IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS--I: FUNDAMENTAL THEORY AND APPLICATIONS,VOL. 43, NO. 6, JUNE 1996

Prof. A. Tesi for helpful comments and advise for preparing themanuscript.

491

Quasi-Synchronization Phenomena in
Chaotic Circuits Coupled by One Resistor
Yoshifumi Nishio and Akio Ushida

REFERENCES

HI I. W. Sandberg and A. N. Willson, Jr., “Some theorems on propertiesof dc equations of nonlinear networks,” Bell Syst. Tech. J., vol. 48, pp.
l-34, 1969.
PI L. 0. Chua and N. N. Wang, “On the application of degree theory
to the analysis of resistive nonlinear networks,” ht. J. Circuit Theory
Applicat, vol. 5, pp. 35-68, 1977.
I31 T. Roska, “Some qualitative aspects of neural computing systems,” inProc. 1988 IEEE ISCAS, Helsinki, 1989, pp. 751-754.
t41 D. Liu and A. N. Michel, “Asymptotic stability of systems operating on
a closed hvuercube.” Svst. Contr. Lett., vol. 19, pp. 281-285, 1992.
[51 M. Forti, S’.‘Manetti, id M. Marini, “Necessary &d sufficient condition
for absolute stability of neural networks,” IEEE Trans. Circuits Syst. Z,
vol. 41, pp. 491494, 1994.
[61 M. Forti, “On global asymptotic stability of a class of nonlinear systems
arising in neural network theory,” J. Differential Equations, vol. 113,
no. 1, pp. 246-264, 1994.
[71 E. Kaszkurewicz and A. Bhaya, “On a class of globally stable neuralcircuits,” IEEE Trans. Circuits Syst. Z, vol. 41, pp. 171-174, 1994.
PI M. ,Forti and A. Tesi, “New conditions for global stability of neural networks with application to linear and quadratic programming problems,”
IEEE Trans. Circuits Syst. I, vol. 42, pp. 354-366, July 1995.
191 D. W. Tank and J. J. Hopfield, “Simple ‘neural’ optimization networks:
An A/D converter, signal decision circuit, and a linear programming
network,” IEEE Trans. Circuits Syst., vol. 33, pp. 533-541, 1986.
UOI L. 0. Chua and L. Yang, “Cellular neural networks: Theory,” IEEE
Trans. Circuits Syst., volr 35, pp. 1257-1272, 1988.
1111J. Li. A. N. Michel. and W. Porod, “Analvsis and synthesis of a classof neural networks: Linear systems operatjng on a Closed hypercube,”
ZEEE Trans. Circuits Syst., vol. 36, pp. 1405-1422, 1989.
[I21 J. Cronin, Fixed Points and Topological Degree in Nonlinear Analysis.
Providence, RI: Amer. Math. Sot., 1964.
[I31 M. M. Green and A. N. Willson, Jr., “(Almost) half of any circuit’s
operating points are unstable,” IEEE Trans. Cinruits Syst. I, vol. 41, pp.
286-293, 1994.
[141 J. J. Hopfield, “Neurons with graded response have collective computational properties like those of two-state neurons,” Proc. Nat Acad. Sci.,
vol. 81, pp. 3088-3092, 1984.
[I51 L. 0. Chua and T. Roska, “Stability of a class of nonreciprocal cellularneural networks,” IEEE Trans. Circuits Syst., vol. 37, pp. 1520-1527,
1990.
Cl61 D. Hershkowitz, “Recent directions in matrix stability,” Linear Algebra
Applicat., vol. 171, pp. 161-186, 1992.
[I71 S. H. Bang, B. J. Sheu, and J. C.-F. Chang, “Search of optimal solutions
in multi-level neural networks,” in Proc. 1994 IEEE ISCAS, London,
England, 1994, vol. 6, pp. 423-426.

