2010 International Workshop on Nonlinear Circuits,Communication and Signal Processing
NCSP'10, Waikiki, Hawaii, March 3-5, 2010

Self-Switching Phenomenon of Synchronizationin Globally Coupled Parametrically Forced Logistic Maps
Hironori Kumeno† and Yoshifumi Nishio††Tokushima University
2-1,Minami-Josanjima, Tokushima, Japan
Phone/FAX:+81-88-656-7470
Email: {kumeno, nishio}@ee.tokushima-u.ac.jp

Abstract

effect of the parametric excitation on synchronization, we focus on a globally coupled system of simple one-dimensional
Various synchronization phenomena in globally coupled maps. A typical scheme for global coupling is given byparametrically forced logistic maps are observed for sevNP
eral coupling intensity when three maps are coupled. Thexi (t + 1) = (1 − ε) f [xi (t)] + Nεf [x j (t)](1)
synchronization phenomena fall into four general categories,
j=1(i=1,2,···,N,)
which are asynchronous, self-switching of synchronization,synchronization of two among the three maps and synchronization of all the maps. In this study, we investigate the self- where ε is the coupling intensity. The globally coupled maps
switching phenomenon of synchronization in detail. In par- are a scheme that an average number of all the maps affectticular, relationship between sojourn time and the coupling each of the maps, and similar to the system that we have studintensity and condition that a synchronization state switch to ied using van der Pol oscillators. The one-dimensional map
used in this study is a logistic map, since the map can be deother synchronization state are investigated.
scribed by a simple discrete equation. Mathematically, thelogistic map is written as
1. Introductionx(t + 1) = αx(t)(1 − x(t)).
(2)
Synchronization is one of the fundamental phenomena in
In this study, firstly, we describe the parametrically forced
nature, and one of typical nonlinear phenomena. Therefore,
logistic
map and synchronization phenomena in the globallystudies on synchronization phenomena of coupled systems
coupledparametrically forced logistic maps. After that, weare extensively carried out in various fields, physics [1], biolinvestigate
the self-switching phenomenon of synchronizaogy [2], engineering and so on. Parametric excitation circuittionindetail.
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, 2. Parametrically forced logistic map
Ref. [3] reports that the almost periodic oscillation occurs if
A parametrically forced logistic map used in this study isnonlinear inductor has saturation characteristic. Additionallydescribedas:the occurrence of chaos is referred in Refs. [4] and [5]. The
network of chaotic elements can be modeled by a system ofx(t + 1) = α f (t)x(t)(1 − x(t)),(3)
coupled one-dimensional maps. Behavior generated in coupled system of chaotic one-dimensional map is investigated inandRefs. [6]-[8] In particular, Coupled Map Lattice (CML) and
Globally Coupled Map (GCM) are well known as mathemat
α , n(τ1 + τ2 ) < t ≤ n(τ1 + τ2 ) + τ1
 1
ical models in discrete-time system. Various kinds of dynamα2 , n(τ1 + τ2 ) + τ1 < t ≤ (n + 1)(τ1 + τ2 ) ,α f (t) = ics are observed in their systems. The research into CML and(n = 1, 2, ...)
(4)
GCM are important for not only modeling of multiple degreeof freedom nonlinear systems but also application to biolog- where α f (t) is a term of the parametric force and timeical networks and engineering. In the past we have investi- varying. The parametric force operation can be described asgated effects of parametric excitation in coupled van der Pol follows: in the time interval n(τ1 + τ2 ) < t ≤ n(τ1 + τ2 ) + τ1 ,oscillators [9]. In this study, for more detailed investigation of the system is driven by a parameter α1 during the duration τ1 ;- 45 -

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. In this study, we assume τ1 = τ2 = τ for simplicity. Figure 1 show a example ofa return map of the parametrically forced logistic maps. For
the original logistic map, two-periodic solution is observedfor α = 3.0. While, three-periodic solution is observed for
α = 3.83. These two solutions are periodic, whereas in thelogistic map involving parametric force, a solution is chaotic
as shown in Fig. 1 when the parameters α1 and α2 are set 3.0
and 3.83. Namely, chaotic solution can be observed in thecombination of two parameters that generate two kinds of periodic solutions. In the following, the parameter values arefixed as α1 = 3.0, α2 = 3.83 and τ = 1.

(a)(b)

Figure 1: Return maps of parametrically forced logistic maps
for α1 = 3.0, α2 = 3.83 and τ = 1.

3. Synchronization(c)
Various synchronization phenomena in the globally coupled parametrically forced logistic map are observed for one
control parameter ε which is a coupling intensity when thenumber of coupling is three. Figure 2 shows examples ofsynchronization phenomena. In Fig. 2, upper figures show
the return maps and lower figures show the phase differences
between the maps. The synchronization phenomena fall
into four general categories, which are asynchronous, selfswitching of synchronization as shown in Fig. 2(a), synchronization of two among the three maps as shown in Fig. 2(b)and synchronization of all the maps as shown in Fig. 2(c).

