International Journal of Bifurcation and Chaos, Vol. 10, No. 7 (2000) 1781–1785c© World Scientific Publishing Company

RICH VARIETY OF BIFURCATIONSAND CHAOS IN A VARIANT OF

MURALI LAKSHMANAN CHUA CIRCUIT

K. THAMILMARAN and M. LAKSHMANANCentre for Nonlinear Dynamics, Department of Physics,

Bharathidasan University, Tiruchirappalli – 620 024, India

K. MURALIDepartment of Physics, Anna University, Chennai – 600 025, India

Received October 1, 1999

A very simple nonlinear parallel nonautonomous LCR circuit with Chua’s diode as its onlynonlinear element, exhibiting a rich variety of dynamical features, is proposed as a variant ofthe simplest nonlinear nonautonomous circuit introduced by Murali, Lakshmanan and Chua(MLC). By constructing a two-parameter phase diagram in the (F–ω) plane, corresponding tothe forcing amplitude (F ) and frequency (ω), we identify, besides the familiar period-doublingscenario to chaos, intermittent and quasiperiodic routes to chaos as well as period-addingsequences, Farey sequences, and so on. The chaotic dynamics is verified by both experimentalas well as computer simulation studies including PSPICE.

1. Introduction

Sometime ago, the simplest nonlinear dissipativenonautonomous electronic circuit consisting of aforced series LCR circuit connected in parallelto the Chua’s diode, which is a nonlinear resis-tor, was introduced by Murali, Lakshmanan andChua [Murali et al., 1994a, 1994b]. The circuitexhibits several interesting dynamical phenomenaincluding period-doubling bifurcations, chaos andperiodic windows. However, in the parameterregimes investigated, it does not exhibit other im-portant features such as quasiperiodicity, intermit-tency, period-adding sequences, and so on. It willbe quite valuable from nonlinear dynamics point ofview to construct a simple electronic circuit whichexhibits a wide spectrum of dynamical phenomena[Lakshmanan & Murali, 1996]. In this paper, wewish to point out that a rich variety of phenom-ena can be realized with a simple variant of theabove MLC circuit, by connecting the Chua’s diodeto a forced parallel LCR circuit instead of the forced

series LCR circuit. The resultant system exhibitsnot only the familar period-doubling route to chaosand windows but also intermittent and quasiperi-odic routes to chaos as well as period-adding se-quences and Farey sequences, to name a few.

2. Circuit Realization of MLCVariant Circuit

The circuit realization of the proposed simplenonautonomous circuit, namely, a variant of thestandard MLC circuit, is shown in Fig. 1. It con-sists of a capacitor (C), an inductor (L), a resis-tor (R), an external periodic forcing voltage sourceand only one nonlinear element (N), namely, theChua’s diode. In the dynamically interesting range,the v–i characteristic of the Chua’s diode is givenby the usual three segment piecewise-linear func-tion [Chua et al., 1987; Kennedy, 1992; Cruz &Chua, 1992]. The nonlinear element is added to thefamiliar forced parallel LCR circuit instead of the

1781

1782 K. Thamilmaran et al.

+−+−+−+−

R

CN

f sin Ωt -

+

-

+

iR

iL iC iN

L v C

Fig. 1. Circuit realization of the simple nonautonomousMLC variant circuit. Here N is the Chua’s diode.

series LCR of MLC circuit [Murali et al., 1994a,1994b]. The resulting circuit can be considered asanother important very simple dissipative second-order nonautonomous nonlinear circuit and a vari-ant of the MLC circuit. By applying Kirchhoff’slaws to this circuit, the governing equations forthe voltage v across the capacitor C and the cur-rent iL through the inductor L are represented bythe following set of two first-order nonautonomous

differential equations:

Cdv

dt=

1

R(f sin(Ωt)− v)− iL − g(v) , (1a)

LdiLdt

= v , (1b)

where g(·) is a piecewise-linear function defined byChua et al. and is given by g(v) = Gbv + 0.5(Ga −Gb)[|v + Bp| − |v − Bp|] which is the mathemat-ical representation of the characteristic curve ofChua’s diode [Kennedy, 1992; Cruz & Chua, 1992].The values of the various circuit parameters cho-sen for our study are as follows: C = 10.15 nF,L = 445 mH, R = 1745 ohms, Ω = 1.116 KHz,Ga = −0.76 ms, Gb = −0.41 ms, and Bp = 1.0 V.

