2007 IEEE Workshop on Nonlinear Circuit Networks
NCN'07, Tokushima, Japan, Dec 7--8, 2007

Switching Phenomena of Coexistence Attractors
in a Chaotic Map and 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

Abstract— Some chaotic systems have two coexistence attractors depending on initial values in same parameters. In thesesystems, changing parameters cause switching phenomena of twoattractors. In this study, we investigate switching phenomena ofcoexistence attractors in an one-dimensional map and a chaoticcircuit. The purpose of this study is a comparison a element ofCML with a element of coupled chaotic circuits. Using resultsof this study, we will investigate the relationship between CML
and coupled chaotic circuits.

tween chaotic elements of CML and coupled chaotic circuits.
Namely, results of this study can make a contribution to thecomparison between CML and coupled chaotic circuits.

I. I NTRODUCTION

xn+1 = ax3n + cxn .

There are many studies of coupled systems. Especially,
coupled chaotic systems attract many researchers’ attentions
because they exhibit a large variety of interesting nonlinear
phenomena. One of famous coupled chaotic systems is theCoupled Map Lattice (CML) proposed by Kaneko [1]. CML
has been investigated by many researchers and many complex
phenomena, for instance, spatio-temporal phenomena, clustering and so on, have been reported. However, this system isa discrete-time system. Some important phenomena including
coupled chaotic systems may be lost by time discretization.
Therefore, CML should be compared with continuous-timesystems. One of the continuous-time system is coupled chaoticcircuits. By the reason that characteristics of circuit elements
are low price and high quality, you can get elements easily
and the repeatability of the experimental results is very high.
Additionally, experiments can be carried out in a short time.
Coupled chaotic circuits are studied actively. In these studies,
phase states or synchronizations are studied mainly.
On the other hands, some chaotic systems have two coexistence attractors depending on initial values in same parameters [2]-[4]. In these systems, changing parameters cause
switching phenomena of two attractors. We consider thatmatching two attractors in coupled systems which have twocoexistence attractors means the phenomenon like a very low
synchronization. By investigating switching phenomena in thesystem, it is expected that some interesting phenomena areobserved in coupled systems which have two coexistence
attractors.
In this study, we investigate switching phenomena of coexistence attractors in an one-dimensional map and a chaoticcircuit. The one-dimensional map described by a third-orderpolynomial function. The chaotic circuit is Shinriki-Mori
circuit[2][3]. The purpose of this study is the comparison be-

In order to investigate switching phenomena, a third-orderpolynomial function are used. Figures 1 show the onedimensional map using in this study. Two coexistence attractors are observed (red and blue.) In the cases of Fig. 1(a)and (b), same parameters and different initial values are used.
By increasing c, switching phenomena are observed in Fig. 1(c). These are symmetric about the origin. Figure 2 shows theone parameter bifurcation diagram of Eq. (1). The route fromperiodic orbits via period doubling bifurcations to chaos areobserved. Switching phenomena are observed from β = 2.60
to β = 3.00. In order to investigate switching phenomena,
we deﬁne region D+ and D− as right and left side attractorsof Figs. 1 respectively. Transitional conditions are shown asfollows.
D+ → D− : when x > 0 and becomes x < 0.
(2)
D− → D+ : when x < 0 and becomes x > 0.

II. S YSTEM M ODELS
A. Chaotic Map
The one-dimensional map using in this study is shown asfollowing equation.
(1)

B. Chaotic Circuit
Figure 3 shows the chaotic circuit used in this study. Thecircuit equation is described as follows.
di
L= v1 ,
dt



dv1
(3)
= −i − id ,
Cdt




dv

 C0 2 = gv2 + id ,dtwhere,
a(v1 − v2 − Vth ), v1 − v2 > Vth ,
0,−Vth ≤ v1 − v2 ≤ Vth ,id =(4)
 a(v1 − v2 + Vth ), v1 − v2 < −Vth .

- 39 -g(a)C0CL(b)Fig. 3.

Shinriki-Mori chaotic circuit.

Id
Vdg1-VthId

Vd
Vth(c)
Fig. 1.
The one-dimensional map described by a third-order polynomial
function. (a) a = −2.75 and c = 2.54. Initial value of x is −0.4. (b)a = −2.75 and c = 2.54. Initial value of x is 0.4. (c) a = −2.75 andc = 2.80.

Fig. 4.

Diodes model.

where,

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

(7)

We carry out computer simulations using this equation.
Figures. 5 show some computer simulation results. Horizontal
axis is z. Vertical axis is y. Two attractors shown in Fig.5 (a)and Fig.5 (b) are observed in same parameters. Here, we deﬁneregion D+ and D− as right and left side attractors of Figs. 5
respectively. Transitional conditions are shown as follows.
Fig. 2. The one parameter bifurcation diagram of Eq. (1). a = −2.75 and1 < c < 3. Red: the initial value of x is 0.4. Blue: the initial value of x is-0.4.

