presentacion - ieee · frexre et al.: periodicity and chaos 241 lv/clll lv/clll l- "2 i l...

,EEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-31, NO. 3, MARCH 1984

Periodicity and Chaos Electronic

231

A/r.vhstracr -In this paper, a simple electronic circuit is analyzed from a qualitative viewpoint. The circuit shows a great variety of dynamical behaviors (equilibrium points, periodic oscillations, chaotic motions.. .) and the analysis proceeds to catalog all of them through a bifurcation study (pitchfork and Hopf bifurcations, flip bifurcations...). This study points out the relevance of qualitative analysis in systems of simple structure but with very complex behavior. The paper includes theoretical study, numerical simulations, and actual circuit experimentation.

I. INTRODUCTION

N ONLINEAR dynamical systems can exhibit a great variety of behavior modes, some of them of a truly

complex nature. To study these complex behaviors the tools of the qualitative theory of dynamical systems are to be used. In this context it is especially interesting the bifurcation theory that allows to define the boundaries between regions of different qualitative behavior.

The complexity of behavior is related more to nonlinear- ity than to dimensionality. As far as electronic circuits include nonlinear components they can show very complicated behavior modes even if its structure is simple, as is the case of the circuit analyzed in this paper.

The behavior modes of a system are strongly related to their long time behavior; that is, to the structure of its attractors. If the attractors are equilibrium points one has the simplest kind of behavior that is found normally in practice. It sometimes happens that the attractors have a more complex structure: they can be closed orbits that correspond to periodic oscillations. Furthermore, attractors exist which are neither single points nor closed orbits, that are called strange attractors; the behavior associated to them is of a truly chaotic nature. These strange attractors cannot occur in linear systems nor in autonomous systems whose dimension is less than three, although they would appear to be quite common in nonlinear system of greater dimension [ 11.

To study the behavior modes real experiments and computer simulation have been used in this paper. Other tools such as Poincare maps and spectral analysis have been also used. In this manner images were obtained on an oscillo-

Manuscript received November 4, 1982; revised February 11, 1983. This work was supported in part, for L. G. Fran uelo and J. Aracil, by the Spanish Ministerio de Education y Ciencia un i er Program “Foment0 a la Investigation”.

E. Freire is with the Department of Mathematics, Escuela Superior de Ingenieros Industriales, University of Seville, Auda Reina Mercedes, Sevilla-12, Spain.

L. G. Franquelo and J. Aracil are with the Department of Automatic Control and Electronics, Escuela Superior de Ingenieros Industriales, University of Seville, Sevilla-12, Spain.

Fig. 1. Basic circuit. R, = R, = 4K7, R =lO K, Cc = 4n7, C =lOO n, L=llOmH:

in an Autonomous System

Nonlinear positive Nonlinear negative conductance conductance

: . ..-......____ ~ 8 6 RAlOO :

EMIL10 FREIRE, LEOPOLD0 G. FRANQUELO, AND JAVIER ARACIL, MEMBER, IEEE

scope screen or on a computer plot. In both cases the images conveyed relevant information about the qualitative behavior of the circuit. The quantitative values taken on by the circuit magnitudes are far less interesting than the qualitative aspects.

Although electrical systems engineers deal mainly with simple behavior mode systems, knowledge of the funda- mental mechanisms of chaotic behavior is of great interest to them as it helps in the design of some systems and allows certain undesirable performances to be corrected. Thus certain effects blamed on noise are really examples of chaotic behavior of a completely deterministic nature, and, therefore, the external noise is not always the cause of an undesirable behavior which is really inherent in the system design [ 11.

Fig. 1 shows the electronic system whose dynamics are to be analyzed. It consists of a resonant circuit and two nonlinear conductances, one negative and another positive. This circuit was proposed by Shinriki et al. [7] as a modified van der Pol oscillator which generates a “random” waveform. The analysis carried out by these authors con- sisted of a numerical simulation and some experiments; no theoretical study of the state equations has been achieved and even the empirical analysis done is incomplete, given that only a few of all the different and complicated behavior modes possible in the dynamics of this circuit were detected.

