Exotic Dynamics of Inflation: Numerical
Simulations
Yosuke Umezuki∗

Masanori Yokoo†

This Version: March 31, 2019

Abstract
This paper develops a deterministic inflation model via a piecewise linear expectations-augmented Phillips curve. Simulations show that the areapreserving property of the model entails complex dynamics including nonattracting but observable periodic or chaotic behaviors. Such behaviors arerelatively uncommon in the economic literature.
JEL Classification Numbers: E31; E32
Keywords: Phillips curve; Piecewise linearity; Inflation dynamics; Chaos;
Area-preserving map∗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†
Faculty of Economics and Graduate School of Humanities and Social Sciences, Okayama
University, Tsushimanaka 3-1-1, Kita-ku, Okayama 700-8530, Japan. e-mail: yokoo@e.okayamau.ac.jp

11Introduction

Inﬂation has always been a crucial topic in macroeconomics, as well as its relationship with the unemployment rate. In this paper, we build a simple inﬂationunemployment model exhibiting complex behaviors that are not commonly observedin the economic literature. The model is composed of simple macroeconomic relations such as the expectations-augmented Phillips curve, static expectations, anda dynamic version of the Okun’s law, all components of which can be found inundergraduate macroeconomic textbooks such as Blanchard (1997).
In addition to these standard relationships, we assume that the Phillips curve ispiecewise linear. This allows us to introduce slight nonlinearity into the model withall other relationships being linear.
A similar inﬂation model1 was analyzed by Soliman (1996a, 1996b), who assumes
a smooth Phillips curve instead of a piecewise linear one. Nonetheless, adopting
piecewise linearity has a distinct advantage over other types of nonlinearities. For
example, Umezuki (2019) shows that, in the same framework as presented here,
the piecewise linearity of the Phillips curve allows the direct calculations of certain
exact analytical results, which could not be obtained under the smooth nonlinearity
assumption. For instance, it can be shown that the model exhibits a transverse
homoclinic point, which implies the existence of a horseshoe, (i.e., a chaotic invariant
set). However, the same could hardly be shown for the Soliman’s model without
relying on numerical methods.
In this paper, we leave aside the detailed analysis and only demonstrate somenumerical results for complex but less common dynamics of the model.
The remainder of this paper is organized as follows. Section 2 describes themodel. Section 3 shows the results of some numerical simulations for several speciﬁc
sets of parameter values. Section 4 deals with the polygonal dynamics with integer
parameters. Section 5 provides some concluding remarks.

2The ModelFirst, we formalize four standard macroeconomic relationships as in Blanchard(1997).
πt = f (ut ) + πte ,1(1)

Yoshida (2015) employs a similar inflation model with a nonlinear Okun’s law and a linear
Phillips curve, and numerically shows that the model exhibits chaotic dynamics.

2
ut+1 = −β(gt − gn ) + ut ,β > 0;

g t = m − πt ,

(2)
(3)

eπt+1= πt .

(4)

Eq.(1) describes the expectations-augmented Phillips curve: the gap between theactual inﬂation rate πt and the expected inﬂation rate πte is related to the unemployment rate ut through the function f . The subscript t denotes time that discretely
extends from 0 to inﬁnity. Eq.(2) represents the dynamic version of the Okun’s law.
According to this formula, if the output growth rate gt is above the natural growthrate gn , then the time variance of unemployment decreases. Eq.(3) presents the equilibrium condition in the money market. We assume here that the nominal growthrate of money, m, is constant over time. We also assume that the expectations are
static as per Eq.(4).
We call the graph of the function f the basic Phillips curve, because if, instead ofEq.(1), we would relate πt to ut directly as πt = f (ut ), we would obtain the Phillips
curve prior to augmentation with expectations. We assume that the basic Phillips
curve has a piecewise linear form represented by{−a1 ut + b1 if ut ≤ θf (ut ) =(5)−a2 ut + b2 if ut > θ,
where either a1 > a2 > 0 and b1 > b2 > 0 when f is convex, or a2 > a1 > 0 andb2 > b1 > 0 when f is concave.2 In either case, f has a kink at
ut = θ =b1 − b2.
a1 − a2Eqs.(1)–(5) can be reduced to the following second-order diﬀerence equation interms of unemployment:
ut+2 = (2 − βai )ut+1 − ut + bi ,(6)

where i = 1 if ut+1 ≤ θ and i = 2 if ut+1 > θ.
Let us introduce new variables as follows:
ut = Cyt + θand yt+1 = xt

withC=2β(a1 b2 − a2 b1 ).
a1 − a2

There have been debates (e.g., Laxton et al., 1999) about the convexity with respect to unemployment of the basic Phillips curve for the United States. Nonetheless, this issue is not much
relevant in our formulation.

