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PAPER

Chaotic Wandering and Its Analysis in Simple Coupled
Chaotic Circuits
Yoshifumi NISHIO†a) and Akio USHIDA†b) , Regular Members

SUMMARY In this paper, four coupled chaotic circuits generating four-phase quasi-synchronization of chaos are proposed.
By tuning the coupling parameter, chaotic wandering over thephase states characterized by the four-phase synchronization occurs. In order to analyze chaotic wandering, dependent variables
corresponding to phases of solutions in subcircuits are introduced.
Combining the variables with hysteresis decision of the phasestates enables statistical analysis of chaotic wandering.
key words: chaos, chaotic wandering, chaos synchronization,
chaotic circuits

1.

Introduction

Spatiotemporal phenomena observed in coupled chaotic
networks, namely coupled systems of many chaotic
cells, have attracted many researchers’ attention. The
studies on coupled chaotic networks are classiﬁed into
two categories; discrete-time systems and continuoustime systems. For discrete-time mathematical models,there have been numerous excellent results. Kaneko’s
coupled map lattice is the most interesting and wellstudied system [1]. He discovered various nonlinear
spatiotemporal chaotic phenomena such as clustering,
Brownian motion of defects and so on. Also Aihara’schaos neural network is the most important chaotic network from an engineering point of view [2]. His study
indicated a new possibility for engineering applicationsof chaotic networks, namely dynamical search of patterns embedded in neural networks utilizing chaotic
wandering. Furthermore, the application of chaos neural networks to optimization problems is widely studied ([3] and references therein). On the other hand,for continuous-time systems, several results on arraysof Chua’s circuits have been reported (e.g. some papers in [4]). However, many of these studies treated
only parameter values for which an isolated chaotic
cell does not generate a chaotic attractor; namely only
multiple stable sinks or multiple stable limit cycles.
Hence, the main subject of many studies has been the
wave propagation phenomenon observed for a given setof initial patterns and there are few studies on spatial patterns observed after vanishing eﬀects of theManuscript received February 28, 2001.
Final manuscript received August 20, 2001.
†
The authors are with the Faculty of Engineering, Tokushima University, Tokushima-shi, 770-8506 Japan.
a) E-mail: nishio@ee.tokushima-u.ac.jpb) E-mail: ushida@ee.tokushima-u.ac.jp

initial patterns. Namely, the pattern switching phenomenon caused by chaotic wandering, as observed in
Aihara’s discrete-time chaos neural network, has not
yet been studied well in continuous-time network models. Therefore, in order to ﬁll the gap between studies of
discrete-time mathematical abstract systems and studies of continuous-time real physical systems, it is important to investigate simple continuous-time coupledchaotic circuits generating chaotic wandering, clustering, pattern switching, and so on.
The authors have proposed continuous-time coupled chaotic circuit systems and have investigated thegeneration of spatial patterns and chaotic wanderingof spatial patterns [5]. Chaotic wandering in the caseof coexistence of asymmetric attractors [6] and controlof generating spatial patterns [7] have been also investigated. An important feature of the coupled circuits
was their coupling structure. Namely, four adjacentchaotic circuits were coupled by one resistor. Becausesuch a coupling exhibited quasi-synchronization withphase diﬀerence [8], [9], various spatial patterns could
be generated. This would be followed by the generationof several complicated spatiotemporal chaotic phenomena similar to those observed in discrete-time mathematical models. Therefore, the network based on the
coupled circuits would be a good model to clarify thephysical mechanism of spatiotemporal chaotic phenomena in continuous-time systems. However, because itis extremely diﬃcult to treat higher-dimensional nonlinear phenomena in continuous-time systems theoretically, we have to develop several tools to reveal theessence of the complicated phenomena.
In this paper, four-phase quasi-synchronization ofchaos and chaotic wandering over the phase states
characterized by the quasi-synchronization are reported
to be observed in simple coupled chaotic circuits.
Although the circuits in this paper are similar to
those proposed in [5], generating four-phase quasisynchronization is more stable and easier to deal withthan the two types of quasi-synchronizations reported
in [5], which were almost impossible to analyze. Furthermore, statistical analysis using dependent variables
corresponding to phases of the solutions in subcircuit
is carried out. We will be faced with a problem wherethe ﬁrst result will not describe the actual phenomenon
well. Careful investigation of the phase states reveals
that numerous inadequate decisions during transitions

