Bifurcation and Basin in Two Coupled
Parametrically Forced Logistic Maps
Hironori Kumeno

Yoshifumi Nishio

Daniele Fournier-Prunaret

Dept. of Electrical and Electronic Eng., Dept. of Electrical and Electronic Eng., LATTIS-INSA, Universite de Toulouse
Tokushima University,
Tokushima University,
135 avenue de Rangueil,
2-1 Minami-Josanjima,
2-1 Minami-Josanjima,
31077 Toulouse, France
Tokushima, 770-8506 JAPANTokushima, 770-8506 JAPAN
Email: daniele.fournier@insa-toulouse.fr
Email: kumeno@ee.tokushima-u.ac.jp
Email: nishio@ee.tokushima-u.ac.jp

Abstract— Two coupled logistic maps whose parameters areforced into periodic varying are investigated. From the investigation of bifurcation in this system, nonexistence of odd periodic orbit except fixed point and existence of many coexisting attractors,
which consist of periodic orbits or chaotic orbits, are observed.
Basins where boundary depends on the invariant manifold of
saddle points are numerically analyzed by considering second
order iteration and using superposition with Newton method,
although the system has discontinuity.

I. I NTRODUCTION
There exists many incomprehensible relationship between
brain mechanism and nonlinear phenomena of neuron andneural network. Although studies by using equipments which
record global brain activity have possibilities, they arepresently difficult. Therefore, we come up with the way whichsolves the problem by using model of brain with nonlinear
systems. Neuron activity can be described by deterministic
dynamical equation. So, it is possible to introduce models toinvestigate nonlinear neuronal activities and mutual influences
between neurons in the neural networks by using nonlinear
maps. When the real model which mimic brain activities canbe analyzed, it does not only make progress on understandingof information processing mechanism of the brain, but also
contributes to realize an information processing brain computer. So the brain activities can be realized on computers orspecial hardwares. It is known that if an autonomic neuron
is influenced by external force, its characteristics are changed
such as the neuron behaves periodic or chaotic [1]. How to give
periodic external force is thought about in various ways, but
we focus on a system whose parameter is forced into periodic
varying and investigate behavior of the system and its pint-size
coupling. The fundamental researches of these small coupled
nonlinear systems are important for the investigation of global
brain activities and the modeling.
As the research for a system with parameter forced into
periodic varying, Ref. [2] reports that the almost periodic
oscillation occurs in a simple oscillator including parametric
excitation if nonlinear inductor has saturation characteristic.
Additionally the occurrence of chaos is referred in Refs. [3]and [4]. Previously, we have investigated synchronization phenomena in three coupled logistic maps involving parametric978-1-4244-9472-9/11/$26.00 ©2011 IEEEforce [5].
In this study, we investigate behavior of orbits which areperiodic or chaotic in two coupled logistic maps whose parameters are forced into periodic varying. The paper is organized
as follows. In the next section, we present the parametricallyforced logistic map and coupling method used in this study.
Bifurcation of the system is investigated in Section 3. In
the investigation, nonexistence of odd periodic orbit except
fixed point is confirmed. Section 4 illustrates basins when theparameter set corresponds to multistability phenomena. In thebasins, fractal structures are confirmed at the corner of eachbasin. In section 5, fixed point depending on the order of basin
is numerically analyzed by using second order projection andits superposition. The last section is devoted to the conclusion.
II. PARAMETRICALLY FORCED LOGISTIC MAP
A parametrically forced logistic map used in this study is
described as:𝑥(𝑛 + 1) = 𝛼𝑓 (𝑛)𝑥(𝑛)(1 − 𝑥(𝑛)),(1)and⎧⎨ 𝛼1 , for each even value of “𝑛”𝛼2 , for each odd value of “𝑛”𝛼𝑓 (𝑛) =,⎩(𝑛 = 0, 1, 2, ...)
(2)
where 𝛼𝑓 (𝑛) is a term of the parametric force and timevarying. In this system, two kinds of parameters, 𝛼1 and
𝛼2 , are alternately replaced every update. Figure 1 shows anexample of a return map of the parametrically forced logistic

Fig. 1. Return map of parametrically forced logistic map for 𝛼1 = 3.0 and
𝛼2 = 3.83.