Fig. 2 shows the current-voltage characteristics of the two nonlinear elements, which are approximated by

i,(q) = - qq+ a,u:, a,>o, a,>0 0098-4094/84/0300-0237$01.00 01984 IEEE

238 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-31, NO. 3, MARCH 1984

Fig. 2. Current-voltage characteristics of the nonlinear elements.

for the negative nonlinear conductance, and by

for the positive nonlinear conductance. These approxima- tions are quite reasonable from the qualitative viewpoint. The state equation is written as

C~=-iL-G2u2-b1(u2-u1)-b3(u2-~1)3

di, Lx=%

with (ui, u2, iL) E R3. The electronic system dealt with in this article has cer-

tain characteristics which make it of special interest in relation to other known systems with chaotic behavior. In the first place it is a continuous system; the study of the chaos generated by discrete systems is simpler and applica- tions in the field of discrete-time feedback control systems have been carried out by Baillieul et al. [2]. Secondly the circuit is autonomous in contrast to the self-oscillatory circuit with periodic excitation studied by Ueda and Akamatsu [3]. In the periodically forced oscillatory systems one has a frequency injected from the outside, which imposes a defined temporal symmetry on such systems. It must finally be emphasized that the dimension of the system is 3 in contrast to the system of coupled oscillators studied by Gollub et al. [4] which introduced a state space of dimension 4. The three-dimensional character is very mteresting from the theoretical point of view, because it is the smallest possible dimension for an autonomous dynamical system with chaotic behavior [5], [6], [S], [9].

II. ANALYSIS OF THE STATES OF EQUILIBRIUM AND THEIR BIFURCATIONS

The study of circuit dynamics is started by analyzing the possible states of equilibrium, its stability, and the bifurcations which appear on varying the parameters corresponding to the circuit elements. The equilibria constitute the simplest cases of steady state, although through the study of its bifurcations it is possible to detect the existence of other more complicated dynamic behavior; the bifurcation

Fig. 3. Bifurcation diagram in the (p, 6) plane.

points of the equilibria in the parameter space act as “organizing centers” for system dynamics [ll].

With the change of notation, x = ur; y = u2; z = i,, the state equation is written as

C,k = (aI - G,)x - u3x3 + b,(y - x)+ b,(y - x)’

Cj=-G2y-z-b,(y-x)-b3(y-x)3 Li= y.

To study the behavior of the system only the parameters G, and G, will be varied, all the other parameters will be kept constant. This is a natural choice, because G, and G, are variable conductances. Due to the small value of b, the condition bf < Co/L is assumed. Let p = G, + b, - a, and 6 = G, + b, be the bifurcation parameters. It is always noted that 6 > 0.

It is clear that (O,O,O) is an equilibrium point for all values of the parameters. In order to analyze its stability and its possible bifurcations the corresponding characteristic exponents must be obtained as function of P and 6; i.e., it is necessary to compute the eigenvalues of the Jacobian matrix:

J(O,O,O)

= I -(G, + b, - d/G WC,

WC -(b,+G,)/C -p/C . 0 l/L 0 1

Fig. 3 shows the bifurcation diagram in the (p, 6) plane. For p > 0 (region I) the three characteristic exponents of (O,O,O) have negative real parts, which implies that the equilibrium (O,O, 0) is hyperbolic and asymptotically stable. For p = 0 (curve BP) this equilibrium is not hyperbolic, as one characteristic exponent is zero. For h > 0 (region II) one characteristic exponent is a positive real and the other two have negative real parts, which implies that the equilibrium (O,O,O) is hyperbolic and unstable. One is faced with a change in stability, which corresponds to a characteristic exponent crossing the imaginary axis; this situation leads one to consider the bifurcation of the equilibrium (O,O, 0) as BP is crossed from region I to region II.