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249


4


dxk


=
β(x
+
y
)
−
z
−
γ
xj
k
k
k



 dτ
j=1
dyk
= α{β(xk + yk ) − zk − f (yk )}


dτ




 dzk = x + ykkdτ(k = 1, 2, 3, 4)

cause the unexpected result. In order to avoid the inadequate decisions, two methods are introduced; introductions of intermediate phase states and hysteresis
decision. Computer simulated results show that thehysteresis decision makes the statistical analysis morereliable. Various characteristics of the chaotic wandering will be clariﬁed through the analysis.
2.

Circuit Model

where

Figure 1 shows the circuit model. In the circuit, four
identical chaotic circuits are coupled by one resistor R.
Each subcircuit is a three-dimensional autonomous one
and consists of three memory elements, one linear negative resistor and one diode. We can regard the diodes aspurely resistive elements, because their operation frequency is not too high. The coupling structure is symmetric in the sense that the exchange of any two subcircuits does not cause any change of the system structure.
Also the coupling is complete in the sense that a signalof one subcircuit can reach the others without passing
through the rest.
At ﬁrst, the i-v characteristics of the diodes are approximated by two-segment piecewise-linear functionsas1(1)
vd (ik ) = (rd ik + E − | rd ik − E | ) .
2
By changing the variables and parameters,CCE xk , ik =E yk , vk = E zk ,
Ik =
L1
L1

t = L1 C τ,

γ=R

C
,
L1

(3)

L1
α=
,
L2

δ = rd


β=r

C
,
L1

C
,
L1

(2)

f (yk ) = 0.5 (δ yk + 1 − | δ yk − 1 | ).

(4)

Note that when the coupling parameter γ, which is in
proportion to R, is equal to zero, the coupling termin (3) vanishes. Figure 2 shows a typical example ofchaotic attractors observed from the isolated subcircuit. Throughout this paper, let us ﬁx all of the circuit
parameters except the coupling parameter (γ and R)
for each subcircuit to produce the chaotic attractor in
Fig. 2; α=7.0, β=0.14 and δ=100.0 for computer calculations and L1 =100.7 mH, L2 =10.31 mH, C=34.9 nF
and r=334 Ω for circuit experiments. Moreover, for allof the computer calculations, the fourth-order RungeKutta method is used with step size h = 0.005.
3.

Four-Phase Quasi-Synchronization of Chaos
and Chaotic Wandering

We can observe four-phase quasi-synchronization ofchaos from the coupled circuits for a relatively wide
range of γ (or R). Because of chaotic oscillations, thesignals cannot synchronize completely. But we canclearly see that the signals from the four subcircuits
are synchronized with about 90◦ phase diﬀerences. Figures 3 and 4 show an example of the observed fourphase quasi-synchronizations of chaos. In the ﬁgures
the phase diﬀerences of x2 , x3 and x4 with respect tox1 are almost 90◦ , 180◦ and 270◦ , respectively.
Because of the symmetry of the coupling structure,

the normalized circuit equations are given as

Fig. 1

Circuit model.

Fig. 2 Typical example of chaotic attractors observed fromeach subcircuit. (a) Computer calculated result. xk vs. zk .
α=7.0, β=0.14, γ=0.0 and δ=100.0. (b) Circuit experimental result. Ik vs. vk . L1 =100.7 mH, L2 =10.31 mH, C=34.9 nF,
r=334 Ω and R=0.0 Ω. H: 0.8 mA/div. V: 1.3 V/div.

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Fig. 3 Four-phase quasi-synchronization of chaos (computer
calculated result). γ=0.30. (a) x1 vs. x2 . (b) x1 vs. x3 . (c) x1
vs. x4 . (d) x1 vs. z1 . (e) Time waveforms.

