May 31, 1996 - If eqn. 13 is true then lim,+-x2 (t) = 0, which is in contradiction to eqn. 10. However, based on eqn. 10 we conclude that the phase.
for each initial condition (x,(0),x2(0)) E M2(0,0). From the preceeding discussion it can be concluded that curve S(X~,X~,A~,) = 0 needs to represent a stable limit cycle of the system of eqn. 5. We can choose a Lyapunov function in the following form:
v = 32 ( 5 1 , 5 2 , A d )
(7)
Its time derivative is =~
( 5 15 2 , &)S(ZI,
5 2 ,Ad)
(8)
= -p22Y(21,22,Ad)U(ICl,ICZ)
We choose a control function u(x,,x,) such that the derivative of the Lyapunov function is a negative semidefinite function. One of the possibilities is U ( ~ I , Z= ~ )axzsgn(s(zi,~z,&))
0
>0
(9)
teed. However, at these points, the velocity of the phase point is tangent to the curve s(x,,x,,A,) = 0 and the phase point remains on the curve s(x,,x,,A,) 0. Experimental results: Finally, we present results from the experimental verification of the theoretically obtained results. The conservative oscillator was created with operational amplifiers, and the control signal ti(x,,x2)was generated by a microprocessor. The results of the experiment are shown in Fig. 2. The oscillator parameters are: r, = 2V, r2 = 2V, T = 5s, o = 0.2. The amplitude deviation for both signals is less than 0.1%. Observing Fig. 20, we can see that the oscillations build-up time is T, = 3.305s. 0 IEE 1996 Electronics Letters Online hro: 19961022
31 May 1996
G. Golo and C . Milosavljevid (Faculty of Electronic Engineering, C'niiwsity of NiS 18 000 .Vi$ Yugoslavia)
In this case eqn. 8 becomes
V = -p.az;I(s(z1,za,Ad)l =-pozzm
(10)
Substituting eqn. 9 into eqn. 5 we finally come to the oscillator equations = px2 i 2 = -ksgn(zl) +ux2sgn(s(zl;x2,&) (11) We show now that there is a finite oscillation build-up time. By solving eqn. 10 for Vwe find
References 1 HOLTZ. J : 'Pulse width modulation- A survey', IEEE Trans., 1992, IE-39, pp. 410420
2 3 4
Suppose that the time in which the phase point reaches this curve s(x,,x2,Ad)= 0 is infinite. This means that lim,,,V(t) = 0. If we substitute this into eqn. 12 we now have
If eqn. 13 is true then lim,+-x2 ( t ) = 0, which is in contradiction to eqn. 10. However, based on eqn. 10 we conclude that the phase point asymptotically converges to the curve s(xI,xzA~,) = 0, so x2(t) cannot converge to zero when t - w . Oscillations build up in finite time, or in other words, there exists T y> 0 such that V(T,) = 0. 31
5 6
KAPLAN. B Z., and TATRASH, Y.: 'New method for generating precise triangular waves and square waves', Electron. Lett., 1977, 13, pp. 71-73 GENIN. R., JEZEQUEL. c., and GENIN, J . : 'Simplified method for generating precise triangular waves', Electron. Lett., 1978, 14, pp. 162-163 SIRA-RMIREz. : 'Harmonic response of variable-structurecontrolled Van der Pol oscillators', ZEEE Tmns., 1987, CAS-34, pp. 103-106 SIRA-RAMIREZ. : 'Differential geometric methods in variablestructure control', Int. J. Control, 1988, 48, (4), pp. 1359-1390 K-IPLAY. B.Z., and TATRASH, Y : 'An 'improved' van-der-Pol equation and some of its possibie applications', Int. J. Electron., 1976, 41. (2), pp. 189-198
Synchronisation of 4D hyperchaotic oscillators A. TamaSeviCius, G. Mykolaitis A. Cenys and A. Namajunas
Indexing twms: Oscillaton, Chaos
-31
"
0
'
"
5
'
10 time,s
" '
" '
15
The synchronisation of two hyperchaotic electronic oscillators, coupled either bi or unidirectionally, is described. Numerical results are presented along with experimental results. The synchronisation i s shown to be achieved via a single variable.
20
cl
Introd~rctioiz:The synchronisation of chaotic systems has attracted
much attention in recent years, see e.g. [l 51 and references therein. As discussed in [2 61 this phenomenon promises interesting applications in secure communications. For higher security, the hyperchaotic systems i.e. those characterised by two or more positive Lyapunov exponents, seem to have an advantage [5, 61 over common chaotic systems with only one positive Lyapunov exponent. Several hyperchaotic oscillators, suitable for secure communications, have been described in the literature e.g. [7 - lo]. In this Letter we present numerical and experimental results on the synchronisation of two extremely simple 4D electronic oscillators. A single 4D oscillator has been described elsewhere [9]. -
~
-0.61 . 8 1 .
0
,
5 b
'
,
10
,
I
15
time,s
'
' 20
m
Fig. 2 Experimental resulis
a Experimentally obtained signals xl(t) and xZ(t) 0 Control signals u(xI,x2)
A
Based on the discontinuity of the control function eqn. 9, the condition that the approach time of the phase point to the trajectory s(x,,x2A
