109

Multimode Chaos in Two Coupled Chaotic
Oscillators with Hard Nonlinearities
Yoshifumi NISHIO and Akio USHIDA
Dept. of Electrical and Electronic Engineering, Tokushima University,
2-1, Minami-Josanjima, Tokushima, 770 JAPAN.
Tel. +81-886-23-2311 Ext. 4615
Fax. +81-886-54-9632
Email nishio@ee.tokushima-u .ac.j p

ABSTRACTIn this study, multimode chaos observed from two coupled chaotic oscillators with hard nonlinearities is investigated. At first, a simple chaotic oscillator with hard
nonlinearities is realized. It is confirmed that in thischaotic oscillator the origin is always asymptotically stable and that the solution, which is excited by giving relatively large initial conditions, undergoes period-doublingbifurcations and bifucates t o chaos. Next, four different modes of oscillations are observed from two coupledchaotic oscillators with hard nonlinearities by both ofcomputer calculations and circuit experiments. One of
the modes of oscillation is a nonresonant double-mode
oscillation and this oscillation is stably generated even in
the case that oscillation is chaotic. Namely, for this oscillation mode, chaotic oscillation and periodic oscillation
can be simultaneously excited. we call this phenomena
as double-mode chaos. Finally, the beat frequency of thedouble-mode chaos is confirmed to be changed by tuning
the value of the coupling capacitor.

INTRODUCTION
Coupled oscillators systems are good model to describe
various nonlinear phenomena in the field of natural science and a number of excellent studies on mutual synchronization of oscillators have been carried out (e.g. [1][4]). Oscillators containing a nonlinear resistor whosev - i characteristics are described by fifth-power nonlinear characteristics are known t o exhibit hard excitation [5][6]. Namely, the origin is asymptotically stable
and an proper initial condition, which is larger than acritical value, is necessary to generate the oscillation.
Such an oscillator is often called as hard oscillator or said
to have hard nonlinearity. Datardina and Linkens haveinvestigated two identical oscillators with hard nonlinearities coupled by a inductor [2]. They have confirmedthat the nonresonant double-mode oscillations, whichcould not occur for the case of third-power nonlinearity,
were stably excited in the coupled system. They havealso confirmed that four different modes coexist for somerange of parameter values; zero, two single-mode, and adouble-mode. Recently, Yoshinaga and Kawakami haveinvestigated the double-mode oscillation observed froman arbitrary number of identical oscillators with hard
nonlinearities coupled by inductors as a ring [4]. They
confirmed the envelopes of the double-mode oscillations
were synchronized. Namely for the case that N oscillators coupled, N-phase synchronization with respect tothe envelopes occurs and the phase shift between theenvelopes of two adjacent oscillators is 27rlN. As thesestudies, coupled systems of hard oscillators exhibits interesting synchronization phenomena of double-mode oscillations as well as single-modes.
On the other hand, many nonlinear dynamical systems in the various fields have been clarified t o exhibits
chaotic oscillations and recently applications of chaost o engineering systems attract many researchers’ attensions [7]. Among the studies on such applications, synchronization of chaotic systems or signals is significant
because such a technique is necessary t o realize communication systems using chaos. Since the chaotic solution
is unstable and small error of initial values are expanded
as time goes, the synchronization of chaos seems to
be extremely interesting phenomena. Further, because
coupled chaotic oscillators are good example of higherdimensional systems which have been drawing recent attentions, investigating what kind of phenomena are observed from such systems would contribute t o develop
the study of nonlinear dynamical systems. We have already studied synchronization phenomena observed fromsome types of coupled chaotic oscillators [8][9]. In thestudies it has been confirmed that quasi-synchronization
occured even if each oscillator exhibits chaos. However,since we did not consider oscillators with hard nonlinearities in the previous studies, we have not obtained any
results about double-mode oscillations.
In this study, we investigate multimode chaos observed from two coupled chaotic oscillators with hard
nonlinearities. At first, a simple chaotic oscillator withhard nonlinearities is realized. Next, various modesof oscillations are investigated observed from two cou-Authorized licensed use limited to: Tokushima Univ. Downloaded on March 14,2010 at 06:47:59 EDT from IEEE Xplore. Restrictions apply.

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pled chaotic oscillators with hard nonlinearities by bothof computer calculations and circuit experiments. Especially, a nonresonant double-mode oscillation is confirmed t o be stably generated even in the case that oscillation is chaotic. Namely, for this oscillation mode,chaotic oscillation and periodic oscillation can be simultaneously excited. we call this phenomena as doublemode chaos. Finally, the beat frequency of the doublemode chaos is confirmed to be changed by tuning thevalue of the coupling capacitor.

In this study we consider two coupled chaotic oscillators with hard nonlinearities as shown in Fig. 2(a). In
this figure two same chaotic oscillators with hard nonlinearities are coupled by a capacitor Co. At first, weapproximate the i - v characteristics of the diode in thecircuit as two-segment piecewise linear function as shownin Fig. 2(b). In this case, the i - v characteristics of thenonlinear resistor including a linear negative resistor andof the nonlinear resistor consisting of six diodes are described by three-segment piecewise linear functions. Thecircuit equation is given as

CIRCUIT MODEL
Fig. l(a) shows the realization of chaotic oscillator withhard nonlinearity. If we remove a resistor Rd and a pairof diodes connected in parallel with Rd, the circuit is
the symmetric version of the chaotic circuit proposed by
Inaba et al. [lo]. We confirmed that this circuit have thefollowing properties. The origin is always stable and it
is surrounded by an unstable limit cycle. Out of the unstable limit cycle, there exists the other stable attractor.
As a parameter -r increases continuously, this attractor bifurcates from one-periodic limit cycle t o chaos asshown in Fig. 1(b) via period-doubling bifurcations.

