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An Experimental Setup for Studying the Effect of Noise on Chua's Circuit - Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 46, NO. 4, APRIL 1999
517
Since by Lemma 2.1 the system states
n(1) and N
g
(1) are nonneg-
ative, we can write (37) as
_
W (X)  0 minfbN
a
+ c; d; 2eg
1
N
a
2(1 0 ) (2n + N
g
) + 12 (N
a
0 N
a
)
2
:
(38)
Thus, we have the linear differential inequality
_
W (t)  0
W (t)
(39)
where
:= minfbN
a
+ c; d; 2eg > 0:
(40)
By a comparison theorem given in [15, p. 2] or [16, p. 3] we conclude
that
W in (39) satises
W (X)  W (X
0
) exp(0
t)
(41)
for all
t  0. Therefore, W (X) tends to zero exponentially, and
consequently
X tends to X
e
.
IV. C
ONCLUSIONS
In this paper, we considered the rate equations of passively
Q-switched lasers. We showed that for nonnegative and bounded
inputs, the laser output is bounded. Furthermore, we showed that
when the input is switched off, the laser output converges to zero
asymptotically.
The dynamics of passively
Q-switched lasers are very well rep-
resented by the system (1), to which step inputs are applied. In
Section II-B, on the one hand, we showed that for some values of
the laser parameters and the input amplitudes, the equilibrium points
of the system (1) can be unstable. On the other hand, we showed
that the system states are bounded, regardless of the values of the
laser parameters and the input amplitudes. The instability of the
equilibrium points and the boundedness of the system states suggest
that the system states can possibly have periodic behavior, which
is known as self-pulsation. We have simulated the system (1) for
longer periods of time and have observed that the system states
behave periodically for certain values of the laser parameters and
the input amplitudes. However, we have not been able to prove
the existence of periodic behavior rigorously, which is a difcult
task. The difculty stems from the fact that almost all existing
mathematical techniques by which the existence of limit cycles
is established, such as PoincareBendixon theorem, are for two-
dimensional (planar) systems. From the practical point of view, it
is important to establish the BIBO stability of the laser, which was
achieved in this paper.
R
EFERENCES
[1] Y.-K. Kuo, M.-F. Huang, and M. Birnbaum, Tunable Cr
4+
: YSO
Q-
switched Cr : LiCAF laser, IEEE J. Quantum Electron., vol. 31, pp.
657663, 1995.
[2] Y.-K. Kuo and M. Birnbaum, Characteristics of ruby passive
Q
switching with a Dy
2+
: Ca F
2
solid-state saturable absorber, Appl.
Opt., vol. 34, pp. 68296833, 1995.
[3] Y. K. Kuo, M. Birnbaum, F. Unlu, and M.-F. Huang, Ho :CaF
2
solid-
state saturable-absorber
Q switch for the 2-m Tm, Cr : Y
3
Al
5
O
2
laser, Appl. Opt., vol. 35, pp. 25762579, 1996.
[4] A. E. Siegman, Lasers.
Mill Valley, CA: University Science, 1986.
[5] A. Yariv, Quantum Electronics, 3rd ed.
New York: Wiley, 1989.
[6] N. B. Abraham, L. A. Lugiato, and L. M. Narducci, Eds., Feature issue
on instabilities in active optical media, J. Opt. Soc. Amer. B, Opt. Phys.,
vol. 2, pp. 5201, 1985.
[7] N. B. Abraham, F. T. Arecchi, and L. A. Lugiato, Eds., Instabilities and
chaos in quantum optics II, in Proc. NATO Advanced Study Institute
Instabilities Chaos Quantum Optics, Il Ciocco, Italy, 1987.
[8] F. T. Arecchi and R. G. Harrison, Eds., Instabilities and Chaos in
Quantum Optics.
New York: Springer-Verlag, 1987.
[9]
, Selected Papers on Optical Chaos (SPIE Milestone Series).
Bellingham, WA: SPIE, 1994.
[10] R. W. Boyd, M. G. Raymer, and L. M. Narducci, Eds., Optical insta-
bilities, in Proc. Int. Meeting Instabilities Dynamics Lasers Nonlinear
Optical Systems.
Cambridge: Cambridge Univ. Press, 1986.
[11] E. Hofelich-Abate and F. Hofelich, Time behavior of a laser with a
saturable absorber as
Q switch, J. Appl. Phys., vol. 39, pp. 48234827,
1968.
[12] Y. I. Khanin, Ed., Principles of Laser Dynamics.
Amsterdam, The
Netherlands: Elsevier, 1995.
[13] P. Mandel, Theoretical Problems in Cavity Optics.
Cambridge: Cam-
bridge Univ. Press, 1997.