Note that one subcircuit should be a reference for thephase diﬀerence since the system is autonomous. It iseasy to observe all of the phase states in (5) by giving
proper initial conditions. (For circuit experiments, we
may have to repeat the on and oﬀ switching operationsof our power supply several times before we observe allof the phase states.)
On increasing γ (or R), we can observe chaotic
wandering over the six phase states of the four-phasequasi-synchronization. For such parameter values, allof the six phase states become unstable and the solution starts wandering over the six phase states. Although the wandering speed depends signiﬁcantly on
the parameter value, we could observe in the circuit
experiments that one phase state switches to anotherwithin one second or after 10 seconds. The wandering
is truly chaotic, i.e. we cannot predict when the next
switching will occur or which phase state will appear
next.
In order to show that the chaotic wandering exists
in the circuit, let us deﬁne the Poincaré section as z1 =0 and x1 < 0 and plot the values of xk (k=1, 2, 3,4) on xk − n (n denotes the number of iterations of
the Poincaré map) plane when the solution hits the
Poincaré section. Figure 5(a) shows time series xk (n)corresponding to the four-phase quasi-synchronizationof chaos in Figs. 3 and 4, while Fig. 5(b) corresponds
to chaotic wandering. In Fig. 5(a), each xk remains
in a certain range, while in Fig. 5(b), we can see xk
often changes its range in a complicated manner. Notethat x1 always remains in a certain range, because of
the deﬁnition of the Poincaré map. In the following
analysis, all of the results are based on the data on the
Poincaré map obtained by computer simulations.
4.

Analysis of Chaotic Wandering

4.1 Introduction of Phase Variables

Fig. 4 Four-phase quasi-synchronization of chaos (circuit experimental result). R=198 Ω. (a) I1 vs. I2 . (b) I1 vs. I3 . (c) I1
vs. I4 . (d) I1 vs. v1 . (e) Time waveforms. (a)–(c) 0.8 mA/div.
(d) H: 0.8 mA/div.
V: 1.3 V/div.
(e) H: 0.1 msec/div.
V: 2.0 mA/div.

six diﬀerent combinations of phase states coexist;
S1 : (0◦ , 90◦ , 180◦ , 270◦ ),S2 : (0◦ , 90◦ , 270◦ , 180◦ ),S3 : (0◦ , 180◦ , 90◦ , 270◦ ),S4 : (0◦ , 180◦ , 270◦ , 90◦ ),S5 : (0◦ , 270◦ , 90◦ , 180◦ ),S6 : (0◦ , 270◦ , 180◦ , 90◦ ).

(5)Although one can see that the phase states in Fig. 5(b)switch in an irregular manner, it is almost impossible
to understand the generating phenomenon completely.
Therefore, we introduce the following independent variables from the discrete data of xk (n) and zk (n) on the
Poincaré map.

zk+1 (n)
π − tan−1x
k+1 (n)
xk+1 (n) ≥ 0






 − tan−1 zk+1 (n)xk+1 (n)ϕk (n) = xk+1 (n) < 0 and zk+1 (n) ≥ 0





zk+1 (n)
2π − tan−1x
k+1 (n)
xk+1 (n) < 0 and zk+1 (n) < 0(k = 1, 2, 3.)(6)

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Fig. 6 Time series ϕk (n) calculated from the data in
Fig. 5. (a) Four-phase quasi-synchronization of chaos. γ=0.30.
(b) Chaotic wandering over diﬀerent phase states. γ=0.46.
Fig. 5 Time series xk (n). (a) Four-phase quasi-synchronization of chaos. γ=0.30. (b) Chaotic wandering over diﬀerent
phase states of four-phase synchronizations. γ=0.46.

Because the attractor observed from each subcircuit
is strongly constrained onto the plane yk = 0 when
the diode is oﬀ, these variables can correspond to thephase diﬀerences between the subcircuit 1 and the others. (Note that the argument of the point (x1 (n), z1 (n))
is always π, because of the deﬁnition of the Poincarémap.)
Figure 6 shows the time evolution of ϕk (n) calculated from the data in Fig. 5. In Fig. 6(a), ϕ1 (n) isalways around π/2, ϕ2 (n) is always around π and ϕ3 (n)is always around 3π/2, while in Fig. 6(b), ϕk (n) changes
its range in a complicated manner.
Using the independent variables in (6), we can givea precise deﬁnition of the six phase states in (5) as
follows,
S1 : ϕ 1 < ϕ 2 < ϕ 3 ,S2 : ϕ 1 < ϕ 3 < ϕ 2 ,S3 : ϕ 2 < ϕ 1 < ϕ 3 ,S4 : ϕ 2 < ϕ 3 < ϕ 1 ,S5 : ϕ 3 < ϕ 1 < ϕ 2 ,S6 : ϕ 3 < ϕ 2 < ϕ 1 .

