心身科学 第 10 巻第 1 号（ 7 14）（2018）

Dynamical scenarios of changes in asymmetric relationships
over time（2）Naohito Chino＊１）
Abstract
This paper is the second part of the revised version of my paper presented elsewhere（Chino,2016）. In this paper we propose a recent version of a set of complex difference equation
models which describes changes in asymmetric relationships among objects over time. Typicalexamples of these objects may be citation frequencies in scientific publications, amountsof trade among nations, connections among neurons, and so on. Supposing that such an
asymmetric relational data matrix or a weighted digraph is observed in an instant of time, we
ﬁrst apply the Chino-Shiraiwa theorem to the matrix and embed objects in a（complex）Hilbert
space. Then we shall apply our recent version of a set of general complex diﬀerence equation
models to the initial conﬁguration of objects embedded in the Hilbert space. As a result, wehave various possible theoretical scenarios of the trajectories of objects in this space. In thispaper, we show some of the possible scenarios of the nonlinear diﬀerence equation models inthe special cases when the number of objects are two and three.
Key words: complex difference equation, Hilbert space, Chino-Shiraiwa theorem, dynamic
weighted digraph, chaos, trade imbalance, neural network.

the following nonlinear diﬀerence equation models

１．Introduction

which include the linear models:In a companion paper, we have reported thefirst part of the revised version of my handout
presented at the 45th annual meeting of the
Behaviormetric Society of Japan（Chino, 2017）,
in which we have described a current difference
equation model and have shown the possible
theoretical scenarios of its trajectories with somespecial linear difference equation models . In thispaper we shall discuss the possible theoreticalscenarios with some nonlinear difference equation
models , as a second part of the revised version.

andAs discussed in the companion paper, we assume＊ 1 ）Department of Psychology, Faculty of Psychological & Physical Science, Aichi Gakuin University.
Requests for reprints should be sent to : chino@dpc.agu.ac.jp
This is the second part of the revised version of my paper titled Dynamical scenarios of changes in asymmetric relationships on a
Hilbert space presented at the 45th annual meeting of the Behaviormetric Society of Japan , Shizuoka, Japan.

―7―Naohito CHINO（ 1 ）to（ 3 ）exhibits excitingly richer dynamical
properties of its solution curve than linear systems.
where m = 1 , 2 , …, q .
Here,denotes the coordinate vector of

member j at time n in a p -dimensional Hilbert
space or an indeﬁnite metric space. Moreover, mdenotes the degree of the vector function,
which is assumed to have themaximum value q , while
i n E q . （ 3 ） a r e complex constants .

Thesemodels are very general and might enable us todescribe various possible changes in asymmetric
relationships among members over time.
２． D y n a m i c a l s c e n a r i o s o f c h a n g e s i n
Fig. 1. Julia set forwith a parabolic fixed point（0.5）with the ʻ・ʼ mark, two repelling
periodic points of period 2（­0.5 ）with the ʻ㲀ʼ mark,and six repelling periodic points of period 3 with theʻ©ʼ mark, and twelve repelling periodic points of three
4-sycles with the ʻ㸡ʼ mark.

asymmetric relationships in complex
diﬀerence equation models
In this section, first we consider the difference
equation models with quadratic terms.
Here, we assume the special case when m= 2 ,p= 1 ,= 0 , and= 0 in Eqs.（ 1 ）to（ 3 ）.
Before we discuss the dynamical scenario of

Such a system is written as

this nonlinear system, we consider the system（4）described by Eq.（ 5 ）, which is naturally induced
from Eq.（ 4 ）. As pointed out earlier, the induced
new system,, which is obtainedThis type of system has a very desirable property

by a linear transformation of the dif ference,in that we can utilize the heritage of the theory, is of the form assumed in discussingof the complex dynamical system developed in

the famous Mandelbrot set. Depending on themathematics directly in classifying its trajectories.

complex constant term,

In fact, defining a new variable,,

dynamical scenarios of this new system, which are

and transforming it linearly, we have a new systemwell known in the area of the complex dynamical（5）, there exist various

system in mathematics（e.g., Milnor, 2000; Peitgen& Richter, 1986; Ueda et al., 1995）.
In examining the dynamical scenarios of, we（K. Shiraiwa, personal

can utilize various theoretical results establishedcommunication, April 25, 2014）. Depending onin the complex dynamical system. Some of thethe value ofkey phrases in these results are the ﬁxed point as

and, we have the Mandelbrot set.
, which determines

well as the periodic point（orbit）, the multiplier of, is one of the eigenvalues of thethese points, and the Fatou set and Julia set（e.g.,

It is interesting to note thatthe value of

Carleson & Gamelin, 1993）.

coefficient matrix of the linear system discussed

Suppose f

in the companion paper. We will see later thatis a holomorphic function , that is,even a quadratic system which is a special

an analytic function in a complex space. Then, zcase of our general model described by Eqs.

is called a fixed point if f

―8―=

. The number

Dynamical scenarios of changes

λ=f

is called the multiplier of the ﬁxed point.

