quite high or low levels. Note that deterministic skeletons of such
stochastic models are an excellent way of developing a better understanding of their complex price dynamics.
Both deterministic and stochastic model versions of our setup may also be used to enhance our knowledge about how
regulatory measures function. For instance, one may incorporate a central authority in our model and study how certain
intervention (feedback) strategies, such as targeting long-run fundamental rules or leaning against the wind rules, influence
the model dynamics. Given the instability of financial markets, this may be a worthwhile endeavor.
Of course, many other extensions of our work are possible. We hope that our paper stimulates further work in the
exciting analysis of bull and bear market models and the rather novel use of piecewise-linear maps to investigate them. In
addition, the study of discontinuous piecewise-linear maps seems to us to be important per se, as many economic problems
entail natural breaks or hard boundaries.

Acknowledgments
We thank two anonymous referees and the editor, Cars Hommes, for helpful comments and suggestions. This work was
performed within the national research Project PRIN-2009NZNM7C “Local interactions and global dynamics in economics
and finance: models and tools”, MIUR, Italy, and under the auspices of COST Action IS1104 “The EU in the new complex
geography of economic systems: models, tools and policy evaluation”.
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