(7)This makes it possible to decide in which phase states
the solution lies. This could be very useful for statisticalanalysis of chaotic wandering. For example, we cancheck when switchings of one phase state to anotheroccur, we can count how many switchings occur in acertain time interval, and so on.
Figure 7 shows how many switchings of phaseFig. 7

Switching number during one million iterations.

states occur during one million iterations of the
Poincaré map. The horizontal axis shows the coupling
parameter γ. As we can see, a large number of switchings are counted for γ > 0.45. Namely, for example, at
γ = 0.47 one switching occurs in every 10 iterations on
average. This number is too large and the result does
not describe the actual phenomenon well.
In order to determine the cause, we investigate
switchings in a certain interval carefully. Figure 8 shows
when switchings occur along with the time series ofϕk (n). In the ﬁgures, each vertical line in the bottom
diagrams indicates one switching. Figures 8(b) and (c)are the magniﬁcations of parts of Fig. 8(a). From thediagrams of ϕk (n) in Fig. 8(a), we can expect the phasestate in the interval [0, 4500] to be as S5 and the phasestate in the interval [6500, 15000] to be S6 . It should

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Fig. 10

Fig. 8 Investigation of switchings of phase states. γ=0.46.
(a) Switchings from S5 to S6 . (b) Magniﬁcation during the transition. (c) Magniﬁcation indicating momentary changes.

Probabilities of solutions being in Si and SIi .

ability that ϕi lies in the corresponding interval. The
left group corresponds to the minimum values of ϕi andthe right group corresponds to the maximum values of
ϕi . We can see from the ﬁgure that the probability
that ϕi remains around 0 is not small. (±30◦ almost
corresponds to the slots 1–5 and 60–64.) Hence, wemust consider that the situation in Fig. 8(b) occurs frequently. Namely, some methods should be applied to
avoid the inadequate decisions caused by the solutions
remaining around the above-mentioned hyper-planes.
4.2 Introduction of Intermediate Phase States
The number of inadequate decisions during transitions
would decrease if we regarded neighborhoods of hyperplanes where the solutions stay during transitions as
new phase states. Hence, the following three intermediate phase states are introduced,

Fig. 9 Probability distribution of ϕi . Horizontal slot m indicates interval [(m − 1)π/32, mπ/32]. γ=0.46.

SI1 : min{2π − ϕ1 , ϕ1 } < θI ∩ |ϕ2 − ϕ3 | < θI
SI2 : min{2π − ϕ2 , ϕ2 } < θI ∩ |ϕ1 − ϕ3 | < θI
SI3 : min{2π − ϕ3 , ϕ3 } < θI ∩ |ϕ1 − ϕ2 | < θI

be noted that it takes a relatively long time to ﬁnish
the transition in the interval [4500, 6500].
In contrast, the diagram of the switching indicatesanother phenomenon. Namely, from the start to theend of the transition a huge number of switchings aredetected. Further, several switchings are detected in
the interval without any transitions. Figure 8(b) showsthat during the transition, the solution stays in theneighborhood of a hyper-plane expressed as ϕ1 = ϕ2
and ϕ3 = 0. The hyper-plane corresponds to theboundary of the phase states S1 , S3 , S5 and S6 . Hence,
small ﬂuctuations around the hyper-plane cause a large
increase of the number of switchings. Figure 8(c) showsthat a momentary change of ϕk causes two consecutive
inadequate decisions.
Figure 9 shows the probability distribution of ϕi .
In the ﬁgure interval [0, 2π] is divided into 64 small intervals equally and the vertical axis indicates the prob-

where θI is a parameter deciding the size of the region of
the intermediate phase states. Note that the conditionsof the decisions corresponding to the original six phasestates in (7) are modiﬁed so that their regions do not
overlap with those of (8).
Figure 10 shows probabilities of the solutions being
in the phase states after introducing the intermediatephase states. Each plot indicates the averaged valueof the probabilities of the corresponding phase states,
which are given by computer simulations (10 million
iterations of the Poincaré map). As θI increases, theprobabilities corresponding to the intermediate phasestates become large. For example, for γ = 0.46 andθI = 45◦ , the sum of the probabilities corresponding to the three intermediate phase states is 3×0.0335,
namely the solution stays in the intermediate phasestates about 10% of the total time.
Figure 11 shows how the switchings change after
introducing the intermediate phase states for diﬀerent(8)