Here,is the ﬁrst derivative of

λ=（e.g., Beardon, 1991）.

with respect to

The neutral point is sometimes called the. The multiplier determines the property of theindifferent point , and the rationally neutral point issometimes called the parabolic point .

fixed point as follows（e.g., Carleson & Gamelin,
1993, p.27）:（ 1 ）Attracting if λ <1. （If λ = 0 , we refer toa superattracting ﬁxed point.）（ 2 ）Repelling if λ >1.
（ 3 ）Rationally neutral if λ = 1 and λn = 1 forsome integer n .
（ 4 ）Irrationally neutral if λ= 1 but λn isnever 1.

Fig. 3. Nearby trajectories which go in or out of theparabolic fixed point.

Let f be the holomorphic function on thecomplex plane, and let the divergent set of its
collection of iterates be I p . The complement of
I p is called the filled-in Julia set K p , and theboundary of K p is called the Julia set . The Fatou
set is the complement of the Julia set. It is known
that a point p 0 belongs to the Julia set if and only
if dynamics in a neighborhood of p 0 displays

Fig. 2. Trajectories of the system with several initialpoints inside the Julia set.

In contrast, the periodic point （periodic orbit ,or cycle ） is defined as follows （e.g., Milnor,

sensitive dependence on initial conditions （e.g.,
Milnor, 2000, p.38）.

2000, p.43）. Let us suppose that we iterate the

It is well known thatholomorphic map m times using the above function（ 1）Every attracting periodic point is

f . We shall denote this m iterates by

are all distinct, and if

.

If

, …,

is equal to z , then the

integer m* 1 is called the period , that is,

contained in the Fatou set of f .
（ 2 ）Every repelling periodic point is containedin the Julia set of f .

（6）（ 3 ）Every rationally neutral（or parabolic）

In this sense, the fixed point is a periodic orbit

periodic point belongs to the Julia set of f ,of period 1. We can say that a periodic point of（ 4 ）Every irrationally neutral periodic point

period m is the ﬁxed point of. Classiﬁcation ofincludes Siegel points （the Fatou set）or

the periodic orbit is the same as that of the ﬁxed

Cremer points（the Julia set）

point discussed above（e.g., Beardon, 1991, p.91;（e.g., Milnor , 2000, pp.43-44, pp.116-131; Ueda

Ueda et al., 1995, p.37）. In this case, however,

et al., 1995, p.12 for the Cremer points）.

the multiplier of the periodic point is defined
as λ=

.

By the Chain Rule, we have―9―

Naohito CHINO

Fig. 4. Trajectories of the two repelling periodic points
with period 2（­0.5 ）, and 6 repelling periodic points
with period 3 which constitute two groups .

Fig. 6. A trajectory after 54 iterations, whose
initial point is almost on the Julia set, that is,
0.700607338144023705­0.1 .

It is also well known that if f maps the Fatou

parabolic periodic points . In fact for a map ofcomponent U onto itself, then there are just four

degree d 㱢 2 the number of parabolic cycles plus

possibilities, as follows: Either U is the immediate

the number of attracting cycles is at most 2 d− 2basin for an attracting ﬁxed point, or for one petal（Milnor, 2000, p.140, p.112）.

of a parabolic ﬁxed point which has multiplier λHereafter, we will discuss the dynamical scenarios= 1 , or else U is a Siegel disk or Herman ring .

of Eq.（ 5 ）in two cases:

However, no polynomial map can have a Herman
Simulation Case 1 is the case whenring（e.g., Milnor, 2000, pp.152-153）.
Regarding the number of the above special sets,
it is known that there always exist inﬁnitely many
repelling cycles . Another important theorem isthat a rational map can have at most finitely many

that is,==0.25,+0.25.

This model is contained in the Mandelbrot set,
and is known as the cauliflower set .
Figure 1 shows the cauliflower set with aparabolic ﬁxed point and several repelling parabolic points. It is clear that all of these repelling
points as well as the parabolic point are containedin the cauliﬂower Julia set from the theorem on therelation between these points and the Julia set.
Figure 2 shows the trajectories of the systemwith several initial points.

It is apparent that

all the trajectories inside the Julia set except
converges to the para-bolic fixed point（0.5）
as time proceeds. To be precise, these curves aretangent to thereal axis. In contrast,

which islocated a bit to the right of the ﬁxed point on thereal axis diverges to infinity on the positive realaxis.
Fig.5. Trajectories of the three 4-cycles whose repelling
periodic points are on the Julia set, i.e., the cauliflower
set.

