reconstruction and synchronization of hyperchaotic ... - Semantic Scholar
International Journal of Bifurcation and Chaos, Vol. 12, No. 10 (2002) 2069–2085 c World Scientific Publishing Company
RECONSTRUCTION AND SYNCHRONIZATION OF HYPERCHAOTIC CIRCUITS VIA ONE STATE VARIABLE MAKOTO ITOH Department of Information and Communication Engineering, Fukuoka Institute of Technology, Fukuoka 811-0295, Japan LEON O. CHUA Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA 94720, USA Received February 14, 2002 In this paper, we demonstrate that some hyperchaotic circuits can be synchronized by using only one state variable. We applied three kinds of synchronization schemes, a continuous synchronization, an impulsive synchronization, and a selective synchronization to these hyperchaotic circuits. Their performance is examined from the viewpoint of synchronization stability and convergence time. Keywords: Synchronization; hyperchaos; Chua’s diode; Chua’s oscillator; Chua’s circuit; hyperchaotic circuit.
1. Introduction Many methods have been proposed to synchronize chaotic systems. The most widely used methods are the continuous synchronization schemes [Pecora & Carroll, 1990], where the driving signals are transmitted continuously to the driven systems. The other synchronization schemes are impulsive synchronization [Panas et al., 1998] and selective synchronization [Itoh et al., 2000]. In an impulsive synchronization scheme, only samples of state variables (or functions of state variables) called synchronization impulses are used to synchronize two chaotic systems. In a selective synchronization scheme, we select only those time periods of driving signals with strong synchronizing effect to the slave system and shut off the driving signals in some other time periods when they show strong desynchronizing effects. These three schemes were applied to several chaotic systems, and all three exhibited
good performance [Panas et al., 1998; Itoh et al., 1999; Itoh et al., 2000; Itoh et al., 2001]. The stability of synchronization is closely connected to the values of Lyapunov components of variational systems, and so the conditions for stable impulsive synchronization in both chaotic and hyperchaotic systems are given in terms of Lyapunov exponents [Pecora & Carroll, 1990; Itoh et al., 1999; Itoh et al., 2000; Itoh et al., 2001]. According to Pyragas [Pyragas, 1993; Brucoli et al., 1999], the minimal number of controlled variables has to be equal to the number of positive Lyapunov exponents of the system. If the hyperchaotic systems have two positive Lyapunov exponents, then, at least two driving signals are needed to synchronize them by the continuous synchronization scheme. In the case of impulsive synchronization, the number of driving signals sequentially transmitted to the slave systems via time-division is greater than or equal to two [Itoh et al., 1999; Itoh et al., 2000;
2069
2070 M. Itoh & L. O. Chua
Itoh et al., 2001; Brucoli et al., 1999]. Almost all models considered up to now seemed to support this assumption. In this paper, we demonstrate that some hyperchaotic circuits can be synchronized by using only one state variable. That is, counter-examples to Pyragas’s conjecture are presented. We applied three kinds of synchronization schemes to the hyperchaotic circuit from [Matsumoto et al., 1986] and to the hyperchaos generator from [Saito, 1990], and succeeded to synchronize them. These circuits have a negative resistor and an eventually strictly monotonically increasing Chua’s diode [Kennedy, 1992].1 Thus, if we choose the voltage across the negative resistor as a driving signal, their variational systems become stable in the regions where the v–i characteristic of the nonlinear resistor is strictly monotonically increasing. In these regions, the driving signal has a strong synchronizing effect. Thus, we can expect that all conditional Lyapunov exponents (CLEs) of the variational system become negative.2 In our computer studies, two hyperchaotic circuits will be synchronized by using three kinds of synchronization schemes: a continuous synchronization, an impulsive synchronization and a selective synchronization. We also discuss their performance from the viewpoints of synchronization stability and convergence time. Furthermore, we show that if the linear part of the system is represented by a ladder circuit,3 we can easily reconstruct all the state variables from the derivatives of one state variable.
2. Preliminaries Consider the nonlinear circuit with the dynamics
i i=f(v) + Chua’s v diode
Y(s)
Fig. 1. Nonlinear circuit with one nonlinear element (Chua’s diode). The admittance of the linear part is given by Y (s). The v–i characteristic of Chua’s diode is given by i = f (v).
(transfer function) of the operator L−1 1 L2 by Y (s), then Y (s) is equal to admittance of the linear circuit (see Fig. 1, which illustrates a two-terminal network representation). That is, Y (s) satisfies the following relation I(s) = Y (s)V (s) =
L1 i = L2 v ,
(1)
where the symbols i and v indicate a current and a voltage, respectively, f is a scalar function, and L1 , L2 are differential operators. The nonlinear element (Chua’s diode) is characterized by the equation i = f (v). If we denote the Laplace transform 1
(2)
where G1 (s) and G2 (s) are the Laplace transforms of L1 and L2 , and I(s) and V (s) are the Laplace transforms of i and v, respectively (under zero initial conditions). Similarly, the impedance Z(s) is defined by G1 (s)/G2 (s) = V (s)I(s)−1 . Here, we assume that the v–i characteristic of the Chua’s diode is eventually strictly monotonically increasing. Next, we define the master and slave circuits as follows: i = f (v) ,
i = f (v) ,
G2 (s) V (s) , G1 (s)
L1 i = L2 v ,
)
master circuit
L1 i0 = L2 v , } slave circuit
(3) (4)
where the voltage across Chua’s diode, that is, v(t), is a driving signal. Then the variational system is given by L1 p = 0 , (5)
A nonlinear resistor is said to be strictly monotonically increasing if, and only if, for each pair of the points (v1 , i1 ) and (v2 , i2 ) on the v–i curve, we observe whenever v1 > v2 , then i1 > i2 . Furthermore, the voltage-controlled Chua’s diode is said to be eventually strictly monotonically increasing, if the v–i characteristic of the nonlinear resistor is strictly monotonically increasing in the region: |v| > v0 > 0 (v0 : some constant). 2 It is well known that the hyperchaotic circuits will synchronize if all conditional Lyapunov exponents of the slave circuits are negative. 3 The circuit elements are combined alternatively in series and in parallel.
Reconstruction and Synchronization of Hyperchaotic Circuits via One State Variable 2071 Z m-2
Z m-1
Y(s)
Ym
Z2
Ym-1
Ym-2
Z1
1 Y(s) = Ym
+ 1
Z m-1 +
Ym-2 +
1 Z m-3 + . . .
1 Z2 + Z1
Z m-2
Z m-1
Y(s)
Ym
Ym-1
Ym-2
Y2
Y1
1 Y(s) = Ym
+ Z m-1 +
1 Ym-2 +
1 Z m-3 + . . .
1 Y2 + Y1
Fig. 2.
