2012 International Symposium on Nonlinear Theory and its Applications
NOLTA2012, Palma, Majorca, Spain, October 22-26, 2012

Multimode Phenomena in Hard Oscillators
Coupled by a Time-Varying Resistor
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 investigate multiple circuits with fifthpower nonlinear characteristics coupled by a time-varyingresistor. In particular, we confirm some different types ofdouble-mode (simultaneous asynchronous) oscillations
which do not occur for the systems with third-power
nonlinear characteristics.
1. Introduction
Coupled oscillator systems can be good models todescribe various nonlinear phenomena in the field of
natural sciences, neural systems and communication
networks. The negative resistor within the oscillator has
fifth-power 𝑣 − 𝑖 characteristics. As a result, the oscillator
is known to exhibit hard excitation. The oscillator
converges on the origin for small initial values because
fifth-power characteristics near the origin act as a positive
resistor. Greater initial values than the origin stableregions are required to generate the oscillation. Such an
oscillator is called a hard oscillator or that with hard
nonlinearity. Datardina and Linkens have investigated two
van der Pol oscillators with hard nonlinearities coupled byan inductor [1]. They have confirmed that four different
modes, which are zero, in-phase synchronous mode, antiphase synchronous mode and double-mode, coexist forsome range of parameter values. These modes are zero(non-oscillation), two single-modes (in-phase and antiphase synchronizations), and double-mode (simultaneousasynchronous oscillations with in-phase and anti-phasefrequency components). In particular, the double-mode
oscillations are stably excited although they do not occur
for the case of the common negative resistor with thirdpower characteristics. Furthermore, Endo and Ohta haveconfirmed that some different types of single-modes anddouble-modes in two, three and four van der Pol
oscillators with hard nonlinearities coupled by inductors
are stably excited [2].
On the basis of these studies, we have proposed two
coupled chaotic circuits with hard nonlinearities for thepurpose of looking for interesting phenomena as doublemode chaotic oscillations [3][4]. We have confirmed
double-mode chaos with one-periodic in-phase andchaotic anti-phase single-modes [3] and that with torus
and chaos [4], respectively. We have also confirmed theother type of double-mode chaos with one-periodic inphase and chaotic three-phase single-modes in three
coupled chaotic circuits with hard nonlinearities [5].
However, we have not found that with chaos and chaos inthese systems. We are considering the reason that these
phenomena do not exist could be caused by frequency
differences between two single-modes.
On the other hand, we have investigatedsynchronization phenomena of two van der Pol oscillators
coupled by a time-varying resistor which means positive
and negative resistors switching periodically [6]. We haveconfirmed that in-phase and anti-phase single-modes arestably excited in these systems at the same parametervalues. We have also confirmed that two chaotic singlemodes coexist in an improved chaos version of thesesystems [7]. The frequency differences between twodifferent single-modes in the systems coupled by a timevarying resistor are possibly smaller than those in thesystems coupled by an inductor or capacitor such as theprevious systems. Therefore, we can expect to improve thefields of the double-mode chaos [3]-[5] if we find doublemode oscillations in hard oscillators coupled by a timevarying resistor. The double-mode oscillation in thesystems coupled by a time-varying resistor has not beenreported yet while the coexistence of the each single-mode
oscillation is confirmed [6][7]. In this study, we propose
multiple van der Pol oscillators with hard nonlinearities
coupled by a time-varying resistor and confirm somedifferent types of double-mode oscillations in thesesystems.
2. Circuit Model
Figure 1 shows the proposed circuit model. In thismodel, 𝑁 identical van der Pol oscillators with hard
nonlinearities are coupled by a time-varying resistor(TVR). The time waveform characteristics of the TVR are
shown in Fig. 2. The parameters 𝜔𝑡 and 𝑝 mean theangular frequency and the duty ratio, respectively. TheTVR is realized by a square wave of switching positiveand negative resistors.
The equations governing the circuit in Fig. 1 aredescribed as follows:- 911 -iR NL

Li R1

I1

INI2

(3)

𝑗=1

(𝑘 = 1, 2, ⋯ , 𝑁)Γ(τ) corresponds to the normalized characteristics of theTVR as shown in Fig. 3.

Lv2

TVRC𝑦̇ 𝑘 = 𝑥𝑘 − Γ(τ) � 𝑦𝑗vNCiR2v1

𝑁CFigure 1: Circuit model (TVR stands for time-varyingresistor).