3The change of variables transforms Eq.(6) into the map F : R2 → R2 in the following
form:
( )
()
x
(2 − βai )x − y + 1
F :
7→,
(7)
yx
where i = 1 if Cx ≤ 0 and i = 2 if Cx > 0.
Note that the map (7) directly depends on the parameters a1 , a2 and β. Theparameters b1 and b2 inﬂuence the dynamics through C, while m and gn have noimpact on the model dynamics. Moreover, the value of C itself does not matter,
while its sign does. By inspection, we see that the case where f is concave andC < 0 is equivalent to the case where f is convex and C > 0. Furthermore, the casewhere f is concave and C > 0 is equivalent to the case where f is convex and C < 0.
Thus, without loss of generality we can focus on the case where f is convex. The
positivity (negativity) of C corresponds to the case where the basic Phillips curve
has the kink at (πt , ut ) = (π ∗ , θ) with π ∗ > 0 ( π ∗ < 0). Fig.1 (a) and (b) depict
examples of the basic Phillips curve when C > 0 and C < 0, respectively.
<<Insert Fig.1 around here>>
Finally, we note that the determinant of the Jacobian matrix of the map (7)is unity wherever the derivative exists. In such a case, the map is referred to asarea-preserving. This implies that no invariant set for F can be an attractor or arepeller. See Guckenheimer and Holmes (1997) for the use of terminology.

3Simulation examples

In this section, we present several ﬁgures, each of which plots a trajectory generated
by the map (7) for a speciﬁc set of parameter values.

<<Insert Figs. 2–5 around here>>Figs.2 and 3 illustrate the cases where |2 − βai | is unity for i = 1, 2, which implies
that all coeﬃcients in Eq.(7) are integers. In both cases, we can ﬁnd a ﬁxed point,
periodic points, and an orbit wandering on the plane in a complex manner while
avoiding the polygonal regions. We can see a triangle and four hexagons in Fig.2 andsix hexagons in Fig.3. In the next section, we explain how such polygonal regions
might arise.

4Figs.4 and 5 depict other complex orbits for cases with non-integer coeﬃcients.
In these cases, the polygons disappear, while elliptic structures arise. Counterintuitively, every orbit depicted in the ﬁgures represents not an attractor due to thearea-preserving property of the map.

4Polygonal dynamics

In Figs.2 and 3, all parameters are integers: (a1 , a2 , β) = (3, 1, 1). In these cases,
we have observed the emergence of polygonal regions on the plane. Devaney (1984)
and Aharonov et al. (1997) study the case of Fig.3. Here, we investigate the case ofFig.2 in more detail. Here, the map (7) turns to( )()x−|x| − y + 1
F :
7→.
(8)
yx
()
The map (8) has a center ﬁxed point at (x, y) = 31 , 13 ∈ R2 . Let J be a Jacobianmatrix of the map (8). In the neighborhood of the ﬁxed point, J is given by)(−1 −1(9)J=
10()
and thus, J 3 = Id. It is easy to see that the maximal neighborhood of 31 , 13 , onwhich J 3 = Id, is a triangle with vertices at (1, 0), (0, 1), and (0, 0). Therefore, allpoints within the triangular region, except the ﬁxed point, have period three. Thus,
the amplitude of the period-three cycle within the triangular region depends on theinitial condition.
Furthermore, the map (8) exhibits the periodic center points of period four at(3, −1), (−1, 3), (−3, −1), and (−1, −3). By the same argument, let J4 be a Jacobianmatrix of the fourth iteration of the map (8). In the neighborhood of the periodic
point (3, −1), for instance, J4 is given by()
1 1
J4 =(10)−1 0
and thus, J46 = Id. Therefore, all points in the neighborhood of each periodic point
have period 24. Again, the amplitude of this cycle depends on the initial conditions.
It is not hard to conclude that such polygonal structures will easily be destroyedwhen the parameters are perturbed from the given integers. Fig.6 provides anexample. In this case, there are inﬁnitely many elliptic orbits instead of the period
three and 24 cycles mentioned above.