Continued fraction of the admittance Y (s). The admittance Y (s) is realized by a ladder circuit.
where p = i − i0 . Therefore, if all eigenvalues of the characteristic equation have negative real parts, then the two systems will be synchronized. That is, if all roots of the function G1 (s) for L1 have negative real parts, then the two circuits will be synchronized. We note that Chua’s oscillator corresponds to this case. In the next section, we study the case where the function G1 (s) has a root with a positive real part. Next, we consider reconstructing all of the state variables by using the derivatives of one state vari-
able. First, we expand the transfer function Y (s) = G2 (s)/G1 (s) into the following form 1
Y (s) = Ym +
,
1
Zm−1 + Ym−2 +
1 Zm−3 + · · ·
1 Z2 + Z1 (6)
2072 M. Itoh & L. O. Chua
or
variables from the derivatives 1
Y (s) = Ym +
.
1
Zm−1 + Ym−2 +
di1 d2 i1 d3 i1 i1 , , ,··· , dt dt2 dt3
1
or
dv1 d2 v1 d3 v1 v1 , , ,··· , dt dt2 dt3
1 Zm−3 + · · · Y2 + Y1 (7)
Then, the transfer function Y (s) can be realized by the ladder circuit as shown in Fig. 2. Furthermore, we have the relations V1 + V2 V3 I3 V4 I4 V5 I5 V6 .. . Im
= (Z2 + Z1 )I1 = V1 + V2 = Y 3 V3 = Z4 I4 = I1 + I3 = V3 + V4 = Y 5 V5 = Z6 I6 . = .. = Y m Vm
I1 + I2 I3 V3 I4 V4 I5 V5 I6 .. . Im
= (Y2 + Y1 )V1 = I1 + I2 = Z3 I3 = Y 4 V4 = V1 + V3 = I3 + I4 = Z5 I5 = Y 6 V6 . = .. = Y m Vm
!
,
(10)
!
.
(11)
It corresponds to a special case of the Takens reconstruction theorem [Takens, 1981]. In this case, we do not need to assume that all roots of the transfer function G1 (s) have negative real parts. Note that similar results hold for the impedance Z(s), too. In this case, we have to consider the currentcontrolled Chua’s diode in place of the voltagecontrolled Chua’s diode.
3. Synchronization of Hyperchaotic Circuits (8)
or
Consider the hyperchaotic circuit from [Matsumoto et al., 1986], whose dynamics are given in Table 1. This circuit contains two capacitors Cj , two inductors Lj with small physical resistance rj , a negative resistor −R, and an eventually strictly monotonically increasing resistor i = g(v)(v = v2 − v1 ) as shown in Fig. 3. The impedance Z(s) for the linear part of the circuit is given by Z(s) = Z1 + Z2 1
=
1
(9)
. 1 sL2 + r2 (12)
where Ij and Vj are the Laplace transform of the current through the impedance Zj (or the admittance Yj ) and the voltage across the impedance Zj (or the admittance Yj ), respectively. Thus, all of the voltages Vj and currents Ij are determined if the current I1 is given in the case of Eq. (8) or if the voltage V1 is given in the case of Eq. (9). Here, we assume that Yj and Zj are polynomial functions of s. Then, all of the currents ij and the voltages vj are described as functions of the derivatives of i1 or v1 , since the symbol s corresponds to the derivative d/dt. That is, we can reconstruct all of the state
The denominator of Y (s) = Z(s)−1 has a root with a positive real part. This implies that the slave circuits will not be synchronized by the driving signal v(t) = v2 −v1 (the voltage across the Chua’s diode). It is well known that the hyperchaotic circuit in [Matsumoto et al., 1986] has two positive Lyapunov exponents (LEs). According to Pyragas, the minimum number of driving signals for synchronizing this chaotic system is equal to two. Here, in order to synchronize the hyperchaotic circuits using one driving signal, we choose the voltage across the negative resistor, that is, the voltage Ri1 , or simply i1 as a driving signal. In the region |v| ≥ d ≥ 1, all eigenvalues of the variational system have negative real parts as shown in Table 2, and so the variational system is completely stable. The symbol d indicates a threshold. In the selective synchronization, d is used as a controlling parameter. In the
sC1 +
1 sL1 + r1 − R
+ sC2 +
Reconstruction and Synchronization of Hyperchaotic Circuits via One State Variable 2073 Table 1.
Hyperchaotic dynamical systems.
Dynamical Systems
Parameters
Hyperchaotic circuit
1 C2 = , 20 L1 = 1.0, 2 L2 = , 3
C1 =
C1
dv1 = g(v2 − v1 ) − i1 , dt
C2
dv2 = −g(v2 − v1 ) − i2 , dt
L1
di1 = v1 − r1 i1 + Ri1 , dt
L2
di2 = v2 − r2 i2 , dt
g(v) = b(v) + 0.5(a − b)(|v + 1| − |v − 1|) .
Driving Signal
1 , 2
i1
1 , 20 1 , 30 −0.2, 3.
R = 1, r1 = r2 = a= b=
Hyperchaos generator C1
dv1 = Gv1 + i1 , dt
C2
dv2 = −i1 − i2 , dt
L1
di1 = v2 − v1 , dt
L2
di2 = v2 − f (i2 ) , dt
f (i) = bi + 0.5(a − b)(|i + 1| − |i − 1|) .
i 1 L1 i
+ Chua’s diode v i=
-
-R
r1 v1
C1 = 1, C2 = 1, L1 = 1,
200 , 3 G = 1.89, a = −1, b = 1.
v2
L2 =
r2
i 2 L2 + v2
+
C2
C1
g(v) Fig. 3.
A hyperchaotic circuit.
region |v| ≤ d ≤ 1, all eigenvalues of the variational system have positive real parts. Thus, the variational system is completely unstable. Since there are two regions with a strong synchronization
effect, we can expect that all conditional Lyapunov exponents (CLEs) of their variational system to become negative. Furthermore, in the case of selective synchronization, we shut off the driving signal in the
2074 M. Itoh & L. O. Chua Table 2.
Hyperchaotic circuit.
Systems
Eigenvalues of the
Maximum LE
Variational Systems
or CLE
Master circuit C1
dv1 = g(v2 − v1 ) − i1 , dt
C2
dv2 = −g(v2 − v1 ) − i2 , dt
L1
di1 = v1 − r1 i1 + Ri1 , dt
L2
di2 = v2 − r2 i2 , dt
g(v) = b(v) + 0.5(a − b)(|v + 1| − |v − 1|) .
0.195
Slave circuit A dv 0 C1 1 = g(v20 − v10 ) − i1 , dt
dv20 = −g(v20 − v10 ) − i02 , dt 0 di2 0 0 L2 = v2 − r2 i2 . dt
C2
if |v20 − v10 | ≤ d ≤ 1, 1.96 ± 4.93i,
1.96
0.430. if |v20 − v10 | ≥ d ≥ 1, −0.233 ± 1.64i,
−0.233
−65.6.
