Intermittent transition between synchronization and desynchronization in multi-regional business cycles

Structural Change and Economic Dynamics 44 (2018) 68–76

Contents lists available at ScienceDirect

Structural Change and Economic Dynamics journal homepage: www.elsevier.com/locate/sced

Intermittent transition between synchronization and desynchronization in multi-regional business cycles Kunihiko Esashi a,1 , Tamotsu Onozaki b,∗ , Yoshitaka Saiki c,d,e , Yuzuru Sato f,g a

Department of Mathematics, Hokkaido University, N10W8, Kita, Sapporo, Hokkaido 060-0810, Japan Faculty of Economics, Rissho University, 4-2-16 Osaki, Shinagawa, Tokyo 141-8602, Japan Graduate School of Commerce and Management, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo 186-8601, Japan d JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan e Institute for Physical Science and Technology, University of Maryland, College Park, 20742 MD, USA f Research Institute for Electronic Science / Department of Mathematics, Hokkaido University, N20W10, Kita-ku, Sapporo, Hokkaido 010-0020, Japan g London Mathematical Laboratory, 14 Buckingham Street, London WC2N 6DF, UK b c

a r t i c l e

i n f o

Article history: Received 4 August 2016 Received in revised form 8 August 2017 Accepted 28 October 2017 Available online 5 November 2017 JEL classification: C61 E32 Keywords: Regional business cycle Nonlinear dynamics Synchronization Globally coupled map Chaotic itinerancy Intermittency Unstable dimension variability

a b s t r a c t Empirical studies often conclude that multi-regional business cycles exhibit intermittent transition between synchronization and desynchronization of each regional fluctuations. In this study, we robustly observe this behavior (called chaotic itinerancy) in a model of multi-regional business cycles, in which all regions of a national economy are homogeneous and connected through producers’ behavior based on the average level announced by the government. Although a producer very slowly adjusts his/her output towards the average level, regional business cycles begin to synchronize because of the entrainment effect. Moreover, when a producer emphasizes the profit maximization more and when puts more weight on the average level in his/her decision-making, the economy is more likely to exhibit such intermittent transition. Further, it is clarified that behind intermittent transition exist cycles among periodic orbits with different number of unstable directions. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Business cycle synchronization has become a topic of growing interest from around the end of the twentieth century (Artis and Zhang, 1999; Imbs, 1999; Selover and Jensen, 1999; Dueker and Wesche, 2003; Süssmuth, 2003, Chap. 5; Baxter and Kouparitsas, 2005; Crowley and Lee, 2005; Savva et al., 2010; Benˇcík, 2011; Yetman, 2011; Antonakakis, 2012).2 The vast empirical literature elucidates that countries with intensified trade linkages have resemblant business cycles, which Selover and Jensen (1999) and Süssmuth (2003) explain by proposing a nonlinear mode-

∗ Corresponding author. E-mail address: [email protected] (T. Onozaki). 1 Present address: NTT DATA Corporation, Toyosu Center Building, 3-3-3 Toyosu, Koto-ku, Tokyo 135-6033, Japan. 2 Synchronization has been paid much attention also in finance after starting the twenty first century. For a survey of related literature, see Huang and Chen (2014). https://doi.org/10.1016/j.strueco.2017.10.005 0954-349X/© 2017 Elsevier B.V. All rights reserved.

locking model. Mode-locking is an inherently nonlinear linkage phenomenon; cycles of different elements are synchronized, that is, attain mode-lock, when the strength of the linkage between oscillating elements reaches a certain threshold. On the other hand, with the exception of a stylized fact that fluctuations in different regions of a national economy are inclined to synchronize with each other (Rissman, 1999; Carlino and Sill, 2001; Clark and van Wincoop, 2001; Kouparitsas and Nakajima, 2006), little is known about the business cycle synchronization across subnational regions within a country, as mentioned by Kouparitsas and Nakajima (2006). One possible hypothesis about subregional synchronization is that it may occur because of common exogenous shocks such as national fiscal and monetary policies, sudden changes in world commodity prices, trends among consumers, and so on. However, several studies find that common shocks do not seem to be the cause of such synchronization (Carlino and DeFina, 1995; Kozlowski, 1995). A different hypothesis is that the synchronization may be caused by trade linkages between different regions