Abstract- In this ‘brief, synchronization phenomena observed from
simple chaotic circuits coupled by one resistor are investigated. A simple
three-dimensional autonomous circuit is considered as a chaotic subcircuit’. By carrying out circuit experiments and computer calculations
for two, three or four subcircuits case, various interesting synchronization phenomena of chaos, which are different types from the results
reported before, are confirmed to be stably generated. Further, quasisynchronization of asmmetric chaos are investigated with attention on
the number of synchronization states.
I. INTRODUCTION

Coupled oscillators systems are good model to describe various
nonlinear phenomena in the field of natural science and a numberof excellent studies on mutual synchronization of oscillators havebeen carried out ([l]-[3]and therein). In [l], Kimura et al. investigated synchronization phenomena observed from two van der
Pol oscillators coupled by resistors and confirmed that the two
oscillators synchronized at the opposite-phase. They considered thatsuch synchronization occurred to minimize the current through thecoupling resistors. Later they investigated three oscillators case andconfirmed the generation of the three-ljhase synchronization [2]. Also,
we investigated the synchronization phenomena of many oscillatorswith strong nonlinearity coupled by one resistor [3].
On the other hand, many nonlinear dynamical systems in thevarious fields have been clarified to exhibits chaotic oscillations andrecently applications of chaos to engineering systems attract many
researchers’ attentions, for example, chaos noise generator ([4] andtherein), control of chaos [5], [6], synchronization of chaos [7]-[lo],and so on. Among the studies on such applications, synchronizationof chaotic systems or signals is extremely interesting, because thechaotic solution is unstable and small error of initial values must
be expanded as time goes. As far as we know, such phenomena
have been firstly reported to be generated in simple real circuits
by a group of Saito [7]. Since Pecora et al. have investigated suchphenomena theoretically [8], many papers have been published until
now. Further, the technique of synchronization of chaos is also
applied to realize secure communication systems using chaos ([l l]and therein). However, almost all studies on synchronization of chaos
treat only the case that chaotic signals generated from two identicalchaotic systems are synchronized at the in-phase. Namely, as far
as we know, another types of synchronizations of chaotic systems
have not been reported at all. Though secure communication systems
do not need another types of synchronizations, the investigation ofvarious synchronizations of chaotic systems will open the way toanother applications of chaos.
In this brief, we investigate quasi-synchronization phenomena
observed from simple chaotic circuits coupled by one resistor. In
this paper we use the term synchronization of chaos as follows.
Two chaotic signals (e.g., currents through inductors) which are
the function of continuous time S1 (t) and S,(t) are said to be
Manuscript received May 2, 1994; revised May 31, 1995. This paper wasrecommended by Associate Editor M. Ogorzalek.
The authors are with the Department of Electrical and Electronic Engineering, Tokushima University, Tokushima, 770 Japan.
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synchronized at the opposite phase, if S1 (t) + Sa (t) 21 0. For anothertypes of synchronization states rigorous definition of synchronization
seems to be extremely difficult. Then, we use the term quasisynchronization. The term means that a chaotic signal from one
subcircuit S;(t) and the other chaotic signal from the other subcircuit
S, (t) ‘have the relation such as IS; (t) k Sj (t - 2’) 1< E where aconstant T represents phase shift between two signals and a constant
E is much smaller than the average amplitude of each chaotic signal.
This is not mathematical definition and is used only for qualitative
explanation of the observed phenomena. We consider a simple threedimensional autonomous circuit as a chaotic subcircuit. This circuit
is a symmetric version of the chaotic circuit proposed by Inaba et al.
[12] and one of the simplest autonomous chaotic circuits. They usedideal piecewise linear model of. diodes in [12], but in this brief theV-I characteristics of the nonlinear resistor consisting of diodes areapproximated by a smooth function. This is more real than piecewise
linear approximation in the sense that every real elements in the natural field are not piecewise linear. By carrying out circuit experiments
and computer calculations for two, three or four subcircuits case,various interesting synchronization phenomena which are different
types from the results reported before, namely opposite-phase quasisynchronization of two chaos, three-phase quasi-synchronization of
three chaos and various kinds of quasi-synchronizations of four chaos,are confirmed to be stably generated. Such types of synchronization ofchaos have never been reported yet. Moreover, quasi-synchronizationof asymmetric chaos are investigated with attention on the numberof synchronization states.

