Double-Mode Oscillation in Chaotic Circuits
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
E-mail: {kojima, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—We investigate two chaotic circuits with fifth power
nonlinear characteristics coupled by a time-varying resistor. Inparticular, we confirm some simultaneous asynchronousoscillations with two different types of chaotic phenomena inthese systems.

I.

INTRODUCTION

Coupled oscillator systems with nonlinear characteristics
have shown some interesting phenomena and brought
improvement to natural science and fundamental research. An
Oscillator with a negative resistor as the fifth-power v – icharacteristics is known to exhibit hard excitation. The
oscillator converges on the origin at the small initial values
because the proposed fifth power characteristics near theorigin are a positive resistor. The initial values beyond origin
stable regions are required to generate the oscillation. Such anoscillator is called a hard oscillator or one with hard
nonlinearity. Datardina and Linkens have investigated two vander Pol oscillators with hard nonlinearities coupled by an
inductor [1]. They have confirmed that four different modes:
Zero, in-phase synchronous mode, anti-phase synchronousmode and double-mode, coexist for some range of parametervalues. These modes are zero (non-oscillation), two singlemodes (in-phase and anti-phase synchronizations), anddouble-mode (simultaneous asynchronous oscillations with inphase and anti-phase frequency components). In particular, thedouble-mode oscillations are stably excited although they do
not occur in the case of common negative resistors with thirdpower characteristics. On the basis of this study, we haveproposed two coupled chaotic circuits with hard nonlinearities
for the purpose of looking for the interesting phenomena as
double-mode chaotic oscillations [2][3]. We have confirmed
double-mode chaos with one-periodic in-phase and chaotic
anti-phase single-modes [2] and that with torus and chaos,respectively [3]. However, we have not found that with chaosand chaos in these systems. We are considering the reason thatphenomena do not exist could be caused by frequency
differences between two single-modes. We have to consider

978-1-4577-1729-1/12/$26.00 ©2012 IEEE.

the double-mode oscillation with much smaller frequency
differences.
On the other hand, we have investigated synchronization
phenomena of two van der Pol oscillators coupled by a timevarying resistor which means positive and negative resistors
switching time-periodically [4]. We have confirmed that inphase and anti-phase single-modes are stably excited in thesesystems at the same parameter values. We have also
confirmed that two chaotic single-modes coexist in an
improved chaos version of these systems [5]. The frequency
differences between two different single-modes in the systems
coupled by a time-varying resistor are possibly smaller thanthose in the systems coupled by an inductor or capacitor suchas the previous systems. Therefore, we can expect to finddouble-mode oscillations with chaotic both in-phase and antiphase frequency components. In this study, we propose twochaotic circuits with hard nonlinearities coupled by a timevarying resistor and look for the expected double-mode chaos.
II. CIRCUIT MODEL
Figure 1 shows the proposed circuit model. In this model,
two identical chaotic circuits with hard nonlinearities arecoupled by a time-varying resistor whose time waveformcharacteristics are shown in Fig. 2.
The equations governing the circuit in Fig. 1 are describedas follows:𝑑𝐼𝑘= −𝑣𝑟 (𝐼𝑘 + 𝑖𝑘 ) − 𝑣𝑘 − 𝑅𝑖 𝐼𝑘 − 𝑅(𝑡)(𝐼1 + 𝐼2 )𝑑𝑡𝑑𝑖𝑘𝐿2(1)= −𝑣𝑟 (𝐼𝑘 + 𝑖𝑘 ) − 𝑣𝑘 − 𝑣𝑑 (𝑖𝑘 )𝑑𝑡𝑑𝑣𝑘
𝐶= 𝐼𝑘 + 𝑖𝑘 (𝑘 = 1, 2)
𝑑𝑡
𝐿1

204

L1

L1i1
v1

I1

I2CTVR

vdvri2v2

L2−𝑎2 − 𝑟1 𝑥 ⋯ �−𝑟2 𝑥 − 𝑎2 +L2Cvdvr

𝑓𝑑 (𝑥 ) =Figure 1. Circuit model. (TVR stands for Time-Varying Resistor)1
1<𝑥<− �𝑎3
𝑎1

𝑟1 + 𝑟2
1⋯ �𝑥 < − �𝑎3𝑎313 ⋯ �𝑥 > �𝑏13𝑏𝑏 ⋯ �|𝑥 | < �𝑏1−3 ⋯ �𝑥 < − �𝑏(5)Their characteristics are shown in Figs. 3 and 4, respectively.