K. Esashi et al. / Structural Change and Economic Dynamics 44 (2018) 68–76

of the economy. Another nonlinear mode-locking model is proposed by Selover et al. (2005) under a scenario in which the cycles of different regions synchronize through interregional trade linkages. In this manner, mode-locking models have been proposed with regard to business cycle synchronization across countries or subnational regions. However, as Süssmuth (2003) properly pointed out, “the idea of ‘mode-locking’ . . . to explain the synchronization of national business cycles . . . is not as “new” as Selover and Jensen might have had in mind at the time of their contribution” (p. 70). In fact, we can cite instances of preceding studies, such as Mosekilde et al. (1992,1993), Larsen et al. (1993), Sterman and Mosekilde (1994), Haxholdt et al. (1995), Krugman (1996), and Brenner et al. (1998) although different appellations such as modelocking, phase-locking, and entrainment are used.. Recent empirical studies often conclude that the regional business cycle exhibits intermittent transition between synchronization and desynchronization of each regional fluctuations (Crowley and Lee, 2005; Savva et al., 2010; Aguiar-Conraria and Soares, 2011; Benˇcík, 2011; Yetman, 2011; Hanus and Vacha, 2016). The trade linkage hypothesis seems to fail in explaining this fact because trades among countries or subnational regions are supposed to depend mainly upon continuous demand. The main focus of the present paper is to explain the fact by the aid of the model presented by Onozaki et al. (2007). Let us outline their model in the rest of this section. Since the discovery of synchronization of coupled pendulum clocks by Huygens in the seventeenth century, it has been wellknown that multiple oscillators may synchronize if they directly interact with each other. In this sense, a scenario of regional business cycle synchronization through interregional trade linkages is probable and realistic but rather straightforward and obvious. In contrast, another mechanism of regional business cycle synchronization than the direct interaction lies in an economic system: if the government announces the average price and production for all regions at each period and if each regional production is determined based on the preceding information announced by the government, each regional production intertemporally affects all regional productions through information on the average. Thus, there may exist the global, or the all-to-all interaction in an economic system. This mechanism is modeled by Onozaki et al. (2007) by utilizing a system of globally coupled maps (GCM). They use nonlinear maps which may behave chaotically depending on parameter values, and illustrate how synchronization occurs and how complex its process is. It would be better to give a short account of the GCM model here, which is first proposed by Kaneko (1990).3 It is represented as follows: ε f (xj (t)), i = 1, . . ., N, N N

xi (t + 1) = (1 − ε)f (xi (t)) +

(1)

j=1

where xi (t) denotes the value of the ith element at discrete time period t, and N the number of elements. A map f(x) describes each element’s endogenous dynamics. Usually a noninvertible map is utilized as f(x) that may exhibit chaotic behavior. The second term on the right-hand side of (1) represents the global interaction of each element through the mean field, that is, a uniform, all-to-all interaction. Therefore, two opposite effects coexist: the all-to-all interaction is inclined to synchronize all elements and the

3 Coupled map lattices (CML), a framework similar to the GCM model, is also first proposed by Kaneko (1992). The main difference between GCM and CML is that the former includes the mechanism of global interaction of all elements and the latter does not include the mechanism of global interaction but that of local interaction. For an application of CML system in finance, see Huang and Chen (2015).

69

chaotic instability in each element tends to desynchronize them. Depending on the value of the coupling parameter ε ∈ (0, 1), that is, the balance between these two effects, the GCM model exhibits a rich variety of complex phenomena including chaotic itinerancy (Kaneko, 1990),4 which we can regard as the cause of intermittent transition between the synchronization and desynchronization mentioned above. The term chaotic itinerancy 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 terminology was coined by its discoverers to denote universal dynamics in a class of high-dimensional dynamical systems (Kaneko and Tsuda, 2000). In an economic context, Yasutomi (2003) studies the emergence and collapse of money in a computer simulation model from the viewpoint of chaotic itinerancy. A supplementary remark should be made on synchronization of chaotic systems. It seems that two separate chaotic systems, even though they are identical, cannot synchronize because of an important property of chaotic systems, that is, the sensitive dependence on initial conditions. However, if a small coupling is introduced, chaotic trajectories of those systems tend to synchronize. Two opposite effects coexist in the same way as in GCM model: the coupling effect leads two elements to synchronize and the chaotic instability in each element leads them to desynchronize. Therefore, a persistent aperiodic switching between synchronized and desynchronized states, called on-off intermittency, may occur in coupled two chaotic systems. The phenomenon was found by Fujisaka and Yamada (1983) and Yamada and Fujisaka (1983), and is sometimes considered as a special case of chaotic itinerancy when N = 2. Bischi et al. (1998), Bischi and Gardini (2000), and Huang and Chen (2014) study synchronization of chaotic oscillators in two-dimensional economic models, all of which exhibit on-off intermittency. In this study, we use and reexamine the model of Onozaki et al. (2007) and enhance the worth of the model by robustly observing chaotic itinerancy, which they did not observe, for various constellations of parameters as well as various complex phases of regional business cycle synchronization. The remainder of the paper is organized as follows. Section 2 describes a regional business cycle model. Section 3 discusses chaotic itinerancy occurring in the model. Section 4 characterizes chaotic itinerancy from a mathematical point of view. The last section concludes the paper.

2. Model Although the model to be analyzed in the present paper was already proposed by Onozaki et al. (2007), we repropose it in this section in a slightly different way for the convenience of the reader. The economy consists of N regions, each with a separate market that is imperfectly competitive. There are n homogeneous producers in each region, and each producer produces homogeneous goods and delivers them only to the market of its own region. Goods are perishable and cannot be carried over to the next period. Consumers are uniformly distributed over all regions and purchase goods from the market they belong to. We assume that there is no interregional trade. Obviously, this assumption is unrealistic, but our main objective is demonstrating that business cycles in different regions may synchronize through producers’ behavior based on information announced by the government even if there is no

4 It is sometimes discussed from the viewpoint of Milnor attractor (Milnor, 1985; Kaneko, 2002).

70

K. Esashi et al. / Structural Change and Economic Dynamics 44 (2018) 68–76

interregional trade.5 The government announces the average price and output for all regions at each period. Each producer does not know prices and productions of other regions but only knows those of his/her own region and therefore considers the announced information when adjusting his/her production level. Thus, all regions are linked via information announced by the government. The producer’s decision-making process consists of the following three steps. First, each producer originally forms a naive (or static) expectation and then considers the announced average price level to some extent. More precisely, at period t, each producer in the ith region expects price at the next period pi (t + 1) to be a linear combination of the actual price pi (t) and the announced average price level N ¯ := (1/N) j=1 pj (t) with a weight ε ∈ (0, 1) for the latter: p(t) ¯ pei (t + 1) = (1 − ␧)pi (t) + ␧p(t) ε pj(t), N N

= (1 − ␧)pi (t) +

(2)

j=1

where superscript e denotes expectations. The parameter ε represents the extent to which a producer puts weight on the average level in his/her decision-making, and in this sense, it indicates coupling strength among regions. Under the price expectation (2), each producer calculates the output level, x˜ i (t + 1), to maximize the expected profit subject to the production cost x2 /2. The resulting amount is x˜ i (t + 1) = pei (t + 1).