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b
(a)

(b)

Cc)

Fig. 1. Circuit model. (a) Chaotic circuits coupled by *one resistor. (b)Synimetric chaotic attractor (T ~650 R ). (b) Coexistence of two asymmetricchaotic attractors (T =554 ? ). Horizontal: 0.4 mA/div. Vertical: 1 V/div.

II. CIRCUIT MODEL :The circuit model is shown in Fig. l(a). In our sysiem N samechaotic circuits are coupled by one resistor. Each chaotic subcircuit
is a symmetric version of the circuit model proposed by Inaba et
al. [12]. It consists of three memory elements, one linear negative
resistor and one nonlinear resistor, which is realized by connectingsome diodes, and is one of the simplest chaotic circuits. This circuit
exhibits bifurcation phenomena which are similar to those reported in[ 131, namely, bifurcation route from one-periodic attractor to chaotic
attractor is explained as follows:
One symmetric one-periodic attractor-(symmetry breaking transition) -+ Two asymmetric one-periodic attractors-(period-doublingroute) + Two asymmetric chaotic attractors~(symmetj
recoveringcrisis) -+ One symmetric chaotic attractor.
‘.
Fig. l(b) and (c) shows typical examples of chaotic attractors
obtained from the chaotic subcircuit. In the following circuit experiments, the values of the inductors and the capacitor in each chaotic
subcircuit are fixed and those values are measured as Li = 204.15
mH f 0.073%, Lz = 9.933 mH f 0.030% and C = 0.034 25 @F
f 0.29%.
At first, we approximate the IV characteristics of the nonlinear
resistor consisting of diodes by the following function.
Od(ik) = m.

(1)

By changing the variables and parameters

vk = azk

the circuit equation is normalized and described as
6k=/r(Zk+Yk)-ik-&Q
j=l

Sk = a{p(zkik+ Yk) -zk -(k = 1,2,.

= Xk + Yk

f(Yk)}. . , N)

(3)

where
(4)

f(Yk) = 6.

For computer calculations, in order to consider the difference ofreal circuit elements, (3) is rewritten as follows:

jZ1

L?ik = @(XLik+ Yk) -= (1 + awk)(xk

zk -

+ Yk)

f(Yk)}(k = 1,2,. . . : N).

(5)

In the following computer calculations, the parameter values corresponding to the inductors and the capacitor are fixed as cy = 20.6 and
AWk = O.OOS(k- 1) and (6) is calculated by using the Runge-Kutta
method with step size At = 0.01.
The circuit in Fig. l(a) can be regarded as a chaotic circuit versionof the coupled van der Pol oscillators in [l]-[3]. Hence, this coupledchaotic circuits are expected to minimize the current through thecoupling resistor (R) and to exhibit various types of synchronization
phenomena.
III.

Two

SUBCIRCUITS CASE

In this section, we consider the case of N = 2, namely only two
chaotic subcircuits are coupled by one resistor.
At first, fix ,L?’= 0.325(r = 650 0) and vary coupling parameter y (A). Fig. 2 shows computer calculated results and theIEEE TRANSACTIONSON CIRCUITS AND SYSTEMS-I:

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(a)

(a)

’

(b)

(1)

Fig. 3. Synchronization of two asymmetric chaos. y = O.lO(R = 190 62).
(a) Symmetric synchronization for p = 0.287 (r = 554 a). (b) Asymmetricsynchronization for p = 0.308 (r = 586 0). (1) zr versus zs. (2) zr
versus zr (3) 22 versus ~1. (4) Circuit experimental results. 11 versus Iz.
0.4 mA/div.

and the loss consumed by the coupling resistor may be negligible.
While for asymmetric synchronization, two chaotic signals cannot be
synchronized completely and hence some power must be consumed
by the coupling resistor. Therefore, in order to generate chaotic
attractors larger power must be supplied to the circuit by the negative
resistor.

