Discussion Paper No.237

Chaotic Itinerancy inRegional Business Cycle Synchronization
Kunihiko Esashi
Department of Mathematics, Hokkaido University
Tamotsu Onozaki
Faculty of Economics, Rissho University
Yoshitaka Saiki
Grauate School of Commerce and Management
Hitotsubashi University

October 2014

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

Chaotic Itinerancy in Regional Business Cycle
Synchronization
Kunihiko Esashi ∗Department of Mathematics, Hokkaido University, Kita 10 Nishi 8, Kita-ku, Sapporo, Hokkaido 060-0810, Japan.

Tamotsu Onozaki †Faculty of Economics, Rissho University, 4-2-16 Osaki, Shinagawa, Tokyo 141-8602, Japan

Yoshitaka Saiki
Graduate School of Commerce and Management, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo 186-8601, Japan

Abstract
Chaotic itinerancy is complex behavior in high-dimensional dynamical systems
characterized by itinerant motion among many diﬀerent ordered states throughchaotic transition. In this study, we robustly observe this behavior in a modelof regional business cycles, in which all regions are homogeneous and connected
through producers’ expectations. Although producers adjust price and output expectations quite slowly toward the average level announced by the government,
regional business cycles begin to synchronize from the entrainment eﬀect. Moreover, the economy is more likely to exhibit chaotic itinerancy when the producers
emphasize the expected profit maximization and when they adjust their expectations slowly toward the average.
Key words: Regional business cycle, Nonlinear dynamics, Synchronization, Globally coupled map, Chaotic itinerancy
JEL Classifications: C61, E32∗
Present address: NTT DATA Corporation, Toyosu Center Building, 3-3-3 Toyosu, Koto-ku, Tokyo
135-6033, Japan.
†
Corresponding author. Phone: +81-3-5487-3021. E-mail address: onozaki@ris.ac.jp.

11Introduction

Business cycle synchronization has become a topic of growing interest from around the endof the 20th century (Antonakakis, 2012; Baxter and Kouparitsas, 2005; Yetman, 2011).
The vast empirical literature elucidates that countries with intensified trade linkages haveresemblant business cycles, which Selover and Jensen (1999) and Süssmuth (2003) explain
by proposing a mode-locking model. However, little is known about the business cycle
synchronization across subnational regions within a country, as mentioned by Kouparitsas
and Nakajima (2006). Selover et al. (2005) also apply a mode-locking model to this topic
under a scenario in which the cycles of disparate regions synchronize through interregional
trade linkages. Onozaki et al. (2007) propose another scenario in which regional businesscycles may synchronize through producers’ expectations based on global information via
a system of globally coupled maps (GCM), first proposed by Kaneko (1990).
Related to a GCM model, complex behavior called “chaotic itinerancy” is sometimes
observed. The term describes the phenomenon of an orbit successively itinerating among
many ordered states through chaotic transitions in dynamical systems. The phenomenon
was independently discovered in a model of optical turbulence by Ikeda et al. (1989), a
globally coupled chaotic system by Kaneko (1990), and nonequilibrium neural networks
by Tsuda (1990). The term “chaotic itinerancy” was coined unanimously by its discoverers to denote universal dynamics in a class of high-dimensional dynamical systems(Kaneko and Tsuda, 2000). To the best of our knowledge, only Yasutomi (2003) has modeled chaotic itinerancy in an economic context. In this study, we reexamine the model of
Onozaki et al. (2007) and robustly observe chaotic itinerancy for various constellationsof parameters. This study is organized as follows: Section 2 presents a regional businesscycle model and Section 3 presents the study on chaotic itinerancy by using the model.
Section 4 presents the conclusion of our study.