(3)

Second, each producer knows that the expected-profitmaximizing amount of production x˜ i (t + 1) does not necessarily become successful and therefore prepares another target as follows: each producer originally expects that at the next period, the same amount would sell under the same price and then considers the announced average production level to some extent. More precisely, at period t, each producer in the ith region sets a naive target of output xˆ i (t + 1) to be a linear combination of the actual xi (t) and N the announced average output level x¯ (t) = (1/N) j=1 xj (t) with a weight ε for the latter: xˆ i (t + 1) = (1 − ␧)xi (t) + ␧¯x(t) ε xj (t). N N

= (1 − ␧)xi (t) +

Fig. 1. Bifurcation diagram of one-dimensional map xt+1 = f(xt ) with respect to . The map can generate chaotic behavior through a period-doubling bifurcation as  increases.

(4)

j=1

Third, each producer determines a final output plan as a linear combination of the naive target xˆ i (t) and the expected-profitmaximizing output x˜ i (t + 1) with a weight  ∈ (0, 1). The resulting formula is as follows: xi (t + 1) = (1 − )ˆxi (t + 1) +  x˜ i (t + 1) ∼

(5)

= (1 − )(1 − ␧) xi (t) + (1 − ) ␧ x¯ (t) +  x i (t + 1). The parameter  represents the extent to which the producer puts weight on profit maximization. Since the sum of the above three coefficients in the last expression is unity, we can paraphrase the producer’s decision making as follows: each producer’s output is determined as a linear combination of xi (t), x¯ (t), and x˜ i (t + 1).

Because there are n producers in each region, the supply in the ith region is represented as nxi (t). The demand for output in the ith region is described by a monotonic inverse demand function as follows: pi (t) =

,

(6)

where yi (t) is the demand in the ith region and  > 0 is the inverse of the price elasticity of the demand. At each period, prices are determined in each market for equilibrating supply and demand: nxi (t) = yi (t).

(7)

By substituting (2), (3), (4), (6), and (7) into (5),we obtain a GCM model (1) with a noninvertible map as follows6 : f (x) = (1 − )x +

 , x

(8)

whose behavior could be chaotic depending on a set of parameters, as well documented by Onozaki et al. (2000): the larger  and  are, the more likely the economy behaves chaotically. To sum up, the regional business cycle model to be analyzed in this study consists of (1) and (8). In what follows, we concentrate on the cases with N = 10 (Section 3) or N = 2 (Section 4). Unless otherwise noted, , , and ε are fixed as 3.5, 0.7, and 0.315, respectively.7 The results of numerical simulation do not depend on initial conditions, and thus, we randomly select xi (0) (i = 1, . . ., N) from the range [0.5, 1.5]. Note that the one-dimensional map x(t + 1) = f(x(t)) can generate chaotic behavior through a period-doubling bifurcation as  increases (Fig. 1). 3. Chaotic itinerancy This section investigates chaotic itinerancy in the regional business cycle model (1) and (8). Chaotic itinerancy is a typical phenomenon occurring in high-dimensional chaotic systems. In its presence, an orbit wanders among different states of complexity (Kaneko and Tsuda, 2000). We first show the time developments of the model in Fig. 2. In coupled systems, separate oscillators sometimes synchronize, that is, a phenomenon called entrainment occurs. A set of synchronized oscillators is called a cluster. Ten types

−



By applying a variable transformation of xi = n 1+ zi , we can cancel out the parameter n from the model. As the transformation is linear, the qualitative behavior of xi remains perfectly preserved. The variable xi appearing in the following is read as zi . 7 Onozaki et al. (2007) show that a very long transient behavior exists when the system dimension N is fixed as 100 and the parameter  is selected from the range [1.0, 8.0]. 6

5 For obvious reasons, we neglect interregional trades as a first step. If we consider them, we have to assume a new economic agent, that is a distributor, who has a behavioral objective completely different from that of a producer. Such extension makes the model much complicated and far more intractable and provides a challenging future task.

1 (yi (t))

K. Esashi et al. / Structural Change and Economic Dynamics 44 (2018) 68–76

71

Fig. 2. Time development of xi (t) (i = 1, . . ., 10) (upper) and its enlargements. The one-cluster regime (lower left) and the 10-cluster regime (lower right) are observed.