Slave circuit B 0 dv2 0 0 0 C2 = −g(v2 − v1 ) − i2 , dt di0 L1 1 = v10 − r1 i01 + Ri01 , dt 0 di2 0 0 L2 = v2 − r2 i2 . dt C1
dv10 = g(v20 − v10 ) − i01 , dt
region |v| ≤ d ≤ 1. Then, two hyperchaotic circuits are run independently without a driving signal. In this case, the maximum real part of the eigenvalues of the variational system in this region, that is, = 1.71, is slightly smaller than that of the continuous synchronization (= 1.96). Thus, the selective synchronization is expected to have a smaller CLE. As we can see from Figs. 4 and 5, all of the CLEs become negative. This implies that the hyperchaotic circuits will be synchronized by using only one control variable. We note that the selective synchronization scheme gives the smallest CLE. 4
if |v20 − v10 | ≤ d ≤ 1, 1.71 ± 4.58i, 0.935 ± 0.851i.
1.71
if |v20 − v10 | ≥ d ≥ 1, −0.0951 ± 2.35i, −65.7,
0.751,
0.751.
However, the convergence time4 is not as quick as shown in Table 3. Note that the convergence time depends on the initial conditions, but the data in Table 3 serves as a reference of the performance. In our laboratory experiment, the synchronization of the hyperchaotic circuits is easily broken by small external noise, if the variational system has completely unstable regions. Thus, we synchronized the hyperchaotic circuits by transmitting the two driving signals via time-division-based impulsive synchronization [Itoh et al., 1997]. In terms of this interpretation, Pyragas’s conjecture may be
We set the convergence condition such that the difference e(t) between the same components is less than 10−12 .
Reconstruction and Synchronization of Hyperchaotic Circuits via One State Variable 2075 0.4
Maximum CLE
0.2
0
-0.2
-0.4
’Hyperchaotic circuit’
-0.6
’Hyperchaos generator’ -0.8
-1 0
1
2
3
4
5
d Fig. 4. Maximum CLE of the hyperchaotic circuit and the hyperchaos generator for selective synchronization. The symbol d indicates the threshold, that is, if |v| > d, then the driving signal is used for synchronizing the circuit. A selective synchronization for d = 0 is equivalent to a continuous synchronization.
0.3
’Hyperchaotic circuit’
0.2
’Hyperchaos generator’
Maximum CLE
0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 0
0.2
0.4
0.6
0.8
1
P/Q Fig. 5. Maximum CLE of the hyperchaotic circuit and the hyperchaos generator for impulsive synchronization. The symbols P and Q indicate the impulse width and the frame length, respectively. An impulsive synchronization for P/Q = 1 is equivalent to a continuous synchronization.
Table 3.
Convergence time of hyperchaotic dynamical systems.
Convergence Time Hyperchaotic circuit Hyperchaos generator
Continuous Synchronization 119 83.8
Selective Synchronization 149 66.5
Impulsive Synchronization (P = 0.05, Q = 0.03) 119 81.1
2076 M. Itoh & L. O. Chua
Table 4.
Hyperchaos generator. Eigenvalues of the Variational Systems
Systems
Maximum LE or CLE
Master circuit C1
dv1 = Gv1 + i1 , dt
C2
dv2 = −i1 − i2 , dt
L1
di1 = v2 − v1 , dt
L2
di2 = v2 − f (i2 ) , dt
f (i) = bi + 0.5(a − b)(|i + 1| − |i − 1|) .
0.223
Slave circuit A
if |i02 | ≤ d ≤ 1,
0 0 = v2 − f (i2 ) .
if |i02 | ≥ d ≥ 1,
dv 0 C2 2 = −i01 − i02 , dt L1 L2
di01 dt di02 dt
= v20 − v1 ,
65.7.
65.7
0.506 ± 3.74i.
−65.7,
−0.506
−0.506 ± 3.74i.
Slave circuit B 0 dv2 0 0 C2 = −i1 − i2 , dt 0 di L1 1 = v20 − v10 , dt 0 di2 0 0 L2 = v2 − f (i2 ) . dt C1
dv10 = Gv10 + i01 , dt
if |i02 | ≤ d ≤ 1, 65.3, 1.04,
65.3
1.10 ± 2.43i. if |i02 | ≥ d ≥ 1, −67.2, −2.49,
2.45
2.45 ± 1.80i.
i
L2
v=-f(i) + Chua’s v diode
C2
Fig. 6.
L1
i1
+
+
v1
v
2
A hyperchaos generator.
C1
-G
Reconstruction and Synchronization of Hyperchaotic Circuits via One State Variable 2077
correct. We guess that the selective synchronization is robust for external noise, since we shut off the driving signal with the strong desynchronization effect. Next, we consider the hyperchaos generator from [Saito, 1990], whose dynamics are given in Table 4. The circuit contains two capacitors Cj , two inductors Lj , a negative conductance −G, and an eventually strictly monotonically increasing Chua’s diode v = g(i) (see Fig. 6). The impedance Z(s) of the linear part of the circuit is given by Z(s) =
G1 (s) G2 (s)
the selective synchronization has an advantage of stable synchronization.
4. Chaotic Circuits In this section, we apply the synchronization schemes to low-dimensional chaotic circuits. We first consider the canonical Chua’s oscillator (see Fig. 7, Tables 6 and 7). The admittance Y (s) has the form Y (s) =
1
= sL2 +
1
sC2 + sL1 +
.
1
= sC1 +
(13)
sL1 + r +
1 sC1 − G
The denominator G2 (s) = sC2 (sL1 (sC1 − G) + 1) + sC1 − G has a root with positive real part. This implies that the slave circuits will not be synchronized by the driving signal i(t). Since the circuit consists of a ladder network and a Chua’s diode, all voltages and currents can be reconstructed from the linear functions of the derivative of v1 : dv1 i1 = C1 − Gv1 , dt d2 v1 dv1 v2 = L1 C1 2 − L1 G + v1 , dt dt d3 v1 d2 v1 i2 = L1 C1 C2 3 + L1 C2 G 2 dt dt dv1 + (C1 − C2 ) − Gv1 . dt
G1 (s) G2 (s) 1 sC2 − G
.
Since Y (s) is expanded into a continued fraction for ladder-networks, all voltages and currents can be reconstructed from the linear functions of the derivative of v2 (t): i1 = −C2
dv2 + Gv2 , dt
d2 v2 dv2 + (rC2 − LG) 2 dt dt + (1 − rG)v2 .