5<<Insert Fig.6 around here>>
For the dynamics outside the polygons, see Umezuki (2019), which shows thepresence of a chaotic invariant set.
<<Insert Fig.7 around here>>5

Concluding remarks

Fig.7 depicts how the actual inﬂation rate and the unemployment rate are related
to the basic Phillips curve. We can observe that the complex trajectory is scatteredaround the basic Phillips curve. Note that the kink of the basic Phillips curve is theonly element of nonlinearity responsible for the complex dynamics in our model.
Finally, the area-preserving property of our economic system is not robust. Infact, we can show that a slight change in the expectations formation can make ourmodel dissipative. As a result, some less common, less intuitive dynamic properties
such as polygonal and elliptic structures, and some sort of sensitivity to initialconditions would easily disappear.

References[1] Aharonov, D., Devaney, R. L., Elias, U., 1997. The dynamics of a piecewiselinear map and its smooth approximation. International Journal of Bifurcation
and Chaos. 7, 351-372.
[2] Blanchard, O., 1997. Macroeconomics, Prentice Hall.
[3] Devaney, R. L., 1984. A piecewise linear model for the zones of instability ofan area preserving map. Physica D. 10, 387-393.
[4] Guckenheimer, J., Holmes, P., 1997. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag.
[5] Laxton, D., Rose, D., Tambakis, D., 1999. The U.S. Phillips curve: The casefor asymmetry. Journal of Economic Dynamics & Control. 23, 1459-1485.
[6] Soliman, A. S., 1996a. Fractals in nonlinear economic dynamic systems. Chaos,Solitons & Fractals. 7, 247-256.

6[7] Soliman, A. S., 1996b. Transitions from stable equilibrium points to periodic
cycles to chaos in a Phillips curve system. Journal of Macroeconomics. 18, 139153.
[8] Umezuki, Y., 2019. Four Essays in Nonlinear Economic Dynamics. Dissertation, Okayama University.
[9] Yoshida, H., 2015. A monetarist model reconsidered: The emergence of chaotic
ﬂuctuations. International Journal of Economic Behavior and Organization. 3,73-76.

7108πt6420-2024681012ut(a) The basic Phillips curve represented by Eq.(5). a1 = 3, a2 = 1, b1 = 10and b2 = 4.

64πt20-2-4-60246810ut(b) The basic Phillips curve represented by Eq.(5). a1 = 3, a2 = 1, b1 = 6and b2 = 1.

Figure 1: Fig.(a): C > 0. Fig.(b): C < 0.

8642

y0-2-4-6-6-4-2024xFigure 2: a1 = 3, a2 = 1, β = 1, and C < 0.

6

986y420-2-4-4-202468x
Figure 3: a1 = 3, a2 = 1, β = 1, and C > 0.

1032y10-1-2-3-3-2-1

0

1

23x
Figure 4: a1 = 3.5, a2 = 1.5, β = 1, and C < 0.

1110y50-5-5

0510x
Figure 5: a1 = 3.5, a2 = 1.5, β = 1, and C > 0.

12642y0-2-4-6-6-4-2

0246x(a) a1 = 3.001, a2 = 1, β = 1 and C < 0.

1.5

fixed point

y1

0.5

0-0.5-0.5

00.5

11.5x(b) Closer look at the square in (a).

Figure 6: Fig.(a) depicts a complex orbit and some elliptic orbits generated by themap slightly perturbed from the case of Fig.2. Fig.(b) depicts an enlargement of
the neighborhood of the ﬁxed point.

13864πt20-2-4-6-8024681012utFigure 7: A trajectory of the actual inﬂation rate and unemployment rate, whichcorresponds to Fig.3. The basic Phillips curve given by Eq.(5) is superimposed.
a1 = 3, a2 = 1, b1 = 10, b2 = 4 and β = m = gn = 1.