2010 International Symposium on Nonlinear Theory and its Applications
NOLTA2010, Krakow, Poland, September 5-8, 2010

Synchronization in Coupled Maps with Triangular Networks
Yoko Uwate† and Yoshifumi Nishio††Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: {uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—In this study, we investigate synchronization
phenomena when the 2-dimensional maps based on neuronmodel are coupled with triangular network property. Furthermore, the diﬀerence of synchronization obtained fromthe coupled maps and the coupled oscillators is compared.

32Xn+1101. Introduction

Pk-1
Xpu

Generally, complex dynamical phenomena can be observed in networks formed by many elements with nonlinearity. Coupled Map Lattice (CML) has proposed by
Kaneko and Bunimovich [1]-[5], to use as general models for the complex high-dimensional dynamics, such as
biological systems, networks in DNA, economic activities, neural networks, and evolutions. We can observe thespatio-temporal patterns in CML. Moreover, coupled oscillatory systems can also produce interesting phase patterns,including wave propagation, clustering, and complex phasepatterns. It is very important to make clear this mechanism of these spatio-temporal patterns for understanding
complex patterns observed in natural science. Usually,
the chaotic maps are used for CML and many interesting
spatio-temporal patterns were observed.
Recently, a discrete map for spiking-bursting neural behavior was proposed by Rulkov [6], [7]. Rulkov map (see.
Fig. 1) in the form of a two-dimensional map can be useful for understanding the dynamical mechanism of oscillators in the large scale networks. And Rulkov map produce
spiking-bursting behavior such as real neurons.
In this study, we investigate synchronization phenomena
when the 2-dimensional maps based on neuron model arecoupled with triangular network property. Furthermore, thediﬀerence of synchronization obtained from the coupledmaps and the coupled oscillators is compared.

-2

Xps-3-3-2-10
Xn

12

3

Figure 1: Rulkov map. The dashed line illustrates a superstable cycle Pk . The stable and unstable fixed points of themap are indicated by x ps and x pu , respectively.

1132432(a) N = 3.

(b) N = 4.
613142(c) N = 5.

53425(d) N = 6.

2. Coupled Maps with Triangular Networks
The several types of coupled maps with triangular networks are shown in Fig. 2.
We consider a chain of coupled maps:
xi,n+1 =yi,n+1 =f (xi,n xi,n−1 , yi,n )1+ g(xi+1,n − 2xi,n + xi−1,n ),2yi,n − µ(xi,n + 1) + µσi(e) General network model with triangular networks.

Figure 2: Several types of coupled maps with triangular
network property.

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1+µ g(xi+1,n − 2xi,n + xi−1,n ),2
i = 1, ..., N,(1)where x and y are the fast and slow dynamical variables,respectively. µ = 10−3 and σi are the parameters of theindividual map and g is the coupling. The function f () has
the following form:α/(1 − xn ) + yn , xn ≤ 0,






0 < xn < α + yn
 α + yn ,(2)f (xn , yn ) = andxn−1 ≤ 0, −1,xn ≥ α + yn or xn−1 > 0,

While, in the case of σ = 0.10, in/anti phase synchronization can be observed as shown in Fig. 5. The first and thesecond maps are synchronized at the in-phase state and theother combinations are synchronized at the anti-phase state.
1.5
10.5
0-0.5
-1-1.5
1.5
10.5
0-0.5
-1-1.5In this simulations, we take α = 3.5 and σi is set for
randomly distributed in the interval [0.15:0.16].

1.5
10.5
0-0.5
-1-1.52.1. Synchronization for N = 3

1.5
10.5
0-0.5
-1-1.5First, we consider the simplest model as shown inFig. 2(a). The three maps are coupled as ring topology.
In this coupled maps model, three phase synchronization
can be observed when the copuling strength is set to g =−0.029.

500

1000

n1500

2000

Figure 4: Two pair of three-phase synchronization (σ =0.24).

