2011 International Workshop on Nonlinear Maps and their Applications
NOMA'11, Evora, Portugal, September 15-16, 2011

Foliated bifurcation structure
in a 2-D coupled logistic map
Hironori Kumeno∗ , Danièle Fournier-Prunaret† , Abdel-Kaddous Taha‡ and Yoshifumi Nishio∗∗ Dept. of Electrical and Electronic Eng., Tokushima University,

2-1 Minami-Josanjima, Tokushima, 770-8506 JAPAN
Email: {kumeno, nishio}@ee.tokushima-u.ac.jp† LAAS-CNRS, 7 avenue du colonel Roche, F-31077 Toulouse Cedex 4, France
Email: daniele.fournier@insa-toulouse.fr‡ LATTIS-INSA, Universite de Toulouse, 135 avenue de Rangueil, 31077 Toulouse, France
Email: taha@insa-toulouse.fr
Abstract—In this study, we investigate bifurcations in coupledlogistic maps whose parameters are forced into periodic varying
in two-dimensional case. On a parameter plane, crossroad areas
centered at fold cusp points regarding to several orders aredetected. Especially, we investigate bifurcation curves regardingorder 2 periodic orbits in detail. From the investigation, a foliatedbifurcation structure is drawn, and existence domains of stableorder 2 cycles with synchronization or without synchronization
are detected. Moreover, evolution of bifurcation curves withrespect to a coupling intensity is analyzed.

I. I NTRODUCTION
The study of information processing in brain using nonlinear
models has attracted extensive attention during the last years.
This attention is based on the consequence that neuron activity
can be described by deterministic dynamical nonlinear equations. Indeed, studies by using equipments which record global
brain activity like electroencephalogram and thermogram arecarried out. However, there are problems depending on capabilities of the equipments. Therefore, we come up with the way
to solve the problem by using model of brain with nonlinear
systems. It is rich worth to introduce models to investigate
nonlinear neuronal activities and mutual influences between
neurons in the neural networks by using nonlinear maps inorder to understand brain activities. 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 invarious ways, but we focus on a system whose parameter isforced into periodic varying and investigate behavior of thesystem and its pint-size coupling. The fundamental researchesof these small coupled nonlinear systems are important for
the investigation of global brain activities and the modeling,
since we can think that general properties of large coupled
nonlinear systems can de deduced from those of the its small
size coupling system.
As the research for a system with parameter forced into
periodic varying, [2] reports that the almost periodic oscillation
occurs in a simple oscillator including parametric excitation if
nonlinear inductor has saturation characteristic. Additionallythe occurrence of chaos is referred in [3] and [4]. Previously,- 41 -we have investigated synchronization phenomena in coupledlogistic maps involving parametric force [5].
This study also deals with the coupled logistic maps whoseparameters are forced into periodic varying. We investigate
bifurcations in two-dimensional cases. As previously said, thislow dimension can give rise to specific phenomena that wecan explain and that are basic phenomena for the behavior ofhigher size neural networks. From the obtained bifurcations,foliated bifurcation structure is drawn, and existence domainof stable order 2 cycles with synchronization or without synchronization are detected. Moreover, evolution of bifurcation
curves with respect to a coupling intensity is investigated. The
bifurcation curves which appear in coupled case spread byincreasing of the coupling intensity. The evolution with respect
to the coupling is analytically proved.
II. S YSTEM
This paper treats a two-dimensional system. This system isobtained from the coupling with mutually influencing schemeof the same one-dimensional system whose parameters areforced into periodic varying. The system is described as:εε x(n + 1) = (1 − 2 )F (x(n), αf (n)) + 2 F (y(n), αf (n))εε,T :
 y(n + 1) = (1 − 2 )F (y(n), αf (n)) + 2 F (x(n), αf (n))(n = 1, 2, ...)

(1)
and
 α1 , for each even value of “n”.
αf (n) = α2 , for each odd value of “n”(n = 1, 2, ...)
(2)
The parameter αf alternately changes from α1 to α2 andε ∈ (0, 1). This system is not continuous in iteration because
αf is periodically changed. Whereas, the twice iteration ofthis system is considered as continuous system since itsperiodicity. This system has the above characteristics. From
this characteristics, non-existence of odd order cycle except
fixed point and special situation α1 = α2 are prouved in[6]. In this study, F is a one-dimensional logistic map and

(a)

(a)

(b)

(b)

Fig. 1. Representation of the parameter plane (α1 , α2 ) for ε = 0.24. (b) isan enlargement of (a).

