2014 RISP International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'14, Honolulu, Hawaii, USA. February 28 - March 3, 2014

Chaotic Circuit Generating Two Main Frequency Band Oscillations
Yasuteru Hosokawa1 and Yoshifumi Nishio2
1

Shikoku University
Furukawa, Ohjin, Tokushima, Japan
Phone:+81-88-665-1300
FAX:+81-88-637-1318
E-mail: hosokawa@keiei.shikoku-u.ac.jp

2

Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470
FAX:+81-88-656-7471
E-mail: nishio@ee.tokushima-u.ac.jp

abstractif(v ) Lvi1In this study, novel chaotic circuit generating two main frequency band oscillations is proposed. We consider that thiscircuit is suitable for finding a novel phenomena related toquasi-synchronization.

-R 1v

1

1

C1

1

2

C22

L2

Figure 1: Proposed circuit.

1. Introduction

Many kinds of interesting phenomena have been observedf(v)in coupled chaotic circuits. One topic of coupled chaotic cirg
f(v)cuits is a chaotic synchronization. A chaotic synchroniza1tion is a phenomenon that orbits of coupled chaotic oscilla-VthV th vtors become same. There are many studies about a chaoticsynchronization. Especially, we pay attention to a chaotic
gquadi-synchronization. In large-scale coupled chaotic cir1
cuits, chaotic quadi-synchronization is observed and some related phenomena are observed. The observed phenomena areinfluenced by a coupling strength, a topology of the system
Figure 2: Bidirectionally coupled diode model.
and characteristics of coupled circuits. The coupling strengthand the topology can be changed easily. However, changing
the characteristics of coupled circuits is not easily, because of
there are not so much the number of chaotic circuits. In or- in Fig. 2. By using this model, the circuit equation is deder to progress the study of coupled chaotic circuit systems, scribed as follows:
many kinds of chaotic circuits are needed.
In this study, novel chaotic circuit generating two main fre1
dv1

quency band oscillations is proposed. Normally, a chaotic=
v1 − i1 − f (v1 ),
C1


dtR
1

oscillation has a continuous frequency band including maindi2
frequency. We consider that this circuit is suitable for findingL1= v1 − v2 ,dta novel phenomena related to quasi-synchronization.
(1)
dv3


C=i−i,
212
dt

2. Proposed circuit



di4


Figure 1 is a proposed circuit in this study. The circuit
 L2 dt = v2 ,consists of two capacitors, two inductors, two diodes and one
negative resistor. By two L-C set, two main bands are generated. Parameters C1 and L1 are set as smaller values than whereparameters C2 and L2 .
In order to derive a circuit equation, bidirectionally couf (v) = gv1 + g(|v1 − Vth | − |v1 + Vth |)/2.
(2)
pled diodes are modeled as a piecewise linear function shownv- 697 -By substituting the variables and the parameters,√v1
L1 i1v2
x1 =, x2 =, x3 =,VthC1 VthVth√
L1 i2
d1x4 =,= ” · ”, τ = √t,
C1 Vth dt
L1 C1√√
1L1C1L1L1α=, β=, γ=and δ = g.
R1 C1C2L2C1equations (1) and (2) are normalized asẋ1 = αx1 − x2 − xd , ẋ2 = x1 − x3 ,ẋ3 = β(x2 − x4 ), ẋ4 = γx3 ,

v1v2
(3)
Figure 3: Waveform observed in the circuit experiment.
R1 = 3.0[kΩ].
(4)
v1v2

where
xd = δx1 + δ(|x1 − 1| − |x1 + 1|)/2.
3. Exact solutions and Poincaré Map(5)Figure 4: Waveform observed in the computer calculation.
α = 0.6.

Since the circuit equations (4) are piecewise-linear, solutions in each linear region can be derived. At first, we define the condition ẋ1 < 0. Namely, S corresponds to the transithree piecewise-linear region as follows.
tional condition from R1 to R2 .
Let us consider the solution starting from an initial point
R1 : x1 > 1.
on S. The solution returns back to S again after wandering
R2 : −1 ≤ x1 ≤ 1.
(6)
R1 , R2 and R3 . Hence, we can derive the Poincaré map as
R3 : x1 < −1.
follows.
T : S → S, x0 7→ T(x0 ).
(10)The eigenvalues in each region are calculated from Eq. (4).
The eigenvalues in each region are described as follows.