(16)
v1 = C2 L1
(14)
This chaotic attractor has two positive Lyapunov exponents. In order to synchronize the hyperchaotic circuit, the voltage across the negative conductance, that is, v2 is used as a driving signal. Then, the variational system becomes asymptotically stable in the regions where the v–i characteristic of Chua’s diode is strictly monotonically increasing. We found that the maximum real part of the eigenvalues for this region is slightly smaller than that of the continuous synchronization. Thus, the selective synchronization scheme is expected to have a smaller CLE. As can be seen from Figs. 4 and 5 and Table 3, the selective synchronization scheme gives the smallest Lyapunov exponent, and so this scheme is the most stable one. Furthermore, the circuits will be synchronized quickly by this scheme. Therefore, we conclude that
(15)
Furthermore, the denominator G2 (s) = C2 L1 s2 + (C2 r − L1 G)s + (1 − Gr) has a root with positive real part. This implies that the slave circuits will not be synchronized by the driving signal v1 (t) (the voltage across the Chua’s diode). Thus, the voltage across the negative conductance, that is, v2 , is used as a driving signal. As can be seen from Figs. 8 and 9, we found that the impulsive synchronization i1
L1
r
i + Chua’s v diode
C1
+
+
v1
v
2
i=-f(v) Fig. 7.
Canonical Chua’s oscillator.
R C2
2078 M. Itoh & L. O. Chua
Table 5.
Chaotic dynamical systems.
Dynamical Systems
Parameters
Driving Signal
Canonical Chua’s oscillator
C1
dv1 = i − f (v1 ) , dt
dv2 C2 = Gv2 − i1 , dt di1 L1 = −v1 + v2 − ri1 , dt f (v) = bv + 0.5(a − b)(|v + 1| − |v − 1|) .
C1 =
C2 = L1 =
G= r= a= b=
25 , 11 1.0, 1, 0.1, −1, 5. 1 , 8
v2
Coupled canonical Chua’s oscillator 25 C2 = , 11 1 C3 = , 5 20 L1 = , 23 20 L2 = , 17 G = 1, 2 r1 = , 23 8 r1 = , 85 a1 = −1, b1 = 5, a2 = −2.5, b2 = 10.
C1 =
C1
dv1 = i1 − f (v1 ) , dt
C2
dv2 = Gv2 − i1 − i2 , dt
L1
di2 = −v1 + v2 − r1 i1 , dt
di2 L2 = −v3 + v2 − r2 i2 , dt f (v) = b1 v + 0.5(a1 − b1 )(|v + 1| − |v − 1|) , g(v) = b2 v + 0.5(a2 − b2 )(|v + 1| − |v − 1|) .
C3
dv3 = i2 − g(v3 ) , dt
1 , 8
v2
Chua’s oscillator C1 = C1
dv1 = G(v2 − v1 ) − f (v1 ) , dt
dv2 C2 = G(v1 − v2 ) + i , dt di L = −v2 − ri , dt f (v) = bv + 0.5(a − b)(|v + 1| − |v − 1|) .
1 , 10
C2 = 1, L=
1 , 15
1 r= , 150 a = −1.27, b = −0.68.
G = 1,
v1
Reconstruction and Synchronization of Hyperchaotic Circuits via One State Variable 2079 Table 6.
Variational systems for the canonical Chua’s oscillator. Eigenvalues of the Variational Systems
Canonical Chua’s Circuit
Maximum LE or CLE
Master circuit C1
dv1 = i − f (v1 ) , dt
C2
dv2 = Gv2 − i1 , dt
di1 L1 = −v1 + v2 − ri1 , dt f (v) = bv + 0.5(a − b)(|v + 1| − |v − 1|) .
0.105
Slave circuit A
C1
dv10 dt
= i − f (v10 ) ,
if |v10 | ≤ d ≤ 1,
6.85, 1.05.
6.85
if |v10 | ≥ d ≥ 1,
di0 L1 1 = −v10 + v2 − ri01 . dt
−0.301
−39.8, −0.301.
Slave circuit B
if |v10 | ≤ d ≤ 1,
0 0 0 = −v1 + v2 − ri1 .
if |v10 | ≥ d ≥ 1,
dv 0 C1 1 = i0 − f (v10 ) , dt dv 0 C2 2 = Gv20 − i01 , dt L1
di01 dt
6.84, 1.34, −0.725.
−39.8,
6.84
−0.0693
−0.0693 ± 0.552i.
0.3 ’Canonical Chua’s oscillator’ ’Coupled canonical Chua’s oscillator’ 0.2
’Chua’s oscillator (v2-drive)’
Maximum CLE
0.1
0
-0.1
-0.2
-0.3
-0.4 0
0.5
1
1.5
2
d Fig. 8. Maximum CLE of chaotic circuits for selective synchronization. The symbol d indicates the threshold. A selective synchronization for d = 0 is equivalent to a continuous synchronization.
2080 M. Itoh & L. O. Chua 0.3
’Canonical Chua’s oscillator’ ’Coupled canonical Chua’s oscillator’
0.2
’Chua’s oscillator (v1-drive)’ ’Chua’s oscillator (v2-drive)’
Maximum CLE
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6 0
0.2
0.4
0.6
0.8
1
P/Q Fig. 9. Maximum CLE of the chaotic circuits for impulsive synchronization. The symbols P and Q indicate the impulse width and the frame length, respectively. An impulsive synchronization for P/Q = 1 is equivalent to a continuous synchronization.
is the most stable scheme. Furthermore, there is no great difference between continuous synchronization (d = 0 in Fig. 8) and selective synchronization (d 6= 0). Thus, there is no advantage for choosing selective synchronization. Note that the CLE is decreasing in the neighborhood of d = 1 in view of the following reason: in selective synchronization, the maximum real part of eigenvalues for the region |v1 | ≤ d ≤ 1 is equal to 6.85, which is close to that of the continuous synchronization (= 6.84). However, if the threshold d is increased from 1, then the maximum CLE for the region |v1 | ≤ d is decreased from 6.85 to 0.105 (= the maximum Lyapunov exponent of the canonical Chua’s oscillator). Thus, the CLE of the selective synchronization is expected to become slightly smaller than the continuous synchronization in the neighborhood of d = 1.
