Globally 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, we propose a parametrically forced
logistic map and investigate its bifurcation. By carrying out computer calculations, unique bifurcations from period to chaos areobserved. Furthermore, we investigate synchronization phenomena in globally coupled logistic maps which involving parametricforce and confirm various kinds of synchronization phenomena.
In the case of two coupled maps, in-phase synchronization oftwo maps is observed. In the case of three coupled maps,
synchronization states are changed with increasing coupling
intensity from asynchronous, self-switching of synchronization,synchronization of two among the three maps, to synchronizationof all the maps. Finally, we investigate the influences of phasedifference of parametric forces on synchronization. The maps can
not be synchronized completely if phase difference exists betweentwo parametric forces of the maps.

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].
In the past we have investigated effects of parametric
excitation of coupled van der Pol oscillators [6]. In this study,
for more detailed investigation of the effect of parametric
excitation on synchronization, we focus on a globally coupledsystem of simple one-dimensional maps. A typical scheme for
global coupling is given byxi (t + 1) = (1 − ε)f [xi (t)] + Nε(i = 1, 2, · · ·, N, )

N

j=1

978-1-4244-4669-8/09/$25.00 ©2009 IEEEf [xj (t)](1)

where ε is the coupling intensity. The globally coupled maps
are a scheme that an average number of all the maps affecteach of the map, 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 dynamicalequations. Mathematically, the logistic map is written asx(t + 1) = αx(t)(1 − x(t)).

(2)

Typical one-parameter bifurcation diagram of the logistic map
is obtained as shown in Fig. 1.
Firstly, we describe behaviors and bifurcations of the parametrically forced logistic map. Next, we investigate synchronization phenomena in the globally coupled parametricallyforced logistic maps.

Fig. 1.

One-parameter bifurcation diagram of logistic map.

II. PARAMETRICALLY FORCED LOGISTIC MAP
A parametrically forced logistic map used in this study isdescribed as:x(t + 1) = αf (t)x(t)(1 − x(t)),

181

(3)

(a)

(b)
Fig. 2.

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

and⎧
⎨ 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 ;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 chaoticsolutions, are possible. In this study, we assume τ1 = τ2 = τ
for simplicity.
III. B IFURCATION

Before investigating synchronization phenomena in theglobally coupled parametrically forced logistic maps, it isnecessary to investigate behavior and bifurcation of one uncoupled parametrically forced logistic map. Figure 2 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. 2(a). Figure 2(b) shows the bifurcation
diagram and the Lyapunov exponents for τ = 2. The different(a)(b)(c)(d)Fig. 3. 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.

types of bifurcation diagrams are obtained. From these figures,
observations of periodic and chaotic attractors are confirmed.
Figure 3 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. 3(a)182when the parameters α1 and α2 are set 3.0 and 3.8. Namely,
chaotic solution can be observed in the combination of two
parameters that generate two kinds of periodic solutions.
IV. S YNCHRONIZATIONIn this section, we investigate synchronization phenomena
by carrying out computer calculations for the cases of twoand three maps. In the following computer calculations, theparameters are fixed as α1 = 3.8, α2 = 4.0 and τ = 1.
(a)A. Two maps caseIn this subsection, we consider the case of N = 2, namely
only two parametrically forced logistic maps are coupled. In
this case, in-phase synchronization of two maps is observed.
Figure 4 shows computer calculated results. As shown in thefigures, with ε increases, chaotic solutions obtained from twomaps become to be synchronized. When ε = 0.450, twochaotic attractors seem to be completely synchronized.

(b)

(a)

(b)
Fig. 4. In-phase synchronization of two maps for α1 = 3.8, α2 = 4.0 andτ = 1. (a) ε = 0.405. (b) ε = 0.450.

(c)

B. Three maps case(d)Fig. 6. Synchronization of three chaos. α1 = 3.8, α2 = 4.0 and τ = 1.
(a) ε = 0.10. (b) ε = 0.30. (c) ε = 0.32. (d) ε = 0.41.

Fig. 5.
maps.

Conceptual structure of three coupled parametrically forced logistic

In this subsection, we consider the case of N = 3. Figure 5
shows a conceptual structure of three coupled parametricallyforced logistic maps. In this case, various kinds of synchronization phenomena are observed with increasing coupling
intensity ε. Figure 6 shows examples of synchronization
phenomena. In Fig. 6, upper figures show the return maps andlower figures show the phase differences between the maps.

First, when the coupling parameter ε is small, the maps
are almost asynchronous as shown in Fig. 6(a). Increasing
the coupling intensity, self-switching phenomenon of synchronization is observed. In this phenomenon, two of the threechaotic maps are synchronized. While, the combination of thetwo synchronized maps switches with time at random. As
increasing the coupling intensity, synchronization of two of
the three maps which generate periodic solutions is observed
as shown in Fig. 6(b). In the figure, the two maps are synchronized, whereas the remain map seem to be synchronized withthe others at the opposite-phase, since solutions of the maps

183are periodic. Moreover, as increasing the coupling intensity
further, synchronization of two of the three maps which
generate chaotic solutions is observed as shown in Fig. 6(c).
In the figure, all return maps behave chaotic, and two maps,
the left return map and the right return map, are synchronized.
While, the shapes of the return maps are different from thatof the uncoupled map. Finally, when the coupling intensity ε
is over 0.4, synchronization of all the maps which generatechaotic solutions is observed as shown in Fig. 6(d). In thefigure, all return maps are chaotic and have the same shapes
as a return map of the uncoupled map.

(a)

(b)

V. E FFECT OF PHASE DIFFERENCE(c)Fig. 8. Return maps and phase differences. α1 = 3.8, α2 = 4.0 and τ = 6.
(a) phase difference = 0. (b) phase difference = 4. (b) phase difference = 6.

Fig. 7. Conceptual structure of Phase difference between parametric forces.

In this section, in order to investigate the synchronous
influence of phase difference of parametric force, we focus on
the two coupled chaotic maps whose parameters are α1 = 3.8,α2 = 4.0 and τ = 6. Figure 7 shows a conceptual structure ofa phase difference between the parametric forces of the maps.
In the above parameters, αf (t) switches every six updates andthe chaotic maps are synchronized if the parametric forces
have no phase differences. With shifting the timing of theswitch of αf (t), we investigate synchronous influence of thephase difference. Figure 8 shows the computer calculatedresults. In the figures, two maps are synchronized if the phasedifference does not exist between the two parametric forcesof the maps (see Fig. 8(a)). Increasing the phase difference,
the two maps become out of synchronization (see Figs. 8(b)and (c)). Additionally, if the phase difference exists between
two parametric forces of the maps, no matter how strong
coupling intensity increase, two maps can not be synchronizedcompletely.
VI. C ONCLUSION
In this study, firstly, we proposed a parametrically forced
logistic map and investigated its bifurcations. By carryingout computer calculations, unique bifurcations from period
to chaos were observed. Furthermore, we investigated synchronization phenomena in the globally coupled parametrically forced logistic maps and observed various kinds ofsynchronization phenomena. In the two maps case, in-phasesynchronization of two maps was observed. In the threemaps case, synchronization states were changed with increasing coupling intensity from asynchronous, self-switching ofsynchronization, synchronization of two of the three maps
to synchronization of all the maps. Finally, we investigated
the influences of phase difference of parametric force onsynchronization. We observed that if phase difference exists
between the two parametric force of the maps, no matter
how strong coupling intensity increase, two maps can not be
synchronized completely.
ACKNOWLEDGMENT
This work was partly supported by Yazaki Memorial Foundation for Science and Technology.
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