Chaotic Dynamics of a Piecewise Smooth Overlapping
Generations Model witha Multitude of Technologies
Yosuke Umezuki ∗ and Masanori Yokoo †This Version: January 31, 2019

Abstract
In this study, we develop a simple overlapping generations model that can exhibit chaotic fluctuation. The key assumption is that a firm’s owner can choose froma continuum of technologies of production. As a result, the model reduces to apiecewise smooth map, which is tractable enough so its dynamics can be analytically investigated in depth. To study chaotic behaviors in the model, we adopt two
approaches: border collision bifurcation and Markov property. The border collisionbifurcation theory characterizes the routes from a globally attracting steady state toother non-stationary behaviors. The Markov property reveals the chaotic dynamics
of the model for a much larger set of parameter values.

JEL Classification Numbers: C62; E32; O14; O41
Key Words: Technology Choice; Piecewise Smoothness; Endogenous Fluctuations;Overlapping Generations Model; Chaotic dynamics∗

Corresponding author: Graduate School of Humanities and Social Sciences, Okayama University,
Tsushimanaka 3-1-1, Kita-ku, Okayama 700-8530, Japan. e-mail: ec19025@s.okayama-u.ac.jp†
Graduate School of Humanities and Social Sciences and Faculty of Economics, Okayama University,
Tsushimanaka 3-1-1, Kita-ku, Okayama 700-8530, Japan. e-mail: yokoo@e.okayama-u.ac.jp

11

Introduction

There is no doubt that the overlapping generations (OLG, hereafter) model is one of themost popular dynamic economic models in the literature. Especially, the OLG setting
plays an important role in constructing an endogenous growth (or business) cycle model,
which exhibits, without external shocks, a perpetual periodic or chaotic (i.e., randomlooking but deterministic) fluctuation. In early literature, a few prominent examples ofsuch perpetual fluctuation in an OLG model can be found in the studies by Benhabib andDay (1982) and Grandmont (1985).
Matsuyama (2007) presents an OLG-type model of credit cycles induced by financial imperfections; he shows that the model has the capability to exhibit several dynamic
growth patterns, including perpetual fluctuations. Asano et al. (2012) focus on a special
case of the model by Matsuyama (2007) and show that, in such a case, the model can
exhibit periodic and non-periodic fluctuations. Iwaisako (2002) investigates a situation inwhich investors can choose from among two production technologies—constant returns
to scale and increasing returns to scale—and shows several dynamic patterns such as perpetual fluctuation and permanent economic growth. It should be noted that, in the above
studies, each model can eventually reduce to a one-dimensional map with discontinuity.
In these models, the set of parameter values for which a complex fluctuation can appear isextremely “thin” even if there exists a complex fluctuation theoretically. In other words,
the chaotic motions cannot be observed virtually.
Along this line of discrete choice in an OLG model, Umezuki and Yokoo (2019) construct an explicit OLG model in which the firm faces a binary choice problem related tothe choice of technology (represented by the Cobb-Douglas production function of constant returns to scale). Similar to the above studies, our model can eventually reduce to apiecewise linear map with discontinuity. Such piecewise-linearization, similar to Asanoet al. (2012), helps us to directly apply to our model some useful results borrowed fromprior studies on mathematical neuron models. The employment of this method completely
characterizes the existence and occurrence of periodic fluctuations in the model. However,
the model also cannot virtually reproduce complex fluctuations, because it has a small setof parameter values for which there exists non-periodic fluctuation in the model.
Of course, we can frequently observe non-periodic, complex fluctuation of economic
variables in the real world. Therefore, it is natural that, along this line, we develop amodel that can virtually reproduce the chaotic fluctuations in the long run. Especially,
this study aims to extend the model in Umezuki and Yokoo (2019). To do this, our idea isto smooth the map with discontinuity through some modifications. If this new piecewise2smooth map has suﬃcient nonlinearity, we can easily conjecture that the model has thecapability to generate observable complex fluctuations.
It must be noted that the approach to smooth the discontinuous model has been discussed in context. Yokoo and Ishida (2008) introduce imperfect observability, which
they call misperception. It is represented by a parameterized random variable in the statevariable to smooth Ishida and Yokoo’s (2004) piecewise linear model with discontinuity, which is a macroeconomic, but not an OLG, model with discrete choice. Asano andYokoo (2019) also adopt this idea into the model in Asano et al. (2012) and investigate
the chaotic dynamics. However, in general, the abovementioned smooth model with imperfect observability is too general to investigate its dynamics in detail. Indeed, thesestudies make a somewhat strict assumption about misperception, in which the model can
reduce to a piecewise linear map. Although such linearization gives us depth and clear
results about the complex dynamics of the model, there is a tradeoﬀ between tractability
and generality in this method.
On the other hand, in this study, we use a more primitive and intuitive way to smoothUmezuki and Yokoo’s (2019) piecewise linear map with discontinuity. There is only one
modification: we assume that the firm faces a continuous choice problem of technology
rather than a binary choice problem. As a result, we can transform, without additional assumption, a piecewise linear model with discontinuity into a continuous piecewise smoothmodel that can virtually reproduce the chaotic motion and is tractable enough to investigate such a fluctuation. In this sense, our model can cope with both generality andtractability. Matsuyama (2013) and Matsuyama et al. (2016) adopt the same method,which assumes a continuous choice instead of discrete choice, to smooth the discontinuous model in Matsuyama (2007). However, our model is simpler than Matsuyama’s interms of the story and motivation behind the modeling.
To study the complex dynamics in continuous piecewise smooth maps, we adopt two
approaches: border collision bifurcation and Markov property. That is, we obtain two
diﬀerent characterizations of chaotic dynamics in the model. The first approach includes
a bifurcation that occurs in piecewise smooth, piecewise linear, and piecewise non-linear
maps. This theory reveals the bifurcation from a globally attracting steady state to othernon-stationary behaviors using only local information in the system. Especially, we cancompletely characterize the bifurcation from an attracting steady state to another nonstationary attractor, such as periodic cycles or chaotic bundles. Gardini et al. (2008)
and Matsuyama et al. (2016) use the border collision bifurcation theory to investigate
the chaotic dynamics in their models. Second, we find that our model can have a special
property—a Markov property for some sets of parameter values. If the model has such3
a property, we can easily establish and characterize its complex dynamics. Yokoo and
Ishida (2008) and Asano and Yokoo (2019) find the Markov property in their piecewise
linear models to analyze the chaotic dynamics. Therefore, from a technical viewpoint,
our model has an advantage against the preceding models because we can characterize
the chaotic fluctuations in the model using two diﬀerent approaches.
The remainder of this paper is organized as follows. Section 2 re-examines the OLG
model with a binary choice of technologies. Section 3 derives the piecewise smoothmodel by introducing a new assumption, a multitude of choices in the technologies, into
the OLG model with a binary choice setting. Section 4 describes the border collisionbifurcation theory characterized by routes from the globally attracting steady state to othernon-stationary cycles, including chaotic fluctuation. Section 5 reveals that the capabilityof the model has a Markov property for some set of parameter values. Section 6 concludes
the study. Some mathematical proofs are delegated to the appendices.