Y11 = sC1 +
Y12 = Y21
We also examined the performance of the synchronization scheme for the coupled canonical Chua’s oscillator (see Fig. 10, Tables 5 and 7). Since the circuit has two Chua’s diodes, the circuit equation can be described as a four-terminal network: "
and
#
"
Ia Y11 Y12 = Ib Y21 Y22
ia = −f (va ) , ib = −g(vb ) ,
#
Va , Vb
(17)
(18)
where Ia , Ib , Va and Vb are the Laplace transforms of the ia , ib , va and vb , respectively. The symbol Y = [yij ] is called as the admittance matrix, which is given by
1 , 1 1 sL1 + r1 + + sC2 − G sL2 + r2
−1 , (sL1 + r1 )(sL2 + r2 )(sC2 − G) + s(L1 + L2 ) + r1 + r2
−1 = , (sL1 + r1 )(sL2 + r2 )(sC2 − G) + s(L1 + L2 ) + r1 + r2
Y22 = sC2 +
#"
1 . 1 1 sL2 + r2 + + sC2 − G sL1 + r1
(19)
Reconstruction and Synchronization of Hyperchaotic Circuits via One State Variable 2081 i1
i
L1
C1
L2
r2
i
+
+
+ Chua’s v diode a
i2
r1
v
v1
2
C2
C3
v3
i b = -g ( v b )
i a = -f ( v a )
Fig. 10.
Table 7.
Coupled canonical Chua’s oscillator.
Variational systems for the coupled canonical Chua’s oscillator.
Coupled Canonical Chua’s Oscillator C1
+ v b Chua’s diode
+ R
Eigenvalues of the Variational Systems
dv1 = i1 − f (v1 ) , dt
dv2 = Gv2 − i1 − i2 , dt di2 L1 = −v1 + v2 − r1 i1 , dt
C2
dv3 = i2 − g(v3 ) , dt
0.0843
di2 L2 = −v3 + v2 − r2 i2 , dt f (v) = b1 v + 0.5(a1 − b1 )(|v + 1| − |v − 1|) , g(v) = b2 v + 0.5(a2 − b2 )(|v + 1| − |v − 1|) .
C3
Maximum LE or CLE
Slave circuit A 0 di2 0 0 L1 = −v1 + v2 − r1 i1 , dt dv 0 C3 3 = i02 − g(v30 ) , dt 0 di2 0 0 L2 = −v3 + v2 − r2 i2 . dt dv 0 C1 1 = i01 − f (v10 ) , dt
if |v10 | ≤ d ≤ 1 and |v30 | ≤ d ≤ 1 , 6.64, 1.25,
12.2
12.2, 0.28. if |v10 | ≥ d ≥ 1 and |v30 | ≥ d ≥ 1, −39.8, −0.347,
−0.153
−49.9, −0.153.
Slave circuit B C1
dv10 = i01 − f (v10 ) , dt
C2
dv20 = Gv20 − i01 − i02 , dt
di02 = −v10 + v20 − r1 i01 , dt 0 dv3 0 0 C3 = i2 − g(v3 ), dt 0 di2 0 0 0 L2 = −v3 + v2 − r2 i2 , . dt L1
if |v10 | ≤ d ≤ 1 and |v30 | ≤ d ≤ 1, 12.2, 6.66, 0.712,
12.2
0.0617 ± 0.863i. if |v10 | ≥ d ≥ 1 and |v30 | ≥ d ≥ 1, −49.9, −39.8, −0.243, 0.0912 ± 0.868i.
0.0912
2082 M. Itoh & L. O. Chua
R
i
+ Chua’s diode v
C1
+
+
v1
v2
L
C2
r i
i=-f(v) Fig. 11.
Table 8.
Chua’s oscillator.
Variational systems for Chua’s oscillator.
Chua’s Oscillator (v1 -drive)
Eigenvalues of the Variational Systems
Maximum LE or CLE
Master circuit C1
dv1 = G(v2 − v1 ) − f (v1 ) , dt
C2
dv2 = G(v1 − v2 ) + i , dt
di L = −v2 − ri , dt f (v) = bv + 0.5(a − b)(|v + 1| − |v − 1|) .
0.294
Slave circuit A dv2 = G(v1 − v20 ) + i0 , dt di L = −v20 − ri0 . dt
C2
−0.55 ± 3.85i
−0.55
Slave circuit B dv10 = G(v20 − v10 ) − f (v10 ) , dt 0 dv2 C2 = G(v10 − v20 ) + i0 , dt 0 di 0 0 L = −v2 − ri . dt
C1
The elements Yij are easily obtained by using the classical circuit theory or the Laplace transform of the circuit equation in Table 5 (under zero initial conditions). In this case, the currents and volt-
if |v10 | ≤ d ≤ 1, 3.86, −1.13 ± 3.09.
3.86
if |v10 | ≥ d ≥ 1,
0.17
−4.64, 0.17 ± 3.19i.
ages cannot be described as linear functions of the derivatives of vj (t) or ij (t), since the linear part of the circuit is described as a four-terminal network. The performance for each scheme is similar to that
Reconstruction and Synchronization of Hyperchaotic Circuits via One State Variable 2083 Table 9.
Variational systems for Chua’s oscillator.
Chua’s Oscillator (v2 -drive)
Eigenvalues of the Variational Systems
Master circuit C1
dv1 = G(v2 − v1 ) − f (v1 ) , dt
C2
dv2 = G(v1 − v2 ) + i , dt
Maximum LE or CLE
di L = −v2 − ri , dt f (v) = bv + 0.5(a − b)(|v + 1| − |v − 1|) .
0.294
Slave circuit A if |v10 | ≤ d ≤ 1,
dv 0 C1 1 = G(v2 − v10 ) − f (v10 ) , dt di0 L = −v2 − ri0 , dt
−0.1, 2.7. if |v10 | ≥ d ≥ 1, −0.1, −3.2.
2.7 −0.1
Slave circuit B if |v10 | ≤ d ≤ 1,
dv 0 C1 1 = G(v20 − v10 ) − f (v10 ) , dt 0 dv2 0 0 0 C2 = G(v1 − v2 ) + i , dt 0 di 0 0 L = −v2 − ri . dt
3.86, if |v10 | ≥ d ≥ 1, −4.64,
G1 (s) G2 (s)
1
= sC1 +
1
R+ sC2 +
.
(20)
1 sL + r
Since Y (s) is expanded into the ladder form, all voltages and currents can be reconstructed from the linear functions of the derivative of i(t): dv2 − ri , dt d2 i di v1 = −C2 LR 2 − (C2 rR + L) dt dt −(R + r)i.
0.17
0.17 ± 3.19i.
of canonical Chua’s oscillator. In this circuit, the selective synchronization is quicker than the other schemes (see Table 10). We also add the examples of Chua’s oscillator for reference (see Fig. 11, Tables 5 and 8). The admittance Y (s) has the form Y (s) =
3.86
−1.13 ± 3.09i,
v2 = −L
(21)
Furthermore, all roots of G2 (s) = C2 LRs2 + (C2 rR + L)s + (R + r) have negative real parts. This implies that the slave circuits will be synchronized by the driving signal v1 (t) (the voltage across the Chua’s diode). In this oscillator, the continuous synchronization is more stable than the impulsive synchronization (see Fig. 9). This is because the variational system is completely stable, and so the driving signal provides a strong synchronization effect to the slave system continuously. This situation is different from the canonical Chua’s oscillator and the coupled canonical Chua’s oscillator. We also studied the case where the voltage v2 (t) is used as the driving signal (Table 9). In this case, the selective synchronization is more effective than the continuous synchronization (see Fig. 8). This is due to the following reason: The variational system without driving signal is not stable, but the two systems converge quickly to the space spanned by the eigenvectors of the two eigenvalues: −1.13 ± 3.09i. Finally, we note that the impulsive synchronization
2084 M. Itoh & L. O. Chua Table 10.
Convergence time of chaotic dynamical systems.
