2009 IEEE Workshop on
Nonlinear Circuit Networks, Tokushima, Dec 11-12, 2009

Bifurcation and Synchronization in Coupled
Parametrically Forced Logistic Maps
Hironori Kumeno

Yoshifumi Nishio

Dept. of Electrical and Electronic Eng.,
Tokushima University,Tokushima, 770-8506 JAPAN
Email: kumeno@ee.tokushima-u.ac.jp

Dept. of Electrical and Electronic Eng.,
Tokushima University,
Tokushima, 770-8506 JAPAN
Email: nishio@ee.tokushima-u.ac.jp

Abstract— In this study, a parametrically forced logistic mapthat a parameter of the logistic map are forced into periodic
varying is suggested. Unique bifurcations from period to chaos
are observed in the map. Then, synchronization phenomena inglobally coupled system of the parametrically forced logistic
map are investigated. When the number of coupling is three,various synchronization phenomena are observed by choosing
a coupling intensity. The synchronization phenomena fall into
three general categories, which are asynchronous, self-switchingof synchronization, synchronization of two among the three maps
and synchronization of all the maps. Further more, relationshipbetween sojourn time and the coupling intensity in the selfswitching of synchronization is investigated. The sojourn timeincrease exponentially with the coupling intensity.

I. I NTRODUCTION
Synchronization is one of the fundamental phenomena innature, and one of typical nonlinear phenomena. Therefore,studies on synchronization phenomena of coupled systems areextensively carried out in various fields, physics [1], biology[2], engineering and so on. However, the issues that should
be investigated for synchronization remain in existence inspite of many researching. In particular, it is necessary toinvestigate synchronization phenomena in special conditions.
There is parametric excitation that an amplitude of oscillation
is increased by periodic varying of a parameter in somesystem. Parametric excitation circuit is one of resonant circuits,
and it is important to investigate various nonlinear phenomena
for future engineering applications. In a simple oscillator
including parametric excitation, Ref. [3] reports that the almost
periodic oscillation occurs if nonlinear inductor has saturation
characteristic. Additionally the occurrence of chaos is referredin Refs. [4] and [5].
By the way, spatiotemporal chaotic phenomena thatspatially-extended systems indicate temporal and spatio complex patterns have been studied. Behavior generated in coupledsystem of chaotic one-dimensional map is investigated inRefs. [6]-[8] In particular, Coupled Map Lattice (CML) andGlobally Coupled Map (GCM) are well known as mathematical models in discrete-time system. Various kinds of dynamics
are observed in their systems. The research into CML andGCM are important for not only modeling of multiple degreeof freedom nonlinear systems but also suggestion to biologicalnetworks and engineering applications. In the past we have- 26 -investigated effects of parametric excitation in coupled van der
Pol oscillators [9]. In this study, for more detailed investigationof effect of the parametric excitation on synchronization, we
focus on a globally coupled system of simple one-dimensional
maps. A typical scheme for global coupling is given byxi (t + 1) = (1 − ε)f [xi (t)] + NεNP
j=1f [xj (t)](1)(i = 1, 2, · · ·, N, )where ε is the coupling intensity. The globally coupled maps
are a scheme that an average number of all the maps affecteach of the maps, and similar to the system that we havestudied using van der Pol oscillators. Hence, we investigate
synchronization phenomena in a globally coupled system of
one-dimensional maps which are forced by periodic parameterchange. The one-dimensional map used in this study is alogistic map, since the map can be described by a simple
discrete equation. The logistic map is a polynomial mapping,often cited as an archetypal example of how complex chaotic
behavior can be arisen from very simple nonlinear dynamical
equations. Mathematically, the logistic map is written asx(t + 1) = αx(t)(1 − x(t)).

(2)

Firstly, we describe behaviors and bifurcations of the parametrically forced logistic map. Next, we investigate synchronization phenomena in the globally coupled parametricallyforced logistic maps.
II. PARAMETRICALLY FORCED LOGISTIC MAP
A parametrically forced logistic map used in this study is
described as:x(t + 1) = αf (t)x(t)(1 − x(t)),
and(3)

 α1 , n(τ1 + τ2 ) < t ≤ n(τ1 + τ2 ) + τ1
α2 , n(τ1 + τ2 ) + τ1 < t ≤ (n + 1)(τ1 + τ2 ) ,αf (t) =(n = 1, 2, ...)
(4)
where αf (t) is a term of the parametric force and timevarying. The parametric force operation can be described asfollows: in the time interval n(τ1 + τ2 ) < t ≤ n(τ1 + τ2 ) + τ1 ,the system is driven by a parameter α1 during the duration τ1 ;

(a)

(b)
Fig. 1.