Fig. 3. (Left): Time development of the effective dimension (ED) of the economy. The economy primarily remains within the one-cluster regime (ED = 1). Once it departs from the regime, it wanders among various regimes of different EDs until returning to the one-cluster regime. (Right): Distribution of duration T for remaining in the one-cluster regime with ED = 1. The distribution for ED = 1 is shown to obey a power law with an exponent estimated to be −3/2 as indicated by the straight line.

of regimes (from one-cluster to 10-cluster regime) appear in the behavior of the model. The one- and 10-cluster regimes are shown in Fig. 2 (lower left, lower right, respectively). To characterize chaotic itinerancy, we define the effective dimension and its mean (Komuro, 2005). The effective dimension (ED) of a point x(t) = (x1 (t), x2 (t), . . ., xN (t)) ∈ RN with the precision ı, denoted by ED(x(t), ı), is defined simply as a number of clusters. If the Euclidian distance between the components xi and xj (i = / j) of a point x ∈ Rn is less than ı, we consider them to belong to the same cluster.8 We rewrite the model (1) and (8) as a map F : RN → RN which is defined as follows: x(t + 1) = F(x(t)).

(ED = 1), which is called a synchronized state. Then, the economy leaves the regime and wanders among various regimes, which are called desynchronized states, until it returns to the synchronized state. This process is repeated for an extended period (Fig. 3 (left)). For relatively low ED, synchronized regional business cycles are observed, and for relatively high ED, desynchronized regional business cycles are observed. We regard this fact as the occurrence of intermittent transition between synchronization and desynchronization among subnational regions’ business cycles. In Fig. 3 (right), the distribution of duration within the regime of ED = 1 is shown to obey a power law with an exponent estimated to be −3/2 (Heagy et al., 1994).9 For other regimes of ED = 2, 5, 10, the same power law distributions are found for shorter periods of time, sug-

Because of entrainment effect, synchronization strengthens and the economy remains for some time within the one-cluster regime

8

The value of ı is fixed as 10−4 throughout this paper.

9 Although this kind of power law distributions are observed in a well-known critical phenomenon called Type III intermittency in low-dimensional systems, they are not robust in the parameter space. In contrast, the distributions in our model are robustly found.

72

K. Esashi et al. / Structural Change and Economic Dynamics 44 (2018) 68–76

Fig. 4. The MEDs of duration  = 105 of the economy with respect to ε. A non-integer MED indicates that the economy wanders among various regimes of different number of clusters (EDs).

itinerancy occurs in Fig. 5 approximately correspond to the chaotic regions (shaded areas) of a single map. We can state an economic meaning of Fig. 5 as follows: when the producer emphasizes the profit maximization more (larger ) and when he/she puts less weight on the average level in his/her decision-making (smaller ε), the economy is more likely to exhibit chaotic itinerancy. Further, Fig. 5 has another powerful implication. It shows that chaotic itinerancy occurs when the coupling parameter ε is relatively small. If ε is large enough, then almost all regions synchronize and the behavior of the whole economy can be understood by looking at the behavior of a certain region. If ε is smaller and chaotic itinerancy occurs, then the behavior of the whole economy may completely differ from the behavior of individual regions. Thus it follows that although all regions (or agents) are economically homogeneous, the behavior of the aggregate cannot be deduced from the behavior of the “representative” region (or agent); all must be considered simultaneously. 4. Mechanisms of intermittent transition To understand intermittent transition mechanisms from a mathematical perspective, let us focus on the following twodimensional coupled maps obtained by substituting N = 2 into (1)13 : ε {f(x1 (t)) + f(x2 (t))}, 2 ε x2 (t + 1) = (1 − ␧)f(x2 (t)) + {f(x1 (t)) + f(x2 (t))}, 2

x1 (t + 1) = (1 − ␧)f(x1 (t)) +

Fig. 5. Parameter region (, ε) of non-integer MED where the economy wanders among various regimes during  = 105 iterations from 10 initial conditions after the transients of 105 iterations.

gesting that each regime has a mechanism to trap an orbit staying in the regime. The mean of the effective dimension (MED) of an orbit {x(t)} with the precision ı for duration  10 is defined as follows: 1 ED(x(t), ı).  −1

MED(x, ı, ) =

t=0

The MEDs of duration  = 105 are calculated after the transients of 105 iterations for various εs as shown in Fig. 4. 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 EDs. In Fig. 5, we calculate MEDs of the economy by changing a set of parameters (, ε) and identify the parameter region (, ε) where MEDs are non-integers, indicating that the economy wanders among various regimes.11 Fig. 5 shows the robustness of chaotic itinerancy with respect to larger  and smaller ε.12 By comparing with the bifurcation diagram of a single map (Fig. 1), it is apparent that, in terms of , the multiple regimes where chaotic

The value of  is fixed as 105 throughout this paper. The MEDs are calculated from 10 randomly chosen initial conditions xi (0) (i = 1, . . ., 10) after neglecting transient of 105 iterations. If the average is a non-integer, the corresponding set of parameters (, ε) is plotted in Fig. 5. These calculations are performed for 105 points in a parameter region (, ε). 12 Although we do not change a parameter value of  in the present paper, we have confirmed the numerical evidence that a chaotic itinerancy is observed robustly in a wide range of parameter constellations of (ε, ). Since the parameter  = 3.5 is not a singular value for which the system has a special symmetry, there should appear a chaotic itinerancy when making at least a slight change in  from 3.5. 10