2

The model under binary technology choice

In this section, for the reader’s convenience, we re-examine the base model of this study.
(See Umezuki and Yokoo (2019) for more details.) We consider a Diamond-type onesector OLG model that is modified further. Time is discrete, that is, t = 0, 1, 2, · · · , andthe agents live for two periods. A young household supplies one unit of labor inelastically.
We keep the utility function, u, of the household as simple as possible; this allows us toassume that it is in the form of a log-linearized Cobb-Douglas production function, thatis,u(cyt , cot+1 ) = (1 − s) log cyt + s log cot+1 , s ∈ (0, 1),(1)where cyt denotes the amount of consumption of the young generation born at time t, andcot+1 denotes the amount of consumption of the old generation living at time t + 1. The
utility given by Eq.(1) is maximized under the following constraints:
cyt + st = wt

andcot+1 = rt+1 st ,(2)

where st , wt , and rt+1 are the amounts of saving, real wage rate, and real gross rate of
return, respectively. The maximization yields
st = swt .

(3)

The final good, Yt , which is perishable, is produced by the firm. Unlike the common
OLG models, we assume that there are two types of production technologies. We also

4
assume that, at the beginning of every period, the firm faces a discrete choice problem
related to the choice of technology. For simplicity, all the technologies are specified as
the Cobb-Douglas of constant returns to scale:
Yt = Fa (Kt , Lt ) = AKta Lt1−aand

Yt = Fb (Kt , Lt ) = BKtb Lt1−b ,where A, B > 0 is the total factor productivity, and 0 < a < b < 1 is the capital share of
the production of technology. In the per-capita form, we can write
yt = fi (kt ) = Fi (kt , 1),(4)where yt = Yt /Lt , kt = Kt /Lt , and i ∈ {a, b}. We assume that the firm’s owner, who belongs
to the old generation, chooses technology that earns the highest return. For simplicity, we
assume that when the highest rates of return are tied among two technologies, technology
1 characterized by (a, A) is chosen. Subsequently, the usual first-order conditions with thetechnology choice are represented by
rt = f J′t (kt ),(5)

wt = f Jt (kt ) − kt f J′t (kt ), and(6)

Jt = arg max f j′ (kt ).

(7)j∈{a,b}

The market clearing condition,
kt+1 = st ,(8)

with Eqs.(3), (6) and (7) generate the dynamic model in the following form:{
kt+1 =sA(1 − a)kta 0 ≤ kt ≤ k̂,sB(1 − b)ktb k̂ < kt ,

where

(9)

[ aA ]1/(b−a)

.
(10)
bB
It is clear that if Eq.(9) has a positive fixed point, it would be globally attracting. As
we are interested in non-stationary behaviors, we examine the case where Eq.(9) has no
positive fixed point. The following lemma shows that there is a parameter set for which
Eq.(9) has no positive fixed point.
k̂ =Lemma 1. Let a, b ∈ (0, 1) with a < b, B > 0, and s ∈ (0, 1) be given. Subsequently,
there exist A > 0 and A > 0 with A < A such that map (9) has no positive fixed point forA ∈ (A, A).

5
Proof. See Umezuki and Yokoo (2019).
Finally, by defining a new variable xt as[]
log kt /s(1 − b)Bk̂b
xt =,log b(1 − a)/a(1 − b)
Eq.(9) can reduce to the following piecewise linear map on the unit interval:{
1 + a(xt − c)
if 0 ≤ xt ≤ c,
xt+1 = τ(xt ) =b(xt − c)
if c < xt ≤ 1,
where

1−b

c = c(s, A) = b−alog [aA/bB] − log s(1 − b)B.
log b(1 − a)/a(1 − b)



(11)

(12)

(13)

It is interesting to note that Eq.(12) can be identified with a simplified version of Caianiello’s equation in the neural networks, whose original model is proposed by Caianielloto describe the behavior of a “model of brain” or “thinking machine”; this model is studied by Nagumo and Sato (1972) in detail. Additionally, Hata (2014) comprehensively
analyzes this type of equation, and we utilize this author’s analysis results for the mathematical neuron model to study economic growth.
When investigating Eq.(12), the position of threshold c(s, A) on the unit interval plays
an important role. The position of the threshold characterizes the dynamical propertyof the model. Generally, for a given value of c, Eq.(12) has either a globally attracting
periodic cycle or a non-periodic attractor.
The following lemma shows that threshold c(s, A) can take any value within the rangeof (0, 1), independently of parameters a and b.
Lemma 2. For any a, b ∈ (0, 1) with a < b and any c∗ ∈ (0, 1), there exist s∗ ∈ (0, 1) andA∗ ∈ (A, A) such that c(s∗ , A∗ ) = c∗ .
Proof. See Umezuki and Yokoo (2019).

Now, we restate the existence of a periodic orbit in the model.
Proposition 1. For each irreducible fraction p/q ∈ (0, 1), there exists a closed interval∆(p/q) ⊂ (0, 1) such that if c(s, A) ∈ ∆(p/q), for any x ∈ (0, 1), the orbit of x converges
to some periodic orbit of period q.
Proof. See Hata (2014), Theorem 4.2 on p.36 and Theorem 10.1 on pp.117–118.

6Thus, we can almost completely grasp the periodic dynamical features of the model byverifying to which closed interval threshold c belongs. Moreover, owing to the piecewise
linearity of the model, we can exactly calculate the left and right endpoints of the closed
interval ∆(p/q) for any p/q. See Umezuki and Yokoo (2019) for more details on thispoint.
Next, we can consider the following set:∪
Γ = [0, 1]\∆(p/q).
p/q∈[0,1]∩Q

Set Γ is the remainder obtained from the unit interval [0, 1] by Deleting the infinite closed
intervals. If c ∈ Γ, Eq.(12) has a non-periodic attractor.
∩ ∪∞kProposition 2. If c(s, A) ∈ Γ, for any x ∈ (0, 1), the ω-limit set of x; that is, ∞n=0
k=n clτ (x),is some compact and totally disconnected uncountable set on the unit interval.
Proof. See Hata (2014), Theorem 7.4 on p.80 and Theorem 8.5 on p.90.