2

Model

We begin by introducing a GCM model represented as follows:ε ∑f (xt (j)),
N j=1
Nxt+1 (i) = (1 − ε)f (xt (i)) +i = 1, · · · , N,(1)where xt (i) denotes the value of the ith element at discrete time period t, and N thenumber of elements. A map f (x) describes each element’s endogenous dynamics. In
this study, we use a noninvertible map as f (x) that can exhibit chaotic behavior. The
second term on the right-hand side of (1) represents the global interaction of each element
through the “mean field,” i.e., the all-to-all interaction. Therefore, two opposite eﬀects
coexist: the all-to-all interaction is inclined to synchronize all elements, and the chaotic
instability in each element tends to desynchronize them. Depending on the value ofε ∈ (0, 1), i.e., the balance between these two eﬀects, the GCM model exhibits a richvariety of complex phenomena (Kaneko, 1990). The remainder of this section shows thatthe model analyzed in this study takes the same form as (1).
The economy has N regions, each with a single producer and a separate market. Each
producer makes homogeneous goods and delivers them only to the market of its own re2

gion. Consumers are uniformly distributed over all regions and purchase goods from themarket they belong to. Since the business cycles of diﬀerent regions may synchronize
through interregional trade, the model is intended to concentrate on factors other thantrade. Therefore, we assume that there is no interregional trade to eliminate its eﬀects.
Government announces the average price and output of all regions in each period, andeach producer acts based on this information; i.e., regions are linked via global information.
Each producer’s decisions involve two stages as follows. First, each producer dislikes
discrepancies between the actual and average levels of price and output, and intends toresolve them in response to the government’s announced averages. At period t, the ith
producer expects that the actual price pt (i) in its market will∑be adjusted adaptively atperiod t + 1 toward the announced average level p̄t := (1/N ) N
j=1 pt (j) such thatε ∑pt (j),
N j=1
N

pet+1 (i) = (1 − ε)pt (i) +(2)where superscript e denotes expectations and ε ∈ (0, 1) is an expectation adjustment
coeﬃcient common among producers. Simultaneously, the ith producer establishes aprovisional target of the output∑x̂t (i) as a weighted average of the actual output xt (i)and average output x̄t := (1/N ) N
j=1 xt (j) with a weight ε, which equals the coeﬃcientof the price expectation adjustment. Thus, x̂t (i) = (1 − ε)xt (i) + εx̄t (i).
Second, each producer calculates the output level maximizing the expected profitx̃t+1 (i) based on its adaptively expected price (2) and the quadratic cost function C(x) =x2 /2 such that x̃t+1 (i) = pet+1 (i). Ultimately, it sets a final output plan as a weighted
average of the provisional target x̂t (i) and the expected profit maximizing output x̃t+1 (i)with a weight ϕ ∈ (0, 1). The resulting formula for each producer’s output is as follows:
xt+1 (i) = (1 − ϕ) x̂t (i) + ϕ x̃t+1 (i)= (1 − ϕ)(1 − ε) xt (i) + (1 − ϕ) ε x̄t + ϕ x̃t+1 (i).
Since the sum of the above three coeﬃcients on the right-hand side is unity, we can paraphrase the producer’s decision making as follows: Each producer’s output is determined
as a weighted average of xt (i), x̄t (i), and x̃t+1 (i).
The demand in each region is assumed to be identical and described by the same
monotonic inverse demand function as follows:1
pt (i) =,(yt (i))η
where yt (i) is the demand of the ith region at period t, and η > 0 is the inverse of theprice elasticity of the demand. By assuming that in each period, prices are determinedin each market for equilibrating supply and demand, we obtain a GCM model (1) witha noninvertible map as follows:f (xt (i)) = (1 − ϕ)xt (i) +ϕ,(xt (i))η

(3)

the behavior of which is well studied by Onozaki et al. (2000) and known to exhibit
chaotic behavior depending on the set of parameters. The larger ϕ and η are, the more
3

10
8

x

6
4
2
0
0.4

0.6

0.8

1

φ

Fig. 1: Bifurcation diagram of one-dimensional map xt+1 = f (xt ) by changing ϕ.
likely the economy behaves chaotically. After all, the regional business cycle model to beanalyzed in this study consists of (1) and (3).
In what follows, the system dimension N is fixed as 10. Unless otherwise noted, η,
ϕ, and ε are fixed as 3.5, 0.7, and 0.315, respectively 1 . We randomly select initialconditions x0 (i)(i = 1, · · · , 10) from the range [0.5, 1.5]. Note that one-dimensional map
xt+1 = f (xt ) can generate chaotic behavior with respect to ϕ around 0.62 through aperiod-doubling bifurcation, as shown in Fig. 1.