Figure 2: Synchronization of three chaos. α1 = 3.0, α2 =3.83 and τ = 1. (a) ε = 0.040. (b) ε = 0.170. (c) ε = 0.200.

In Fig. 3(a), firstly, map 1 and map 2 are synchronized. However, after a time, the synchronous state breaks up and map 1and map 3 are synchronized. As seen above, the synchronousstates switch with time in sequence.
4.1. Sojourn time

A sojourn time of the self-switching is related to ε. The sojourn time is short when ε is small, whereas the sojourn time
The self-switching phenomenon of synchronization is ob- is long when ε is big as shown Figs. 3(a) and (b). Figure 4
served when the ε is set around 0.04. The phenomenon is that shows sojourn time of the self-switching between ε = 0.04
two among the three maps are synchronized and the combi- and ε = 0.0405. The sojourn time increases exponentiallynation which maps are coupled changes with time. Figure 3 with ε.
shows time series of differences of x(t) between two maps.
Areas where the amplitudes of the time series are small corre- 4.2. Second iterate of the map
spond to in-phase synchronization in the figure. Additionally,
To investigate the self-switching phenomenon of synchrothe difference between synchronized maps is smaller than 0.1.
4. Self-switching phenomenon- 46 -

(a)
(a)
(b)
Figure 3: Time series of differences between two maps. α1 =3.0, α2 = 3.83 and τ = 1. (a) ε = 0.0400. (b) ε = 0.0415.
(b)
Figure 5: Return maps of the second iterate of the map . α1 =3.0, α2 = 3.83 and τ = 1. (a) Second iterate of the map foreach even number of t. (b) Second iterate of the map for each
odd number of t

Figure 4: Sojourn time of the self-switching. α1 = 3.0, α2 =3.83 and τ = 1. Horizontal axis: ε, vertical axis: the meanvalue of the sojourn time.

nization, a second iterate of the map is used. xi (1) is calculated with α f (0) = α1 in the first update. After that, xi (2) iscalculated with α f (1) = α2 . The above calculations are alternately repeated. Thus, a second iterate of the map for eachodd number of t is different from a second iterate of the mapfor each even number. Here, The two kinds of the second iterate of the map are termed as:
MAP-A: the second iterate of the map for each even numberof t,
MAP-B: the second iterate of the map for each odd numberof t.

tion of map 1 and map 3, for combination of x1 , x2 andx3 . Here, we assume that xi (0) is xi (t) before the switchingfor convenience. Fixing difference between x1 (0) and x2 (0),combinations of x1 (0) and x3 (0) that the synchronization stateswitch to other synchronization states are shown in Fig. 6. If
the difference between x1 (0) and x2 (0) is smaller than 0.05,
the switching of synchronization state does not occur. InFig. 6, the upper figures show combinations of x1 (0) and x3 (0)that the switching occurs on MAP-A. While, the right figures
show the combinations of x1 (0) and x3 (0) on MAP-B. In thefigures, the green area shows combinations of x1 (0) and x3 (0)that map 1 and map 3 are synchronized after the switching.
While the red area shows combinations of x1 (0) and x3 (0) thatmap 2 and map 3 are synchronized after the switching. The
switching is easy to occur in MAP-B, by comparison withthe case of MAP-A. Moreover, with increasing the difference
between x1 and x2 , the combinations of x1 and x3 that theswitching occurs is increased.

Return maps of the MAP-A are shown in Fig. 5(a). From the 5. Conclusionfigure, it can be seen xi (2t) is projected in 0.41 < xi (2t) <0.96. On the other hand, return maps of the MAP-B are
In this study, the self-switching phenomenon of synchroshown in Fig. 5(b). From the figure, it can be seen thatnization in globally coupled parametrically forced logisticxi (2t − 1) is projected in 0.12 < xi (2t − 1) < 0.75.
maps was investigated. In the investigation of the relationship between sojourn time and the coupling intensity, it was
4.3. Switching values
observed that the sojourn time increases exponentially withIn this subsection, condition that synchronization state the coupling intensity. In the investigation of the condition
switches to other synchronization states in the self-switching that a synchronization state switches to other synchronizaare investigated. When map 1 and map 2 are synchronized, tion states, it was observed that the combinations which thethe synchronization state change to other synchronization switching of synchronization state occurs is increased withstates, synchronization of map 2 and map 3 or synchroniza- increasing a difference between synchronized maps.

- 47 -(a)

(b)
(1)

(2)

(3)

Figure 6: Combinations of x1 and x3 that switching of synchronization state occurs for α1 = 3.0, α2 = 3.83 and τ = 1. (a) t iseven number. (b) t is odd number. (1) x2 (0) − x1 (0) = 0.05. (2) x2 (0) − x1 (0) = 0.07. (3) x2 (0) − x1 (0) = 0.09.

Acknowledgment

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