Phase Synchronization in a Ring of Chaotic Circuits
Yoshifumi NISHIO and Akio USHIDA
Dept. of Electrical & Electronic Engineering, Tokushima University,
2-1 Minami-Josanjima, Tokushima 770-8506, Japannishio@ee.tokushima-u.ac.jp

Abstract— In this study, phase synchronization
observed in a ring of chaotic circuits coupled by resistors is analyzed. For the case that an odd number(2N +1) of the circuits are coupled, a kind of frustration is caused and (2N +1)-phase synchronization ofchaos is stably generated.
I. Introduction
Recently, phase synchronization of chaotic oscillators
attract many researchers’ attentions [1]. Because it
seems to be impossible to give rigorous deﬁnition ofphase for chaotic signals, theoretical analysis of thephenomenon is quite diﬃcult and hence much still
remains to be done to understand the phenomenon
completely. Further, there have been few results onphase synchronization based on real physical systems
such as electrical circuits. On the other hand, the authors have been working on coupled chaotic circuits
and have found some of them could produce similar
phenomenon [2]-[4].
In this study, we propose a ring of chaotic circuits
coupled by resistors. Because the system has the coupling structure such that the energy consumed by thecoupling resistors becomes minimum when the adjacent two circuits are synchronized at anti-phase, a kindof frustration is caused for the case that an odd number of the circuits are coupled. In that case, we canobserve the phase synchronization. The phenomenon
is observed from both of circuit experiments and computer calculations. For any odd number 2N +1, weobserve (2N +1)-phase synchronization of chaos. Inorder to analyze the phenomena, we deﬁne phase ofchaotic signal using the projections of the orbits of thechaotic circuits. Although the deﬁnition cannot giveexact meaning of the phase of chaotic signal, that canmake it possible to give qualitative discussions on thephase synchronization.

too high. Figure 2 shows typical example of chaoticattractors observed from each subcircuit.

CvkIi
L2

L1-r

vd(i)

Figure 1: Chaotic Subcircuit.

Figure 2: Typical example of chaotic attractors observed from subcircuit. (a) Computer calculated result. x vs. z. α=7.0, β=0.14 and δ=100.0. (b) Circuit experimental result. I vs. v. L1 =100.7mH,
L2 =10.31mH, C=34.9nF and r=334Ω. H: 0.8mA/div.
V: 1.3V/div.

II. Circuit Model
Figure 1 shows the chaotic subcircuit. Each subcircuit is three-dimensional autonomous one and consistsof three memory elements, one linear negative resistor
and one diode. We can regard the diodes as pure resistive elements, because operation frequency is not

In this study we consider a ring of the circuits asshown in Fig. 3. In the circuit adjacent two subcircuits
are coupled by one resistor R. Because such couplingsystems tend to minimize the energy consumed by thecoupling resistors [2][4], every two adjacent subcircuits

2L 1

I Lk

I Rk

2L 1

2L 1

I L(k+1) I R(k+1)

ik
C

vk

2L 1

i k+1
CL2

vk+1

L2

R-r

R

vd(ik)-rvd(ik+1)

Figure 3: Ring of chaotic circuits.
tend to synchronize with anti-phase. For the case thatthe total number of the subcircuits is even, the abovementioned phase state does not cause any frustration.
However, for the case that the total number is odd,
such a apparently stable phase state does not exist
and the system gets frustrated.
At ﬁrst, the i−v characteristics of the diodes are approximated by two-segment piecewise-linear functions
as1
vd (ik ) = (rd ik + E − | rd ik − E | ) .
(1)
2
By changing the variables and the parameters,


CC


IRk =
E xRk , ILk =E xLk ,
L1
L1



C ik =E yk , vk = E zk ,L1(2)CL1
t = L1 C τ, α =, β=r,
L2L1CC γ=R, δ = rd,
L1
L1
the normalized circuit equations are given as
dxRk



dτ




 dxLk


 dτ










==

1
{β(xRk + xLk + yk ) − zk2−γ(xRk + xL(k+1) )1{β(xRk + xLk + yk ) − zk2−γ(xLk + xR(k−1) )dykdτ= α{β(xRk + xLk + yk ) − zkdzkdτ

= xRk + xLk + yk−f (yk )}( k=1, 2, 3, · · ·, N )(3)wheref (yk ) =1
(δ yk + 1 − | δ yk − 1 | )2(4)andxL(N +1) = xL1 , xR0 = xRN .