In the proposition, clA represents closure of set A. We can conclude that, for c(s, A) ∈Γ, Eq.(12) has a global non-periodic attractor; it implies that once such a threshold value
is chosen, the economy would fluctuate in a non-periodic manner in the long run for any
initial condition. However, there is the fact that set Γ is extremely “thin.” The Hausdorﬀ
dimension of set Γ is zero. This implies that this base model cannot virtually reproduce
non-periodic fluctuations in the long run.

3

The model under continuum technology choice

In what follows, we modify the base model examined in the previous section to construct a
simple, tractable, OLG model with technology choice to virtually reproduce non-periodic
fluctuations in the long run.
To do this, we make only one modification. We assume that the firm faces a continuumof technologies. That is, there is a continuum of production functions,
Yt = Fα (Kt , Lt ) = AKtα Lt1−α ,α ∈ [a, b],

and the firm can choose this before the production of a good. Here, we also assume thatthe parameter of total factor productivity, given as A and B in the binary choice setting, isconstant through all technologies.
Therefore, we face the dynamic model in the following form:{
kt+1 = sA(1 − α)ktα ,{}∗(14)
α = arg maxα∗ ∈[a,b] Aα∗ ktα −1 , and k0 > 0.

7
It is easily shown that α is given by


a,




1
−
,
α=
log kt



 b,

kt
≤ e− a ,11e− a < kt ≤ e− b ,1e− b < kt .
1

if

Subsequently, we have the following piecewise-smooth dynamic model:1β(1 − a)kta ,kt≤ e− a ,) − log1 k
 (
1
1
1
kt+1 = β1+
kt t , if e− a < kt ≤ e− b ,

log
k

t

1

 β(1 − b)kb ,e− b < kt ,twhere β = sA.
By defining a new variable yt = log kt , Eq.(16) turns out to be
yt ≤ − a1 ,
 h1 (yt ) = ayt + log1β(1 − a),h2 (yt ) = log(1 + yt ) + log β − 1, if − 1a < yt ≤ − b1 ,yt+1 = h(yt ) =  h3 (yt ) = byt + log β(1 − b),− 1b < yt .

(15)

(16)

(17)

)
(
Fig.1 depicts the graph of return map of yt , which has a fixed point in − a1 , − b1 .
<<insert Fig.1 around here>>(] [)The graphical argument shows that if Eq.(17) has a fixed point on −∞, − 1a or − b1 , ∞ ,it would be globally attracting. Elementary algebra shows the following proposition that
detects the set of parameter β = sA, for which the model exhibits the globally attracting
steady state.
Proposition 3. Let
(
)
1
1−u
β (u, v) =exp −, 0 < u, v < 1.
1−v
u
∗

(
]
For any a < b, if β < β∗ (a, a) or β∗ (b, b) < β, map (17) has a fixed point on −∞, − a1 or[)− b1 , ∞ , respectively.
Here, it should be noted that, for any a < b, we have β∗ (a, a) < β∗ (b, b).
As we are interested in non-stationary behaviors,we) examine the case wherein Eq.(17)(11
has a positive steady state on the interval − a , − b . That is, we mainly focus on thefollowing case:β∗ (a, a) < β < β∗ (b, b).

8Let us further classify the model. It is evident
that every] trajectory generated by (17)[1eventually enters the trapping interval T = h2 (− b ), h1 (− a1 ) and never leaves it. Thus, tostudy the long-run behavior of the model in this situation, it is suﬃcient to focus on the
dynamics of the trapping interval, I. It must be noted that, depending on parameter values
a and b, we face the following two scenarios.
1. If

a−b
(1 − b)> log, then β∗ (b, a) < β∗ (a.b). We have the following cases:ab(1 − a)• If β∗ (a, a) < β < β∗ (b, a), then T contains the right threshold only.
• If β∗ (b, a) < β < β∗ (a, b), then T contains all the thresholds.
• If β∗ (a, b) < β < β∗ (b, b), then T contains the left threshold only.

2. If

a−b
(1 − b)< log, then β∗ (a, b) < β∗ (b.a). We have the following cases:ab(1 − a)• If β∗ (a, a) < β < β∗ (a, b), then T contains the right threshold only.
• If β∗ (a, b) < β < β∗ (b, a), then T contains no thresholds.
• If β∗ (b, a) < β < β∗ (b, b), then T contains the left threshold only.

If T contains only the right (left) threshold, the model can be regarded as a unimodal
map. On the other hand, if T contains both thresholds, the model reduces to a bimodal
map on the unit interval. Finally, if T contains no thresholds, the model can be regarded
as a monotonically decreasing map, which can have a periodic cycle at most period two.
It is reasonable that we omit such an insignificant case.
As we will see later, both unimodal and bimodal maps in our model are capable ofgenerating chaotic behaviors. We investigate the chaotic dynamics of the unimodal andbimodal cases in sections 4 and section 5, respectively.

4 Chaotic dynamics: Border collision bifurcation
In this study, we adopt two approaches to investigate the chaotic behaviors in the model.
In this section, we deal with border collision bifurcation to investigate the dynamical
property of the unimodal case. In this case, we can locally, but almost completely, characterize the bifurcation from a globally attracting steady state to non-stationary behaviors;that is, bifurcation from a globally attracting steady state to a periodic cycle or a chaotic
bundle. Roughly speaking, a one-dimensional piecewise map may undergo border collision bifurcation when the 45 degree line touches the graph of the map at its non-smoothboundaries. Therefore, in our model, border collision bifurcation occurs when β exceeds9β∗ (a, a) as well as β approaches β∗ (b, b). To avoid increasing the length of the paper, we
focus on the former case, in which the model can be represented by a single peaked unimodal map. Of course, the same analytical method is adopted for the latter case, in which
the model can be represented by a V-shaped unimodal map. In section 4.1, we briefly discuss some mathematical results for border collision bifurcation. In section 4.2, we adopt
the results to our OLG model with technology choice.

4.1

Some mathematical results from border collision bifurcation

All results in this section follow from Nusse and Yorke (1995).
Before we examine the piecewise smooth map (17), we consider the following piecewise linear map:{Fν1 ,ν2 =ν1 x + 1 x ≤ 0,
ν2 x + 1 0 < x,(18)

with 0 < ν1 < 1 and ν2 < −1. For convenience, let
D = {(ν1 , ν2 ) : 0 < ν1 < 1 andν2 < −1}.

