2013 International Symposium on Nonlinear Theory and its Applications
NOLTA2013, Santa Fe, USA, September 8-11, 2013

Double-mode Chaos with Hard Nonlinearities Coupled with an Inductor
Masaaki Kojima† ‡ , Yoko Uwate† and Yoshifumi Nishio††Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, 770-8506, Japan‡Engineering Administration Department, NHK
2-2-1 Jinnan, Shibuya-ku, Tokyo, 150-8001, Japan
Email: kojima, uwate, nishio@ee.tokushima-u.ac.jp
Abstract—We propose two chaotic circuits with hard
nonlinearities coupled with an inductor and investigate
some interesting phenomena in these systems. At first, welook into that several types of synchronization modes arestably excited. Next, we confirm that the synchronization
modes are chaotically vibrated by observation of attractors
and Poincare maps in the circuit systems. In particular,
we aim to find double-mode oscillations with two diﬀerent
types of chaotic phenomena in these systems.

respectively. However, we have not found that with chaosand chaos in these systems. We are considering the reason
that phenomena do not exist could be caused by their single circuit architecture. Bifurcation phenomena to chaos
depend on the circuit architecture.
In this study, we propose two chaotic circuits with hard
nonlinearities coupled with an inductor and look for theexpected double-mode chaos. Especially, we have adopted
a diﬀerent type of single circuit from the previous studies[2][3].

1. Introduction
Coupled oscillator systems with nonlinear characteristics have shown some interesting phenomena and brought
improvement to natural science and fundamental research.
It is known that two oscillators coupled with an inductor excite two diﬀerent modes: In-phase single-mode andanti-phase single-mode coexist for some range of parameter values. Both of negative resistors within these oscillators are normal. Such an oscillator is called a soft oscillator or one with soft nonlinearity. In comparison withsoft oscillator, hard oscillator has a nonlinear negative resistor whose i − v characteristics contain both negative andpositive. The characteristics near the origin act as a positive resistor. Therefore the oscillator converges on the origin for small initial values. Greater initial values than theorigin stable regions are required to generate the oscillation. Such an oscillator is called hard oscillator or one withhard nonlinearities. Two oscillators with hard nonlinearities coupled with an inductor excite four diﬀerent modes:
Zero (non-oscillation), in-phase single-mode, anti-phasesingle-mode and double-mode coexist for some range ofparameter values [1]. The double-mode shows simultaneous asynchronous oscillations with in-phase and anti-phasefrequency components. In particular, the double-mode oscillations are stably excited although they do not occur inthe case of common negative resistors with soft nonlinearities.
On the basis of this study, we have proposed two coupledchaotic circuits with hard nonlinearities for the purposeof looking for the interesting phenomena as double-mode
chaotic oscillations [2][3]. We have confirmed doublemode chaos with one-periodic in-phase and chaotic antiphase single-modes [2] and that with torus and chaos [3],2. Circuit Model
Figure 1 shows the proposed circuit model. Two chaotic
circuits with hard nonlinearities are coupled with an inductor in the model. Each single circuit consists of an capacitor, inductors, diodes and nonlinear negative resistors. Thei-v characteristics of the diodes are shown in Fig 2.
{vd (i) =rd i · · · (i 5 V/rd )V · · · (i > V/rd )(1)

On the basis of the characteristics of the diodes, the nonlinear negative resistors and fourteen diodes can be shownin Figs 3 and 4, respectively.
The number of series-connected diodes is proportional
to amplitude of each oscillation. Six diodes were adopted
in the previous studies [2][3]. An attractor in this model
is more dented than the previous model [2][3] though eachL0v1

vr- 146 -i1C

L2v2vDi2

C

L1

Rd−r

I1

L2

L1

Rdvr−r

I2vD

Figure 1: Circuit model.

‫ݒ‬௥ ሺ݅ሻ‫ݒ‬ௗ ሺ݅ሻܸ݅݅െ‫ݎ‬‫ݎ‬ௗ‫ݒ‬ௗ ሺ݅ሻܴௗ

݅‫ݒ‬௥ ሺ݅ሻʹܴௗ ൅ ‫ݎ‬ௗܸܴௗ ‫ݎ‬ௗܴ݅ௗ ‫ݎ‬ௗെ‫ݎ‬ʹܴௗ ൅‫ݎ‬ௗ

Figure 2: i − v characteristics of a diode.

Figure 3: i − v characteristics of a negative resistor.

஽ ሺሻhard oscillator has to avoid exciting on the origin stable regions as the positive resistor. Therefore the fourteen diodes
have been adopted in this model.
V( − ri · · · (i )> J)


Rd rd

− r i · · · (|i| 5 J)
vr (i) = 
2Rd + rd −V − ri · · · (i < −J),

7
଻
ଶௗ

஽ ሺሻ()2Rd + rdwhere J =V ,Rd r d
7V · · · (i > 2V/rd ) 7vD (i) = rd (i) · · · (|i| 5 2V/rd )2 −7V· · · (i < −2V/rd ).