3

Chaotic Itinerancy

This section studies chaotic itinerancy in the regional business cycle model (1) and (3).
Chaotic itinerancy is typical in high-dimensional chaos. In its presence, an orbit wanders
among the diﬀerent states of complexity (Kaneko and Tsuda, 2000). We first show thetime developments of the model in Fig. 2 2 . In coupled systems, separate oscillators
sometimes synchronize, a phenomenon called “entrainment”. A set of synchronizing
oscillators is a cluster. Ten types of regimes (from one-cluster to 10-cluster regime)
appear in the model’s behavior. The one- and 10-cluster regimes are shown in Fig. 3(left, right).
To definitely characterize chaotic itinerancy, let us define the eﬀective dimension andits mean (Komuro, 2005). The eﬀective dimension (ED) of a point x ∈ RN with theprecision δ, denoted by ED(x, δ), is defined as a number of clusters. Here, variables
within a distance δ are considered to belong to the same cluster 3 . We rewrite the model(1) and (3) as a map F : RN → RN defined as follows:(xt+1 (1), · · · , xt+1 (N )) = F (xt (1), · · · , xt (N )).

(4)

Then the mean of the eﬀective dimension (MED) of a point x ∈ RN with the precision δ
1

Onozaki et al. (2007) show that the very long transient behaviors exist when the system dimension
N is fixed as 100 and the parameter η is selected from the range [1.0, 8.0].
2
To obtain this figure, we neglect the first 104 iterations as transient behaviors and use 105 iterations.
3
The value of δ is fixed as 10−4 in this paper.

4

4

x

3
2
1
0
45000

45500
t

46000

4

4

3

3

2

2

x

x

Fig. 2: Time development of xt (i)(i = 1, · · · , 10).

11045000 45005 45010 45015 45020 45025
t045800 45805 45810 45815 45820 45825
tFig. 3: Enlarged view of Fig. 2: Detailed observation of time development of xt (i) (i = 1, · · · , 10) in anorbit (one-cluster regime (left), 10-cluster regime (right)).

5

ED

10
9
8
7
6
5
4
3
2
1
0

5000

10000

15000

t

Fig. 4: Time development of eﬀective dimension (ED) of the economy. It primarily remains within a
one-cluster regime (ED = 1). Once it departs from the one-cluster regime, it wanders among various
regimes of eﬀective dimensions before returning to the former regime.

for duration τ

4is defined as follows:
1∑
ED(F i (x), δ).
τ i=0
τ −1

MED(x, δ, τ ) =Because of entrainment eﬀect, synchronization strengthens and the economy remains
within the one-cluster regime (ED = 1) as a laminar state for some time. Then, theeconomy leaves the regime and wanders among various regimes as a bursting state. This
process is repeated for an extended period (Fig. 4).
In Fig. 5, the distribution of duration time in a regime of ED (= 1) is shown toobey a power-law with exponents estimated to be −3/2. This implies that extended
trapping in such a regime is less infrequent than in cases involving popular exponential
distribution 5 . Conversely, the exponential distribution is confirmed for the duration
time in a regime of ED (= 2, 5, 10) 6 . MEDs of duration τ = 105 are calculated after
the transients of 105 iterations for various εs, as shown in Fig. 6. An integer MED
indicates that the economy remains in the same regime over 105 iterations. A non-integer
MED indicates that the economy wanders among various regimes of diﬀerent eﬀective
dimensions. Chaotic itinerancy is considered to occur in such cases.
As shown in Fig. 7, we calculate MEDs of the economy by changing a set of parameters(ϕ, ε), and identify the parameter region (ϕ, ε) where MEDs are non-integers, indicatingthat the economy wanders among various regimes 7 . Figure 7 shows the robustness ofchaotic itinerancy with respect to large ϕ and small ε. The larger ϕ is and the smaller εis, i.e., when the producers emphasize the profit maximization and when they adjust their
4