11

where f(x) is the same map as (8). We investigate chaotic itinerancy especially in terms of unstable dimension variability in Section 4.2 (Kostelich et al., 1997; Viana and Grebogi, 2000, 2001; Bonatti et al., 2005). This is because complex-behavior wandering among multiple states is observable even in a two-dimensional map, that is, on-off intermittency (Fujisaka and Yamada, 1983; Yamada and Fujisaka, 1983; Pikovsky and Grassberger, 1991; Glendinning, 2001). Examples of two dimensional chaos synchronization model in economics are Bischi et al. (1998), Bischi and Gardini (2000), and Huang and Chen (2014), all of which exhibit on-off intermittency. 4.1. Phenomenology of the two dimensional model Time series of x1 − x2 are depicted in Fig. 6 (left), showing the on-off intermittency between a synchronized state (ED = 1) and a desynchronized state (ED = 2). In the former state, a point (x1 , x2 ) stays in the diagonal line (x1 = x2 ), and in the latter state, a point (x1 , / x2 ) (Fig. 6(right)). x2 ) deviates from the diagonal line (x1 = Next, we shed light on the duration of synchronized and desynchronized states as seen in Fig. 3(right) for the ten-dimensional map. The distribution of synchronized duration in a regime of ED = 1 is shown to obey a power law with an exponent estimated to be −3/2 (Fig. 7(left)). For ED = 2, the same distribution can be found for a shorter period of time. The results are qualitatively the same as the cases for the ten-dimensional map case. Fig. 7(right) shows a parameter region where the system shows chaotic itinerancy. This also implies that the chaotic itinerancy in the two-dimensional map is qualitatively the same as that in Fig. 5 for the ten-dimensional map. Although it is said that chaotic itinerancy is a typical phenomenon mainly in high-dimensional chaos, these facts enable us to focus on the low-dimensional map in this section.

13 This is identical to the ten-dimensional map (N = 10), where the initial conditions are given as x1 (0) = x3 (0) = x5 (0) = x7 (0) = x9 (0) and x2 (0) = x4 (0) = x6 (0) = x8 (0) = x10 (0). If (x1 (t), x2 (t)) is a solution, (x2 (t), x1 (t)) is also a solution because of the symmetry within the system.

K. Esashi et al. / Structural Change and Economic Dynamics 44 (2018) 68–76

73

Fig. 6. Time series of x1 − x2 (left). Phase space dynamics in (x1 , x2 ) plane (right). On-off intermittency between a synchronized state and a desynchronized state is observed.

Fig. 7. (Left): Distributions of duration T for remaining in a regime of ED = 1. The distribution for ED = 1 is shown to obey a power law with an exponent estimated to be −3/2 as indicated by the straight line. (Right): Parameter region (, ε) where the economy wanders among various regimes during  = 105 iterations from 10 initial conditions after the transients of 105 iterations.

4.2. Classification of periodic orbits Finally, we investigate the transition from a desynchronized state to a synchronized state and that from a synchronized state to a desynchronized state in terms of unstable dimension variability. Unstable dimension variability is characterized by the existence of cycles among sets of periodic orbits with different number of unstable directions.14 Note that in the two-dimensional model, a periodic orbit can be an attractor, saddle, or repellor, depending on the number of unstable directions being 0, 1, or 2, respectively. Tsuda (2009) conjectured that unstable dimension variability can be a key to understand the dynamics of chaotic itinerancy. In this subsection, we confirm that unstable dimension variability is observed in on-off intermittency in the two-dimensional coupled maps. We identify unstable periodic orbits with different number of unstable directions and confirm the existence of periodic orbits connecting one state to the others. Periodic orbits play an important role to characterize dynamical systems that show chaotic synchronization (Zhao et al., 2005). If all points of a periodic orbit stay in a synchronized state, we call it a synchronized periodic orbit. If at least one point on a periodic orbit goes away from a synchronized state, we call it a desynchronized periodic orbit. We also classify periodic orbits based on the number of unstable directions (see Table 1). A number of periodic orbits with periods up to 20 are depicted in Fig. 8. The upper graph shows all the points of detected peri-

14 Homoclinic tangency and unstable dimension variability (heterodimensional cycles) are thought to be typical structures that break hyperbolicity (Bonatti et al., 2005).

Table 1 Types of unstable periodic orbits. Type of Periodic orbits

State

# of unstable directions

Synchronized saddle Synchronized repellor Desynchronized saddle Desynchronized repellor

Synchronized Synchronized Desynchronized Desynchronized

1 2 1 2

odic orbits, which almost cover chaotic attractors (Fig. 6 (right)). It consists of desynchronized repellors (Fig. 8 (lower left)), desynchronized saddles (Fig. 8 (lower right)), synchronized repellors, and synchronized saddles. For synchronized periodic orbits near the diagonal line (x1 = x2 ), distributions of synchronized saddles (Fig. 9(left)) and synchronized repellors (Fig. 9 (right)) are depicted. It is found that repellors tend to distribute around a repelling fixed point (x1 , x2 ) = (1, 1). Figs. 8 and Fig. 9 show that saddles and repellors are distributed in both synchronized and desynchronized states. We can expect that a trajectory wondering close to the synchronized state tends to leave the state near (x1 , x2 ) = (1, 1) due to the transversal instability. In order to investigate how close a chaotic attractor is to those four types of periodic orbits, we calculate distances between a chaotic trajectory and a period 4 desynchronized saddle, a period 6 desynchronized repellor, a period 10 synchronized saddle, and a period 4 synchronized repellor (Fig. 10). Distances between a chaotic orbit x(t) showing on-off intermittency and a point x0 on a periodic orbit is defined as d(t) = mint dist(x(), x0 ), where the function dist(·, ·) denotes the Euclidean distance between two points. The results show that a chaotic trajectory tends to approach

74

K. Esashi et al. / Structural Change and Economic Dynamics 44 (2018) 68–76

Fig. 8. Distributions of periodic orbits of all four types with periods up to 20 (upper), desynchronized repellors (lower left), and desynchronized saddles (lower right).