Figure 3: Normalized TVR characteristics.
3. Multi-Mode Oscillations
3.1. Two Coupled Oscillators

Figure 2: TVR characteristics.
𝐶𝑑𝑣𝑘= −𝑖𝑘 − 𝑖𝑅𝑅 (𝑣𝑘 )𝑑𝑑𝑁𝑑𝑖𝑘𝐿= 𝑣𝑘 − R(t) � 𝑖𝑗
𝑑𝑑(1)

𝑗=1

(𝑘 = 1, 2, ⋯ , 𝑁)𝑖𝑅𝑘 (𝑣𝑘 ) indicates 𝑣 − 𝑖 characteristics of the fifth-power
nonlinear resistor. It is approximated by the following
function:𝑖𝑅𝑘 (𝑣𝑘 ) = 𝑔1 𝑣𝑘 − 𝑔3 𝑣𝑘3 + 𝑔5 𝑣𝑘5 (𝑔1 , 𝑔3 , 𝑔5 > 0). (2)By changing the variables and the parameters,4𝑣𝑘 = �
𝛽=

𝑔1𝑥 ,
5𝑔5 𝑘

𝐶 4 𝑔1
𝑖𝑘 = � �𝑦 ,𝐿 5𝑔5 𝑘𝐶𝐿
3𝑔3 𝑔1, γ = r� , 𝜀 = 𝑔1 � ,�𝐿𝐶𝑔1 5𝑔5t = √𝐿𝐿τ, ω =(1) is normalized as

1√𝐿𝐿𝜔t , " ⋅ " =𝑑,𝑑𝑑

11
𝑥̇ 𝑘 = −𝑦𝑘 − 𝜀 �𝑥𝑘 − 𝛽𝑥𝑘3 + 𝑥𝑘5 �35(3)In this section, we mention two coupled oscillators withhard nonlinearities in the case of 𝑁 = 2 in Fig. 1. In thissystem, four different oscillation modes (zero, in-phasesingle-mode, anti-phase single-mode and double-mode)
coexist. These modes are determined by the initial values.
Figures 4 and 5 show some simulation results on eachmode attractor and time waveform, respectively. The
parameter values in Figs. 4 and 5 are 𝛽 = 3.2, 𝛾 =0.1, 𝜀 = 0.01, 𝜔 = 1.5 and 𝑝 = 0.5.
Zero means 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.
In-phase single-mode means that the two oscillators aresynchronized with the in-phase. As the simulation resultsin Figs. 4-(1) and 5-(1) show, they are stably excited witha 0 phase difference.
Anti-phase single-mode means that the two oscillators
are synchronized with the opposite phase. As thesimulation results in Figs. 4-(2) and 5-(2) show, they arestably excited with a π phase difference.
Double-mode means that the above two single-mode
oscillations
areexcited
simultaneously
andasynchronously. As the simulation results in Figs. 4-(3)and 5-(3) show, they are simultaneous asynchronous
oscillations with in-phase and anti-phase single-modes atthe stable. The envelopes of the double-mode time
waveform are synchronized with the opposite phase in
Fig. 5-(3).
3.2. Three Oscillators
Here is the case of 𝑁 = 3 in Fig. 1. It is also four
different oscillation modes (zero, in-phase single-mode,- 912 -

2.5

2.5

2.5

-2.5
-2.5

(a) (𝑥1 − 𝑥2 )

2.5

-2.5
-2.5

(c) (𝑥2 − 𝑦2 )(b) (𝑥1 − 𝑦1 )(1) In-phase single-mode(a) (𝑥1 − 𝑥2 )(d) (𝑥1 − 𝑦1 )(b) (𝑥1 − 𝑦1 )(c) (𝑥2 − 𝑦2 )(a) (𝑥1 − 𝑥2 )(2) Anti-phase single-mode(a) (𝑥1 − 𝑥2 )(c) (𝑥2 − 𝑦2 )(b) (𝑥1 − 𝑦1 )(3) Double-mode
Figure 4: Attractors of three different oscillation modes(𝑁 = 2).
(a) (𝑥1 − 𝑥2 )(b) (𝑥1 − 𝑥3 )(c) (𝑥2 − 𝑥3 )(e) (𝑥2 − 𝑦2 )(f) (𝑥3 − 𝑦3 )(1) In-phase single-mode(b) (𝑥1 − 𝑥3 )(c) (𝑥2 − 𝑥3 )

3.0

𝑥1

(d) (𝑥1 − 𝑦1 )(e) (𝑥2 − 𝑦2 )(f) (𝑥3 − 𝑦3 )(2) Three-phase single-mode-3.0
3.0𝑥2-3.00302010(1) In-phase single-modeτ40(a) (𝑥1 − 𝑥2 )𝑥1𝑥20102030(2) Anti-phase single-mode40
τ

𝑥1𝑥20400

800

1200(3) Double-mode
Figure 5: Time waveforms vs amplitude (𝑁 = 2).