െ7Figure 4: i − v characteristics of a negative resistor.
(3)

The equations governing the circuit in Fig. 1 are described as follows:
()
dIk
L1k+1
L1
= −vr + Ik + ik − (I2 − I1 ) · (−1) ) − vk
dt
L0()
dik
L1k+1
L2= −vr + Ik + ik − (I2 − I1 ) · (−1)− vk − vD (ik )
dt
L0
vk
L1
C = Ik + ik − (I2 − I1 ) · (−1)k+1(4)dtL0(k = 1, 2)
By changing the variables and parameters,√√CC
Exk , ik =Eyk , vk = Ezk ,Ik =L1L1√
dt = L1Cτ, ” · ” =dτ√L1CL1
α= , β=r, γ= ,L2
L1L0√√
Rd rdrdCC
a=, b=.
2Rd + rd L12L1(k = 1, 2)

(2)(1) is normalized as follows:()ẋk = − fr xk + yk − γ(x2 − x1 ) · (−1)k+1 − zk{}()ẏk = α − fr xk + yk − γ(x2 − x1 ) · (−1)k+1 − zk − fD (yk )żk = xk + yk − γ(x2 − x1 ) · (−1)k+1(6)(k = 1, 2)
The functions fr and fD correspond to vr (nonlinear negative resistor) and vD (fourteen diodes), respectively, and arerepresented as follows:
1 − γx · · · (x > 1/a)
(a − γ)x · · · (|x| 5 1/a)fr (x) =  −1 − γx · · · (x < −1/a),(7)
7 · · · (x > 1/b)

7bx · · · (|i| 5 1/b)fD (x) =  −7 · · · (x < −1/b).

(8)

3. Multi-mode Chaos(5)

Four diﬀerent oscillation modes (zero, in-phase singlemode, anti-phase single-mode and double-mode) coexist inthe circuit model of the Fig. 1.

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

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% ߚ ߬
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Figure 5: Attractors and Poincare maps of three diﬀerent oscillation modes. (a) x1 - x2 Attractors. (b) x1 - z1 Attractors. (c) x1- x2 Poincare maps. (A) β = 0.175. (B) β = 0.2. (1) In-phasesingle-mode. (2) Anti-phase single-mode. (3) Double-mode.

[ [ 

߬

Each mode is determined by the initial values. Theinitial values (x1 , x2 , y1 , y2 , z1 , z2 ) of in-phase singlemode, anti-phase single-mode and double-mode are(6.3, 6.5, 1.8, 1.5, 6.4, 6.5), (4.0, −4.1, 0.6, −0.5, 4.4, −4.2)and (4.0, 5.9, 1.6, −1.5, −5.9, 6.0), respectively (Figs. 5 and6). In this section, we show some simulation results ofsome attractors and Poincare maps on each mode to investigate its behavior when the parameter β as the absolutevalues of the negative resistor increases (Fig. 5). The otherparameters are fixed as α = 26.5, γ = 0.28, a = 1.15, b =40 (Figs. 5 and 6).

% ߚ Figure 6: Time waveforms vs amplitude of three diﬀerent oscillation modes. (A) β = 0.175. (B) β = 0.2. (1) In-phasesingle-mode. (2) Anti-phase single-mode. (3) Double-mode.

Zero means that neither of the two oscillators is excited.
This state is always stable in some range of small initialvalues. We do not mention this state in the rest of thismanuscript because it is trivial.

- 148 -In-phase single-mode means that the two oscillators aresynchronized with the in-phase. Although they are exactly
the quasi-in-phase synchronization, we see it as the inphase in this study. The in-phase single-mode is chaotic oscillation at β = 0.175 and 0.2 by the attractors and Poincare
maps. When β increases, we can see that the in-phasesingle-mode is more chaotic.
Anti-phase single-mode means that the two oscillators
are synchronized with π phase diﬀerence. Although they
are not synchronized with exact π phase diﬀerence, we also
see it as the anti-phase in this study. The anti-phase singlemode is chaotic oscillation at β = 0.175 and 0.2 by the attractors and Poincare maps. When β increases, we can seethat the anti-phase single-mode is more chaotic.
Double-mode means that the above two single-mode oscillations (in-phase and anti-phase) are excited, simultaneously and asynchronously. The double-mode is chaotic oscillation at β = 0.175 and 0.2 by the attractors and Poincare
maps. When β increases, we can see that the Poincare mapsof the double-mode is more chaotic because the Poincare
maps more spread out and breakdown.
Figure 6 shows time waveforms of three diﬀerent oscillation modes. The frequency of the anti-phase single-mode
is higher than that of the in-phase. The envelope periodsof the double-mode time waveforms show the frequency
diﬀerences between the in-phase and anti-phase singlemodes. When β increases, we can also see that the threemodes are more chaotic from the time waveforms.
4. Conclusions
We proposed two identical chaotic circuits with hard
nonlinearities coupled with an inductor and investigated thedouble-mode chaos in these systems by looking at some attractors and Poincare maps. As a result, we confirmed theinteresting phenomena as double-mode chaos with chaos(in-phase) and chaos (anti-phase) by computer simulations.
We will analyze the bifurcation of these three modes in thenear future.
References[1] S. P. Datardina and D. A. Linkens, “Multimode oscillations in mutually coupled van der Pol type oscillators with fifth-power nonlinear characteristics,” IEEETrans. Circuits Syst., vol. 25, no. 5, pp. 308-315, May
1978.
[2] Y. Nishio and A. Ushida, “Multimode chaos in two
coupled chaotic oscillators with hard nonlinearities,”
IEICE Trans. Fundamentals, vol. E79-A, no. 2, pp.
227-232, Feb. 1996.
[3] M. Kojima, M. Yamauchi, Y. Nishio and A. Ushida,“Simultaneous asynchronous oscillation of chaos andtorus in coupled chaotic circuits,” Proc. of NOLTA ’00,
vol. 2, pp. 567-570, Sep. 2000.

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