1.5
10.5

X1,n

00-0.5-1-1.5

1.5
10.5
0-0.5
-1-1.5

1.5
10.5

X2,n0-0.5
-1

1.5
10.5
0-0.5
-1-1.5-1.5
1.5
10.5

X3,n0-0.5

1.5
10.5
0-0.5
-1-1.5-1-1.5
0500

1000

1500

2000

nFigure 3: Three-phase synchronization (g = −0.029).

1.5
10.5
0-0.5
-1-1.52.2. Synchronization for N = 4
Next, the model of coupled maps with two triangular networks as shown in Fig. 2 (b) is considered. We observe twotypes of synchronization states dependence on the value ofσ. When σ is fixed with 0.24, two pair of three phase synchronization is obtained. The time wave forms of each mapare shown in Fig. 4. From this figure, the first, the second
and the third maps are synchronized with three-phase state.
Also, the first, the second and the fourth maps synchronize at three-phase. Furthermore, we confirm that the third
and the fourth maps are synchronized with in-phase state.

0500

1000

n1500

2000

Figure 5: In/Anti-phase synchronization (σ = 0.10).

3. Comparison with Coupled Oscillatory System
In this section, we compare synchronization phenomena
between the coupled maps and the coupled oscillators.

- 497 -3.1. Circuit Model for N = 3

whereThe circuit model of three coupled van der Pol oscillator
as ring topology is shown in Fig. 6. This circuit model
corresponds to the three coupled maps shown in Fig. 2 (a).
2nd oscillator
ia22Li R2√LCτ, vk =2L

rm

ib2

√

g1
xk ,3g3
√
√
√
√
g1
C
g1
C
iak =
yak , ibk =ybk ,3g3 L
3g3 L
√
√
√
LCC
ε = g1, γ=R, η = rm,CLL
t=(k = 1, 2, 3).

Crmv22Lrm

In this equations, γ is the coupling strength and ε denotes the nonlinearity of the oscillators. For the computersimulations, γ and ε are fixed with 0.1, 0.1, respectively.
For the computer simulations, we calculates Eqs. (4)-(6)using a fourth-order Runge-Kutta method with the step size
h = 0.005. Figure 7 shows the three time wave forms obtained by each oscillator. We can see that the three oscillators are synchronized at the three-phase state.

2LR

ib1

2L

rm

ia1

2L
rm

ib3

ia3
rm

i R1

iR3
v1
CRCv3

1.5

3rd oscillator

1st oscillator

x1

x2

x3

15002000
τ1Figure 6: Circuit model (N = 3).
x1, x2, x3

0.5

We assume that the vk − iRk characteristics of the nonlinear resistor in each oscillator is given by the following thirdorder polynomial equation,

0-0.5

-1

(3)

-1.5

0500

1000

(k = 1, 2, 3, 4).

4000

3.2. Circuit Model for N = 4
Here, we consider the two coupled triangle oscillatory
networks sharing a branch. The circuit model of two coupled triangle oscillatory networks sharing the branch isshown in Fig. 8.
ic1

3L

3L

i R1

rm

3L

ia3
ib3

rm

ic3

R3L3LR

rm

ib2

ia2

3L

3L

iR4
Cv4

R

4th oscillator

ic2

rm

i R33L

ia4

rm

R(5)

rm

Cv1

ib4 ic4

3L
rm

ia1

rm

rm

1st oscillator
ib1

iR2
C

[Third oscillator]
(dx31 2)

=
ε1−x3 x3 − (ya3 + yb3 )



dτ3


}
 dya3 1 {=x3 − ηya3 − γ(ya3 + yb1 )


dτ2


}
dy1{
b3
=x3 − ηyb3 − γ(ya2 + yb3 )
dτ
23500

Figure 7: Three phase synchronization obtained from coupled oscillatory system.