In Eq. (1), f is the amplitude and Ω is the angu-lar frequency of the external periodic signal. Rescal-ing Eq. (1) as v = xBp, iL = GyBp, G = 1/R,ω = ΩC/G, and t = τC/G and then redefining τas t, the following set of normalized equations areobtained:

x = F sin(ωt)− x− y − g(x) , (2a)

0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16

0.45

0.35

0.25

0.15

Qusiperiodicity

1T1T

2T 4T

9T

8T

7T

6T

5T

4T

3T

6T

5T

2T

9T

5T

8T

3T

4T

7T

4T 9T 6T

6T

chaos

chaos

F

ωFig. 2. Two parameter phase diagram in the (F–ω) plane.

A Variant of Murali–Lakshmanan–Chua Circuit 1783

y = βx ,

(· = d

dt

)(2b)

where β = C/LG2, F = f/Bp, G = 1/R. Ob-viously g(x) = bx + 0.5(a − b)(|x + 1| − |x − 1|).Here a = Ga/G, b = Gb/G. Note that the two cou-pled first-order ordinary differential equation givenby Eqs. (2) can also be written as a single second-order differential equation of the Lienard’s type inthe form

y + y + βg

(y

β

)+ βy = βF sin(ωt) . (3)

We note at this point that chaos via torus break-down generated in a piecewise-linear forced van derPol equation of the form (3) with asymmetric non-linearity has been studied by Inaba and Mori [1991]sometime ago. However the corresponding circuituses more nonlinear elements than the present cir-cuit. Now the dynamics of Eq. (2) or equivalently(3) depends on the parameters a, b, β, ω and F .Then for the above chosen experimental circuit pa-rameter values, we have β = 0.05, a = −1.121,b = −0.604 and ω = 0.105. We use the amplitude

F of the external periodic forcing as the controlparameter and carry out a numerical simulation ofEqs. (2) either by integrating Eqs. (2) or by solvingEq. (3) analytically and numerically, for increasingvalues of F . For the above choice of parametersthe numerical simulation of Eqs. (2) exhibits noveldynamical phenomena.

To begin with we have carried out an experi-mental study of the dynamics of the circuit givenby Fig. 1. The driving amplitude (f) is slowly in-creased from 0 V and the response of the systemis observed to progress through a series of tran-sition from periodic motion to aperiodic motion.When the driving amplitude f = 0 (correspond-ing to the autonomous case), a period-1 limit cy-cle is observed. By increasing the amplitude fromzero upwards, the circuit of Fig. 1 is found to ex-hibit a quasiperiodic (torus) attractor which thentransits to chaos via torus breakdown, followedby periodic windows, period-adding sequences etc.We have confirmed these results also by usingthe standard fourth-order Runge–Kutta integrationroutine and carrying out a numerical analysis of

0.09 0.095 0.1 0.105 0.11 0.115 0.12 0.125 0.13 0.135 0.14−0.1

−0.05

0

0.09 0.095 0.1 0.105 0.11 0.115 0.12 0.125 0.13 0.135 0.14−1.5

−1

−0.5

0

0.5

1

1.5

ω

ω

λ

x

(a)

(b)

Fig. 3. (a) One-parameter bifurcation diagram in the (ω–x) plane and (b) maximal Lyapunov spectrum (λmax). The valueof the forcing amplitude has been fixed at F = 0.28.

1784 K. Thamilmaran et al.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−2

−1

0

1

2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.15

−0.1

−0.05

0

0.05

F

F

λ

x

(a)

(b)

Fig. 4. (a) One-parameter bifurcation diagram in the (F–x) plane and (b) maximal Lyapunov spectrum (λmax). The valueof the forcing frequency has been fixed at ω = 0.105.