Continuous Synchronization
Synchronization Schemes Canonical Chua’s oscillator
Selective Synchronization
94.6
97.4
92.3
Coupled circuit
312
184
312
Chua’s oscillator (v1 -drive) Chua’s oscillator (v2 -drive)
68.0 271
— 202
70.5 181
R1
ia
+ Chua’s diode va
C1
+
+
v1
v2
R2 L1 C2
R3 L2
r
r
i1
i2
C4
i a = -f ( v a )
5. Miscellaneous Hyperchaotic Systems It is well known that the coupled Chua’s circuits in Fig. 12 have two positive Lyapunov exponents [Brucoli et al., 1999]. The circuit consists of two Chua’s diodes and linear elements. Thus, the dynamic can be described as Ib
#
"
=
+
+
+
v4
v3
v b Chua’s diode
C3
Coupled Chua’s oscillators.
is not so stable (as we can see from Fig. 9), but the convergence is quick. From computer studies, we conclude that the performance of the synchronization schemes depends on the chaotic circuits.
Ia
ib
i b = -f ( v b ) Fig. 12.
"
Impulsive Synchronization (P = 0.05, Q = 0.03)
Y11
Y12
Y21
Y22
#"
Va Vb
#
,
(22)
and ia = −f (va ) , ib = −f (vb ) ,
forms of ia , ib , va and vb , respectively. Since the circuit is a four-terminal network and Yij 6= 0, the currents and voltages of this circuit are not reconstructed from a linear combination of the derivatives of one state variable. Furthermore, even if we choose any variable as a driving signal, at least one eigenvalue of the difference system has a positive real part. Thus, the circuits cannot be synchronized by transmitting only one state variable. However, the circuits can be synchronized if we transmit the two driving signals via timedivision-based impulsive synchronization [Brucoli et al., 1999]. Next, we study the R¨ossler hyperchaos oscillator [R¨ossler, 1979]
(23)
where the symbol Y = [Yij ] is the admittance matrix, and Ia , Ib , Va and Vb are the Laplace trans-
dx dy = −y − z, = x + 0.25y + w , dt dt dz dw = 3 + xz, = −0.5z + 0.05w . dt dt
(24)
In this oscillator, the state variables are reconstructed from w as
Reconstruction and Synchronization of Hyperchaotic Circuits via One State Variable 2085
z = −2
dw d2 w − 0.1 +3 2 dt x = dt , dw 2 − 0.1w dt 2
dw + 0.1w , dt
d2 w d2 w dw d3 w d3 w dw + 0.2 2 − 0.2 3 w + 0.01 2 w 4 3 dw dt dt dt dt y=2 . − 0.1w − dt dt 2 dt dw 2 − 0.1w dt
−
d2 w −4 dt2
!2
dw − 0.01 dt
However, two state variables are needed to synchronize the R¨ossler hyperchaotic oscillators.5
6. Conclusions We have shown that some hyperchaotic circuits can be synchronized by using only one state variable. In particular, we presented two counter-examples to Pyragas’s conjecture. We also examined the performance of the continuous, impulsive and selective synchronization schemes.
Acknowledgments This research was partially supported by Fukuoka Institute of Technology, and by ONR grant N0001401-1-0741.
References Brucoli, M., Cafagna, D. & Camimeo, L. [1999] “Synchronization method of hyperchaotic circuits based on a scalar transmitted signal,” Proc. European Conf. Circuit Theory and Design, Stresa, pp. 365–368. Itoh, M., Wu, C. W. & Chua, L. O. [1997] “Communication systems via chaotic signals from a reconstruction viewpoint,” Int. J. Bifurcation and Chaos 7(2), 275–286. Itoh, M., Yang, T. & Chua, L. O. [1999] “Experimental study of impulsive synchronization of chaotic and hyperchaotic circuits,” Int. J. Bifurcation and Chaos 9(7), 1393–1424. Itoh, M., Tauchi, T., Yang, T. & Chua, L. O. [2000] “Synchronization of chaotic and hyperchaotic
5
2
−6
dw 2 − 0.1w dt
2
d2 w dw + 0.3 2 dt dt
(25)
.
systems,” Proc. 2000 IEEE Int. Conf. Industrial Electronics, Control, and Instrumentation, Nagoya, pp. 2407–2412. Itoh, M., Yang, T. & Chua, L. O. [2001] “Conditions for impulsive synchronization of chaotic and hyperchaotic systems,” Int. J. Bifurcation and Chaos 11(2), 551–560. Kennedy, M. P. [1992] “Robust OP-AMP implementation of Chua’s circuit,”Frequenz 46(3–4), 66–80. Matsumoto, T., Chua, L. O. & Kobayashi, K. [1986] “Hyperchaos: Laboratory experiment and numerical confirmation,” IEEE Trans. Circuits Syst. 33(11), 1143–1147. Panas, A. I., Yang, T. & Chua, L. O. [1998] “Experimental results of impulsive synchronization between two Chua’s circuits,” Int. J. Bifurcation and Chaos 8(3), 639–644. Pecora, L. & Carroll, T. [1990] “Synchronization in chaotic systems,” Phys. Rev. Lett. 64(8), 821–824. Peng, J. H., Ding, E. J., Ding, M. & Yang, W. [1996] “Synchronizing hyperchaos with a scalar transmitted signal,” Phys. Rev. Lett. 76(6), 904–907. Pyragas, K. [1993] “Predictable chaos in slightly perturbed unpredictable chaotic systems,” Phys. Lett. A181, 203–210. R¨ ossler, O. E. [1979] “An equation for hyperchaos,” Phys. Lett. A71(2 & 3), 155–157. Saito, T. [1990] “An approach toward higher dimensional hysteresis chaos generator,” IEEE Trans. Circuits Syst. 37(3), 399–409. Takens, F. [1981] Detecting Strange Attractors in Turbulence, Lecture Notes in Mathematics, Vol. 898 (Springer-Verlag, Berlin), pp. 366–381.
It is possible to synchronize two R¨ ossler hyperchaotic oscillators by using one scalar signal [Peng et al., 1996].
