2014 International Symposium on Nonlinear Theory and its Applications
NOLTA2014, Luzern, Switzerland, September 14-18, 2014

Bifurcation and Basin in Two Coupled Cubic Maps
Takuya Nishimoto∗ , Yoko Uwate∗ , Yasuteru Hosokawa∗∗ ,Yoshifumi Nishio∗ , Danièle Fournier-Prunaret† and Abdel-Kaddous Taha‡∗

Dept. of Electrical and Electronic Eng., Tokushima University,
2-1 Minami-Josanjima, Tokushima, 770-8506 Japan
Email: {nishimoto, uwate, nishio}@ee.tokushima-u.ac.jp∗∗
Dept. of Media and Information Systems, Shikoku University,Furukawa, Ohjin, Tokushima, 771-1192, Japan
Email: hosokawa@keiei.shikoku-u.ac.jp†LAAS-CNRS, INSA, University of Toulouse, France
Email: daniele.fournier@insa-toulouse.fr‡
INSA, University of Toulouse, France
Email: taha@insa-toulouse.fr
Abstract—Some complex nonlinear phenomena havereported on the coupled chaotic system included double
scroll family. It is considered that investigating bifurcation
structures of the system is extremely important to clarify
high-order complex nonlinear phenomena. In this study,
we investigate bifurcation structures on two coupled cubic maps which belongs to double scroll family. By carrying out computation, bifurcation curves and basins areobtained.
1. Introduction
Chaotic phenomena observed on large-scale coupledchaotic systems have been attracted attentions in various
science fields since it can be regarded as models of real
physical system. For instance, chaos phenomena have beenreported in engineering, biology, economics, astronomy
and so on. Therefore investigation of chaotic system onhigh-order chaotic systems are very important to grasp essentials of the complex phenomena observed in natural system. On the other hand, some chaotic system generate
multi-scroll attractor [1]. Complex nonlinear phenomena
have reported in chaotic system which belongs to multiscroll family. Moreover, the relationships between bifurcation structure and synchronization phenomena have reported [2] [3]. Therefore investigating bifurcation structures of multi-scroll family is useful to clarify complex
nonlinear phenomena.
In this study, we investigate bifurcation structure andbasins in a coupled cubic maps which generates double
scroll attractor. By carrying out computation, complex bifurcation curves can be obtained. Be based on the bifurcation curves, we investigate the basins of multistability offixed point. This paper is organized as follows. In Section
2, we propose coupled cubic maps. In Section 3, bifurcation curves on the system are investigated. Be based on
Section 3, multistability of fixed point are shown. Finally,Section 4 concludes this work.
2. Coupled Cubic Map
In this study, we use a cubic map as the one-dimensional
subsystem in the coupled chaotic system. A cubic map
used in this study is described as:x(n + 1) = ax(n)(1 − x(n)2 ).

(1)

Figure 1 shows chaotic orbit which can be observed in thesystem described with Eq. (1). The attractors are classified with colors. One-parameter bifurcation diagram andthe Lyapunov exponent of the cubic map is obtained asshown in Fig. 2. By increasing the parameter a, pitchfork
bifurcation and flip bifurcation can be observed in the onedimensional map. The cubic map shown in Fig. 1 is cou-- 435 -Figure 1: Chaotic orbit observed in a cubic map.

(1)

(2)

Figure 2: (1) One-parameter bifurcation diagram. (2) The Lyapunov exponent. Horizontal axis: a.

Figure 3: Parameter bifurcation diagram corresponds to thepitchfork bifurcation. Blue, green and black lines show stable node, saddle and unstable node respectively. ε = 0.3.

respectively. Figure 4 (a) shows a representation of theparameter plane and its enlargement. Each colored partshows the existence of stable cycle. In the figure somediscontinuous boundaries of bifurcation are observed. The
phenomena are caused due to multistability behaviors andNeı̈mark-Sacker bifurcation. Diﬀerent attractors can coexist, depending on the choice of diﬀerent initial conditions ina certain parameter region. Figure 4 (b) shows bifurcation
curves corresponding to pitchfork bifurcation curves andflip bifurcation curves of order 2 and 4 cycles which can bederived with the eigenvalue of Jacobian matrix. The bifurcation curves regarding order 2 and 4 cycles are obtained
analytically and given by:Λ̄12(0) : a(ε) = 1

pled with the scheme of GCM which is described as:ε ∑f [x j (n)]N j=12xi (n + 1) = (1 − ε) f [xi (n)] +i = 1, 2

1ε−11ε+2Λ̄32(0) : a(ε) = −2ε−1
Λ12 : a(ε) = 2
Λ̄22(0) : a(ε) = −(2)where ε ∈ [0 : 1] is the coupling intensity. Then, the map
T is described as:ε
x1 (n + 1) = (1 − )ax1 (n)(1 − x1 (n)2 )


2

ε
2
ax+
2 (n)(1 − x2 (n) )2T = (3)ε2x(n+1)=(1−)ax(n)(1−x(n))
222
2ε
2

+ax

1 (n)(1 − x1 (n) )2
3. Bifurcation
In this section, bifurcations are investigated with respect
to parameter plane (a, ε). In the map T, pitchfork bifurcations of supercritical case and subcritical case are coexisting as shown in Fig. 3. Blue, green and black lines
are corresponding to stable node, saddle and unstable node