Fig. 2. Bifurcation curves of order 2, 4 and 8 cycles indicating only the foldand flip curves according to boundaries between existence domains of stable
cycles.

is described as:
F (x(n), αf (n)) = αf (n)x(n)(1 − x(n))
F (y(n), αf (n)) = αf (n)y(n)(1 − y(n)).

(3)

III. B IFURCATIONS
Here, we consider the two-dimensional case, two parametrically forced logistic maps are coupled.
Figure 1 shows bifurcation diagrams for ε = 0.24 andits enlargement. Each colored domain indicates existence ofat least one stable cycle whose order corresponds to upper
colored squares in the figure. Figure 2 shows bifurcation
curves of order 2, 4 and 8 cycles indicating only the curves
according to boundaries between existence domains of stable cycles. Bifurcation curves are obtained using analytical-- 42 -numerical methods [7]. In the figure, Λj(k)0 , Λ̄j(k)0 , Λjk andΓk denote fold, pitchfork, flip and Neı̈mark-Sacker bifurcation
curves, respectively. The index j is a number characterizing
the considered order k cycle. Ckj denotes a cusp point ofan order k cycle. The bifurcation curves well correspond tothe boundaries between colored domains in Fig. 1. On theparameter plane, crossroad areas (in the sense of [8]) centeredat fold cusp points regarding to several orders can be detected.
A. Foliated bifurcation structure of order 2 cycle
Here, bifurcations regarding order 2 cycle are considered.
Figure 3 shows fold, flip, pitchfork and Neı̈mark-Sacker bifurcation curves. In the figure, arrowed lines correspond tocascades of order 2 cyclic points associated with fold andFig. 3.

2Bifurcation curves of order 1 cycle of Toddfor ε = 0.24.

pitchfork bifurcations at which heads of the arrowed lines
point. Christcrosses on the arrowed lines indicate that the order2 cyclic points on the cascades undergo flip bifurcations. For
instance, an order 2 cycle on the arrowed line starting from
Λ3(2)0 in left upper side of the figure undergoes flip bifurcations
Λ12 and Λ22 , and finally reaches to a fold bifurcation Λ1(2)0 at
which the head of the arrowed line points. The cycle on thecascade is not associated with other bifurcations like Λ̄1(2)0 ,00
Λ32 , Λ12 , Λ22 and left side of Λ1(2)0 through which the arrowed
line passes. On the other hand, an order 2 cycle on the arrow
line starting from Λ3(2)0 in right lower side of the figure
00
undergoes flip bifurcations Λ12 and Λ22 , and finally reaches
to a fold bifurcation Λ1(2)0 at which the head of the arrowed
line points. The cycle on the cascade is not associated withother bifurcations. From these relationships, Λ1(2)0 , Λ3(2)0 , Λ12 ,00
Λ22 , Λ12 and Λ22 organize a crossroad area centered at a fold01cusp C2 . In the same way, Λ2(2)0 , Λ4(2)0 , Λ32 and Λ32 organizea crossroad area centered at a fold cusp C22 . Λ2(2)0 , Λ̄1(2)0 , Λ42 ,0
Λ42 organize a communication area including codimension-2
points. Here, it is detected that arrow lines starting from Λ4(2)0and Λ̄1(2)0 reach to Λ2(2)0 . This is because Λ2(2)0 is associatedto four cycles.
In order to understand relationships between bifurcation
curves and appearance or disappearance of cycles and theirstabilities, we consider a cascade when α1 is fixed (α1 = 4)
and α2 is increasing. Relationships between bifurcation curves
and appearance or disappearance of cycles and their stabilities
on the cascade are represented in Fig. 4. In the figure, dots
indicate that cycles are related to bifurcations denoted in the
top of the figure. At the points, appearance or disappearance ofcycles occur and their stability changes. Because we focus onorder 2 cycle, we do not consider unstable order 4 cycles andindicate only appearance of stable order 4 cycles in the figure.

- 43 -Fig. 4. Representation of bifurcations of order 2 cycles of T and its stabilities
when α1 is fixed and α2 increases. We do not consider unstable order 4 cycle.
SNkj , SAjk , U Nkj and U Skj indicate stable node, saddle, unstable node andunstable spiral of order k cyclic point, respectively. The index j characterizes
the permutation of the order k cycle. 0 is fixed point at (x, y) = (0, 0).