where, x0 is an initial point on S, while T(x0 ) is the point at
which the solution starting from x0 hits S again. T(x0 ) can
R1 and R3 : λ11 , λ12 , σ11 ± jω11 .
(7) be derived by using the exact solutions. The Jacobian matrix
R2 :σ21 ± jω21 , σ22 ± jω22 .
DT of the Poincaré Map T can be also derived.
We can calculate the largest Lyapunov exponent byThe eigenvalues in each region are obtained from the 4th order eigenequation of Eq. (4). The equilibrium points of eachN
1 ∑region are calculated by putting the right side of Eq. (4) to beµ = limlog |DTj · ej |.
(11)
N →∞ Nequal to zero. Then, the solutions in each linear region can be
j=0
described. We omit the description, because it is complicatedhere ej is a normalized base.
enough to write here.
Now, we are deriving the Poincaré map and the Jacobianmatrix.
4. Circuit Experiments And Computer Calculations
Let us define the following subspace
Figures 3, 4 and 5 are a circuit experiment results and a
S = S1 ∩ S2 .
(8) computer calculation results. In circuit experiments, circuit
parameters are fixed as C1 = 0.015[µF], L1 = 5[mH], C2 =where,0.10[µF] and L2 = 100[mH]. Corresponding parameters of
S1 : x1 = 1,
(9) computer calculations are fixed as β = 0.15, γ = 0.05 andS2 : x2 > α.
δ = 10. Control parameters are set as R1 and α.
The subspace S1 corresponds to the boundary condition beWaveforms are shown in Figs 3 and 4. Long and shorttween R1 and R2 , while the subspace S2 corresponding to wavelengths are observed.

- 698 -Projections of attractors onto v1 and v2 are shown inFig. 5 (a). Projections of attractors onto x1 and x3 and theirPoincaré maps are shown in Fig. 5 (b) and (c), respectively.
Rows of Fig. 5 are corresponding to each others.
By changing control parameters, periodic orbits are observed in Figs. 5 (1), (2), (4) and (5). Chaotic phenomena
are observed in Figs. 5 (3) and (6).
One-parameter bifurcation diagram and the calculated
largest Lyapunov exponent are shown in Fig. 6 and Fig. 7, respectively. Bifurcation phenomena of periodic orbits, chaos,
window and so on are observed. For 1.40 < α < 7.80,
the largest Lyapunov exponent becomes positive though somelarge windows are observed. By using this result, We can say
the generation of chaos is confirmed numerically. By changing other parameters, period doubling bifurcation phenomena, tours and so on are also observed in this circuit.

(1)

α = 0.20

R1 = 1654[Ω]α = 0.30

R1 = 707[Ω]α = 0.40

R1 = 562[Ω]α = 0.54(2)(3)

5. Conclusions
In this study, a novel chaotic circuit generating two mainfrequency band oscillations has been proposed. By circuit
experiments, chaotic attractors have been observed and byusing a linearized model, the generation of chaos has been
confirmed numerically.
We consider that this circuit is suitable for finding a novel
phenomena related to quasi-synchronization. Therefore, it isexpected that this circuit contributes to studies of large scale
coupled chaotic circuits.

R1 = 2210[Ω](4)(5)

References

R1 = 530[Ω](a)α = 0.77(c)[1] L. M. Pecora and T. L. Carrol, “Synchronization inChaotic Systems,” Physical Review Letters, vol. 64, pp.
821–824, 1990.
Figure 5: Circuit Experimental Results and Computer Calculation Results. (a) Circuit Experiments. v1 -v2 . (b) Attractors.
[2] T. Endo, and S. Mori, “Mode Analysis of a Ring of a x -x . (c) Poincaré maps. v -v .
1 21 3Large Number of Mutually Coupled van der Pol Oscillators,” IEEE Trans. Circuit and Systems, vol. CAS-25, no.
5, pp. 7–18, 1978.
[6] Y. Hosokawa, R. Tsujioka and Y. Nishio, “Relation between Synchronous Rate and Small Variations[3] D. A. Linkens, “Analytical solution of large numbers
on an Asymmetrical Coupled Chaotic System,” Proc.
of mutually-coupled nearly-sinusoidal oscillators,” IEEENOLTA’05, pp. 166–169, 2005.
Trans. Circuits and Systems, vol. CAS-21, pp.294–300,[7] Y. Hosokawa and Y. Nishio, “Asymmetrical Chaotic CouMar, 1974.
pled System Using Two Different Parameter Sets,” Proc.
NOLTA’06, pp. 543–546, Sep. 2006.
[4] H. Sekiya, S. Moro, S. Mori, and I. Sasase, “Synchronization Phenomena on Four Simple Chaotic Oscillators
Full-Coupled by Capacitors,” IEICE Trans. Fundamentals, vol. J82-A, No. 3, pp.375–385, 1999.
[5] M. Miyamura, Y. Nishio and A. Ushida, “Clustering in
Globally Coupled System of Chaotic Circuits,” IEEE
Proc. ISCAS’02, vol. 3, pp. 57–60, 2002.

- 699 -(b)

1.0

0.0

-1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 6: One parameter bifurcation diagram. Vartical axis: x3 . Horizontal axis: α. 0.01 ≤ α ≤ 0.80. Parameter Step:0.001. Plotted time: 10000[τ ] par. one parameter.

0.5

0.0

-0.5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Figure 7: Largest Lyapunov exponent. Horizontal axis: α. 0.01 ≤ α ≤ 0.80. Parameter Step: 0.001.

- 700 -0.8