Comparison of Switching Phenomena
in Chaotic Maps or a Chaotic Circuit
Tomoko Yamada

Yasuteru Hosokawa

Yoshifumi Nishio

Dept. Information Science,
Shikoku University
Email: s1839074@keiei.shikoku-u.ac.jp

Dept. Information Science,
Shikoku University
Email: hosokawa@keiei.shikoku-u.ac.jp

Dept. Electrical and Electronic Eng.,
Tokushima University
Email: nishio@ee.tokushima-u.ac.jp

Summary— Some chaotic systems have two coexisting attractors depending on initial values in same parameters. In thesesystems, changing parameters causes switching phenomena ofcoexisting attractors. In this study, we investigate switching
phenomena of coexisting attractors in one-dimensional maps ora chaotic circuit. The purpose of this study is to compare aelement of CML with a element of coupled chaotic circuits. By
using results of this study, we will investigate the relationshipbetween CML and coupled chaotic circuits.

Figures 3 and 4 show one parameter bifurcation diagramsof Eq. (1) and Eq. (2), respectively. Switching phenomena
are observed between 2.60 < a < 3.00 and 2.00 < a <3.00. In Fig. (1), period doubling phenomena are observed. InFig. (2), chaotic phenomena are observed in 1.00 < a < 3.00.
In order to investigate switching phenomena, we deﬁne region
D+ and D− as right and left side attractors of Figs. 1 and 2.
Transitional conditions are shown as follows.

I. I NTRODUCTION

D+ → D− : when x > 0 and becomes x < 0.
D− → D+ : when x < 0 and becomes x > 0.

Some chaotic systems have two coexisting attractors depending on initial values in same parameters [1]-[3]. In thesesystems, changing parameters causes switching phenomena ofcoexisting attractors. We consider that matching two attractors
in coupled systems which have two coexisting attractors means
the phenomenon like a very weak synchronization. By investigating switching phenomena in the system, it is expected thatsome interesting phenomena are observed in coupled systems
which have two coexistence attractors.
In this study, we investigate switching phenomena in onedimensional maps and a chaotic circuit. The chaotic circuit
is Shinriki-Mori circuit [1][2]. The purpose of this study isthe comparison between chaotic elements of CML [4] andcoupled chaotic circuits. Namely, results of this study canmake a contribution to the comparison between CML andcoupled chaotic circuits.

(a)

(b)

(3)

(c)

Fig. 1. The one-dimensional maps described by Eq. (1). (a) a = 2.6 andx0 = −0.4. (b) a = 2.6 and x0 = 0.4. (c) a = 2.8 and x0 = 0.4.

II. S YSTEM M ODELS
A. Chaotic Map
In order to investigate switching phenomena, two kindsof one-dimensional maps are investigated in this study. The
functions of two maps are shown as follows.
xn+1 = −ax3n + axn .

(1)1
1xn+1 = −axn + a|xn + | − a|xn − |.
(2)
22Figures 1 and 2 show the one-dimensional maps derived byEqs. (1) and (2), respectively. Two coexisting attractors areobserved (red and blue.) Same parameters and different initialvalues are used in Figs. 1 (a) and 1 (b) and Figs. 2 (a) and2 (b), respectively. By increasing a, switching phenomena areobserved in Figs. 1 (c) and 2 (c). These are symmetric about
the origin.

(a)(b)(c)Fig. 2. The one-dimensional maps described by Eq. (2). (a)a = 1.97. andx0 = −0.40. (b)a = 1.97. and x0 = 0.40. (c)a = 2.60. and x0 = 0.40

B. Chaotic Circuit
Figure 5 shows the chaotic circuit used in this study. Thecircuit equation is described as follows.

- 55 -Id
Vdg1
Vd-VthVthId

Fig. 6.

Diodes model.

Changing variables and parameters as follows,Fig. 3. The one parameter bifurcation diagram of Eq. (1). 1 < a < 3. Red:
the initial value of x is 0.4. Blue: the initial value of x is -0.4.

1x=Vth√√L· i,C1
1· v1 , z =· v2VthVth√
CCLL, β=g, γ=,α=aC
C0 C
C01dτ=√
t,=”·”dτLC
y=(6)Normalized circuit equation is described as follows.

Fig. 4. The one parameter bifurcation diagram of Eq. (2). a = −2.75 and1 < c < 3. Red: the initial value of x is 0.4. Blue: the initial value of x is-0.4.

diLdt
dv1Cdtdv C0 2dt=v1 ,=−i − id ,=gv2 + id ,(4)ẋ = y,ẏ = −x − f (y − z),ż = βz + γf (y − z),(7)where,

α(y − z − 1), y − z > 1,0,−1 ≤ y − z ≤ 1,f (y − z) = α(y − z + 1), y − z < −1.

(8)

Using this equation, computer simulations are carried out.

where,

a(v1 − v2 − Vth ), v1 − v2 > Vth ,0,−Vth ≤ v1 − v2 ≤ Vth ,id = a(v − v + V ), v − v < −V .
12th12th(5)(a)

Two diodes are modeled as a piece-wise linear function
shown in Fig. 6.

gFig. 5.

C0CL(b)(c)

Computer simulation results of the chaotic circuit. α = 10.0 andFig. 7.
γ = 2.605. (a) β = 0.54. (b) β = 0.54. (c) β = 0.56. Differential initialvalues are used in (a) and (b).