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253

Fig. 11 Switchings of phase states with three intermediate
states. γ=0.46. (a) θI = 60◦ . (b) θI = 75◦ . (c) θI = 90◦ .

values of θI . As we can see from the ﬁgures, the intermediate phase states make the situation worse with respect to the expectations. Undoubtedly, inadequate decisions during the transition (in [4500, 6500]) decrease
considerably as θI increases. However, the switchingnumber in the other intervals increases considerably.
As a result, the total number of switchings becomesless meaningful.
We have to conclude that introducing new boundaries of the phase states causes an increase of the inadequate decisions around the new boundaries. However, the fact that the solutions stay for a relatively
long time in the intermediate phase states is useful forunderstanding the phenomenon correctly.
4.3 Introduction of Hysteresis Decision
In order to reduce the inadequate decisions during transitions without inﬂuence on the other interval, we introduce a hysteresis feature into the decisions of the six
phase states in (7). Namely, we do not count switchings

Fig. 12 Switchings of phase states with hysteresis decision.
γ=0.46. (a) θH = 30◦ . (b) θH = 45◦ . (c) θH = 60◦ .

if the angle between any two phases of four is smaller
than θH . In other words, we make decisions of thephase states only if the following is satisﬁed;
min{ϕ1 , ϕ2 , ϕ3 , 2π − ϕ1 , 2π − ϕ2 , 2π − ϕ3 ,|ϕ1 − ϕ2 |, |ϕ1 − ϕ3 |, |ϕ2 − ϕ3 |} > θH

(9)

Figure 12 shows how the switchings change after
introducing the hysteresis decision for diﬀerent valuesof θH . We can see that even for small θH , the inadequate decisions decreases considerably. The hysteresis
decision can eliminate the inadequate decisions by the
momentary change as well as those during transitions
successfully.
Figure 13 shows how the switching numbers decrease by introducing the hysteresis decision. We canobserve from Fig. 13(b) that slopes of the curves changearound θH = 50◦ regardless of the coupling parameter.
Although the theoretical reason why the curves havethe turning points has not been clariﬁed, it may becaused by the existence of the two inadequate decisions
as shown in Figs. 8(b) and (c).

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Table 1 shows the probabilities of sojourn time ineach phase state for diﬀerent values of θH , which are
given by computer simulations (10 million iterations of
the Poincaré map). For example, for θH = 0◦ (i.e.
no hysteresis decision) the probability that the sojourntime is 1 is 78.2%. This means that the solution entering a phase state will exit to another phase state after
one step of the Poincaré map with probability 78.2%.
Furthermore, for θH = 0◦ the probability that the sojourn time is more than 1000 is 0.1%. Clearly, thisresult does not explain the phenomenon correctly. (Seethe horizontal scale of Fig. 5.) For θH ≥ 45◦ , the eﬀectof the hysteresis is clear. Especially, for θH ≥ 60◦ theprobability that the sojourn time is less than 100 is zeroand for θH = 75◦ the probability that the sojourn time
is more than 10000 is 37.8%. The distribution of theseprobabilities of the sojourn time is depicted in Fig. 14.
The slots in the horizontal axis of the ﬁgure denote

the ranges of the sojourn time and are summarized inTable 2.
Finally, Table 3 shows the averages of the sojourntime in each phase state for diﬀerent values of θH , which
are given by computer simulations. The average numbers of iterations are converted into average time using
the approximated value of the period (0.36 [msec]) in
Fig. 4(e). Although the corresponding data obtained
from real circuit experiments are not available, the results for θH ≥ 45◦ do not appear to be far from theobserved phenomenon.
5.