Figure 3 zooms in the parabolic ﬁx point to show
trajectories going in or out of it. This is the very
feature of the parabolic fixed point which is well

― 10 ―

Dynamical scenarios of changes

Fig. 7. Values of the real axis of the trajectory of the system whose initial value is almost
on a Julia set（cauliflower set）.

known elsewhere（e.g., Carleson & Gamelin, 1993,
p.37）.

Figure 7 depicts 54 values of the real axis of
the trajectory of the system whose initial value

Figure 4 shows trajectories of the two repelling
periodic points with period 2 （­0.5

is almost on a Julia set（cauliflower set）. It is）, and six

apparent that these points exhibit a random

repelling periodic points with period 3. It is apparent

behavior at a glance, reflecting the sensitivethat these six periodic points constitute two groups,

dependence on initial conditions. Values of theone being in the upper part and the other in theimaginary axis also behave randomly like those of

relatively lower part of this figure, both of which

the real axis.

constitute 3-cycles.

To sum up, deﬁning a new variable,

Figure 5 shows the three 4 -cycles whose twelve
repelling points are on the Julia set, that is, the, and considering a new system which is
obtained by transforming

cauliﬂower set.

linearly, that is,（the cauliflower set）
, its

Figure 6 is a trajectory after 54 iterations,whose initial point is almost on the Julia set,

solution curves can be classified as follows,
depending on the initial value:that is, 0.700607338144023705­0.1 . We definesuch a point as the one just before the trajectory（ 1 ）In the case when the initial point isfalls inside the Julia set visually, and is actuallyoutside the Julia set（the cauliflower）:

computed by a program with high precision like

The solution curves diverge to inﬁnity.

MATLAB.

Fig. 8. Julia set for
0.7400145000 .

（ 2 ）In the case when the initial point is on=­0.1200138225+

Fig. 9. Trajectories of the one repelling 2-cycle, one
attracting 3-cycle, and one repelling 3-cycle, with two
repelling fixed points.

― 11 ―Naohito CHINO

the Julia set:

system.

2 - 1 ）when it is on the parabolic fixed

above case is merely one of the examples of thepoint（0.5）:

It should be noticed further that theMandelbrot set. Depending on the parameter

The solution curves continue tostay in the ﬁxed point.

of Eq.（ 5 ）, we encounter inﬁnitely many dynamical
scenarios of solution curves of this equation which

2 - 2 ）when it is on the repelling periodic
points with period 2 or 3 :is the transformed quadratic system of the original
dyadic system, Eq.（ 4 ）.

The solution curves continue
to go around the corresponding
periodic points.
（In Figs. 4 to 5 we depicted up to
4 -cycles. Theoretically, infinitely
many repelling points exist.）
2 - 3 ）when it is on the Julia set excluding
the parabolic fixed point and therepelling periodic points:
The solution curves continue tomove irregularly on the Julia set.

Fig. 11. Trajectory starting from an initial point in asmall bud inside the Julia set, which is located almost at
the bottom of this figure.

Simulation Case 2 is the case when=

­0.1200138225+0.7400145000 , that is,=­0.1200138225+0.7400145000 （Figure 8 ）.
This system is almost the same as the case where=­0.12+0.74 , which is described by Peitgen

and Richter（1986, p.10, Fig. 4 ）. This system
has two repelling fixed points, thus these points
are on the Julia set. Moreover, it has one 2 -cycle
composed of two repelling periodic points. It has
two 3 -cycles, one of which is a 3 -cycle composedof three repelling periodic points, and the otherFig. 10. Trajectories of the three repelling 4-cycles.

of which is a 3 -cycle which is composed of three（ 3 ）In the case when the initial point is inside
the Julia set:

attracting periodic points. Figure 9 shows thesepoints. Furthermore, it has three 4 -cycles each

The solution curves converge to theof which are composed of four repelling periodic

parabolic fixed point in a direction

points. Figure 10 shows this. Figure 11 shows thetangent to the real axis.

trajectory starting from an initial point in a small
bud inside the Julia set, which is located almost at

It should be noticed from Simulation Case 1

the bottom of this ﬁgure.

that nonlinear systems exhibit extremely richer
behaviors of the solution curve than linear systems
discussed earlier, even in a quadratic dyadic― 12 ―

3 ．Discussion
In this paper we have examined some of theDynamical scenarios of changes

possible scenarios of our nonlinear differ-rence

Aizenberg, I. N., Aizenberg, N. N., & Vandewalle,equation models, especially those models on

J.（2000）. Multi-valued and universal binary
neurons ‒ Theory, Learning and Applications .

dyadic relations with a quadratic term described by
Eq.（ 4 ）in section 2. These models are composed

Boston: Kluwer Academic Publishers.

of two diﬀerence equations with a quadratic term

Barabási , A-L. & Albert, R.（1999）. Emergence ofin addition to linear terms, each of which describes

scaling in random networks. Science, 286, 509-

the changes in the coordinate of the corresponding

512.

object in a Hilbert space over time.