The value of τ is fixed as 105 in this paper.
Power-law distributions are observed in a well-known critical phenomenon called “Type I intermittency” in a low-dimensional system, but they are not robust in a parameter space. In contrast, thedistributions in our model are robustly found.
6
Exponential distribution is confirmed for each regime of ED (= 2, · · · , 10).
7The MEDs are calculated from 10 randomly chosen initial conditions x0 (i) (i = 1, · · · , 10) afterneglecting transitions of 103 iterations. If the average is a non-integer, the corresponding set of parameters(ϕ, ε) is plotted in Fig. 6. These calculations are performed for 105 points in a parameter region (ϕ, ε).
5

6

104

104
#

106

#

106

102

100 0
10

102

10

1

10

2

10

3

10

4

10

100 0
10

5

101

102

T

103

104

105

103

104

105

T

104

104
#

106

#

106

102

100 0
10

102

10

1

10

2

10

3

10

4

10

100 0
10

5

101

T

102
T

10
9
8
7
6
5
4
3
2
1

10

10-MED

MED

Fig. 5: Distribution of duration time T for remaining in a regime of eﬀective dimension ED (= 1 (upper
left), 2 (upper right), 5 (lower left), and 10 (lower right)). The distribution of ED (= 1) is shown toobey a power-law and that of ED (= 2, 5, 10) to obey exponential.

1

0.1

0.01
0

0.1

0.2

0.3

0.4

0.5

0

ε

0.02

0.04

0.06

0.08

0.1

ε

Fig. 6: MEDs of duration τ = 105 of the economy with respect to ε. A non-integer MED indicates thatthe economy wanders among various regimes of diﬀerent clusters (EDs) (left). The graph of 10−MED
for small values of ε (right: logarithmic scale). The value of 10−MED for small ε is found not to be 0,
implying that the orbit wanders among various regimes.

7

1

10

-1

0.8
10-4
ε

ε

0.6
0.4

10-7

0.2
0
0.4

0.5

0.6

0.7
φ

0.8

0.9

10

1

-10

0.6

0.7

0.8
φ

0.9

1

Fig. 7: Parameter regions (ϕ, ε) where the economy wanders among various regimes in τ = 105 iterations
from 10 initial conditions after the transients of 105 iterations (η is still fixed as 3.5.) (left). Enlarged
view of the left figure (right). The economy for various parameters (ϕ, ε), including the small ε region,shows chaotic itinerancy.

expectations slowly toward the average announced by the government, the more likely
the economy will exhibit chaotic itinerancy. By comparing the bifurcation diagram of asingle map (Fig. 1) with Fig. 7, it is apparent that the model shows chaotic itinerancy ofcomplex behaviors with multiple regimes around bifurcation points and chaotic regionsof a single map. This phenomena is closely related to the structural stability (Robinson,2013): The single map (3) is nonhyperbolic or less hyperbolic in these parameter regions,and thus the GCM (1) can easily destroy the respective dynamics of ED (= 10) andshow chaotic itinerancy by the extremely small perturbation described by the coupling
parameter ε.

4Conclusion

In this study, we have robustly observed chaotic itinerancy in a model of regional business cycles coupled through producers’ expectations derived from global information.
The economy wanders among various regimes featuring diﬀerent numbers of clusters forparticular constellations of parameters. Only a small coupling eﬀect is required for thisphenomenon to occur. This implies that although all regions or agents are economicallyhomogeneous, the situation should not compel attention to a “representative” region or
agent; all must be considered simultaneously. Furthermore, in this study, we have shownthat when the producers emphasize the profit maximization and when they adjust their
expectations slowly toward the average level announced by the government, the more
likely the economy will exhibit chaotic itinerancy.
The dependency of chaotic itinerancy on the system dimension remains an important
issue for future work. Especially, the relation between chaotic itinerancy and on-oﬀ
intermittency, sometimes observed in two-dimensional coupled systems, appears to beinteresting for understanding the mechanism of chaotic itinerancy.

8

Acknowledgements
The authors would like to thank Professors Ichiro Tsuda and Yuzuru Sato for theirhelpful information about chaotic itinerancy. This work was partly supported by JSPS
KAKENHI (Grant Number 24653050, 23330086, 23740065) and by the Collaborative
Research Program for Young Scientists of ACCMS and IIMC, Kyoto University.

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