Thus maintaining 6 constant, the characteristic exponent hr(p) changes from being negative for p > 0 to positive for p < 0 with A,(O) = 0. The other two characteristic exponents AZ(p), h3(p) stay in the left semiplane of the complex plane. At this point it is necessary to verify the Hopf

FREXRE et al.: PERIODICITY AND CHAOS 241

lV/Clll lV/Clll L-

"2 i L

IV/Cm 1 mA/cm

Fig. 8. Phase-space oscilloscope views of T, 2T, and 4T periodic oscillations.

kHz appear, which arise from the simultaneous Hopf bifurcations of the symmetrical equilibrium points (& X, f p, + Z). As the value of the parameter p decreases, always within region III, both amplitude and period increase.

If the value of F continues decreasing, curve Fl will be crossed giving rise to a “flip” bifurcation of closed orbits [13]. This bifurcation is analyzed in the case of the closed orbit corresponding to the equilibrium (+ X, + J, + Z); because of the existing symmetry the situation for the other closed orbit corresponding to the equilibrium (- X, - J, - Z) will be completely analogous.

When the bifurcation curve Fl is crossed, the T-periodic orbit in region III losses its stability, because a Floquet multiplier passes out of the unit circle through - 1, and a stable periodic orbit with approximately twice the period T arises. When studying periodic orbits the use of Poincare maps on a transversal section is normal. In this case, the plane y = 0 has been taken as the transversal section S; from the numerical solution of the state equation it is possible to compute the Poincare map on S. Thus, a point (x0,0, z,,) E S is taken, in such a way that at said point

jt > 0; with this point as the initial condition the corresponding orbit y(x,,O, zO) is obtained numerically. In the periodic behavior situation, the orbit y(x,,O, zO) will again intersect the section S at a point (x1,0, zr) with j, > 0; therefore, for the Poincare map P on S it will be had that fYx,,o, zo> = (x1,0, ZJ

Fig. 6(a) shows the projection in the plane z = 0 of a stable closed orbit of double period for ,ii = - 0.06 (region III,). Fig. 6(b) shows a representation of the discrete dynamics generated by the corresponding Poincare map; that is from an initial point (x0, 0, zO) the successive images P”(x,,,O, zO), n = 1,2, . . . , are obtained.

The stable periodic orbit in region III, which arose from the “flip” bifurcation, now experiences a new “flip” bifurcation on the curve F2 (see Fig. 5). That is, as p decreases the system pass from region III, to region III, crossing the bifurcation curve F2; this leads to the loss of stability of the closed orbit of period 2T and the consequent appearance of a stable closed orbit of period 4T approximately. For F = -0.061 (region III, near to F2), Fig. 7 shows the situation. Experimental results are shown in Fig. 8.


(4 Fig. 12. 6T-periodic oscillation.

"1

iv/cm t “2

lV/Crn

"1

lV/cm I- iL

1 mA/cm

Fig. 13. Phase-space oscilloscop: views. (a) Chaotic oscillation. (b) 3T- .t¶ :.A--. --


(4 (b) Fig. 15. Oscilloscope views. (a) Chaotic oscillation. (b) ST-periodic

oscillation.

of greater intensity, and the absence of sound corresponding to the frequencyf. It should be noted that we are faced with a “deterministic noise” produced by a chaotic attrac- tor of a dynamical system governed by ordinary differen- tial equations.

CONCLUSIONS

The analysis of an autonomous electronic circuit with a three-dimensional state space has been presented. The different dynamical regimes (periodic and chaotic), which are obtained when the parameters of the circuit are changed, has been described. The point of view adopted emphasized the qualitative aspects of the dynamics and takes bifurcations as a reference framework.

The study of the circuit has been based on experimental results and theoretical analysis. On the experimental side actual circuit experimentation and numerical simulation have been developed. In both cases pictures illustrating the dynamical behavior have been obtained. The work includes spectral analysis and Poincare maps and has been guided by the bifurcation analysis that allows the classification of the regions of different qualitative behavior. This example

is an excellent illustration of the interest and power of the qualitative methods for the study of physical systems.