Eqs. (2) or equivalently Eq. (3) with the rescaledcircuit parameters of Fig. 1 as given above. Wehave observed a series of appearence and disap-pearence of the quasiperiodic (torus) and phaselocking (periodic) attractors, and period-adding se-quences alternatively by varying the amplitude Fof the external source at fixed frequency. Illus-trated in Fig. 2 is the two-parameter phase diagramwhich is plotted in the (F–ω) plane. In partic-ular, quasiperiodic motions and period-adding se-quences besides the standard bifurcation sequenceshave been observed in regions of the lower drive am-plitude (F ) and frequency (ω) values. For example,typical quasiperiodic attractors exist in the rangeF = (0.15, 0.2), and ω = (0.09, 0.16). Similarly, aperiod-adding sequence exists for F = (0.36, 0.38)and ω = (0.1, 0.115), a period-doubling bifurca-tion sequence for F = 0.39, and ω = (0.09, 0.1),and also type I intermittency has been identifiedfor F = 0.38 and ω = (0.10636). In addition forF = (0.24, 0.27) and ω = (0.138, 0.146), as wellas for F = (0.26, 0.29), ω = (0.128, 0.138) and

for F = (0.28, 0.33), ω = (0.11, 0.125), Farey se-quences [Kaneko, 1986] exist. The regions of chaosare also indicated in Fig. 2. Finally Fig. 3(a) rep-resents the one-parameter bifurcation in the (ω–x)plane, for F = 0.28 which consists of quasiperi-odicity, chaos, windows, period-adding sequencesand the familiar period-doubling bifurcation se-quence, intermittency and so on. In Fig. 3(b) itscorresponding maximal Lyaponov spectrum in (ω–λmax) is plotted. Also Fig. 4(a) represents theone-parameter bifurcation in the (F–x) plane, forω = 0.105, Fig. 4(b) depicts the corresponding max-imal Lyaponov spectrum (F–λmax). Further, theexperimental results obtained for a choice of cir-cuit parameters were also confirmed by using thePSPICE [Roberts, 1997] deck available to simulatethe behavior of the circuit in Fig. 1. In Fig. 5, wehave included for comparison the chaotic attractorcorresponding to f = 0.389 V in Eq. (1a) for ex-perimental and SPICE analysis and correspondingnumerical simulation for amplitude F = 0.383 inEq. (2a).

A Variant of Murali–Lakshmanan–Chua Circuit 1785

(a)

Fig. 5. (a) Chaotic attractor corresponding to f = 0.389 V,Ω = 1.116 KHz in Eq. (1): (a) experimental (b) PSPICE and(c) corresponding numerical simulation of Eq. (2).

3. Conclusion

In this letter, we have presented a very simplesecond-order dissipative nonlinear nonautonomouscircuit which is a variant of the MLC circuit and iscarried out. It is shown to possess a very rich va-riety of dynamical phenomena. In view of the ap-pearance of several ubiquitous routes to chaos suchas quasiperiodicity, intermittency, period-doubling,period-adding and Farey sequences in such a singlebut simple circuit, one can make use of the circuit invaried investigations on chaotic dynamics and appli-cations, including spatiotemporal studies. A moredetailed analysis of these aspects will be publishedelsewhere.

Acknowledgments

The work by M. Lakshmanan forms part of theDepartment of Science and Technology, Govern-ment of India research project. We are grateful toMr. A. Venkatesan for his assistance in the numer-ical analysis.

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and Nonlinear Circuits (McGraw-Hill, NY).Cruz, J. M. & Chua, L. O. [1992] “A CMOS IC nonlin-

ear resistor for Chua’s circuit,” IEEE Trans. CircuitsSyst. I 39, 985–995.

Inaba, N. & Mori, S. [1991] “Chaos via torus break downin a piecewise-linear forced van der Pol oscillator witha diode,” IEEE Trans. Circuits Syst. I 38, 398–409.

Kaneko, K. [1986] Collapse of Tori and Genesis of Chaosin Dissipative Systems (World Scientific, Singapore).

Kennedy, M. P. [1992] “Robust op amp realization ofChua’s circuit,” Frequenz 46, 66–80.

Lakshmanan, M. & Murali, K. [1996] Chaos in NonlinearOscillators: Controlling and Synchronization (WorldScientific, Singapore).

Murali, K., Lakshmanan, M. & Chua, L. O. [1994a] “Thesimplest dissipative nonautonomous chaotic circuit,”IEEE Trans. Circuits Syst. I 41, 462–463.

Murali, K., Lakshmanan, M. & Chua, L. O. [1994b]“Bifurcation and chaos in the simplest dissipa-tive nonautonomous circuit,” Int. J. Bifurcation andChaos 4, 1511–1524.

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