Note that the above map is typologically conjugated with the following map:{
Fµ =

ν1 x + µ x ≤ 0
ν2 x + µ 0 < x,

(19)

for all µ > 0, which undergoes border collision bifurcation when µ = 0.
First, we state the existence of the periodic orbit in map (18).
Theorem 1. (Nusse and Yorke, 1995)
For integer m ≥ 2, let{}
1 − ν1m−1
1−m
Pm = (ν1 , ν2 ) ∈ D : −ν1 < ν2 < m−1.
ν1 − ν1m−2
If (ν1 , ν2 ) ∈ Pm , Fν1 ,ν2 has a period-m attractor. Moreover, the set{x : ω-limit set of x is not the period-m attractor}
Has Lebesgue measure zero.
Proof. See Nusse and Yorke (1995).

101−νm−1Let γm be the intersection of ν2 = −ν11−m and ν2 = νm−1 −ν1 m−2 . Note that, for any integer1
1
m ≥ 2, γm+1 < γm . Moreover, if ν1 ≤ γN for some N,−ν11−(m+1) <1 − ν1(m+1)−1< −ν11−m <(m+1)−2

ν1(m+1)−1 − ν1

1 − ν1m−1
ν1m−1 − ν1m−2

for any m < N. Therefore, the family of sets Pm is pairwise disjoint. Fig.2 depicts theP2 , P3 , P4 , and P5 on the (ν1 , ν2 )-plane.
<<insert Fig.2 around here>>∪
Next, it is reasonable that we examine the case wherein (ν1 , ν2 ) ∈ D\ m≥2 Pm B Σ.
That is, if (ν1 , ν2 ) ∈ Σ, map Fν1 ,ν2 has a chaotic attractor in the sense that there exist no
periodic attractors, and, for any x, the trajectory of x has(a positive
Lyapunov exponent.
)
0 0Hereafter, we call set Σ the chaotic region. Moreover, if v1 , v2 ∈ Σ, we call such a pairthe chaotic pair.
()
We discuss the chaotic region for more details. Let v01 , v02 be any given chaotic pair.
( )Define L ν10 for the intersection of the vertical line and chaotic region, that is,}( ) {L ν10 = (ν1 , ν2 ) ∈ Σ : ν1 = ν10 .
( )()
Let m be the integer for which the “top” of the component of L ν10 that contains v01 , v02( )( )is the boundary of set Pm . Moreover, let Cm ν10 be this component of L ν10 . It is clear( )that there are two cases wherein Cm ν10 is bounded or unbounded. Fig.3 depicts someexamples of set Cm on the (ν1 , ν2 )-plane.
<<insert Fig.3 around here>>
As we
the
( will) see later, in our model, we only have to consider( )( case) wherein m = 2000
and Cm ν1 is unbound. Therefore, we simply define Cm ν1 as C ν1 . Considering this,we state the existence of the chaotic orbit in map (18) as follows.
Theorem 2. (Nusse and Yorke, 1995, with some modifications1 )Let (ν1 , ν2 ) ∈ C be a chaotic pair and assume that (ν1 , ν2 ) ∈ C(ν1 ). There exist (ν1 , ν̄2 ) ∈C(ν1 ) and (ν1 , ν̂2 ) ∈ C(ν1 ), for which ν̄ < ν̂ < −v−1
1 such that the following hold.
k+1
1. If ν̂ < ν2 < −v−1
-piece1 , map F ν1 ,ν2 has a 2( √chaotic) attractor for some k ≥ 1 (depend1
ing on ν1 and ν2 ). In addition, if ν12 < 2 5 − 1 , then k = 1.
1

( )Here, we only consider the case wherein m = 2 and Cm ν10 is unbounded.

11
2. If ν̄ < ν2 < ν̂, then Fν1 ,ν2 has a two-piece chaotic attractor.
3. If ν2 < ν̄, then Fν1 ,ν2 has an one-piece chaotic attractor.
Proof. See Nusse and Yorke (1995).

We have seen the periodic and chaotic dynamics generated from border collision bifurcation on the piecewise linear map (18). The following theorem translates the above results to general border collision bifurcation results for piecewise, smooth, one-dimensional
nonlinear maps.
Theorem 3. (Nusse and Yorke, 1995)
Let G : R × J → R be a piecewise smooth map. Assume that G has the isolated crossing
fixed-point property at (x0 , µ0 ). Let m ≥ 2 and k ≥ 2 be any integers. Define ν1 =lim x↑x0 ∂G(x, µ0 ) and ν2 = lim x↓x0 ∂G(x, µ0 ). Then,∂x∂x(i) if (ν1 , ν2 ) ∈ Pm , G exhibits border-collision bifurcation from a fixed-point attractor toa period-m attractor at (x, µ) = (x0 , µ0 ).
{}(ii) if (ν1 , ν2 ) ∈ Σk , where Σk = Int (ν1 , ν2 ) ∈ Σ : Fν1 ,ν2 has a k-piece chaotic attractor ,G exhibits a border-collision bifurcation from a fixed point attractor to a k-piecechaotic attractor at (x, µ) = (x0 , µ0 ).
Proof. See Nusse and Yorke (1995).

The property called the isolated crossing fixed-point property may be a somewhat unfamiliar notion; however, it is not an unusual property. Indeed, our model has this property
and can be adopted to Theorem 3.
()
Proposition 4. Map (17) has the isolated crossing fixed-point property at (y, β) = − 1a , β∗ (a, a) .
Proof. See Appendix.

4.2



Border collision bifurcation on the OLG model.

In this section, we detect the periodic and chaotic dynamics generated from the bordercollision bifurcation in our model.
We now consider the following unimodal map:{h1 (yt ) = ayt + log β(1 − a)yt ≤ − a1 ,yt+1 = h(yt ) =(20)
1h2 (yt ) = log(1 + yt ) + log β − 1 − 1a < yt ≤ − b1 .