Previous published work has dealt mainly with forced circuits. The system considered is autonomous and three dimensional, and so it belongs to the simplest class of dynamical systems in which chaotic motions may occur.

A lot of the descriptions carried out comes from direct experimentation with the circuit or from the numerical simulation using the state equations; it would be helpful to convert the numerical calculations into strict mathematical proofs, which leads us to some unsolved problems. The electronic system analyzed is, therefore, a good candidate for further study both theoretically and experimentally.

ACKNOWLEDGMENT

The authors are grateful to Emique Ponce for his help in preparing most of the numerical computations included in the paper.

REFERENCES [l] A. I. Mees and C. T. Sparrow, “Chaos,” Proc. Inst. Elect. Eng., vol.

121 128, pp. 201-205, Sept. 1981. J. Baillieul, R. W. Brockett and R. B. Washburn. “Chaotic motion in nonlinear feedback systems,” IEEE Trans. bcuits Spt., vol. CAS-21, pp. 990-991, 1980.

FREIRE et al.: PERIODICITY AND CHAOS 241

131

141

[51

PI [71

PI 191

WV [ill

WI u31

P41

P51

WI

[I71

WI

1191

WI

tions ‘;f Mach. Monographs, vol. 12, Amer. Math. RI, 1964, ch. IV.

Soc:,~Providence,

M. C. Irwin, Smooth Dynamical Systems. London, England: Academic, 1980. M. Shimiki. M. Yamamoto. and S. Mori. “Multimode oscillations in a modified van der Pol oscillator containing a positive nonlinear conductance,” Proc. IEEE, vol. 69, p. 394-395, Mar. 1981. M. W. Hirsch and S. Smale, Di ferentral E Systems and Linear Algebra.

p glM!ions, Dynamical New York: Aca emrc, 1974.

N. RouchC, P. Habets, and M. Laloy, Stability Theory by Liapunov’s ~ire;;e~~th‘$. New York: Spnnger, 1977.

AnaIis~s cuahtativo y de bifurcaciones en sistemas dinamico~,” Doctoral thesis, Univ. Seville, Seville, Spain, 1982. P. J. Holmes and J. Marsden, “Bifurcation to divergence and flutter in flow-induced oscillations: An infinite dimensional analysis,” Au- tomatica, vol. 14, pp. 367-384, 1978. V. Arnold, Equations Differentielles Ordinaires, MIR, Moscow, USSR: 1974. J. Guckenheimer, “Patterns of bifurcation,” in Proc. Engineering Foundation Conf. on New Ajproaches to Nonlinear Problems in Dynamics, SIAM, 1980. J. P. Gollub and S. V. Benson, “Phase Locking in the Oscillations Leading to Turbulence,” in Pattern formation by Dynamic Systems and Pattern Recognition, Springer, 1919. J. Guckenheimer, “Bifurcations of dynamical systems,” in Dynami- cal Systems, CIME Lectures, Birkhaiiser 1980. J. Guckenheimer, G. Oster and A. Ipaktchi, “The dynamics of density dependent population models,” J. Math. Biology, vol. 4, pp. 101-147, 1977. T. Li and J. A. Yorke, “Period three implies chaos,” Am. Math. Monthly, vol. 82, pp. 985-992, 1975. R. May and G. Oster, “Bifurcations and dynamic complexity in simple ecological models,” The American Naturalist, vol. 110, pp. 573-599,1916. M. J. Feigenbaum, “Universal behavior in nonlinear svstems.” Los Alamos Science, pp. 4-27, summer 1980. _I I D. Ruelle, “Sensitive deuendence on initial condition and turbulent behavior of dynamical s$stems,)) Ann. New York Academy of Scien- ces, vol. 316, pp. 408-416, 1978.

Javier Aracil (A’69-M’70) was born in Alcoy, Alicante, Spain,. in 1941. He received the In- geniero Industrial and the Doctor Ingeniero In- dustrial degrees, both from the Universidad Po- litecnica de Madrid, Madrid, Spain, in 1965 and 1969, respectively.