0Figure 2. TVR characteristics.

By changing the variables and parameters,

Figure 3. Normarized fifth-power characteristics of a negative resistor.

𝐶𝐶𝐼𝑘 = � 𝐸𝑥𝑘 , 𝑖𝑘 = 𝐿1 � 𝐸𝑦𝑘 , 𝑣𝑘 = 𝐸𝑧𝑘 ,𝐿1
𝐿1𝛼=𝐿1𝐶𝐶𝐶, β = r� , 𝛾 (𝜏) = 𝑅 (𝑡)� , 𝛿 = 𝑅𝑖 � ,𝐿2
𝐿1
𝐿1
𝐿1𝑡 = �𝐿1 𝐶𝜏,(1) is normalized as

𝜔=1

�𝐿1 𝐶

(2)

𝜔𝑡 ,𝑑𝑥𝑘= −𝑓𝑟 (𝑥𝑘 + 𝑦𝑘 ) − 𝑧𝑘 − 𝛿𝑥𝑘 − 𝛾 (𝜏)(𝑥1 + 𝑥2 )𝑑𝜏𝑑𝑦𝑘= 𝛼 {−𝑓𝑟 (𝑥𝑘 + 𝑦𝑘 ) − 𝑧𝑘 − 𝑓𝑑 (𝑦𝑘 )}(3)𝑑𝜏𝑑𝑧𝑘= 𝑥𝑘 + 𝑦𝑘𝑑𝜏The functions 𝑓𝑟 and 𝑓𝑑 correspond to 𝑣𝑟 (fifth-power negativeresistor) and 𝑣𝑑 (six diodes), respectively, and are represented
as follows:
𝑟1 + 𝑟2
1𝑟2 𝑥 + 𝑎2 −⋯ �𝑥 > �𝑎3𝑎3
𝑓𝑟 (𝑥 ) =𝑎2 − 𝑟1 𝑥 ⋯ �

11<𝑥< �𝑎1𝑎3(𝑎1 𝑎2 − 𝑟1 )𝑥 ⋯ �|𝑥 | <1�
𝑎1(4)Figure 4. Normarized characteristics of six diodes.

The characteristics of the Figs. 3 and 4 realize the oscillations
with hard nonlinearities and the chaotic oscillations,respectively. In this study, the parameter values of thesecharacteristics are always fixed as 𝑎1 = 2.0 , 𝑎2 = 0.1 ,𝑎3 = 0.01, 𝑟1 = 0.15, 𝑟2 = 0.1.

III. MULTI-MODE CHAOS
Four different oscillation modes (zero, in-phase singlemode, anti-phase single-mode and double-mode) coexist in thecircuit model of the Fig. 1. Each mode is determined by theinitial values. In this section, we show some simulation resultsof some attractors and Poincare maps on each mode toinvestigate its behavior when the parameter α increases. The
other parameters are fixed as β = 0.15, γ = 0.1, δ = 0.05 ,dτ = 0.001, ω = 1.88. (Fig. 5)

205

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

4.0(C-a)In-phase single-mode means that the two oscillators aresynchronized with the in-phase. The in-phase single-mode is
one-periodic oscillation at α = 4.0 , 8.0 and 12.0 , chaotic
oscillation at α = 16.0 by the attractors and Poincare maps.