12
As we have (seen in Proposition
4, map (20) has the isolated crossing fixed-point prop)1∗
erty at (y, β) = − a , β (a, a) , and therefore we can adopt Theorem 3 into the model. Now,
we have the following:∂h(x, β∗ (a, a)) = a andy↑− 1a ∂y

(21)

a2
∂h
(x, β∗ (a, a)) = −.
1−ay↓− a1 ∂y(22)

ν1 = lim

ν2 = lim

It should be noted that, in our model, we cannot take ν1 and ν2 independently. Fig.4
depicts line
ν2(23)
ν2 = − 1 ,1 − ν1on the (ν1 , ν2 )-plane. We have only to examine the parameter set of (ν1 , ν2 ) on this line.
In the figure, the left white circle represents intersections of two lines, ν2 = −1 and (23).
On the other hand, the right white circle represents intersections of two lines, ν2 = − ν11and (23). The ν1 coordinate of the left and the right whitecircles are given by ν∗ and ν∗∗ ,√respectively. Subsequently, we easily see that ν∗ = −1+2 5 ≈ 0.618, and ν∗∗ ≈ 0.682 which
is a root of the equation ν13 + ν1 − 1 = 0 in [0, 1].
<<insert Fig.4 around here>>First, we can immediately state the periodic cycle in our model.
Proposition 5. If a ∈ (ν∗ , ν∗∗ ), our model exhibits border collision bifurcation from anattracting fixed point to a period-two attractor at (y, β) = (−1/a, β∗ (a, a)).
We now examine the chaotic dynamics in the model, which is the main focus of thissection. It is easy to see that that our model exhibits the chaotic attractor for a > ν∗∗ .
Moreover, we see that we have to only consider sets C2 (a) for a > ν∗∗ , which are unbounded. According to Theorems 2 and 3, we expect that the model exhibits a 2k+1 -piecechaotic attractor for some k ≥ 1 (depending on ν1 and ν2 ), a two-piece chaotic attractor,
or a one-piece chaotic attractor.
Fortunately, Sushko et al. (2015) comprehensively analyze the border collision bifurcation in piecewise linear map (18). Their results are too technical to refer to all of them
here; however, we need to refer to a few results to investigate the chaotic dynamics in ourmodel. Now, we define the following sets:}{
Q1 = (ν1 , ν2 ) ∈ Σ : ν1 ν22 − ν1 + ν2 < 0 ,13{}
Q2 = (ν1 , ν2 ) ∈ Σ : ν1 ν22 − ν1 + ν2 > 0, ν12 ν23 + ν1 − ν2 < 0 ,}{Q4 = (ν1 , ν2 ) ∈ Σ : ν12 ν23 + ν1 − ν2 > 0, ν16 ν26 − ν1 + ν2 < 0 .

andSubsequently, we have the following.
Theorem 4. For k = 1, 2, and 4, if (ν1 , ν2 ) ∈ Qk , map (18) has the k-piece chaotic
attractor.
Proof. See Sushko et al. (2015).



Fig.5 depicts the sets Q1 , Q2 , and Q4 on the (ν1 , ν2 )-plane. Black circles represent theν2intersections of line ν2 = − 1−ν1 1 and the boundaries of Qk (k = 1, 2, 4). The ν1 coordinateof these intersections, represented by σ∗ and σ∗∗ , are as follows:• σ∗ ≈ 0.705 is a root of the equation ν18 − ν13 + 2ν12 − ν1 = 0 in [0, 1].
• σ∗∗ ≈ 0.724 is a root of the equation ν15 + ν12 − ν1 = 0 in [0, 1].
<<insert Fig.5 around here>>Now, we can completely characterize the chaotic dynamics generated from border collision bifurcation in our OLG model.
Proposition 6. For map (20), we have the following three scenarios of bifurcation fromthe attracting fixed point to a chaotic attractor.
1. For a ∈ (ν∗∗ , σ∗ ), our model exhibits border collision bifurcation from the attracting
fixed point to a four-piece chaotic attractor at (y, β) = (−1/a, β∗ (a, a)).
2. For a ∈ (σ∗ , σ∗∗ ), our model exhibits border collision bifurcation from the attracting
fixed point to a two-piece chaotic attractor at (y, β) = (−1/a, β∗ (a, a)).
3. For a ∈ (σ∗∗ , 1), our model exhibits border collision bifurcation from the attracting
fixed point to an one-piece chaotic attractor at (y, β) = (−1/a, β∗ (a, a)).
On a k-piece chaotic attractor, the trajectory goes through each interval every k-th
period. Of course, when a trajectory returns to the same interval, it never takes the samevalue. Interestingly, when observing such a trajectory, one may regard the trajectory as ak-cycle with random noises. Fig.6 (a) depicts the bifurcation diagram of map (20) respect
to β, where the model has the capability to exhibit four-piece chaotic attractor. Fig.6 (b)depicts time series obtained from map (20) with four-piece chaotic attractor.

14<<insert Fig.6 around here>>In this section, we observed the chaotic dynamics where β ∈ (β∗ (a, a), β∗ (a, a) + ϵ) and,eventually, β ∈ (β∗ (b, b) − ϵ, β∗ (b, b)). In other words, we have investigated the beginningand end of the chaotic dynamics in the OLG model with technology choice. In the next
section, we examine the case wherein the β is not suﬃciently near β∗ (a, a) or β∗ (b, b).

5

Chaotic dynamics : Markov property

In this section, we show that our model has the Markov property for some sets of parameters, wherein the model can reduce to a bimodal map on the unit interval. If the map hassuch a property, we can easily reproduce chaotic dynamics. In what follows, we assume
a−b
1−b
> log
ab
1−a

and

β∗ (b, a) < β < β∗ (a, b),

where the model can be reduce to a bimodal map on the unit interval.
To simplify the analysis of the dynamics, we first use a variable change to reduce ourmodel to the bimodal map in the unit interval.
By defining a new variable xt as
xt =yt + 1 − log β(1 − b),log(1 − a) − log(1 − b)

Eq.(16) can be transformed into the following piecewise smooth map:f (x ) = 1 + a(xt − θL ),xt ≤ θ L ,


 L t
ϕ(xt ),
θ L < xt ≤ θR ,xt+1 = f (xt ) =  f (x ) = b(x − θ ),
θ < x,
R

t

t

R

R

t

where
[

g(xt )(1 − b)(g(xt ) − 1)
(1 − a)λ = log,(1 − b)g(xt ) = λxt + log β(1 − b),ϕ(xt ) = log

]1
λ

,

and
(1 − a) + a log β(1 − b),λa
(1 − b) + b log β(1 − b)
.
= −
λb

θL = −
θR

with

(24)

15
Now, we briefly discuss some property of map (24). It is easily shown thatdϕ(x)a2(θL ) = −
and
dx
1−a
dϕ(x)
b2
(θR ) = −.
dx
1−bMoreover, we have the following.
Proposition 7. Map ϕ(x) is a monotonically decreasing and concave function on (θL , θR ).


Proof. See Appendix.
2

2

a
b
Therefore, the derivatives of ϕ have lower bound − 1−b
and upper bound − 1−afor x ∈
[θL , θR ].