(1)

(2)

(3)

(4)

Cd)

Fig. 2. Synchronization of two symmetric chaos. /3 = 0.325 (r = 650 a).
(a) y = 0.0 (I? = On). (b) y = 0.10 (R = 48R). (c)y = 0.22 (I3 = 2920). (d) y = 0.35 (R = 5400). (1) zr versus
2s. (2) zr versus zr. (3) 5s versus 2s. (4) Circuit experimental results.
II versus 12. 0.4 mA/div.

corresponding circuit experimental results. As shown in the figure
two subcircuits generate symmetric chaos for this parameter value.
However, as r(R) increases, chaotic signals obtained from two
subcircuits become to be synchronized at the opposite-phase. Wheny = 0.35 (R = 540 n), two chaotic signals seem to be completely
synchronized at the opposite-phase.
Second, let us investigate the synchronization of asymmetric chaos.
In this case there exist two different types of synchronization ofchaos as shown in Fig. 3. In the Fig. 3(a) two asymmetric chaos
located symmetrically with respect to the origin are completely
synchronized. We call this type of synchronization as symmetricsynchronization. While, in the Fig. 3(b) two chaos tends to besynchronized at the opposite-phase but they cannot completely,
because two asymmetric attractors are not located symmetrically.
We call this type of synchronization as asymmetric synchronization.
For small y values (namely when coupling is small), asymmetricsynchronization of chaos can coexist with symmetric synchronizationof chaos. However, for relatively larger y values as Fig. 3, the valueof ,8 at which asymmetric synchronization stably exist must be larger
than the value for symmetric synchronization. Namely, symmetricand asymmetric synchronizations cannot coexist for such y values.
We can explain this reason physically as follows. For symmetricsynchronization, two chaotic signals can be synchronized completely.
Hence, the current through the coupling resistor is extremely small

IV. THREE SUBCIRCUITS CASE
In this section, we consider the case of N = 3. In this case,three-phase quasi-synchronization of chaos can be observed.
Fig. 4 shows the computer calculated results and the corresponding
circuit experimental results. Each subcircuit exhibits symmetric chaos
as shown in Figs. l(b) and 2(d(2)(3)) for these parameter values. In
the two subcircuits case two symmetric chaos can be synchronizedcompletely for larger y value, while three symmetric chaos cannot be
synchronized completely even if y is large. The reason is explained
as follows. The circuit tends to be synchronized at the three-phase inorder to minimize the current through the coupling resistor. However,the sum of chaotic signals cannot be zero even if the phase-difference
between two signals is equal to 2~13, because each chaotic signal
is not completely sinusoidal. The three-phase quasi-synchronization
has two different phase-states, namely the phase of one phase-stateis arranged as { 1, 2, 3}, while the phase of the other phase-stateis arranged as { 1, 3, 2). We have also confirmed that one-periodic
attractors bifurcate to chaotic attractors by varying the value of p
keeping the three-phase quasi-synchronization as well as the two
subcircuits case.
Next, let us consider three-phase quasi-synchronization of three
asymmetric chaos. Because each subcircuit has two asymmetricchaotic attractors located symmetrically with respect to the origin
as shown in Fig. l(c), there exist 23 I= 8 different combinations of
attractors coexist for three subcircuits. Further, each synchronization
has two different phase-states as well as symmetric attractors case({ 1, 2, 3) and { 1, 3, 2)). Totally, 16 different synchronization states
coexist. Fig. 5 shows two different synchronization states for threephase quasi-synchronization. Note that when three attractors obtained
from three subcircuits have the same shape, attractor is one-periodic
as Fig. 5(a). There coexist four such synchronization states. While
attractor is chaotic for the other 12 cases as Fig. 5(b). This reason
is explained as well as the asymmetric synchronization for two