One-parameter bifurcation diagrams (left) and the Lyapunov exponents (right) for α1 = 3.8. Horizontal axis: α2 . (a) τ = 1. (b) τ = 2.

while in the interval n(τ1 + τ2 ) + τ1 < t ≤ (n + 1)(τ1 + τ2 ),the system is driven by a parameter α2 during the duration τ2 .
Namely, in this system, two kinds of parameters are replaced
alternately by the number of updates. Then, a parameter giving
a periodic solution and a parameter giving another periodic
solution can be combined. Of course, other combinations,for instance two parameters giving a periodic solution and achaotic solution or two parameters giving two kinds of chaotic
solutions, are possible. In this study, we assume τ1 = τ2 = τfor simplicity.

(a)

(b)

(c)

(d)

III. B IFURCATION
Before investigating synchronization phenomena in the
globally coupled parametrically forced logistic maps, it isnecessary to investigate behavior and bifurcation of one uncoupled parametrically forced logistic map. Figure 1 shows
computer calculated results. Fixing a parameter α1 = 3.8
and τ = 1 and varying a parameter α2 , the one-parameterbifurcation diagram and the Lyapunov exponents are obtained
as shown in Fig. 1(a). Figure 1(b) shows the bifurcation
diagram and the Lyapunov exponents for τ = 2. The different
types of bifurcation diagrams are obtained. From these figures,
observations of periodic and chaotic attractors are confirmed.
Figure 2 show some examples of the return maps of theparametrically forced logistic maps. For the original logistic
map, two-periodic solution is observed for α = 3.0. While,
three-periodic solution is observed for α = 3.8. These twosolutions are periodic, whereas in the logistic map involving
parametric force, a solution is chaotic as shown in Fig. 2(a)when the parameters α1 and α2 are set 3.0 and 3.8. Namely,- 27 -Fig. 2. Return maps of parametrically forced logistic maps for τ = 1. (a)α1 = 3.0 and α2 = 3.8. (b) α1 = 3.8 and α2 = 4.0. (c) α1 = 3.0 and
α2 = 4.0. (d) α1 = 3.5 and α2 = 4.0.

chaotic solution can be observed in the combination of two
parameters that generate two kinds of periodic solutions.
IV. S YNCHRONIZATION
Synchronization phenomena generated in the globally coupled logistic map involving parametric force are investigatedfor one control parameter ε which is a coupling intensity when(a)Fig. 3.
maps.

Conceptual structure of three coupled parametrically forced logistic(b)

Fig. 4. Lyapunov exponents in globally coupled parametrically forced logistic
maps for α1 = 3.0, α2 = 3.8 and τ = 1. Horizontal axis: ε.

the number of coupling is three. Figure 3 shows a conceptualstructure of teh three coupled parametrically forced logistic
maps. In the following computer calculations, the parameters
are fixed as α1 = 3.0, α2 = 3.8 and τ = 1.
The three-dimensional Lyapunov exponents obtained at
α1 = 3.8, α2 = 4.0 and τ = 1 are shown in Fig. 4. In Fig. 4,
λ1 is maximal Lyapunov exponent. Figure 5 shows examplesof synchronization phenomena. In Fig. 5, upper figures show
the return maps and lower figures show the phase differences
between the maps.
First, when the coupling parameter ε is small, three lyapunov exponents are almost same and the maps are almost
asynchronous as shown in Fig. 5(a). Increasing the coupling intensity over 0.0400, the lyapunov exponents decrease
rapidly. Then a self-switching phenomenon of synchronization
is observed. As increasing the coupling intensity, λ1 becomes
negative and attractors of the maps become periodic. Then,
two of the three maps are synchronized as shown in Fig. 5(b).
Moreover, as increasing the coupling intensity over 0.100,
all the λ become positive and the attractors become chaotic
as shown in Fig. 5(c). Although all the maps become to beasynchronous. After that, λ3 comes close to zero. Then twoof the three maps are synchronized as shown in Fig. 5(d). Inthe figure, all attractors behave chaotic, and map 1 and map
3 are synchronized. While, the shapes of the return maps are- 28 -(c)(d)(e)Fig. 5. Synchronization of three chaos. α1 = 3.0, α2 = 3.8 and τ = 1.
(a) ε = 0.000. (b) ε = 0.085. (c) ε = 0.100. (d) ε = 0.170. (e) ε = 0.200.

different from that of the uncoupled map. Finally, when thecoupling intensity ε is over 0.2, λ2 and λ3 become negative.
Then, all the maps are synchronized as shown in Fig. 5(e). In
the figure, all attractors are chaotic and have the same shapes
as a attractors of the uncoupled map.