5.1

Mathematical definition of the Markov propertyFirst, we give some mathematical definitions of the Markov property. See, for example,
Boyarsky and Góra (1997) for more details. Let I = [0, 1] and ψ : I → I be a map on I
onto itself. Let P be a finite partition of I and Ik (k = 1, 2, . . . , n) be subintervals of thepartition. Moreover, let ψk be the restriction of ψ to Ik . Now, we can define the following.
Definition 1. If each ψk is a homeomorphism from Ik onto some connected union of subintervals Im , then ψ is said to be Markov. Partition P is said to be a Markov partition withrespect to ψ.
To examine complex dynamics in the piecewise smooth map, we also need to discuss
the following property. Here, let ψ be a piecewise smooth map, which is not necessarily
Markov, and let ψn be an n-th iterate of map ψ. Then, we define the following.
nDefinition 2. If there is an integer n ≥ 1 such that inf dψdx(x) > 1 on each Ik whenever thederivative exists, then ψ is said to be eventually expanding.
It is readily seen that the eventual expansion implies that the map has no attracting
periodic cycles and exhibits chaotic dynamics with long-run observability.

5.2

Markov partition on the OLG model

We first show that our model (24) can exhibit the following periodic cycle for some set ofparameter values:
0 = f 5 (0) < f 2 (0) = θL < f 4 (0) = θR < f (0) = c < f 3 (0) = 1.

(25)

Fig.7 depicts the graph of return map of xt with the periodic cycle (25). We can show thatsuch a specific cycle implies chaotic dynamics.

16<<insert Fig.7 around here>>
Proposition 8. (Observable chaos on a period-5 Markov partition)
There exist aL and aR with < 0aL < aR < 1. If a ∈ (aL , aR ), there exists b ∈ (a, 1) and
β > 0 such that for (a, b, β), we have a period-5 Markov partition of the unit interval withrespect to f . Moreover, f is eventually expanding and thus exhibits observable chaos.
Proof. To calculate the period-5 cycle given by (25), θL and θR must solve the following
equations,fR (1) = θR and fR ( fL (0)) = θL .
A straightforward calculation shows that
[
]
1
bλ
1−b
β=
exp −
+
and
1−b
1+b
b
(b − a)(ab + 1)(1 + b)
λ=
.
a2 b3

(26)
(27)

First, we check that
[
]
bλ
1−b
1
exp −
+
< β∗ (a, b).
β (b, a) <
1−b
1+b
b
∗

(28)

We see that the inequality
[
]
1
bλ
1−b
β (b, a) <
exp −
+
1−b
1+b
b
∗

λ
holds if and only if 0 < 1+b. Thus, clearly, this inequality holds. Moreover, we see thatthe inequality
[
]
bλ
1
1−b
exp −
+
< β∗ (a, b)
1−b
1+b
b
1−b
1+b
holds if and only if log 1−a
< a−b
. Let
ab b

η1 (b) = log

1−b
1−a

and

η2 (b) =a−b1+b.
abb

It is easily seen that η1 (b) is a monotonically decreasing and concave function in [0, 1].
Moreover, we have
1+adη2 (a)= − 3 < 0 anddbad2 η2 (a) 2(a + 2)
=
> 0.
db2
a4

(29)
(30)

17
Therefore, there exists ϵ > 0 such that for any a ∈ (0, 1), η2 (b) is a monotonically decreasing and convex function for b ∈ (a − ϵ, a + ϵ). Note that η1 (a) = η2 (a) anddη1 (a)
1
=−
db
1−a

and

dη2 (a)1+a=− 3 .
dbaConsidering the concavity of η1 and convexity of η2 for neighborhood a, if−

1
1+a< − 3 ⇒ a3 + a2 − 1 > 0,
1−a
a

1−b
1+b
there exists a < ā such that log 1−a< a−b
for b ∈ (a, ā). Let aL ≈ 0.755 be a root of
ab b32
the equation a + a − 1 = 0 in (0, 1). We conclude that if a ∈ (aL , ā), then inequality (28)holds.
Next, we examine Eq.(27). Let

φ1 (b) = log

1−a
1−b

and

φ2 (b) =(b − a)(ab + 1)(1 + b).
a2 b3

Clearly, φ1 (a) = φ2 (a), and we can easily see that φ1 (b) is a monotonically increasing
convex function and limb=1 φ1 (b) = ∞. Moreover, we have2(1 − a2 )is bounded for a ∈ (0, 1) anda2dφ2 (a) a3 + a2 + a + 1
=
> 0.
db
a5

φ2 (1) =Therefore, if
1
a3 + a2 + a + 1
dφ1 (a)=<⇒ a5 + a4 − 1 < 0,
db
1−a
a5
φ1 (b) intersects with φ2 (b) at least one point b ∈ (a, 1). Let â ≈ 0.857 be a root of theequation a5 + a4 − 1 = 0 in (0, 1). We conclude that if a < â, there exists b ∈ (a, 1) suchthat, for a and b, Eq.(27) holds.
Let aR = min {ā, â}. Now, we can state that if a ∈ (aL , aR ), there exists b ∈ (a, 1) and
β > 0 such that for (a, b, β), we have a period-5 Markov partition of the unit interval withrespect to f .
Finally, we show that f is eventually expanding. Let I1 = [0, θL ], I2 = [θL , θR ], I3 =∪[θR , c], I4 = [c, 0], and I¯k be an interior of Ik . Note that every point x ∈ kk=1 I¯k will visit I2∪
at least once every fourth iteration. Therefore, for x ∈ kk=1 I¯k ,a5( f ) (x) ≥ a |ϕ | ≥ −
,
1−a
4 ′

3

′

18
and ifa5−> 1 ⇒ a5 + a − 1 > 0,
1−a

(31)

map f is eventually expanding. Since we can easily check whether aL < a, inequality(31) holds, which completes the proof.
Proposition 8 can be immediately extended to a more general case of a period-(2n + 3)
Markov partition for n ≥ 1.
Proposition 9. (Observable chaos on a period-(2n + 3) Markov partition)
For any n ∈ N, there exist a2n+3,L and a2n+3,R with 0 < a2n+3,L < a2n+3,R < 1. If a ∈(a2n+3,L , a2n+3,R ), there exists b ∈ (a, 1) and β > 0 such that for (a, b, β), we have a period(2n + 3) Markov partition of the unit interval with respect to f . Moreover, f is eventually
expanding and, hence, chaotic.
Proof. See Appendix.