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(a)(b)Fig. 2. Representation of the parameter plane (𝛼1 , 𝛼2 ). Each colored part corresponds to the existence of a stable cycle (periodic point), the order 𝑘(period) of which is given by the upper colored squares. The black color corresponds to 𝑘 ≥ 15 or to chaotic behavior. The white region corresponds to thenonexistence of attracting set, with chaotic transient toward infinity. Horizontal axis: 𝛼1 . Vertical axis: 𝛼2 . (b) Enlargement of (a)map. For the original logistic map, two-periodic solution
is observed for 𝛼 = 3.0, while, three-periodic solution isobserved for 𝛼 = 3.83. These two solutions are periodic,
whereas in the logistic map involving parametric force, asolution is chaotic as shown in Fig. 1 when the parameters
𝛼1 and 𝛼2 are set 3.0 and 3.83, respectively. Namely, chaotic
solution can be observed in the combination of two parametersthat generate two kinds of periodic solutions. This map iscoupled with following coupling method:𝑥𝑖 (𝑛 + 1) = (1 − 𝜀)𝑓 [𝑥𝑖 (𝑛)] + 𝑁𝜀𝑖 = 1, 2.

2∑
𝑗=1

𝑓 [𝑥𝑗 (𝑛)]

(a)

(b)
(3)

where 𝜀 ∈ [0, 1] is the coupling intensity. Then, the map 𝑇 is
described as:⎧𝜀
𝑥1 (𝑛 + 1) = (1 − )𝛼𝑓 𝑥1 (𝑛)(1 − 𝑥1 (𝑛))
2
𝜀
⎨
+ 𝛼𝑓 𝑥2 (𝑛)(1 − 𝑥2 (𝑛))2𝑇 :. (4)𝜀)𝛼𝑓 𝑥2 (𝑛)(1 − 𝑥2 (𝑛))𝑥(𝑛+1)=(1−2
2𝜀⎩+ 𝛼𝑓 𝑥1 (𝑛)(1 − 𝑥1 (𝑛))2This study deals with the case that two parametrically forced
logistic maps are coupled.
III. B IFURCATIONIn this section, bifurcations are investigated with respect
to 𝛼𝑓 . Figure 2 shows a two-parameter bifurcation diagramand its enlargement when 𝜀 is fixed at 0.04 and 𝛼1 and 𝛼2are changed. In the parameter (𝛼1 , 𝛼2 ) plane, the existence
domain of attracting set at finite distance is given by theFig. 3. return maps and phase differences in coexistence of period six orbit
and chaotic orbit depending on initial values. 𝛼1 = 2.94, 𝛼2 = 3.84 and𝜀 = 0.04.

colored part of Fig. 2. Only even periodic orbits are observed
except fixed point and the case of 𝛼1 = 𝛼2 . In the figure,some discontinuous boundaries of bifurcation are observed,
for instance smooth boundary between green region andblack region is intercepted by protuberant green region inFig. 2(b). This discontinuity is caused due to multistability andinitial conditions of calculation. In some regions, multistability
behaviors can be observed. Different attractors can coexist,
depending on the choice of different initial conditions. An
example of the coexistence is shown in Fig. 3 when 𝛼1 = 2.94,
𝛼2 = 3.84 and 𝜀 = 0.04. In this figure, period 6 orbit andchaotic orbit coexist, these orbits correspond to different initialvalues of (𝑥1 , 𝑥2 ).

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(a)
Fig. 4.

(b)(c)

Basin regarding to 𝑇 for 𝛼2 = 3.84 and 𝜀 = 0.04. (a) 𝛼1 = 2.94. (b) 𝛼1 = 2.97. (c) 𝛼1 = 2.98. Horizontal axis: 𝑥1 . Vertical axis: 𝑥2 .

In the next section, basin which is an initial value set
converging to a given orbit is investigated for parameter values
corresponding to multistability.
IV. BASIN
Depending on the property of the system, 𝛼𝑓 is changed
every update periodically. Figure 4 shows basin when theparameters 𝛼2 and 𝜀 are considered as having fixed values,
𝛼1 varies with increasing values. Each basin of attraction,
represented by a different color, corresponds to the period oforbit. Basins in Fig. 4(a) consist of the region having period
6 orbit, represented by cyan, and the region having chaoticorbit, represented by black. Increasing 𝛼1 , basins are changing
because there is coexistence of a period 12 orbit and a chaoticorbit. More increasing 𝛼1 , we obtain three kinds of orbits,
which are period 6 orbit, period 12 orbit and chaotic orbit.
In the basins, fractal structures are confirmed at the cornerof each basin. Figures 5 show the enlargement around a cornerof basin in Fig. 4(a) and its enlargements. With repeating
enlargement, similar structure continues.