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(4

(2)

(3)
(b)

@I

Fig. 4. Three-phasequasi-synchronizationof three symmetric chaos. (a)Computer calculated results. p = 0.305 and y = 0.05. (al) ~1 versus~2.
(a2) ~1 versus~3. (a3) Time-waveforms.(b) Circuit experimentalresults.T =650 s1 and R = 50.8 0. (bl)(b2) 0.4 mA/div. (b3) Horizontal: 0.2 ms/div.
Vertical: 1 mA/div.

Fig. 6. Quasi-synchronizationof four symmetricchaos(computercalculated
results)./3 = 0.30. (a) In and opposite-phasesquasi-synchronizationfor y =0.40. (b) Two pairs of opposite-phasequasi-synchronizationsfor and y =0.10.

opposite phase. There coexist three different’ phase-states for thistype of quasi-synchronization.
Two pairs of opposite-phase quasi-synchronizations are shownin Fig. 6(b). Two of four subcircuits are almost synchronized at
the opposite-phase. Also the remaining two subcircuits are almost
synchronized at the opposite-phase. However, two opposite-phase(4
quasi-synchronizations are independent. Before we carried out experiments, we had considered that the number of the combination of
subcircuits must be three because of the symmetry of the coupling,
namely (1, 2) and (3, 4}, {1,3 } and (2, 4) and (1, 4) and (2,3). However, the combination seems to be decided by the differenceof real circuit elements and we observed only one combination for.both of cbmputer calculations and circuit experiments. For computer(b)
Fig. 5. Three-phasequasi-synchronizationof three asymmetricchaos(com- calculations x1 and .XZ are synchronized at the opposite phase and ~3
puter calculated results). p = 0.287 and y = 0.05. (a) One-period. (b) and ~4 are also synchronized, but 21 and ~3 are independent. Other
Chaos.
combination states can be observed only in the long transient states
and they are eventually attracted to the state in Fig. 6(a). For circuit
experiments 11 and 1s are synchronized at the opposite phase and 12
subcircuits case. Namely, because the current through the couplingand 14 are also synchronized, but 11 and 1~ are independent. Other
resistor for Fig. S(a) is larger than that for Fig. 5(b). As y increases,
combination states are not observed.
the synchronization states shown in Fig. 5(a) also bifurcates to chaos.
Self-switching of three opposite-phase quasi-synchronizations is
For such parameter values, the attractor corresponding to Fig. 5(b)that three phase-states corresponding to the two pairs of oppositebecomes to be unstable and there exist only four synchronization
phase quasi-synchronizations are switched alternately. The order ofstates.
the appearance of three phase-states is truly chaotic. Further switching
period is also chaotic, namely a state may be switched to the next
V. FOUR SUBCIRCUITS CASE
state instantly and a state may be switched after about one second.
In this section, we consider the case of N = 4.
Because of the difference of real circuit elements, stability of threeAt first, let us investigate the quasi-synchronization phenomena phase-states must be different and the difference must influence theof symmetric chaos. In this case we observed three different types switching period. We also observed such quasi-synchronizations fromof quasi-synchronization phenomena; in and opposite-phases quasi- computer calculations. The detailed results on this phenomenon will
be discussed elsewhere.
synchronization, two-pairs of opposite-phase quasi-synchronizations,
Next, let us investigate the quasi-synchronization phenomena ofand self-switching of three opposite-phase quasi-synchronizations.
asymmetric chaos. In this case we observed only the two-pairs ofThese three types of q&i-synchronizations
were observed for different parameter values and they cannot coexist.
opposite-phase quasi-synchronization. The in and opposite-phases
In and opposite-phases quasi-synchronization is observed for relor the self-switching cannot be observed. As well as symmetricatively larger y values. This quasi-synchronization is shown in chaos, the combination ‘the synchronized subcircuits seems to be
decided by the difference of real circuit elements and we observed
Fig. 6(a). If we take the subcircuit one as reference circuit, one ofremaining threk subcircuits is almost synchronized at the in-phase only one combination; { 1, 2) and (3, 4). Because two asymmetricand the remaining two subcircuits are almost synchronized at the attractors coexist in each subcircuit, we had considered that there