Fig. 9. Histograms of frequencies of synchronized saddles (left) and frequencies of synchronized repellors (right). The former does not distribute around a fixed point (x1 , x2 ) = (1, 1) but the latter does. The bin width is 0.1.

Fig. 10. Distances between a chaotic orbit x(t) showing on-off intermittency and a point x0 on a periodic orbit in the four periodic orbits d(t) = mint dist(x(), x0 ). The function dist(·, ·) denotes the Euclidean distance between two points. A chaotic trajectory approaches to these periodic orbits.

K. Esashi et al. / Structural Change and Economic Dynamics 44 (2018) 68–76

a point on each of the four periodic orbits. These numerical results give us an evidence of the existence of unstable dimension variability, or heterodimensional cycles, which are characterized by cycles connecting invariant sets with different number of unstable directions. In summary, both saddles and repellors are embedded in both synchronized and desynchronized states and the connecting orbits among them play an important role in synchronized desynchronized transition in chaotic itinerancy in a two-dimensional dynamical system. We conjecture that this scenario works for chaotic itinerancy in high-dimensional dynamical systems as well. 5. Conclusion In this study, we have robustly observed chaotic itinerancy in a model of multi-regional business cycles, emerging through coupling of producers’ behavior based on information announced by the government, and investigated its synchronized desynchronized transition mechanisms. The following results are obtained: (i) The economy exhibits chaotic itinerancy, that is, it wanders among various regimes featuring different numbers of clusters for particular constellations of parameters. This fact is regarded as the occurrences of intermittent transition between synchronization and desynchronization in regional business cycles that are often documented in the literature. (ii) When a producer emphasizes the profit maximization more and when puts more weight on the average level in his/her decision-making, the economy is more likely to exhibit chaotic itinerancy. (iii) Chaotic itinerancy occurs even if the coupling effect is relatively weak. This implies that although all regions (or agents) are economically homogeneous, the behavior of the aggregate cannot be deduced from the behavior of the “representative” region (or agent); all must be considered simultaneously. As for the standard macroeconomic approach based on the representative agent, many authors, for example, Kirman (1992), Davis et al. (1996), Gallegati and Kirman (1999), Solow (2004), Aoki and Yoshikawa (2007), Stiglitz (2010), Delli Gatti et al. (2011), and Bischi and Cerboni Baiardi (2015) have been criticized on the ground that the complex behavior of an economy consisted of heterogeneous agents cannot be represented by any agent’s behavior. Although such criticism is quite reasonable, simultaneously, our result has stronger implication: even though all regions (or agents) are economically homogeneous, the assumption of representative region (or agent) may fail to explain the behavior of the aggregate in nonlinear systems. (iv) Behind chaotic itinerancy, unstable dimension variability (heterodimensional cycles) may exist. This is identified by finding the unstable periodic orbits with the different number of unstable directions embedded in an attractor showing chaotic itinerancy and the cycles among them. Finally, we remark on the policy implication of this study. The model we studied is very simple and does not have any explicit policy parameters. There are three key parameters, ε, , and . The parameter  represents the extent to which a producer puts weight on profit maximization, that is, it depends on the entrepreneur spirit, and the parameter  is the inverse of price elasticity of demand. Therefore, it seems difficult for the government’s policy to have an influence on these two parameters at least in the short run. As for the parameter ε, it represents the extent to which a producer puts weight on the average level in his/her decision-making, and it indicates coupling strength among regions. Thus the government is able to vary the strength of synchronization of regional business