(a)

(b)

(c)

(d)

V. P ERIODIC ORBITSThe parametrically forced logistic map is not continuous in
iteration because 𝛼𝑓 is periodically changed. Then, in order toobtain periodic orbits of 𝑇 , that are all even, we can consider
𝑇 2 . An order 𝑘 periodic orbit of 𝑇 2 will be an order 2𝑘periodic orbit of 𝑇 . So we will obtain all even periodic orbitsof 𝑇 by calculating all periodic orbits of 𝑇 2 . To do this, weintroduce 𝑇𝑒𝑣𝑒𝑛 and 𝑇𝑜𝑑𝑑 . We define the trajectory startingfrom the even number of 𝑛 as 𝑇𝑒𝑣𝑒𝑛 , and the trajectory startingfrom the odd number of 𝑛 as 𝑇𝑜𝑑𝑑 . All order 𝑘 periodic orbits
22of 𝑇 2 will be obtained by considering 𝑇𝑒𝑣𝑒𝑛and 𝑇𝑜𝑑𝑑
and by2applying Newton method to the equations 𝑇𝑒𝑣𝑒𝑛 (𝑋) = 𝑋 and222(𝑋) = 𝑋. 𝑇𝑒𝑣𝑒𝑛and 𝑇𝑜𝑑𝑑
are continuous. Thus, Newton𝑇𝑜𝑑𝑑
method can be applied. In Figure 6, dots indicate order 3
periodic orbits of 𝑇 2 which are detected by superimposing the22and 𝑇𝑜𝑑𝑑. Purple
order 3 periodic orbits calculated on 𝑇𝑒𝑣𝑒𝑛dots correspond to stable order 3 periodic orbit of 𝑇 2 , red

Fig. 5. Enlargement around a corner of basin. 𝛼1 = 2.94, 𝛼2 = 3.84 and𝜀 = 0.04. (a) Enlargement in [0.3, 0.4]. (b) Enlargement in [0.35, 0.38]. (c)Enlargement in [0.369, 0.365]. (d) Enlargement in [0.3615, 0.3625]. Fractal
structures are shown.

points correspond to saddle, and yellow dots correspond tounstable node. Saddles are plotted on the corner of boundaries
between basins. Then, it appears that stable manifold of saddle
can be part of boundaries of basins.
VI. C ONCLUSION
In this study, we investigated a two-dimentional coupled logistic map whose parameters are forced into periodic varying.
From the investigations of bifurcation, we confirmed that all1325

Fig. 6. Order 3 periodic orbits of 𝑇 2 . 𝛼1 = 2.94, 𝛼2 = 3.84 and 𝜀 = 0.04.
Purple dots correspond to stable order 3 periodic orbit of 𝑇 2 , red points
correspond to saddle, and yellow dots correspond to unstable node.

the orbits correspond to only even order cycles except fixed
point and the special case that 𝛼1 equals to 𝛼2 . Moreover,existence of coexisting attractors, which consist of cycle orbit
or chaotic orbit, is confirmed. Then, basin has been investigated when the coexistence occurs. In the basins, fractal structure appears. Periodic orbits and their basins are numericallycalculated by using second order iteration and superposition
with Newton method. By calculating its eigenvalues, it wasconfirmed that saddles are on the corner of boundaries between
basins.
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Differential Equations,” Proc. of the Ninth International Conference onNonlinear Oscillations, pp. 1-8, 1981.
[4] M. Inoue, “A Method of Analysis for the Bifurcation of the Almost Periodic Oscillation and the Generation of Chaos in a Parametric Excitation
Circuit,” Trans. of IEICE, vol. J68-A, no. 7, pp. 621-626, 1985.
[5] H. Kumeno and Y. Nishio, “Globally Coupled Parametrically Forced Logistic Maps,” Proc. of Asia Pacific Conference on Postgraduate Researchin Microelectronics and Electronics (PrimeAsia’09), pp. 181-184, Nov.
2009.

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