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(4

Cd)

(e)

Fig. 7. Two pairs of opposite-phase quasi-synchronizations of four asymmetric chaos (computer calculated results). /3 = 0.285 and y = 0.10. (a) and (b)Two pairs of asymmetric synchronizations. (c)-(f) Two pairs of symmetric synchronizations.

exist 24 = 16 different synchronization states. However, we observed only six different synchronization states for both of computer
calculations and circuit experiments as shown in Fig. 7. The six
different synchronization states are divided into two groups, namely
two pairs of asymmetric synchronizations [Fig. 7(a) and (b)] and twopairs of symmetric synchronizations [Fig. 7(c)-(f)]. As well as two
subcircuits, these two different types of quasi-synchronizations cannot
coexist for larger y values.
VI.

three or four subcircuits case, we confirmed that various kinds ofquasi-synchronization phenomena of chaos were stably observed.
We would like to emphasize that almost synchronization phenomena in this paper had not been reported yet. Moreover such interesting
phenomena have been observed from real circuit model made up
easily.
We enumerate the synchronization phenomena introduced in thispaper. The numbers in ( ) denote the number of the coexisting
synchronization states.
2 subcircuits casel

CONCLUSION

In this brief, we investigated quasi-synchronization phenomena
observed from simple chaotic circuits coupled by one resistor. By
carrying out circuit experiments and computer calculations for two,

4
b)
cl

Opposite-phase synchronization of symmetric chaos (1)Symmetric synchronization of asymmetric chaos (2)Asymmetric synchronization of asymmetric chaos (2)IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-~ FUNDAMENTAL THEORY AND APPLICATIONS,VOL.43, NO. 6, JUNE 1996496

l

3 subcircuits case4

Three-phase quasi-synchronization of symmetric chaosb)

Three-phase quasi-synchronization of asymmetric chaos(same shape 4, not same shape 12)(2)l

4 subcircuits case
4
b)
c)
d)
e)

In and opposite-phases quasi-synchronization of symmetric chaos (3)Two pairs of opposite-phase quasi-synchronizations ofsymmetric chaos (1)Self-switchingof
three opposite-phase quasisynchronizations of symmetric chaos (1)Two pairs of asymmetric synchronization of asymmetricchaos (2)
Two pairs of symmetric synchronization of asymmetricchaos (4)For five or more subcircuits case, we could not observe any
synchronization phenomena of chaos. We have confirmed that fiveor more van der Pol oscillators could not be synchronized when
the nonlinearity is weak [3]. Because the chaotic signals obtained
from the present subcircuit are similar to sinusoidal wave, five or
more chaotic subcircuits could not be synchronized. If we use circuits
generating chaotic waveform looks like rectangular wave, they may
be synchronized.
Though we omit to introduce some results for another chaotic
circuits in this paper, we have carried out circuit experiments for some
types of chaotic circuits and have confirmed the generation of similartypes of quasi-synchronization of chaos. Some examples of the asymmetric synchronization and the three-phase quasi-synchronization
for another chaotic circuit can be seen in [14]. Hence, various
interesting quasi-synchronization phenomena introduced in this brief
are considered to be generated in various coupled systems. Namely,
quasi-synchronization phenomena of chaos are not special phenomena
observed from only a few systems, but common phenomena as wellI as chaos.
We consider that dimension of chaotic attractors must be deeply
related with quasi-synchronization of chaos and that it will bevery useful to classify quasi-synchronization’ of chaos. Hence, theanalysis of Lyapunov exponents of quasi-synchronization of chaos isextremely important as our future research. Further, we must establish
the method for theoretical analysis of the quasi-synchronizations ofchaos. We hope that our study would motivate the establishment ofanalyzing method for-quasi-synchronizations of chaos and that thephenomena in this brief would be applied to various engineeringsystems.
ACKNOWLEDGMENT