75

cycles if its policy can affect the coupling parameter ε. This may contribute to mitigating the impact of severe recession. Since the time of Mitchell (1927) and Keynes (1936), it has been well known that business cycles exhibit asymmetric shape across expansion and recession phases: recessions are briefer, sharper, and generally more violate processes than expansions. From the viewpoint of regional business cycle synchronization, this fact is translated to mean that during the recession process, the coupling effect is so strong that many regions are synchronized together and during the expansion process, the coupling effect is so weak that few regions tend to synchronize.15 Therefore, during severe recession process, if the government can decrease the coupling parameter ε through some policy instruments, the impact of recession will be diminished. The model we have studied includes multiple regions which are assumed to behave homogeneously in the sense that they have the same key parameters, ε, , and  in common. Since this assumption is strong, it should be weakened in future work by introducing different values for different regions. Although such heterogeneity is supposed to make dynamics of the model more and more complicated, introducing different ε’s especially will be of assistance to investigate the policy implication of the model stated above. Furthermore, the dependency of occurrences of chaotic itinerancy on system dimensions remains an important issue for future work. The mechanism of the robustness of chaotic itinerancy is to be clarified to understand economic dynamics. Acknowledgements The authors would like to thank Professor Ichiro Tsuda for his helpful information about chaotic itinerancy. This work was partly supported by JSPS KAKENHI (Grant Numbers 17K05360, 26610034, 24653050, 23330086, 23740065), JST PRESTO grant JPMJPR16E5, and the Collaborative Research Program for Young Scientists of ACCMS and IIMC, Kyoto University. Yuzuru Sato is supported by the External Fellowship at London Mathematical Laboratory. References Aguiar-Conraria, L., Soares, M.J., 2011. Business cycle synchronization and the euro: a wavelet analysis. J. Macroecon. 33, 477–489. Antonakakis, N., 2012. Business cycle synchronization during US recessions since the beginning of the 1870. Econ. Lett. 117, 467–472. Aoki, M., Yoshikawa, H., 2007. Reconstructing Macroeconomics: A Perspective from Statistical Physics and Combinatorial Stochastic Processes. Cambridge University Press, New York. Artis, M.J., Zhang, W., 1999. Further evidence on the international business cycle and the ERM: is there a European business cycle? Oxf. Econ. Pap. 51, 120–132. Baxter, M., Kouparitsas, M.A., 2005. Determinants of business cycle comovement: a robust analysis. J. Monet. Econ. 52, 113–157. Benˇcík, M., 2011. Business Cycle Synchronization between the V4 Countries and the Euro Area. National Bank of Slovakia, Working Paper 1. Bischi, G.I., Cerboni Baiardi, L., 2015. Fallacies of composition in nonlinear marketing models. Commun. Nonlinear Sci. Numer. Simul. 20, 209–228. Bischi, G.I., Stefaninia, L., Gardini, L., 1998. Synchronization, intermittency and critical curves in a duopoly game. Math. Compute. Simul. 44, 559–585. Bischi, G.I., Gardini, L., 2000. Global properties of symmetric competition models with riddling and blowout phenomena. Discrete Dyn. Nat. Soc. 15, 149–160. Bonatti, C., Díaz, L.J., Viana, M., 2005. Dynamics beyond Uniform Hyperbolicity. Springer, Berlin. Brenner, T.W., Weidlich, W., Witt, U., 1998. International Co-Movement of Business Cycles in a ‘Phase-Locking’ Model. Max-Plank-Institute of Economics,

15 In our upcoming empirical study, we have investigated multi-regional business cycle synchronization and found that business cycles in more regions tend to behave similarly in the recession stage, whereas those in less regions do in the recovery-expansion stage. This result is obtained by applying cross-wavelet analysis to monthly data (from Jan. 1977 to Dec. 2011) of Japan’s index of industrial production by prefecture (original index, 2005 average = 100) for all of the 47 prefectures and cross-wavelet analysis.

76

K. Esashi et al. / Structural Change and Economic Dynamics 44 (2018) 68–76

Papers on Economics and Evolution, No. 9802 (Revised 2002 as Metroeconomica 53, 113–138). Carlino, G.A., DeFina, R., 1995. Regional income dynamics. J. Urban Econ. 37, 88–106. Carlino, G.A., Sill, K., 2001. Regional income fluctuations: common trends and common cycles. Rev. Econ. Stat. 83, 446–456. Clark, T.E., van Wincoop, E., 2001. Borderds and business cycles. J. Int. Econ. 55, 59–85. Crowley, P.M., Lee, J., 2005. Decomposing the Co-Movement of the Business Cycle: A Time-Frequency Analysis of Growth Cycles in the Euro Area. Bank of Finland, Discussion Paper 12. Davis, S.J., Haltiwagner, J.C., Schuh, S., 1996. Job Creation and Destruction. MIT Press, Cambridge, MA. Delli Gatti, D., Desiderio, S., Gaffeo, E., Cirillo, P., Gallegati, M., 2011. Macroeconomics from the Bottom-Up. Springer, Italia. Dueker, M., Wesche, K., 2003. European business cycles: new indices and their synchronicity. Econ. Inq. 41, 116–131. Fujisaka, H., Yamada, T., 1983. Stability theory of synchronized motion in coupled-oscillator systems. Prog. Theor. Phys. 69, 32–47. Gallegati, M., Kirman, A. (Eds.), 1999. Beyond the Representative Agent. Edward Elgar Publishing Ltd., Cheltenham. Glendinning, P., 2001. Milnor attractors and topological attractors of a piecewise linear map. Nonlinearity 14, 239–257. Hanus, L., Vacha, L., 2016. Business Cycle Synchronization within the European Union: A Wavelet Cohesion Approach. arXiv:1506.03106 [q-fin.EC]. Haxholdt, C., Kampmann, C., Mosekilde, E., Sterman, J.D., 1995. Mode-locking and entrainment of endogenous economic cycles. Syst. Dyn. Rev. 11, 177–198. Heagy, J.F., Platt, N., Hammel, S.M., 1994. Characterization of on-off intermittency. Phys. Rev. E 49, 1140–1150. Huang, W., Chen, Z., 2014. Modeling regional linkage of financial markets. J. Econ. Behav. Organ. 99, 18–31. Huang, W., Chen, Z., 2015. Heterogeneous agents in multi-markets: a coupled map lattices approach. Math. Comput. Simul. 108, 3–15. Ikeda, K., Otsuka, K., Matsumoto, K., 1989. Maxwell-Bloch turbulence. Prog. Theor. Phys. Suppl. 99, 295–324. Imbs, J., 1999. Co-Fluctuations. Center for Economic Policy Research, Discussion Paper No. 2267 (Revised 2003). Kaneko, K., 1990. Clustering, coding, switching, hierarchical ordering, and control in network of chaotic elements. Physica D 41, 137–172. Kaneko, K., 1992. Overview of coupled map lattices. Chaos 2, 279–282. Kaneko, K., 2002. Dominance of Milnor attractors in globally coupled dynamical systems with more than 7 ± 2 degrees of freedom. Phys. Rev. E 66, 055201. Kaneko, K., Tsuda, I., 2000. Complex Systems: Chaos and Beyond. Springer, Berlin (English translation from the original Japanese version: Fukuzatsukei no Kaosu-teki Shinario, Asakura Publishing Co., Ltd., 1996). Keynes, J.M., 1936. General Theory of Employment, Interest, and Money. Macmillan, London. Kirman, A.P., 1992. Whom or what does the representative individual represent? J. Econ. Perspect. 6, 117–136. Komuro, M., 2005. Kiso karano Rikigakukei (Introduction to Dynamical Systems. Newly-Revised). Saiensu-Sha Co., Ltd. Publishers, Tokyo (in Japanese). Kostelich, E.J., Kan, I., Grebogi, C., Ott, E., Yorke, J.A., 1997. Unstable dimension variability: a source of nonhyperbolicity in chaotic systems. Physica D 109, 81–90. Kouparitsas, M.A., Nakajima, D.J., 2006. Are U.S. and seventh district business cycles alike? Federal Reserve Bank of Chicago. Econ. Perspect. 30, 41–60. Kozlowski, P.J., 1995. Money and interest rates as predictors of regional economic activity. Rev. Reg. Stud. 25, 143–157.