Conclusions

In this study, chaotic wandering observed in four coupled chaotic circuits has been reported and analyzed.

Fig. 14 Probability distribution of sojourn time. γ=0.46.
Ranges of slots are in Table 2. (a) θH = 30◦ . (b) θH = 45◦ .
(c) θH = 60◦ . (d) θH = 75◦ .
Table 2
Slot
1
2
3
4
5
6
7
8

Fig. 13 Switching number with hysteresis decision.
(a) Ordinary scale. (b) Logarithmic scale.

Table 1

Ranges of slots in Fig. 14.

Sojourn time (n)
1−2
3−5
6−10
11−20
21−50
51−100101−200
201−500

Probabilities of sojourn time for diﬀerent θH . γ=0.46.

Sojourn time (n)

0◦

15◦

30◦

1
1–10
1–100
1–10001–10000
1–∞

0.782
0.931
0.977
0.999
1.000
1.000

0.554
0.693
0.959
0.985
1.000
1.000

0.284
0.382
0.832
0.951
0.994
1.000

θH

45◦

60◦

75◦

0.000
0.022
0.177
0.611
0.919
1.000

0.000
0.000
0.000
0.137
0.761
1.000

0.000
0.000
0.000
0.017
0.622
1.000

Slot
9
10
11
12
13
14
15
16

Sojourn time (n)501−1000
1001−2000
2001−5000
5001−10000
10001−20000
20001−50000
50001−100000
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255
Table 3

Average of sojourn time for diﬀerent θH . γ=0.46.
θH
0◦
15◦
30◦
45◦
60◦
75◦

Average of sojourn time
iteration (n)time [sec]12.00.0043250.4
0.0181307.9
0.111
2697.80.971
7079.22.55
9915.9
3.57

In order to analyze chaotic wandering, dependent variables corresponding to phases of solutions in subcircuits
were introduced. Combining the variables with hysteresis decision of the phase states enabled statisticalanalysis of chaotic wandering.
Although the results obtained by introducing theintermediate phase states were not good, we will continue to work with the idea of the intermediate phasestates in the future, because the simulation results suggest that the solution stays for a long time in the intermediate phase states.
This is the ﬁrst step in the statistical analysis ofchaotic wandering in simple continuous-time circuits.
However, we believe this result would contribute to understanding the higher-dimensional nonlinear phenomena, because the phenomenon analyzed in this paper
would be observed in various other coupled chaotic circuits. At the same time, we feel that more detailed statistical analysis on the phenomenon should be carriedout and more complicated phenomena in larger sized
coupled circuits should also be investigated. Furthermore, some theoretical supports would make the results
more reliable, though this may be diﬃcult.
Acknowledgements
The authors would like to thank Professor Martin
Hasler of the Swiss Federal Institute of Technology,
Lausanne and Professor Takashi Hikihara of Kyoto University for their valuable comments.
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Yoshifumi Nishio
received the B.E.,
M.E., and Ph.D. degrees in electrical engineering from Keio University, Yokohama
Japan, in 1988, 1990, and 1993, respectively. In 1993, he joined the Department of Electrical and Electronic Engineering at Tokushima University, Tokushima Japan, where he is currently an
Associate Professor. From May 2000 hespent a year in the Laboratory of Nonlinear Systems (LANOS) at the Swiss Federal Institute of Technology Lausanne (EPFL) as a visiting professor. His research interests include analysis and application ofchaos in electrical circuits, analysis of synchronization in coupled oscillators circuits, development of analyzing methods fornonlinear circuits and theory and application of cellular neuralnetworks. Dr. Nishio is a member of the IEEE.

Akio Ushidareceived the B.E.
and M.E. degrees in electrical engineering from Tokushima University in 1961
and 1966, respectively, and the Ph.D. degree in electrical engineering from University of Osaka Prefecture in 1974. He wasan associate professor from 1973 to 1980
at Tokushima University. Since 1980 hehas been a Professor in the Department
of Electrical Engineering at the same university. From 1974 to 1975 he spent one
year as a visiting scholar at the Department of Electrical Engineering and Computer Sciences at the University of California,
Berkeley. His current research interests include numerical methods and computer-aided analysis of nonlinear system. Dr. Ushida
is a member of the IEEE.