3L[Second oscillator]
(dx21 )= ε 1 − x2 2 x2 − (ya2 + yb2 )



dτ3


}

 dya2 1 {=x−ηy−γ(y+y)2a2a2b3


dτ
2


}


dy1{
b2

=x2 − ηyb2 − βγ(ya1 + yb2 )
dτ
23000

3L

The normalized circuit equations governing the circuit
are expressed as[First oscillator]
(dx11 2)
=ε1−x1 x1 − (ya1 + yb1 )

dτ3

{}

 dya1 1(4)=x1 − ηya1 − βγ(ya1 + yb2 )

dτ2
}


dyb1 1 {=x1 − ηyb1 − γ(ya3 + yb1 )
dτ
2

2500

rm

(g1 , g3 > 0),3L

iRk = −g1 vk + g3 vk

3v3

3rd oscillator

RCv2

2nd oscillator

Figure 8: Circuit model (N = 4).

(6)

The parameters of this circuit model are fixed as ε = 0.1,γ = 0.1, η = 0.0001.

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Figure 9 shows the time wave form of the voltage
charged at the capacitance of each oscillator. From thisfigure, we can see that the first and the second oscillators are synchronized at in-phase (phase diﬀerence: 0 degree). While, the other combination oscillators synchronize with anti-phase (phase diﬀerence: 180 degree). Furthermore, the amplitude of between the first/second andthe third/fourth oscillators has small diﬀerence. The phaseplane of each combination oscillator is shown in Fig. 10.

5. Conclusions
In this study, we have investigated synchronization phenomena when the 2-dimensional maps based on neuronmodel are coupled with triangular network property. Furthermore, the diﬀerence of synchronization obtained fromthe coupled maps and the coupled oscillators was compared.

x1

1.5
10.5
0-0.5
-1-1.5This work was partly supported by JSPS Grant-in-Aidfor Scientific Research 22500203.

x2

1.5
10.5
0-0.5
-1-1.5

References

x3

1.5
10.5
0-0.5
-1-1.5[1] K. Kaneko, “Spatiotemporal Intermittency in Coupled Map Lattice,” Prog. Theor. Phys., vol.75, no.5,pp.1033–1044, 1985.

x4

Acknowledgment

1.5
10.5
0-0.5
-1-1.5
01000

2000

τ

3000

4000

[2] L. A. Bunimovich and Ya. G. Sinai, “SpacetimeChaos in Coupled Map Lattices,” Nonlineality 1,no.4, pp.491–516, 1988.

5000

Figure 9: Time wave form of the voltage charged at thecapacitance of each oscillator.

[4] K. Kaneko, “Spatiotemporal Chaos in One- andTwo- Dimensional Coupled Map Lattice,” Physica D,
vol.37, pp.60–82, 1989.

Anti-phase synchronization(2-3, 3-1, 1-4, 4-2 oscillators)

3rd osc.

x1
x3

4th osc.

x1

x2

x3

[5] K. Kaneko, “Simulating Physics with Coupled Map
Lattice - Pattern Dynamics, Information Flow, andThermodynamics of Spatiotemporal Chaos,” Formation, Dynamics, and Statistics of Patterns, World Sci.,
pp.1–52, 1990.

x4

1st osc.

x1x3[6] N. F. Rulkov, “Modeling of Spiking-Bursting Neural Behavior using Two-dimensional Map,” PhysicalRev., E, vol. 65, 041922, 2002.

x2

2nd osc.

x2[3] K. Kaneko, “Pattern Dynamics in SpatiotemporalChaos,” Physica D, vol.34, pp.1–41, 1989.

In-phase synchronization(1-2 oscillators)[7] N. F. Rulkov, I. Timofeev and M. Bazhenov, “Oscillations in Large-Scale Cortical Networks: Map-BasedModel,” Journal of Computational Neuroscience vol.
17 pp. 203–223, 2004.

Figure 10: Phase plane of each oscillator.

4. Discussions
In this section, we compare the synchronization phenomena between the coupled maps system and the coupledoscillators system. In the case of that the number of coupled elements is N = 3, the three-phase synchronization
can be observed from the both systems. When the number of coupled elements is fixed as N = 4, we observe
the in/anti-phase synchronization from the both systems.
Furthremore, the two pair of three-phase synchronization
can be observed from the coupled maps system.

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