1600τ(b) (𝑥1 − 𝑥3 )(c) (𝑥2 − 𝑥3 )(d) (𝑥1 − 𝑦1 )(e) (𝑥2 − 𝑦2 )(f) (𝑥3 − 𝑦3 )(3) Double-mode
Figure 6: Attractors of three different oscillation modes(𝑁 = 3).

three-phase single-mode and double-mode) coexist.
Figures 6 and 7 show some simulation results on eachmode attractor and time waveform, respectively. The
parameter values in Figs. 6 and 7 are 𝛽 = 3.2, 𝛾 =0.07, 𝜀 = 0.01, 𝜔 = 1.5 and 𝑝 = 0.5.

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3.0

the stable. The envelopes of the double-mode time
waveform are synchronized with a 2π/3 phase differencein Fig. 7-(3).

𝑥1-3.0
3.0

4. Conclusions

𝑥2-3.03.0𝑥3-3.00102030(1) In-phase single-modeτ40𝑥1

References𝑥2𝑥30102030(2) Thee-phase single-modeτ

40𝑥1𝑥2𝑥30

We proposed two and three oscillators with hard
nonlinearities coupled by a time-varying resistor andlooked for the parameter values that two different singlemodes and a double-mode could coexist and be stablyexcited. As a result, we confirmed that these modes
coexisted and were stably excited by computer
simulations. Especially, the double-mode oscillations forthe systems coupled by a time-varying resistor were thefirst report. We will also investigate chaotic circuits withhard nonlinearities coupled by a time-varying resistor in
the near future.

4008001200(3) Double-mode
Figure 7: Time waveforms vs amplitude (𝑁 = 3).

1600
τ

Zero means that three oscillators are not excited.
In-phase single-mode means that the three oscillators
are synchronized with the in-phase. As the simulation
results in Figs. 6-(1) and 7-(1) show, they are stably
excited with a 0 phase difference.
Three-phase single-mode means that the threeoscillators are synchronized with a 2π/3 phase difference.
As the simulation results in Figs. 6-(2) and 7-(2) show,
they are stably excited with a 2π/3 phase difference.
Double-mode means that the above two single-mode
oscillations
areexcited
simultaneously
andasynchronously. As the simulation results in Figs. 6-(3)and 7-(3) show, they are simultaneous asynchronous
oscillations with in-phase and three-phase single-modes at[1] S. P. Datardina and D. A. Linkens, “Multimodeoscillations in mutually coupled van der Pol typeoscillators with fifth-power nonlinear characteristics,”
IEEE Trans. Circuits Syst., vol.25, no.5, pp.308–315, May1978.
[2] T. Endo and T. Ohta, “Multimode oscillations in acoupled oscillator system with fifth-power nonlinear
characteristics,” IEEE Trans. Circuits Syst., vol.27, no.4,pp.227-283, Apr. 1980.
[3] Y. Nishio and A. Ushida, “Multimode chaos in two
coupled chaotic oscillators with hard nonlinearities,”
IEICE Trans. Fundamentals, vol.79-A, no.2, pp.227–232,
Feb. 1996.
[4] M. Kojima, M. Yamauchi, Y. Nishio and A. Ushida,“Simultaneous asynchronous oscillation of chaos andtorus in coupled chaotic circuits,” Proc. NOLTA ’00, vol.2,pp.567–570, Sep. 2000.
[5] M. Kojima, Y. Nishio and A. Ushida, “Simultaneousasynchronous chaotic oscillation of in-phase and threephase in coupled chaotic circuits,” Proc. ITC-CSCC ’01,
vol.2, pp.716–719, July 2001.
[6] Y. Uwate and Y. Nishio, “Synchronization phenomena
in van der Pol oscillators coupled by a time-varyingresistor,” International Journal of Bifurcation and Chaos,vol.17, no.10, pp.3565–3569, Oct. 2007.
[7] Y. Uwate and Y. Nishio, “Switching phase states ofchaotic circuits coupled by a time-varying resistor,” Proc.
of ISCAS ’07, pp.209–212, May 2007.

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