Anti-phase single-mode means that the two oscillators aresynchronized with π phase difference. The anti-phase singlemode is quasi-periodic oscillation at α = 4.0 , one-periodic
oscillation at α = 8.0 , chaotic oscillation at α = 12.0 and
16.0 by the attractors and Poincare maps.
4.0

(A-a)

- 4.0

4.0

4.0

- 4.0

- 4.0

- 2.5

(A-c) 2.5

- 3.0

2.0
2.0

2.5

(1)

(B-a)

(B-b)

3.0

(2)

- 4.0
3.5

1.0

(3)

- 4.0

- 4.0

4.0

4.0

4.0

- 4.0

4.0

- 4.0

- 4.0

- 2.5

4.0

(B-c) 2.5

- 3.0

0

2.5

(1)

- 3.5
3.0 2.5

- 4.0

- 4.0

4.0

(C-c) 2.5

- 3.0

0

- 3.5
3.0 2.5

2.5

(1)

(D-a)

3.0

(2)

B: α = 8.0

C: α = 12.0

(D-b)

5.0

(3)

- 4.0

4.0

- 4.0

1.0

4.0

- 4.0

- 4.0

- 4.0

4.0

4.0

4.0

4.0

- 4.0

- 4.0

4.0

- 4.0

5.0

4.0

4.0

4.0

- 4.0

- 4.0
3.5 - 3.0

3.0

(2)

4.0

4.0

- 4.0

- 4.0

3.0

- 2.5

4.0

(D-c) 2.5

- 3.0

0

- 4.0
3.5 - 3.0

2.0
2.0

4.0

- 4.0

3.0

2.0
2.0

4.0

4.0

- 4.0

- 4.0

- 4.0

4.0

- 4.0

4.0

4.0

- 4.0

4.0

4.0

- 4.0

- 4.0

- 4.0

4.0

- 2.5

0

4.0

4.0

- 4.0

4.0

3.0

4.0

- 4.0

A: α = 4.0

4.0

4.0

- 4.0

2.0
2.0

4.0

- 3.5
3.0 2.5

- 4.0

4.0

- 4.0

- 4.0

3.0

- 4.04.0

4.0

- 4.0

4.0

- 4.0

- 4.0

- 4.0

- 4.0

4.0

(A-b)

- 4.0

4.0

- 4.0

(C-b)

- 4.0

4.0

- 4.0

4.0

4.0

4.0

- 4.0

4.0

- 4.0

4.0

4.0

1.0

(3)

2.5

(1)

- 3.5
3.0 2.5

3.0

(2)

- 4.0
3.5 - 3.0

1.0

(3)

5.0

D: α = 16.0

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) α = 4.0. (B) α = 8.0. (C) α = 12.0. (D) α = 16.0.
(1) In-phase sigle-mode. (2) Anti-phase single-mode. (3) Double-mode.
Red arrows point to the one-periodic Poincare maps: (A-c)(1), (B-c)(1)(2)
and (C-c)(1).

Double-mode means that the above two single-mode
oscillations (in-phase and anti-phase) are excited,
simultaneously and asynchronously. The double-mode isquasi-periodic oscillation at α = 4.0 , chaotic oscillation atα= 8.0, 12.0 and 16.0 by the attractors and Poincare maps.
5.0
Although the oscillations of double-mode chaos in the system
coupled by a capacitor have been reported [2][3], theseoscillations coupled by a time-varying resistor have not been206

reported yet. Especially, the double-mode chaos with chaos(in-phase) and chaos (anti-phase) such as the results at
α = 16.0 could not be found in the previous studies [2][3].

Figure 6 shows time waveforms of three different
oscillation modes. The frequency differences between two
single-modes are smaller than those of the previous studies[2][3]. We consider that this possibly affects the bifurcation
phenomena of each mode.
4.0IV.

CONCLUSIONSWe proposed two identical chaotic circuits with hard
nonlinearities coupled by a time-varying resistor andinvestigated the double-mode chaos in those systems bylooking at some attractors and Poincare maps. As a result, we
confirmed the interesting phenomena as double-mode chaoswith chaos (in-phase) and chaos (anti-phase) by computer
simulations. We will analyze the bifurcation of this doublemode chaos in the near future.
ACKNOWLEDGMENT

x1 0This work was partly supported by JSPS Grant-in-Aid for
Scientific Research 22500203.

- 4.0x2

REFERENCES01020(1) In-phase single-mode

3040

τ

4.0

[1]

[2]x1 0- 4.0[3]x20103020(2) Anti-phase single-mode

40

τ

[4]
[5]

4.0x1 0- 4.0x20200400(3) Double-mode

600

τ
800

Figure 6. Time waveforms vs amplitude (α = 16.0).

207

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