The authors would like to thank Professor S. Mori of Keio
University and Associate Professor T.- Saito of Hosei University for
their valuable comments and encouragement.
REFERENCES[I]

H. Kimura and K. Mano, “Some properties of mutually synchronizedoscillators coupled by resistances,” Trans. IECE, vol. 48, no. 10, pp.
1647-1656, Oct. 1965.
[2] , “Three-phase synchronization in resistively coupled oscillators,”
Tech. Rep. IECE, vol. 2, no. 21, pp. l-23, Feb. 1965.

[31 S. Moro, Y. Nishio, and S. Mori, “Synchronization phenomena inoscillators coupled by one resistor,” IEICE Trans. Fundamentals, vol.
E78-A, no. 2, pp. 244-253, Feb. 1995.
r41 K. Suzuki, Y. Nishio, and S. Mori, “Design of the noise generator using
chaotic circuit,” in Proc. ECCTD’93, Sept. 1993, pp. 849-854.
[51 E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev.
L&t., vol. 64, pp. 11961199, 1990.
WI E. R. Hunt, “Stabilizing high-period orbits in a chaotic system: the dioderesonator,” Phys. Rev. Len., vol. 67, pp. 1953-1955, 1991.
[71 Y. Ohmori, M. Nakagawa, and T. Saito, “Mutual coupling of oscillatorswith chaos and period doubling bifurcation,” in Proc. ISCAS’86, May
1986, pp. 61-64.
@I L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,”
Phys. Rev. Lett., vol. ,64, no. 8, pp. 821-824, 1990.
[91 V. S. Anishchenko, T. E. Vadivasova, D. E. Postnov, and M. A.
Safonova, “Synchronization of chaos,” ht. J. Bijiuc. Chaos, vol. 2, no.
3, pp. 633-644, 1992.
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Circuits Syst. I, vol. 40, pp. 693-699, Oct. 1993.
H11 T. L. Carroll, “Communicating with use of filtered, synchronized,
chaotic signals,” IEEE Trans. Circuits Syst. I, vol. 42, pp. 105-110,
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WI N. Inaba and S. Mori, “Chaotic phenomena in circuits with a linear
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v31 Y. Nishio, N. Inaba, S. Mori, and T. Saito, “Rigorous analyses ofwindows in a symmetric circuit,” IEEE Trans. Circuits Syst., vol. 37,
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637-642.

Robustness of Pole-Clustering
for Structured Perturbation

in a Ring
Systems

Chun-Hsiung Fang and Chun-Lin Lu

Abstract-A simple approach is proposed to ensure that all poles of theuncertain system are clustered in a prescribed ring. The explicit bounds
on linear time-invariant structured perturbations are obtained. Under
these allotiable highly structured perturbations, both stability robustness
and certain performance robustness will thus be ensured. In the literature,
as far as we are aware, little effort has been devoted to investigating
pole-clustering robustness in such region.
NOTATIONSFirst of all, we introduce some notations which will be usedthroughout this brief.
field of complex numbercthe ith eigenvalue of M E C” xnk(M)spectral radius of matrix M E C- ’ nP(M)determinant of matrix M E C”‘”magnitude of z E C
the (i, j) th element of matrix M
Manuscript received March 24, 1993; revised August 25, 1994. This work
was supported by the National Science Council of Taiwan, R.O.C. underGrant NSC-82-0404-E-151-025. This paper was recommended by Editor J.
A. Nossek.
The authors are with the Department of Electrical Engineering, National
Kaohsiung Institute of Technology, Kaohsiung 807, Taiwan, R.O.C.
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