Figure 7 shows some computer simulation results. Horizontal
axis is z. Vertical axis is y. Two attractors shown in Fig. 7 (a)and Fig. 7 (b) are observed in same parameters. Here, wedeﬁne region D+ and D− as right and left side attractorsof Fig. 7 respectively. Transitional conditions are shown asfollows.
D+ → D− : when y − z > −1 and becomes y − z < −1.
D− → D+ : when y − z < 1 and becomes y − z > 1.
(9)

Shinriki-Mori chaotic circuit.

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Increasing β, coexistence of attractors are observed shownas Fig. 7 (c). Figure 8 shows the one parameter bifurcation
diagram of the chaotic circuit. The case that the initial value is0.11 is shown as red. The case that the initial value is −0.11
is shown as blue. Switching phenomena are observed fromβ = 0.56 to β = 0.68.

Fig. 9.

The magniﬁcation of a third-order polynomial function.

C. Comparison

Fig. 8. The one parameter bifurcation diagram of the chaotic circuit. α =−2.75 and 0.4 < β < 0.68.

III. S WITCHING P HENOMENA
A. Chaotic Map
Figure 9 shows the magniﬁcation of a third-order polynomial√ function as shown in Eq. (1). In the case of 1 < x <2 3
9 a, switching are occurred. In the case of 0 < x < 1,switching are not occurred. In order to carry out the theoreticalanalysis, we assume that the distribution of x is homogeneous
distribution. Using this assumption, we can described theswitching rates SRi as follows.
√3 3
SRi =.
(10)
2a
In the case of Eq. (2), the switching rates SRt can be also
described as follows.
2SRt = .
(11)
aFigures 10 and 11 show relationships between parameter aand switching rates. Yellow lines show SRi and SRt , respectively. Blue and red lines show switching rates calculated bycomputer simulations. Blue and red lines are the same because
this map is symmetric about the origin. Some local maximums
are observed by periodic orbits called as a window. In thecase of Fig. 10, switching rates differ from SRi . We consider
that the reason is the distribution of x is not homogeneous.
However, these gradients are similar. In the case of Fig. 11,switching rates is very close with SRt , because the distributionof x is similar to homogeneous.
B. Chaotic Circuit
Figure 12 shows the relationship between parameter β andswitching rates of the chaotic circuit. Horizontal axis shows
β and vertical axis shows switching rates. This system is not
symmetric in the origin. Therefore, red and blue lines are not
the same.

Major difference between the maps and the circuit is thatchaotic maps are symmetry systems and the chaotic circuit
is asymmetry system. However, increasing parameter a or β
increase switching rates. These results show that these systems
are similar. In the cases of Figs. 10 and 12, some peaks of rates
are observed by windows. In the case of Fig. 11, any peak are
not observed because there is no window as shown in Fig. 4.
As a result of this comparison, we can say that as follows.
• Eq. (1) and the chaotic circuit can observe similar phenomena.
• Switching rate of Eq. (1) is not much to Eq. (10).
• Switching rate of Eq. (2) is much to Eq. (11).
Therefore, in order to investigate coupling systems, Modifying
Eq. (1) or detecting a chaotic circuit whose observed phenomena is similar to Eq. (2).
IV. C ONCLUSIONS
We have investigated switching phenomena of coexisting
attractors in two one-dimensional maps and a chaotic circuit.
As a result, closing phenomena which is observed in chaotic
maps and a circuit is needed in order to compare the chaotic
map and the chaotic circuit. By closing observed phenomena,
we will investigate the switching phenomena of CML andcoupled chaotic circuits.
R EFERENCES[1] M. Shinriki, M. Yamamoto and S. Mori, “Multi-mode Oscillations ina Modiﬁed van der Pol Oscillator Containing a Positive Nonlinear
Conductance,” Proc. IEEE, vol. 69, pp. 394-395, 1981.
[2] N. Inaba, T. Saito and S. Mori, “Chaotic Phenomena in a Circuit witha Negative Resistance and an Ideal Switch of Diodes,” Tran. IEICE,
vol. E70, no. 8, 1987.
[3] Y. Nishio, N. Inaba, S. Mori and T. Saito, “Rigorous Analyses of Windows
in a Symmetric Circuit,” IEEE Tran. Circuits Syst., vol. 37, no. 4, 1990.
[4] K. Kaneko, “Spatiotemporal Intermittency in Coupled Map Lattices,”
Prog. Theor. Phys. vol. 74, no. 5, pp. 1033-1044, 1985.
[5] M. Wada, K. Kitatsuji and Y. Nishio, “Spatio-Temporal Phase Patterns
in Coupled Chaotic Maps with Parameter Deviations,” Proc. NOLTA’05,pp. 178-181, Oct. 2005.

- 57 -Fig. 10.

Switching rates of the chaotic map described by Eq. (1). Red: D+ → D− . Blue: D− → D+ . Green: Calculating results of Eq. (10).

Fig. 11.

Switching rates of the chaotic map described by Eq. (2). Red: D+ → D− . Blue: D− → D+ . Green: Calculating results of Eq. (11).

Fig. 12.

Switching rates of the chaotic circuit. α = 10.0, γ = 2.605 and 0.56 < β < 0.68.

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