6Concluding remarks

Based on the model studied by Umezuki and Yokoo (2019), we develop a piecewisesmooth OLG model. By assuming a continuous choice instead of discrete choice of thetechnologies, the model has the capability to exhibit several dynamic growth patterns,
including chaotic dynamics, which cannot be observed virtually for the model in Umezuki
and Yokoo (2019). Moreover, in spite of its generality, the model is tractable enough toanalytically investigate such complex dynamics in depth.
First, by using the border collision bifurcation theory, we investigated the routes froma globally attracting steady state to other non-stationary behaviors. Then, we completely
characterized the chaotic dynamics around the bifurcation point; that is, we detected fourpiece, two-piece, and one-piece chaotic attractors. Second, we calculated the parametervalues for which the model has the Markov property. Then, we characterized the chaotic
dynamics, which are diﬀerent from those in the case of border collision bifurcation. However, one might argue that sets of the parameter value, for which the map is Markov, are
too restricted. For future research, it is a natural task to check the robustness of the chaotic
dynamics by characterizing the dynamics near the parameter values for which the map is
Markov.
Finally, it should be pointed out that, compared with the output elasticity for capital(or labor) estimated in the real world, parameter values a and b, for which the model can
exhibit chaotic dynamics, are relatively high. Although the intuition behind this remains

19unclear, from the technical view point, it could be explained by considering an inverse
problem as follows. In this study, we assume that the total factor productivity of thetechnology, represented by A, is constant for all technologies. However, in general, we
can consider diﬀerent total factor productivity for each technology. In that case, the model
has more flexibility, and various new problems might appear, including the following. “A
set of which type of technologies is needed to reduce the model to a continuous piecewise
linear map?” “For a given piecewise smooth map, is there a set of technologies such thatthe model can be reduced to the given piecewise smooth map?” “If so, what features does
the set of technologies have?” In the future, we hope that our model is investigated moredeeply with additional settings without losing its tractability and generality.

Appendix A

Proof of Proposition 4

Proof. For a general statement of the isolated crossing fixed-point property,( see Nusse and)Yorke (1995). Map (20) has the isolated crossing fixed-point property at − 1a , β∗ (a, a) if
the following conditions hold.
}{()
1. There exists − a1 , β : β ∈ J0 , J0 is an open interval containing β∗ (a, a) , and for any()
β ∈ J0 , map (20) is not diﬀerentiable at the point − a1 , β .
2. The{(set {(x(β),β) : }h(x(β)) = x(β), β (∈ J0 } )is a continuous curve and intersects curve)
1− a , β : β ∈ J0 transversally at − 1a , β .
3. For β ∈ J0 , dh(y)> 0 for y < − 1a and dh(y)< 0 for y > − 1a . Moreover, the left limitdydylimy↑− a1 dh(y)and right limit limy↓− 1a dh(y)exist.
dydy( )dh(− 1 )4. h − 1a is a smooth function of β ∈ J0 , and for β = β∗ (a, a), dβ a > 0.
It is obvious that map (20) satisfies these conditions.

Appendix B

Proof of Proposition 7

Proof. We easily show that
1
dϕ(x)
=−
.
dx
g(x)[g(x) − 1]b
Since λ > 0 and g(θR ) = − 1−b< 0, g(x) < 0 for [θL , θR ]. Therefore, we have

dϕ(x)
< 0.
dx



20
Moreover, we have[]
λ 2g(x) − 1
d2 ϕ(x)=< 0,
dx2
g(x)2 [g(x) − 1]2
for x ∈ [θL , θR ]. This completes the proof.

Appendix CProof of Proposition 9

Proof. First, we show that there exists the following 2n + 3 cycle:
0 = f 2n+3 (0) < f n+1 (0) = θL < f 2n+2 (0) = θR < · · · < f n+3 (0) < f (0) < 1 = f n+2 (0).
(C.1)
To calculate the cycle given by (C.1), θL and θR must solve the following equations:fRn (1) = θR

andfRn ( fL (0)) = θL .

A straightforward but tedious calculation shows that
[ n
]
1
bλ
1−b
β=
exp − n k +and
1−bΣk=0 bb(b − a)(1 + abn )Σnk=0 bk
λ=.
a2 b2n+1

(C.2)
(C.3)

First, we check that
[ n
]
bλ
1
1−b
exp − n k +< β∗ (a, b).
β (b, a) <1−b
Σk=0 bb∗

We see that the inequality
[ n
]
1
bλ
1−b
β (b, a) <exp − n k +1−bΣk=0 bb∗holds if and only if

(
0<λ

bn
Σnk=0 bk).

Clearly, this inequality holds. Moreover, we also see that the inequality
[ n
]
1
bλ
1−b
exp − n k +< β∗ (a, b)1−bΣk=0 b
b
Σn bk

1−b
k=0
holds if and only if log 1−a
< a−b
. Let
ab
bn

η1 (b) = log

1−b
1−a

and

η̂2 (b) =a − b Σnk=0 bk
.
ab
bn

(C.4)

21
It is easily seen that η1 (b) is a monotonically decreasing and concave function in [0, 1].
Moreover, we haveΣn akdη̂2 (a)= − k=0< 0 anddb
an+2
d2 η̂2 (a) (1 + n)(1 + an+1 ) + 2Σnk=1 (n − k + 1)ak
=
> 0.
db2
an+3

(C.5)
(C.6)

Therefore, there exists ϵ > 0 such that for any a ∈ (0, 1), η2 (b) is a monotonically decreasing and convex function for b ∈ (a − ϵ, a + ϵ). Note that η1 (a) = η̂2 (a), anddη1 (a)
1
=−
db
1−a

and

Σn ak
dη̂2 (a)= − k=0.
db
an+2

Considering the concavity of η1 and convexity of η̂2 for neighborhood a, if
−

Σn ak
1
< − k=0⇒ an+2 + an+1 − 1 > 0,1−aan+2
Σn b k

1−b
k=0
< a−b
for b ∈ (a, ā2n+3 ). Let a2n+3,L be athere exists a < ā2n+3 such that log 1−a
ab
bn
n+2
n+1
root of the equation a + a − 1 = 0 in (0, 1). We conclude that if a ∈ (a2n+3,L , ā2n+3 ),inequality (C.4) holds.

Next, we examine Eq.(C.3). Let
φ1 (b) = log

1−a
1−b

and φ̂2 (b) =(b − a)(1 + abn )Σnk=0 bk.
a2 b2n+1

Clearly, φ1 (a) = φ2 (a). We can easily see that φ1 (b) is a monotonically increasing convex
function and limb=1 φ1 (b) = ∞. Moreover, we have(n + 1)(1 − a2 )is bounded for a ∈ (0, 1) anda2kdφ2 (a) Σ2n+1
k=0 a= 2n+3 > 0.
dba

φ2 (1) =Therefore, if

Σ2n+1 ak1
dφ1 (a)=< k=0⇒ a2n+3 + a2n+2 − 1 < 0,
db
1−a
a2n+3
φ1 (b) intersects φ̂2 (b) at least one point b ∈ (a, 1). Let â2n+3 be a root of the equation
a2n+3 + a2n+2 − 1 = 0 in (0, 1). We conclude that if a < â2n+3 , there exists b ∈ (a, 1) suchthat, for a and b, Eq.(27) holds. Let a2n+3,R = min {ā2n+3 , â2n+3 }. Now, we can state that
if a ∈ (a2n+3,L , a2n+3,R ), there exists b ∈ (a, 1) and β > 0 such that for (a, b, β), we have aperiod-2n + 3 Markov partition of the unit interval with respect to f .