RECONSTRUCTION AND SYNCHRONIZATION OF HYPERCHAOTIC CIRCUITS VIA ONE STATE VARIABLE MAKOTO ITOH Department of Information and Communication Engineering, Fukuoka Institute of Technology, Fukuoka 811-0295, Japan LEON O. CHUA Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA 94720, USA Received February 14, 2002 In this paper, we demonstrate that some hyperchaotic circuits can be synchronized by using only one state variable. We applied three kinds of synchronization schemes, a continuous synchronization, an impulsive synchronization, and a selective synchronization to these hyperchaotic circuits. Their performance is examined from the viewpoint of synchronization stability and convergence time. Keywords: Synchronization; hyperchaos; Chua’s diode; Chua’s oscillator; Chua’s circuit; hyperchaotic circuit.
1. Introduction Many methods have been proposed to synchronize chaotic systems. The most widely used methods are the continuous synchronization schemes [Pecora & Carroll, 1990], where the driving signals are transmitted continuously to the driven systems. The other synchronization schemes are impulsive synchronization [Panas et al., 1998] and selective synchronization [Itoh et al., 2000]. In an impulsive synchronization scheme, only samples of state variables (or functions of state variables) called synchronization impulses are used to synchronize two chaotic systems. In a selective synchronization scheme, we select only those time periods of driving signals with strong synchronizing effect to the slave system and shut off the driving signals in some other time periods when they show strong desynchronizing effects. These three schemes were applied to several chaotic systems, and all three exhibited
good performance [Panas et al., 1998; Itoh et al., 1999; Itoh et al., 2000; Itoh et al., 2001]. The stability of synchronization is closely connected to the values of Lyapunov components of variational systems, and so the conditions for stable impulsive synchronization in both chaotic and hyperchaotic systems are given in terms of Lyapunov exponents [Pecora & Carroll, 1990; Itoh et al., 1999; Itoh et al., 2000; Itoh et al., 2001]. According to Pyragas [Pyragas, 1993; Brucoli et al., 1999], the minimal number of controlled variables has to be equal to the number of positive Lyapunov exponents of the system. If the hyperchaotic systems have two positive Lyapunov exponents, then, at least two driving signals are needed to synchronize them by the continuous synchronization scheme. In the case of impulsive synchronization, the number of driving signals sequentially transmitted to the slave systems via time-division is greater than or equal to two [Itoh et al., 1999; Itoh et al., 2000;
2069
2070 M. Itoh & L. O. Chua
Itoh et al., 2001; Brucoli et al., 1999]. Almost all models considered up to now seemed to support this assumption. In this paper, we demonstrate that some hyperchaotic circuits can be synchronized by using only one state variable. That is, counter-examples to Pyragas’s conjecture are presented. We applied three kinds of synchronization schemes to the hyperchaotic circuit from [Matsumoto et al., 1986] and to the hyperchaos generator from [Saito, 1990], and succeeded to synchronize them. These circuits have a negative resistor and an eventually strictly monotonically increasing Chua’s diode [Kennedy, 1992].1 Thus, if we choose the voltage across the negative resistor as a driving signal, their variational systems become stable in the regions where the v–i characteristic of the nonlinear resistor is strictly monotonically increasing. In these regions, the driving signal has a strong synchronizing effect. Thus, we can expect that all conditional Lyapunov exponents (CLEs) of the variational system become negative.2 In our computer studies, two hyperchaotic circuits will be synchronized by using three kinds of synchronization schemes: a continuous synchronization, an impulsive synchronization and a selective synchronization. We also discuss their performance from the viewpoints of synchronization stability and convergence time. Furthermore, we show that if the linear part of the system is represented by a ladder circuit,3 we can easily reconstruct all the state variables from the derivatives of one state variable.
2. Preliminaries Consider the nonlinear circuit with the dynamics
i i=f(v) + Chua’s v diode
Y(s)
Fig. 1. Nonlinear circuit with one nonlinear element (Chua’s diode). The admittance of the linear part is given by Y (s). The v–i characteristic of Chua’s diode is given by i = f (v).
(transfer function) of the operator L−1 1 L2 by Y (s), then Y (s) is equal to admittance of the linear circuit (see Fig. 1, which illustrates a two-terminal network representation). That is, Y (s) satisfies the following relation I(s) = Y (s)V (s) =
L1 i = L2 v ,
(1)
where the symbols i and v indicate a current and a voltage, respectively, f is a scalar function, and L1 , L2 are differential operators. The nonlinear element (Chua’s diode) is characterized by the equation i = f (v). If we denote the Laplace transform 1
(2)
where G1 (s) and G2 (s) are the Laplace transforms of L1 and L2 , and I(s) and V (s) are the Laplace transforms of i and v, respectively (under zero initial conditions). Similarly, the impedance Z(s) is defined by G1 (s)/G2 (s) = V (s)I(s)−1 . Here, we assume that the v–i characteristic of the Chua’s diode is eventually strictly monotonically increasing. Next, we define the master and slave circuits as follows: i = f (v) ,
i = f (v) ,
G2 (s) V (s) , G1 (s)
L1 i = L2 v ,
)
master circuit
L1 i0 = L2 v , } slave circuit
(3) (4)
where the voltage across Chua’s diode, that is, v(t), is a driving signal. Then the variational system is given by L1 p = 0 , (5)
A nonlinear resistor is said to be strictly monotonically increasing if, and only if, for each pair of the points (v1 , i1 ) and (v2 , i2 ) on the v–i curve, we observe whenever v1 > v2 , then i1 > i2 . Furthermore, the voltage-controlled Chua’s diode is said to be eventually strictly monotonically increasing, if the v–i characteristic of the nonlinear resistor is strictly monotonically increasing in the region: |v| > v0 > 0 (v0 : some constant). 2 It is well known that the hyperchaotic circuits will synchronize if all conditional Lyapunov exponents of the slave circuits are negative. 3 The circuit elements are combined alternatively in series and in parallel.
Reconstruction and Synchronization of Hyperchaotic Circuits via One State Variable 2071 Z m-2
Z m-1
Y(s)
Ym
Z2
Ym-1
Ym-2
Z1
1 Y(s) = Ym
+ 1
Z m-1 +
Ym-2 +
1 Z m-3 + . . .
1 Z2 + Z1
Z m-2
Z m-1
Y(s)
Ym
Ym-1
Ym-2
Y2
Y1
1 Y(s) = Ym
+ Z m-1 +
1 Ym-2 +
1 Z m-3 + . . .
1 Y2 + Y1
Fig. 2.
Continued fraction of the admittance Y (s). The admittance Y (s) is realized by a ladder circuit.
where p = i − i0 . Therefore, if all eigenvalues of the characteristic equation have negative real parts, then the two systems will be synchronized. That is, if all roots of the function G1 (s) for L1 have negative real parts, then the two circuits will be synchronized. We note that Chua’s oscillator corresponds to this case. In the next section, we study the case where the function G1 (s) has a root with a positive real part. Next, we consider reconstructing all of the state variables by using the derivatives of one state vari-
able. First, we expand the transfer function Y (s) = G2 (s)/G1 (s) into the following form 1
Y (s) = Ym +
,
1
Zm−1 + Ym−2 +
1 Zm−3 + · · ·
1 Z2 + Z1 (6)
2072 M. Itoh & L. O. Chua
or
variables from the derivatives 1
Y (s) = Ym +
.