Krugman, P., 1996. The Self-Organizing Economy. Blackwell Publishers, Massachusetts. Larsen, E.R., Morecroft, J.D.W., Thomsen, J.S., Mosekilde, E., 1993. Devil’s staircase and chaos from macroeconomic mode interaction. J. Econ. Dyn. Control 17, 759–769. Milnor, J., 1985. On the concept of attractor. Commun. Math. Phys. 99, 177–195. Mitchell, W.C., 1927. Business Cycles: The Problem and its Setting. National Bureau of Economic Research, New York. Mosekilde, E., Larsen, E.R., Sterman, J.D., Thomsen, J.S., 1992. Nonlinear mode-interaction in the macroeconomy. Ann. Oper. Res. 37, 185–215. Mosekilde, E., Larsen, E.R., Sterman, J.D., Thomsen, J.S., 1993. Mode-locking and nonlinear entrainment of macroeconomic cycles. In: Day, R.H., Chen, P. (Eds.), Nonlinear Dynamics and Evolutionary Economics. Oxford University Press, Oxford, pp. 58–83. Onozaki, T., Sieg, G., Yokoo, M., 2000. Complex dynamics in a cobweb model with adaptive production adjustment. J. Econ. Behav. Organ. 41, 101–115. Onozaki, T., Yanagita, T., Kaizoji, T., Toyabe, K., 2007. Regional business cycle synchronization through expectations. Physica A 383, 102–107. Pikovsky, A.S., Grassberger, P., 1991. Symmetry breaking bifurcation for coupled chaotic attractors. J. Phys. A: Math. Theor. 24, 4587–4597. Rissman, E.R., 1999. Regional employment growth and the business cycle. Federal Reserve Bank of Chicago. Econ. Perspect. 23, 21–39. Savva, C.S., Neanidis, K.C., Osborn, D.R., 2010. Business cycle synchronization of the euro area with the new and negotiating member countries. Int. J. Finance Econ. 15, 288–306. Selover, D.D., Jensen, R.V., 1999. ‘Mode-locking’ and international business cycle transmission. J. Econ. Dyn. Control 23, 591–618. Selover, D.D., Jensen, R.V., Kroll, J., 2005. Mode-locking and regional business cycle synchronization. J. Reg. Sci. 45, 703–745. Solow, R.M., 2004. Introduction: the Tobin approach to monetary economics. J. Money Credit Bank. 36, 657–663. Sterman, J.D., Mosekilde, E., 1994. Business cycles and long waves: a behavioral disequilibrium perspective. In: Semmler, W. (Ed.), Business Cycles: Theory and Empirical Methods. Springer, New York, pp. 13–51. Stiglitz, J., 2010. Needed: a new economic paradigm. Financial Times, August 20. Süssmuth, B., 2003. Business Cycles in the Contemporary World. Physica-Verlag, Heidelberg. Tsuda, I., 1990. Possible biological and cognitive functions of neural networks probabilistically driven by an influence of probabilistic release of synaptic vesicles. Proceedings of the 12th Annual International Conference, IEEE/EMBS, 1772–1773. Tsuda, I., 2009. Hypotheses on the functional roles of chaotic transitory dynamics. Chaos 19, 015113. Viana, R.L., Grebogi, C., 2000. Unstable dimension variability and synchronization of chaotic systems. Phys. Rev. E 62, 462–468. Viana, R.L., Grebogi, C., 2001. Riddled basins and unstable dimension variability in chaotic systems with and without symmetry. Int. J. Bifurcat. Chaos 11, 2689–2698. Yamada, T., Fujisaka, H., 1983. Stability theory of synchronized motion in coupled-oscillator systems. II: The mapping approach. Prog. Theor. Phys. 70, 1240–1248. Yasutomi, A., 2003. Itinerancy of money. Chaos 13, 1148–1164. Yetman, J., 2011. Exporting recessions: international links and the business cycle. Econ. Lett. 110, 12–14. Zhao, L., Lai, Y.C., Shih, C.W., 2005. Transition to intermittent chaotic synchronization. Phys. Rev. E 72, 036212.