22Finally, we show that f is eventually expanding. Let Ik (k = 1, 2, . . . n − 1) be subintervals of the period-2n + 3 Markov partition and I ∗ = [θL , θR ]. Moreover, let I¯k be an interior∪of Ik . A minor consideration shows that every point x ∈ kk=1 I¯k will visit I ∗ at least once∪
every 2n + 2th iteration. Therefore, for x ∈ kk=1 I¯k ,( f 2n+2 )′ (x) ≥ a2n+1 |ϕ′ | ≥ −and if
−

a2n+3
,
1−a

a2n+3
> 1 ⇒ a2n+3 + a − 1 > 0,
1−a

(C.7)

map f is eventually expanding. We can easily see that if a2n+3,L < a, inequality (C.7)
holds. This completes the proof.


References[1] Asano, T., Kunieda, T., and Shibata, A., 2012. Complex Behaviour in a PiecewiseLinear Dynamic Macroeconomic Model with Endogenous Discontinuity. Journal ofDiﬀerence Equations and Applications 18, 1889-1898.
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[3] Benhabib, J., Day, R.H., 1982. A Characterization of Erratic Dynamics in the Overlapping Generations Model. Journal of Economic Dynamics and Control 4, 37-55.
[4] Boyarsky, A., Góra, P., 1997. Laws of Chaos: Invariant Measures and Dynamical
Systems in One Dimension, Springer.
[5] Gardini, L., Sushko, I., and Naimzada A. K., 2008. Growing through Chaotic Intervals. Journal of Economic Theory 143, 541-557.
[6] Grandmont, J., 1985. On Endogenous Competitive Business Cycles. Econometrica
53, 995-1045.
[7] Hata, M., 2014. Neurons: A Mathematical Ignition. Singapore: World Scientific
Publishing.
[8] Ishida, J., Yokoo, M., 2004. Threshold Nolinearities and Asymmetric Endogenous
Business Cycles. Journal of Economic Behavior and Organization 54, 175-189.

23[9] Iwaisako, T., 2002. Technology Choice and Patterns of Growth in an Overlapping
Generations Model. Journal of Macroeconomics 24, 211-231.
[10] Matsuyama, K., 2007. Credit Traps and Credit Cycles. American Economic Review
97, 503-516.
[11] Matsuyama, K., 2013. The Good, the Bad, and the Ugly: An Inquiry into the Causes
and Nature of Credit Cycles. Theoretical Economics 8, 623-651.
[12] Matsuyama, K., Sushko, I., and Gardini, I., 2016. Revisiting the Model of Credit
Cycles with Good and Bad Projects. Journal of Economic Theory 163, 525-556.
[13] Nagumo, J., Sato, S., 1972. On a Response Characteristic of a Mathematical Neuron
Model. Kybernetik 10, 155-164.
[14] Nusse, H. E., Yorke, J. A., 1995. Border-Collision Bifurcation for Piecewise Smooth
One-Dimentional Maps. Journal of Bifurcation and Chaos 5, No.1, 189-207.
[15] Sushko, I., Avrutin, V., and Gardini, L., 2015. Bifurcation Structure in the Skew Tent
Map and Its Application As a Border Collision Normal Form, Journal of Diﬀerence
Equations and Applications 21, 1-48.
[16] Umezuki, Y., Yokoo, M., 2019. A Simple Model of Growth Cycles with Technology
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24

-0.6

-0.8

h(y)

-1

-1.2

-1.4

-1.6
-1.6

-1.4

-1.2

-1

-0.8

-0.6

y
(
)
Figure 1: Piecewise smooth map (17) which has the fixed point in − 1a , − b1 . a = 0.7, b =0.85 and β = 4.

25

0
P2
-5

ν2

γ2

γ3
γ4
P3

-10

γ5
-15
P4

P5

-20
0

0.2

0.4

0.6

0.8

1

ν1

Figure 2: The sets P2 , P3 , P4 and P5 on the (ν1 , ν2 )-plane. If (ν1 , ν2 ) ∈ Pk , the map (18)
has an attracting period-k cycle.

26

0
P2

ν2

-5

C2(νL)

P3-10

C2(νR)
C3(νL)

-15
P4
-20
0

C4(νL)P5
νL 0.5νR

1

ν1Figure 3: Some examples of the set Cm on the (ν1 , ν2 )-plane. The sets C2 (νL ), C3 (νL ) andC4 (νL ) are bounded, while the set C2 (νR ) is unbounded. Note that L(νR ) = C2 (νR ) andL(νL ) = C2 (νL ) ∪ C3 (νL ) ∪ C4 (νL ) ∪ . . . .

27

0
P2

ν2

-5

P3

-10

-15
P4
-20

P5
∗ ∗∗
0.5 ν ν
ν1

0

ν2

1

Figure 4: The line ν2 = − 1−ν1 1 on the (ν1 ν2 )-plane. For a > ν∗∗ , our model has thecapability to generate chaotic dynamics induced by border collision bifurcation.

28

-1
P2

Q4

Q2

ν2

-1.5

-2

Q1

-2.5
P3

-3
0.5

∗

σ σ

∗∗

1

ν1

Figure 5: Classification of the chaotic region. Depending the value a, our model exhibit
four-piece, two-piece and one-piece chaotic attractor which relate to the region Q4 , Q2
and Q1 , respectively.

29

-1.4
-1.42
-1.44

y

-1.46
-1.48
-1.5
-1.52
-1.54
1.96 1.98

2

2.02 2.04 2.06 2.08 2.1β(a) Bifurcation diagram of map (20) with respect to β..

-1.43
-1.44

y

-1.45
-1.46
-1.47
-1.48
0

5

10 15 20 25 30 35 40 45 50
time(b) Time series obtained from map (20) with four-piece chaotic attractor.

Figure 6: Panel (a) depicts the bifurcation diagram of map (20) respect to β, where a =0.684 ∈ (ν∗∗ , σ∗ ). Panel (b) depicts time series obtained from map (20) with a = 0.684
and β = 2.06.

30

xt+1

1

0
0

θL

θR

c

1

xt

Figure 7: The map (24) with a period-5 Markov parition. a = 0.8, b ≈ 0.902 and β ≈6.507.