With increasing α2 , cycles appear through fold and pitchfork
bifurcations, and the number of cycles finally becomes 15. It
confirms that Λ2(2)0 relates to four distinct cycles.
From the above results, a foliated bifurcation structure,
three-dimensional representation, is drown up in Fig. 5 as
introduced in [8]. The foliated bifurcation structure is organized by five organizations that each organization has afold cusp and the organizations overlap. Organization 2 andorganization 2’ have the same structures. Organization 3 andorganization 3’ also have the same structures. Organization 3and organization 3’ appear from Λ2(2)0 in the center sheet oforganization 1 and construct shapes like pockets. In the figure,a blue-slushed domain corresponds to existence of a stable
fixed point (x, y) = (0, 0). Red-slushed domains correspondto existence of stable order 2 cycles. It can be seen that stableorder 2 cycles exist in organization 1, organization 3 andorganization 3’. All coordinates of order 2 cycles belonging
to organization 1 correspond to x = y. Therefore, in theparameter domain according to red-slushed domains in theorganization 1, synchronized order 2 cycles exist. On the other
hand, all coordinates of order 2 cycles belonging to the otherorganizations correspond to x 6= y. Therefore, in the parameterdomain according to red-slushed domains in the organization
3 and the organization 3’, unsynchronized order 2 cycles exist.
B. Evolution of bifurcation curves for εHere, evolution of bifurcation curves with respect to thirdparameter ε is investigated. Figure 6 shows bifurcation curves
with different values of ε. When ε is small, Λ4(2)0 is closeto Λ3(2)0 , Λ2(2)0 and Λ̄1(2)0 are close to Λ1(2)0 , Λ22 and Λ32 are0000
close to Λ12 , Λ22 and Λ32 are close to Λ12 , Λ42 and Λ42 are

(a)(b)Fig. 6. Evolution of bifurcation curves with respect to third parameter ε.
(a)ε = 0.05. (b)ε = 0.70.

(c)Λ3(2)0 and Λ4(2)0 are described as

Fig. 5.

Λ3(2)0 : α1 α2 = 1

Three dimensional representation of Fig. 3.

Λ4(2)0 : α1 α2 =0close to parts of Λ12 and Λ12 . With increasing ε, Λ2(2)0 , Λ4(2)0 ,000
Λ̄1(2)0 , Λ22 , Λ22 , Λ32 , Λ32 , Λ42 and Λ42 spread and finally tend0
toward infinity when ε = 1, while Λ1(2)0 , Λ3(2)0 , Λ12 and Λ12 do
not move. Here, we prove these obtained results, analytically.

- 44 -.
11 − 2ε + ε2(4)From these equations, Λ3(2)0 is not related to ε and Λ4(2)0 isrelated to ε and tends toward infinity at ε = 1. It is difficult tosolve other bifurcation curves analytically. Then, we consider
crossing points where fold and pitchfork bifurcation curves
cross the diagonal of (α1 , α2 ) parameter plane. We denote thecrossing points where pitchfork bifurcation curves Λ̄1(2)0 crossthe diagonal as ∗Λ̄1(2)0 . The crossing points are described as
C21 : α1 = α2 = 3√−1 + ε − ε2 − 2ε + 4
C22 : α1 = α2 =.
−1 + ε11∗Λ̄(2)0 : α1 = α2 = 2 +1−ε(5)From these equations, C21 is constant with respect to ε, andC22 and ∗Λ̄1(2)0 relate to ε and tend toward infinity at ε = 1.
Only the crossing points between bifurcation curves and thediagonal can be solved, analytically. However, the crossing
points are parts of bifurcation curves and each bifurcation
curve is continuous. Therefore, it is clear that the evolutionof the crossing points is correlated with the evolution of theirwhole bifurcation curves.
IV. C ONCLUSION
In this study, we investigated bifurcations in coupled logistic
maps whose parameters are forced into periodic varying in
two-dimensional case. From the investigation of bifurcation
diagrams and bifurcation curves, existence domains of several
stable cycles and chaotic orbits have been detected. Thebifurcation curves construct crossroad areas centered at foldcusp points. Especially, we investigated bifurcation curves
regarding to order 2 cycle in detail. From the investigation, afoliated bifurcation structure has been drawn, and existence ofsynchronized and non-synchronized cycles have been derived.
Moreover, evolution of bifurcation curves with respect to acoupling intensity was investigated. The bifurcation curves
caused by the coupling spread by increasing of the coupling
intensity and tend toward infinity for the coupling intensity
beeing equal to 1. The evolution with respect to the couplinghas been analytically proved.
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