Two diodes are modeled as a piece-wise linear function shownin Fig. 4. Changing variables and parameters as follows,√1L1
1x=· i, y =· v1 , z =· v2
Vth C
VthVth√√CCdL
L, β=g, γ=,=”·”
α=aC
C0 CC0dt(5)Normalized circuit equation is described as follows.
ẋ = y,ẏ = −x − f (y − z),(6)ż = βz + γf (y − z),

D+ → D− : when y − z > −1 and becomes y − z < −1.
D− → D+ : when y − z < 1 and becomes y − z > 1.
(8)
Increasing β, coexistence of attractors are observed shownas Fig. 5 (c). Figure 6 show 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. In this area, phenomena do not dependon the initial value.
III. S WITCHING P HENOMENA
A. Chaotic Map
Figure 7 shows the magniﬁcation of a third-order polynomial function using in this study. In the case of q < x < p,switching are occurred. In the case of 0 < x < q, switching are
not occurred. In order to carry out the theoretical analysis, we- 40 -orbits shown as Fig. 9 (b) and (d). We consider that the reasonof differences between SRI and switching rates are that thedistribution of x is not homogeneous. However, these gradientsare similar. We will investigate the distribution of x in this map
and improve the SRI .

(a)(b)

Fig. 7.

(c)Fig. 5.
Computer simulation results of the chaotic circuit. α = 10.0 and
γ = 2.605. (a) β = 0.54. (b) β = 0.54. (c) β = 0.56. Differential initialvalues are used in (a) and (b).

Fig. 6. The one parameter bifurcation diagram of the chaotic circuit. α =−2.75 and 0.4 < β < 0.68.

assume that the distribution of x is homogeneous distribution.
Using this assumption, we can described the switching rates
SRI as follows.

B. Chaotic Circuit
Figure 10 shows the relationship between parameter β
and switching rates of the chaotic circuit. Horizontal axis
shows β and vertical axis shows switching rates. Figures 11
show some computer simulation results. Some periodic orbits(b)(c)(d) are observed. In these areas, switching phenomena
are stable states. In another areas, switching rates are increased
by increasing β. This system is not symmetric in the origin.
Therefore, red and blue lines are not the same. However, verysimilar results are conﬁrmed. For instance, blue and red lines
are corresponding to blue and red in Figs. 5 and Figs. 9.
C. Comparison
Both of them, periodic orbits, period doubling bifurcation,chaos, switching phenomena are observed. Increasing parameter c or β increase switching rates. These results show that twosystem are similar. Major difference is that the chaotic map issymmetry system. In order to investigate coupling system, wewill closer this map to the chaotic circuit.
IV. C ONCLUSIONS
We have investigated switching phenomena of coexistence
attractors in an one-dimensional map and a chaotic circuit.
Using obtained results, we will investigate the switching
phenomena of CML and coupled chaotic circuits.

√c√−aq3 3√
SRI = 1 − = 1 −.=1−.
p2c2c −c33a

The magniﬁcation of a third-order polynomial function.

R EFERENCES(9)

Figure 8 shows the relationship between parameter c andswitching rates. Green line shows SRI , Blue and Red lines
show switching rates calculated by computer simulations.
Figures 9 show computer simulation results. In Fig. 8, blue andred lines are the same. Because this map is symmetric about
the origin. Some local maximums are observed by periodic[1] K. Kaneko, “Spatiotemporal Intermittency in Coupled Map Lattices,”
Prog. Theor. Phys. vol. 74, no. 5, pp. 1033-1044, 1985.
[2] M. Shinriki, M. Yamamoto and S. Mori, “Multi-mode Oscillations ina Modiﬁed van der Pol Oscillator Containing a Positive NonlinearConductance,” Proc. IEEE, vol. 69, pp. 394-395, 1981.
[3] 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.
[4] 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.
[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.

- 41 -                                          Fig. 8.

Switching rates of the chaotic map. a = −2.75, 2.60 < c < 3.00

Fig. 10. Switching rates of the chaotic circuit. α = 10.0, γ = 2.605 and0.56 < β < 0.68.

(a) c = 2.65(b) c = 2.70(a) β = 0.561(b) β = 0.588(c) c = 2.76(d)c = 2.84(c) β = 0.620(d) β = 0.655(e) β = 0.666(e) c = 3.00
Fig. 9.

Computer simulation results of the chaotic map. a = −2.75

Fig. 11. Computer simulation results of the chaotic circuit. α = 10.0,
γ = 2.605- 42 -