(a)served. Then, synchronization phenomena in globally coupled system of the parametrically forced logistic map wereinvestigated. For the number of coupling is three, various
synchronization phenomena are observed by choosing a coupling intensity. The synchronization phenomena fall into three
general categories, which are asynchronous, self-switching ofsynchronization, synchronization of two among the three maps
and synchronization of all the maps. Further more, relationshipbetween sojourn time and the coupling intensity in the selfswitching of synchronization was investigated. The sojourntime increase exponentially with the coupling intensity.

(b)This work was partly supported by Yazaki Memorial Foundation for Science and Technology.

V. S ELF - SWITCHING PHENOMENONACKNOWLEDGMENT

Fig. 6. Time series of differences between two maps. α1 = 3.0, α2 = 3.8
and τ = 1. (a) ε = 0.0400. (b) ε = 0.0415.

Fig. 7. Sojourn time of the self-switching. α1 = 3.0, α2 = 3.8 and τ = 1.
Hertical axis: ε, vertical axis: the mean value of the sojourn time.

The self-switching phenomenon of synchronization is observed when the ε is set around 0.04. The phenomenon isthat two among the three maps are synchronized and thecombination which maps are coupled changes with time.
Figure 6 shows time series of differences of x(t) between
two maps. Areas where the amplitudes of the time series aresmall correspond to in-phase synchronization in the figure. InFig. 6(a), firstly, map 1 and map 2 are synchronized. However,after a time, the synchronous state breaks up and map 1 andmap 3 are synchronized. As seen above, the synchronous statesswitch with time in sequence. Additionally, a sojourn time ofthe self-switching is related to ε. The sojourn time is shortwhen ε is small, whereas the sojourn time is long when ε isbig as shown Figs. 6(a) and (b). Figure 7 shows sojourn timeof the self-switching between ε = 0.04 and ε = 0.0405. The
sojourn time increases exponentially with ε.
VI. C ONCLUSION
In this study, parametrically forced logistic map was suggested. Unique bifurcations from period to chaos were ob-- 29 -R EFERENCES[1] I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer, “Synchronization andgraph topology,” Int. J. Bifurcation and Chaos, vol.15, no.11, pp.34233433, Nov. 2005.
[2] J. Cosp, J. Madrenas, E. Alarcon, E. Vidal and G. Villar, “Synchronizationof nonlinear electronic oscillators for neural computation,” IEEE Trans.
Neural Networks, vol.15, no.5, pp.1315-1327, Sep. 2004.
[3] C. Hayashi, “Nonlinear Oscillations in Physical Systems,” Chap. 11,
McGraw-Hill, New York (1964).
[4] C. Hayashi, M. Abe, K. Oshima and H. Kawakami, “The Method ofMapping As Applied To the Solution for Certain Types of Nonlinear
Differential Equations,” Proc. of the Ninth International Conference onNonlinear Oscillations, pp. 1-8, 1981.
[5] 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.
[6] K. Kaneko, “Spatiotemporal intermittency in couppled map lattice,” Prog.
Theor. Phys., vol.74, no.5, pp.1033-1044, Nov. 1985.
[7] K. Kaneko, “Pattern dynamics in spatiotemporal chaos,” Physica D,
vol.34, nos.1-2, pp.1-41, Jan. 1989.
[8] K. Kaneko, “Clustering, coding, switching, hierarchical ordering, andcontrol in a network of chaotic elements,” Physica D, vol.41, nos.2,
pp.137-172, March. 1990.
[9] H. Kumeno and Y. Nishio, “Synchronization Phenomena in Coupled Parametrically Excited van der Pol Oscillators,” Proceedings of International
Symposium on Nonlinear Theory and its Applications (NOLTA’ 08), pp.
128- 131, Sep. 2008