I.

LL
Ic"dvdt= ik(IC = 1,2)

(covo= C(v1 +vz))

where the functions V D and v r means the i - v characteristics of the nonlinear resistors and are represented asfollows.

{IU-(3)The circuit equation (1) is calculated by using theRunge-Kutta method after normalizing the variables
and parameters.

NUMERICAL A N D EXPERIMENTAL
RESULTS

4 i

Lo.ShA/dIVI<h)Fig. 1 (a) Realization of chaotic oscillator withhard nonlinearity. (b) Typical example of chaotic
attractor observed from this circuit (L1 = 100mH,
L2 = 200mH, C = 0.068pF, R = 50R, Rd = 2kR,
r = 1.44kO).

We carried out computer calculations and circuit experiments. Parameters are fixed as CO = 0.34pF, L1 =100mH, L2 = 200mH, C = 0.068pF, R = 500 and
Rd = 2 k 0 and only r is varied as a control parameter.
For any r we obtained four different oscillation states;
zero, in-phase single-mode, anti-phase single-mode, anddouble-mode. For the in-phase single-mode synchronization, the attractor is always one-periodic. For the antipha.se single-mode synchronization, the attractor bifurcates to chaos while holding quasi-synchronization via
the following route. 1-period with symmetry
1period with asymmetry
2-period with asymmetry--

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111tal--(h)Fig. 2 (a) Two coupled chaotic oscillators with hard nonlinearities. (b) Piecewise linear
approximation of the i - v characteristics of the diode.

-2n period with asymmetry
chaos with asymmetry
chaos with symmtery. For the double-mode oscillation, namely two single-mode oscillations (in-phaseand anti-phase) simultaneously exist, attractor chenges
its character according to the character of contained
single-mode oscillation. Namely, if the in-phase oscillation is periodic and the anti-mode oscillation is chaotic,
the double-mode oscillation must contain both of periodic and chaotic oscillation.
Fig. 3 shows examples of numerical and experimental results. Figs. (a) and (b) show in-phase and antiphase single-modes oscillation, respectively. For this parameter value, anti-phase oscilation is chaotic. Fig. (c)shows double-mode chaos; periodic in-phase oscillationand chaotic anti-phase oscillations simultaneously exist.
By changing initial conditions we can obtain one of thesefigures or remaining trivial stable state; zero.
Finally, the beat frequency of double-mode chaos canbe changed by varying the value of the coupling capacitor Co. Fig. 4 shows the results obtained by using another values of Co.

CONCLUSIONSIn this study, we have investigated multimode chaos observed from two coupled chaotic oscillators with hard
nonlineaxities. In this circuit model, in-phase singlemode was confirmed t o be always one-periodic. If inphase is chaotic as well as anti-phase, double-mode oscillation must contain two different kind of chaos. Searching such phenomena is one of our future researchs.

ACKNOWLEDGEMENT
This work was partially supported by Research Aid ofInoue Foundation for Science.

Oscillators”, IEEE Trans. on Circuits Syst., Vol.
CAS25, No. 1, pp. 7-18, Jan. 1978.
[2] S . P. Datardina and D. A. Linkens,“Multimode Oscillations in Mutually Coupled van der Pol Type
Oscillators with Fifth-Power Nonlinear Characteristics”, IEEE Trans. on Circuits Syst., Vol. CAS25,
No. 5, pp. 308-315, May 1978.
[3] T. Endo and T. Ohta, “Multimode Oscillations in aCoupled Oscillator System with Fifth-Power Nonlinear Characteristics”, IEEE Trans. on Circuits
Syst., Vol. CAS27, No. 4, pp. 277-283, Apr. 1980.
[4] T. Yoshinaga and H. Kawakami, “SynchronizedQuasi-Periodic Oscillations in a Ring of Coupled
Oscillators with Hard Characteristics”, Trans. IEICE Japan, Vol. J75-A, No. 12, pp. 1811-1818, Dec.
1992 (in Japanese).
[5] C. Hayashi, Nonlinear Oscillations in Physical Systems, Princeton Univ. Press (1984), p. 367.
[6] V. 1. Arnold, Geometrical Methods in the Theoryof Ordinary Diferential Equataons, Springer-Verlag(1988), pp. 270-272.
[7] For example, the papers in the special session “Advances in Theory and Applications of Chaotic Circuits” in Proc. of ECCTD’93.

[8] Y. Nishio, K. Suzuki, S. Mori and A. Ushida,“Synchronization in Mutually Coupled Chaotic Circuits”, Proc. of ECCTD’93, pp. 637-642, Aug. 1993.
[9] Y. Nishio and A. Ushida, “Synchronization Phenomane in Chaotic Oscillators Coupled as a Ring”,
Proc. of NOLTA’93, pp. 613-616, Dec. 1993.
[lo] N. Inaba and S. Mori, “Chaotic Phenomena in Circuits with a Linear Negative Resistance and an IdealDiode”, Proc. 31st Midwest Synpoium on Circuitsand Systems, pp. 211-214, Aug. 1988.

REFERENCES[l]T. Endo and S. Mori, “Mode Analysis of a Ring ofa Large Number of Mutually Coupled van der Pol

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I-i,Fig. 3 Three different modes of oscillations obtained numerically and experimentally for
T = 1.37kR. (a) In-phase single-mode. (b) Anti-phase single-mode chaos. (c) Doublemode chaos. (For the lissajous H=V=0.5mA/div. For time waveforms V=lmA/div.,
H=0.2ms/div. in (a)(b) and H=2ms/div. in (c). )+tFig. 4 Change of the beat frequency of double-mode chaos. (a) CO= 0.408pF. (b) CO=0.204pF. (V=lmA/div., H=2ms/div.)

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