(5)

Note that when the coupling parameter γ, which is inproportion to R, is equal to zero, the coupling termin (3) vanishes. For all of computer calculations, thefourth-order Runge-Kutta method is used with step
size h = 0.005.
III. Simulation Results
We carried out computer simulations for the case of
N = 3 ∼ 15 and circuit experiments for N = 3 ∼ 5.
Figure 4 shows the computer simulated result forthe case of N = 7. We can see the adjacent subcircuits
are almost synchronized with anti-phase. However, because of the boundary condition of the ring structure,
the phase diﬀerence between the adjacent subcircuits
is not around π but around π − π/7. The margin π/7
is accumulated along the ring and makes π phase difference to compensate the frustration. Namely, in thiscase 7-phase synchronization of chaos appears in thering.
Figure 5 indicates the changes of the amplitudes by
20 levels of gray scale. The phase synchronization canbe observed more clearly.
In order to investigate the observed phase synchronization, let us deﬁne the Poincaré section as z1 = 0
and xR1 +xL1 < 0 and plot the solutions of the subcircuits on (xRk +xLk ) – zk plane. The data are shown inFig. 6. Because the attractor observed from each subcircuit is strongly constrained onto the plane yk = 0
when the diode is oﬀ, we can see the phase diﬀerence
from the data. The data of the subcircuit 1 are notvisible, because the points (xR1 (n)+xL1 (n), z1 (n)) arealways on the Poincaré section.

Figure 5: Phase synchronization for N = 7. α=7.0,β=0.14, γ=0.1 and δ=50.0. τ vs. xRk + xLk . k=1, 2,3, · · ·, 7.

Figure 4: Computer simulated result for N = 7.
α=7.0, β=0.14, γ=0.1 and δ=50.0. Upper ﬁgures:xRk + xLk vs. zk . Middle ﬁgures: xRk + xLk vs.
xR(k+1) + xL(k+1) . Lower ﬁgures: τ vs. xRk + xLk .
k=1, 2, 3, · · ·, 7.

Further, we introduce the following independent
variables from the discrete data of xRk (n) + xLk (n)and zk (n) on the Poincaré map.

Figure 6: Poincaré map for N = 7. α=7.0, β=0.14,γ=0.1 and δ=50.0. xRk (n) + xLk (n) vs. zk (n). k=1,2, 3, · · ·, 7.
IV. Conclusions
In this study, a ring of chaotic circuits coupled by resistors has been proposed and the phase synchronization
caused by a kind of frustration of the coupled system
has been analyzed.
We consider that it is interesting to investigate theeﬀect of the frustration of coupled systems.

zk+1 (n)−1 π − tanx(n)+ xL(k+1) (n)R(k+1) · · · xR(k+1) (n) + xL(k+1) (n) ≥ 0
zk+1 (n)− tan−1xR(k+1) (n) + xL(k+1) (n)ϕk (n) =· · · xR(k+1) (n) + xL(k+1) (n) < 0
and zk+1 (n) ≥ 0
zk+1 (n)
2π − tan−1x(n)+ xL(k+1) (n)R(k+1)
· · · xR(k+1) (n) + xL(k+1) (n) < 0and zk+1 (n) < 0References[1] ”Phase Synchronization in Regular and ChaoticSystems,” A. Pikovsky, M. Rosenblum and J.
Kurths, International Journal of Bifurcation andChaos, vol. 10, no. 10, pp. 2291-2305, 2000.
(6)[2] ”Synchronization in Mutually Coupled ChaoticCircuits,” Y. Nishio and S. Mori, Technical Report of IEICE, no. NLP92-100, pp. 176–181, Mar.
1993.

These variables can correspond to the phase diﬀerences
between the subcircuit 1 and the others. (Note thatthe argument of the point (xR1 (n) + xL1 (n), z1 (n)) isalways π, because of the deﬁnition of the Poincarémap.) The distribution of the ϕk (n) for N = 7 isshown in Fig. 7.
Finally, the computer simulated results for the caseof N = 15 are shown in Figs. 8 and 9. We can see
15-phase synchronization of chaos is stably generated.

[3] ”On a Ring of Chaotic Circuits Coupled by Inductors,” Y. Nishio and A. Ushida, IEICE Transactions on Fundamentals, vol. E78-A, no. 5, pp.
608–617, May 1995.

( k=1, 2, 3, · · ·, N − 1. )[4] ”Quasi-Synchronization Phenomena in ChaoticCircuits Coupled by One Resistor,” Y. Nishio andA. Ushida, IEEE Transactions on Circuits andSystems I, vol. 43, no. 6, pp. 491-496, June 1996.

  


α=7.0, β=0.14, γ=0.1 and δ=50.0.

τ vs.

  


Figure 9: Distribution of ϕk (n) for N = 15. α=7.0, β=0.14, γ=0.1 and δ=50.0.

Figure 8: Phase synchronization for N = 15.
k=1, 2, 3, · · ·, 15.

Figure 7: Distribution of ϕk (n) for N = 7. α=7.0, β=0.14, γ=0.1 and δ=50.0.

 


xRk + xLk .