[14] C. O. Weiss and R. Vilaseca, Dynamics of Lasers.
New York: VCH,
1991.
[15] D. Bainov and P. Simeonov, Integral Inequalities and Applications.
Dordrecht, The Netherlands: Kluwer, 1992.
[16] V. Lakshmikantham, S. Leela, and A. A. Martynyuk, Stability Analysis
of Nonlinear Systems.
New York: Marcel Dekker, 1989.
[17] H. K. Khalil, Nonlinear Systems, 2nd ed.
Upper Saddle River, NJ:
PrenticeHall, Inc., 1996.
[18] L. Perko, Differential Equations and Dynamical Systems, 2nd ed.
New
York: Springer-Verlag, 1996.
An Experimental Setup for Studying the
Effect
of Noise on Chuas Circuit
Esteban S´anchez, Manuel A. Mat´as, and Vicente P´erez-Munuzuri
Abstract The analog simulation of nonlinear dynamical systems is
advantageous in some cases, i.e., when compared with the study using
digital computers and, in particular, when one wishes to investigate the
role of noise in these systems. In the present work we introduce two
different methods of introducing a noise component in the most widely
used chaotic circuit, namely, Chuas circuit, and apply these methods to
study the effect of noise on identically driven chaotic circuits.
Index TermsChaos, Chua circuit, noise, nonlinear circuits, stochastic
systems.
I. I
NTRODUCTION
The study of the behavior of nonlinear dynamical systems under
the inuence of noise is usually a difcult task [1]. Normally, one
performs this type of study through digital simulation. However, noise
usually interacts with numerical methods in a complicated manner,
Manuscript received December 3, 1997; revised March 10, 1998. This work
was supported in part by DGES (Spain) under Grants PB95-0570 and PB96-
0937 and by Xunta de Galicia and Junta de Castilla y Le´on under Grants
XUGA-20602B97 and SA81/96. This paper was recommended by Associate
Editor V. P´erez.
E. S´anchez is with Escuela T´ecnica Superior de Ingenier´a Industrial,
Universidad de Salamanca, E-37700 B´ejar (Salamanca), Spain.
M. A. Mat´as is with F´sica Te´orica, Facultad de Ciencias, Universidad de
Salamanca, E-37008 Salamanca, Spain.
V. P´erez-Munuzuri is with the Group of Nonlinear Physics, Faculty
of Physics, University of Santiago de Compostela, E-15706 Santiago de
Compostela, Spain.
Publisher Item Identier S 1057-7122(99)02759-2.
10577122/99$10.00
©
1999 IEEE 518
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 46, NO. 4, APRIL 1999
often posing a difcult challenge in the study of these stochastic
differential equations [2], while the generation of high-quality random
numbers in digital devices may prove difcult [3]. One conclusion is
that the results of these studies with digital devices are not always
clear cut. Thus, in many cases, analog simulations have practical
advantages [4] as noise comes from a physical (random) process and
has the appropriate correlation properties, and no ambiguities in the
formulation of the system exist as one is studying a real physical
process. This fact also implies that there are no doubts in the reality
of the phenomenon under consideration, its robustness against small
parameter mismatches, etc. Moreover, one can easily study the effect
of parameter variation by simply tuning a knob.
A topic that has received some attention in the last few years is
that of the effect of noise in (uncoupled) chaotic systems. When
trying to study this effect in the case of Chuas circuit, we found that
introducing an external source of noise, e.g., in the voltage across
one of the capacitors of the circuit, was less trivial than expected,
although we were able to develop various techniques for this purpose.
Our research on this topic started after Maritan and Banavar [5]
suggested that identical chaotic systems that are subject to the same
noise do synchronize. In this paper, we shall show our results obtained
from experiments with analog circuits (Chuas circuits) subjected to
external noise [6].
II. E
XPERIMENTAL
S
ETUP
The aim of this work is to present some suggestions regarding the
implementation of two electronic setups used to introduce noise in
analogue circuits. In this case we want to introduce the same noise
in two identical chaotic circuits to observe if there is synchronization
between them. Our experiments are based on Chuas circuit, a
well-known paradigm of nonlinear analog circuits exhibiting chaotic
behavior [7]. It is dened by the following evolution equations:
C
1
dV
1
dt =
1
R (V
2
0 V
1
) 0 h(V
1
)
C
2
dV
2
dt =
1
R (V
1
0 V
2
) + i
L
Ldi
L
dt = 0V
2
0 r
0
i
L
(1)
where
V
1
and
V
2
; the voltages across C
1
and
C
2
; respectively, and
i
L
; the current through L; are the three variables that describe the
dynamical system. The three-segment piecewise linear characteristic
of the nonlinear resistor is dened by
h(V