From 1965 to 1969 he was successively Assis- tant and Professor Adjunto at the Departamento de Automatica of the Universidad Politecnica de Madrid. Since 1969 he has been in the Escuela Superior de Ingenieros Industriales of the Uni-

1974 he was appointed Director of the Departa- :n

mento de Automatica y Electronica. His research interest are in the areas of the theory and philosophy of modeling and systems theory, with emphasis in the application of qualitative methods to so&-economic and to electrical systems. He is the author of numerous papers and has coauthored the book Practice of Zntegrated Automation, North-Holland, Amsterdam, The Netherlands, 1975.

He is member of the Real Academia de Medicina de Sevilla and of numerous scientific Societies such as IFAC Sociedad EspaIrola de Siste- mas Generales (president of the Andaloussian Section), SIAM, and others. He is an editor of the journal Regulation y mando automirtico (journal of the IFAC Spanish Committee).

+

sets, mathematical mod lems in dynamical syste

Emilio Freire was born in Reus, Tarragona, Spain, on March 16. 1952. He received the Ineeniero Industrial and Doctor Ingeniero Indust&l degrees from the University of Seville, Spain, in 1975 and 1982, respectively.

Since 1978, he has been with the Department of Mathematics at the Escuela Superior de In- genieros Industriales of the University of Seville, where he was successively Assistant Professor and Professor Agregado. His research interests lie in the fields of multicriteria analysis, fuzzy

eling of circuits and systems and nonlinear prob- ‘ms.

+

Leopoldo Garcia Franquelo was born in Malaga, Spain, on April 14, 1954. He received the In- geniero Indu&al and the Doctor Ingeniero In- dustrial degrees from the University of Seville. Seville, Spa&, in 1977 and 1980, respectively.

From 1978 he was successively Assistant and Professor Adjunto at the Department of Auto- matic Control and Electronics in the Escuela Superior de Ingenieros Industriales of the Uni- versidad de Sevilla. His current interest includes mathematical modeling of circuits, computer ap-

ning techniques.

A Chip for Real-Time Generation of Chaotic Behaviors 1739

(a)

(b)

Fig. 1. (a) Chip architecture; (b) Experimental setup showing oscilloscope, chip with four tuning resistors, and the batterypack.

Int.

J. Bi

furc

atio

n Ch

aos 1

997.

07:1

737-

1773

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntifi

c.co

mby

UN

IVER

SITY

OF

SEV

ILLE

LIB

RARY

on

03/0

5/13

. For

per

sona

l use

onl

y.


Fig. 5. Experimental Lissajous figures, state waveforms, and power spectrum of the x1 variable for Icont2 = 1.0 µA, Icont3 =2.35 µA.

Int.

J. Bi

furc

atio

n Ch

aos 1

997.

07:1

737-

1773

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntifi

c.co

mby

UN

IVER

SITY

OF

SEV

ILLE

LIB

RARY

on

03/0

5/13

. For

per

sona

l use

onl

y.

1744 M. Delgado-Restituto et al.

Fig. 6. Experimental Lissajous figures, state waveforms, and power spectrum of the x1 variable for Icont2 = 1.04 µA,Icont3 = 2.35 µA.

Int.

J. Bi

furc

atio

n Ch

aos 1

997.

07:1

737-

1773

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntifi

c.co

mby

UN

IVER

SITY

OF

SEV

ILLE

LIB

RARY

on

03/0

5/13

. For

per

sona

l use

onl

y.



Int.

J. Bi

furc

atio

n Ch

aos 1

997.

07:1

737-

1773

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntifi

c.co

mby

UN

IVER

SITY

OF

SEV

ILLE

LIB

RARY

on

03/0

5/13

. For

per

sona

l use

onl

y.

presentacion - ieee · frexre et al.: periodicity and chaos 241 lv/clll lv/clll l- "2 i l...

Documents