1 3ε − 42 ε−1√0.25ε − 0.5 − 0.25 25ε2 − 36ε + 36
3
Λ2 : a(ε) =ε−11ε−44Λ2 : a(ε) =2ε−125
Λ2 : a(ε) = −ε−1√
1 16 − 2ε2 + 4ε
6
Λ2 : a(ε) =2√
ε−1
Λ14 : a(ε) = 5√
1 18ε2 − 36ε + 20
2
Λ4 : a(ε) = −2 √ε−1120−
4ε − 2ε2Λ34 : a(ε) = −2√ε−154
Λ4 : a(ε) = −ε−1
Λ22 : a(ε) = −- 436 -(4)

(a)

6

Λ25

Λ24

Λ4
Λ

3

Λ4423

Λ2

4

Λ4
12

Λ4

Λ2(b)

36

Λ4
Λ

Λ

24

Λ2

125

Λ23

Λ (2)0Λ42(2)0Λ

Λ22411

Λ (2)0Λ43Λ22Λ2(1)(2)

Figure 4: Representation of the parameter plane (a, ε). Each colored part corresponds to the existence of a stable cycle(periodic point), the order k (period) of which is given by the upper side colored squares. The black color corresponds tok > 15 or to chaotic behavior. The white region corresponds to the nonexistence of attracting set, with chaotic transient
toward infinity. Horizontal axis: ε. Vertical axis: a. (a) Existence of stable periodic orbit. (b) Bifurcation curves of (1).
(2) Enlargement of (1).
where a pitchfork bifurcation curve is noted Λ̄kj , and a flip
bifurcation curve is noted Λkj . Note that the k is the order ofcycle and j diﬀerentiates cycles of same order Λkj denotes
flip bifurcation curves [5] [6]. From the representation andbifurcation curves, we found that a lot of multistability region are coexisting. In the next section, the basin which
is an initial value set converging to a given orbit is investigated for parameter values corresponding to multistability.

4. Basin
Figure 5 shows basin when T has multistability. Eachbasin of attraction represented by diﬀerent colors are corresponding to the stable periodic orbit. In Fig. 5 (1), twostable period 1 orbits are coexisting. Blue dots and black
dots are corresponding to stable fixed point and unstable
fixed point respectively. In another parameter sets, 2 stable
period 2 orbits, a stable period 3 orbit, and another stable
period are coexisting as shown in Fig. 5 (2). Similar shapeof basin as shown in Fig. 5 (1) and complicated basin withfractal structure can be observed in the parameter set. The
stable orbits observed in Fig. 5 (2) are shown in Fig. 6.

- 437 -(1)(2)

Figure 5: Basin regarding to T . (1) a = 1.99, ε = 0.39. (2) a = 2.46, ε = 0.16.
5. Conclusion
In this study, we investigated a two-dimensional coupledcubic map which belongs to double scroll family. In theinvestigation of bifurcation curves, coexisting supercriticaland subcritial pitchfork bifurcation are confirmed. Moreover, bifurcation curves regarding to order 2 pitchfork bifurcation and flip bifurcation curves and the basins are obtained on the parameter plane. In our future work, we will
investigate Neı̈mark-Sacker bifurcation curve and foliatedstructure of the coupled cubic map.

[6] J.P. Carcassès, “An algorithm to determine the natureand the transitions of communication areas generated
by a one-dimensional map”, in Proc. European Conference on Iteration Theory (ECIT 1991), J.P Lampreia, J.
Llibre et al. (Eds.), World Scientific, Singapore (1992),
27―38.

References[1] L.O. Chua, M. Komuro and T. Matsumoto, “Thedoulbe scroll family”, IEEE Transactions on Circuitand Systems, vol. 33, no. 11, pp. 1073-1118, Oct. 1986.
[2] Alatriste, F. and Mateos, J. L. “Phase synchronizationin tilted deterministic ratchets”, Physica , A 372, 263―271. 2006.
[3] Fournier-Prunaret, D. and Leonel Rocha, J. andCaneco, A. and Fernandes, S, and Gracia, C., “Synchronization and Basins of Synchronized States in
Two-Dimensional Piecewise Maps via Coupling Three
Pieces of One-Dimensional Maps”, Int. J. Bifurcation
and Chaos , vol. 23, No. 8, 1350134, 2013.
[4] H. Sekiya, S. Mori and I. Sasase, “Synchronization ofSelf-Switching phenomena on Full-Coupled ChaoticOscillators”, IEICE Trans., vol.J83-A, no. 11 pp. 12641275, 2000.
[5] J.P. Carcassès, “An algorithm to determine the natureand the transitions of communication areas generated
by a one-dimensional map”, in Proc. European Conference on Iteration Theory (ECIT 1991), J.P Lampreia, J.
Llibre et al. (Eds.), World Scientific, Singapore (1992),
27―38.

Figure 6: Coexisting stable period. The horizontal axis: x1 .
The vertical axis: x2 . (1) a = 1.99, ε = 0.39 (2) a = 2.46,
ε = 0.16.

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