C h e s s o n , P. L . & Wa r n e r, R . R . （ 1 9 8 1 ）.

It is easy to show that these two equations canEnvironmental variability promotes coexistence

be converted into a complex diﬀerence equation,

in lottery competitive systems. American

Naturalist , 117, 923-943.

that is, Eq.（ 5 ）in section 2 , with a quadratic term
and a complex constant by taking the difference

Chino, N. （1978）. A graphical technique forbetween the coordinates of two objects. Since Eq.

representing the asymmetric relationships（ 5 ）includes a Mandelbrot set （e.g., Mandelbrot,

between N objects. Behaviormetrika , 5 , 23-40.

1977）as a special case（Chino, 2014）depending

Chino, N. （2014）. A general non-Newtonian

on the value

n-body problem and dynamical scenarios of, this system exhibits various

in the previous section, we can utilize various

solutions. Proceedings of the 42th Annual
Meeting of the Behaviormetric Society of Japan .

theoretical results established in the complex
dynamical system in examining these scenarios of

Chino, N.（2000）. Complex differential sy stem

scenarios of its solution curves.

As discussed

Sendai, Japan.
models of social interaction. Bulletin of theFaculty of Letters of Aichi Gakuin University ,.
We conducted a small simulation study inwhich two cases were examined. One is the casewhere

30, 43-54.

= 0.25 which constitutes the so-called

Chino, N. （2016）. A general non-Newtonian

cauliflower set, and the other is the case wheren-body problem and dynamical scenarios= ­0.1200138225+0.7400145000 which isof solutions. Paper presented at the 31th
International Congress of Psychology .

frequently cited elsewhere（e.g., Peitgen & Richter,1886）. As shown in the simulation study, even in

Yokohama, Japan.

the dyadic case, possible scenarios of an aspect

Chino, N., & Shiraiwa, K.（1993）. Geometricalof the special cases of our nonlinear difference

structures of some non-distance models forequation models are innumerable in contrast withasymmetric MDS. Behaviormetrika , 20, 35-47.

the linear diﬀerence equation models which were

Elaydi, S. N.（1996）. An introduction of diﬀerence

equations . New York: Springer.

discussed in a companion paper.

Harary, F.（1969）. Graph theory. Massachusetts:

Acknowledgements

Addison-Wesley.
Hirose, A.（1992）. Continuous complex-valued
back-propagation learning. Electronics Letters ,The author is indebted to Gregory L. Rohe for
proofreading of an earlier version of this paper.

28, 1854-1855.
Mandelbrot, B. B.（1977）. The fractal geometry

Referencesof nature . San Francisco: W. H. Freeman and

Aizenberg, N. N., Ivaskiv, Y. L., & Pospelov, D. A.

McCann, K., Hastings, A., & Huxel, G. R.（1998）.

（1971）. A certain generalization Of threshold

Weak trophic interactions and the balance offunctions. Dokrady Academii Nauk SSSR , 196,

nature. Nature , 395, 794-798.

Company.

1287-1290.

Ott, E., Grebogi, C., & Yorke, J. A. （1990）.

― 13 ―Naohito CHINO

Controlling chaos. Physical Review Letters, 64,
1196-1199.
Peitgen, H.-O., & Richter, P. H. （1986）. The

beauty of fractals . New York: Springer- Verlag.
Putzer, E. J. （1966）. Avoiding the Jordan
canonical form in the discussion of linear
systems with constant coefficients. American

Mathematical Monthly , 73, 2 -7.
S u k s m o n o , A . B . , & H i r o s e , A . （ 2 0 0 5 ）.
Beamforming of ultra-wideband pulses by a complex-valued spatio-temporal multilayer neuralnetwork. International Journal of Neural

Systems , 15, 85-91.
Takane, Y.（2015）. Analysis of brain connectivity
through fMRI data: Dynamic GSCA and GCANO.

Paper given as the Yanai Lecture at the 43 th
Meeting of the Behaviormetric Society of Japan .
September 3.
Watts, D. J. & Strogatz, S. H.（1998）. Collective
dynamics of ʻsmall-worldʼ networks. Nature ,393, 440-442.
（Final version submitted on December 26 , 2017）― 14 ―