1
Zm−1 + Ym−2 +
di1 d2 i1 d3 i1 i1 , , ,··· , dt dt2 dt3
1
or
dv1 d2 v1 d3 v1 v1 , , ,··· , dt dt2 dt3
1 Zm−3 + · · · Y2 + Y1 (7)
Then, the transfer function Y (s) can be realized by the ladder circuit as shown in Fig. 2. Furthermore, we have the relations V1 + V2 V3 I3 V4 I4 V5 I5 V6 .. . Im
= (Z2 + Z1 )I1 = V1 + V2 = Y 3 V3 = Z4 I4 = I1 + I3 = V3 + V4 = Y 5 V5 = Z6 I6 . = .. = Y m Vm
I1 + I2 I3 V3 I4 V4 I5 V5 I6 .. . Im
= (Y2 + Y1 )V1 = I1 + I2 = Z3 I3 = Y 4 V4 = V1 + V3 = I3 + I4 = Z5 I5 = Y 6 V6 . = .. = Y m Vm
!
,
(10)
!
.
(11)
It corresponds to a special case of the Takens reconstruction theorem [Takens, 1981]. In this case, we do not need to assume that all roots of the transfer function G1 (s) have negative real parts. Note that similar results hold for the impedance Z(s), too. In this case, we have to consider the currentcontrolled Chua’s diode in place of the voltagecontrolled Chua’s diode.
3. Synchronization of Hyperchaotic Circuits (8)
or
Consider the hyperchaotic circuit from [Matsumoto et al., 1986], whose dynamics are given in Table 1. This circuit contains two capacitors Cj , two inductors Lj with small physical resistance rj , a negative resistor −R, and an eventually strictly monotonically increasing resistor i = g(v)(v = v2 − v1 ) as shown in Fig. 3. The impedance Z(s) for the linear part of the circuit is given by Z(s) = Z1 + Z2 1
=
1
(9)
. 1 sL2 + r2 (12)
where Ij and Vj are the Laplace transform of the current through the impedance Zj (or the admittance Yj ) and the voltage across the impedance Zj (or the admittance Yj ), respectively. Thus, all of the voltages Vj and currents Ij are determined if the current I1 is given in the case of Eq. (8) or if the voltage V1 is given in the case of Eq. (9). Here, we assume that Yj and Zj are polynomial functions of s. Then, all of the currents ij and the voltages vj are described as functions of the derivatives of i1 or v1 , since the symbol s corresponds to the derivative d/dt. That is, we can reconstruct all of the state
The denominator of Y (s) = Z(s)−1 has a root with a positive real part. This implies that the slave circuits will not be synchronized by the driving signal v(t) = v2 −v1 (the voltage across the Chua’s diode). It is well known that the hyperchaotic circuit in [Matsumoto et al., 1986] has two positive Lyapunov exponents (LEs). According to Pyragas, the minimum number of driving signals for synchronizing this chaotic system is equal to two. Here, in order to synchronize the hyperchaotic circuits using one driving signal, we choose the voltage across the negative resistor, that is, the voltage Ri1 , or simply i1 as a driving signal. In the region |v| ≥ d ≥ 1, all eigenvalues of the variational system have negative real parts as shown in Table 2, and so the variational system is completely stable. The symbol d indicates a threshold. In the selective synchronization, d is used as a controlling parameter. In the
sC1 +
1 sL1 + r1 − R
+ sC2 +
Reconstruction and Synchronization of Hyperchaotic Circuits via One State Variable 2073 Table 1.
Hyperchaotic dynamical systems.
Dynamical Systems
Parameters
Hyperchaotic circuit
1 C2 = , 20 L1 = 1.0, 2 L2 = , 3
C1 =
C1
dv1 = g(v2 − v1 ) − i1 , dt
C2
dv2 = −g(v2 − v1 ) − i2 , dt
L1
di1 = v1 − r1 i1 + Ri1 , dt
L2
di2 = v2 − r2 i2 , dt
g(v) = b(v) + 0.5(a − b)(|v + 1| − |v − 1|) .
Driving Signal
1 , 2
i1
1 , 20 1 , 30 −0.2, 3.
R = 1, r1 = r2 = a= b=
Hyperchaos generator C1
dv1 = Gv1 + i1 , dt
C2
dv2 = −i1 − i2 , dt
L1
di1 = v2 − v1 , dt
L2
di2 = v2 − f (i2 ) , dt
f (i) = bi + 0.5(a − b)(|i + 1| − |i − 1|) .
i 1 L1 i
+ Chua’s diode v i=
-
-R
r1 v1
C1 = 1, C2 = 1, L1 = 1,
200 , 3 G = 1.89, a = −1, b = 1.
v2
L2 =
r2
i 2 L2 + v2
+
C2
C1
g(v) Fig. 3.
A hyperchaotic circuit.
region |v| ≤ d ≤ 1, all eigenvalues of the variational system have positive real parts. Thus, the variational system is completely unstable. Since there are two regions with a strong synchronization
effect, we can expect that all conditional Lyapunov exponents (CLEs) of their variational system to become negative. Furthermore, in the case of selective synchronization, we shut off the driving signal in the
2074 M. Itoh & L. O. Chua Table 2.
Hyperchaotic circuit.
Systems
Eigenvalues of the
Maximum LE
Variational Systems
or CLE
Master circuit C1
dv1 = g(v2 − v1 ) − i1 , dt
C2
dv2 = −g(v2 − v1 ) − i2 , dt
L1
di1 = v1 − r1 i1 + Ri1 , dt
L2
di2 = v2 − r2 i2 , dt
g(v) = b(v) + 0.5(a − b)(|v + 1| − |v − 1|) .
0.195
Slave circuit A dv 0 C1 1 = g(v20 − v10 ) − i1 , dt
dv20 = −g(v20 − v10 ) − i02 , dt 0 di2 0 0 L2 = v2 − r2 i2 . dt
C2
if |v20 − v10 | ≤ d ≤ 1, 1.96 ± 4.93i,
1.96
0.430. if |v20 − v10 | ≥ d ≥ 1, −0.233 ± 1.64i,
−0.233
−65.6.
Slave circuit B 0 dv2 0 0 0 C2 = −g(v2 − v1 ) − i2 , dt di0 L1 1 = v10 − r1 i01 + Ri01 , dt 0 di2 0 0 L2 = v2 − r2 i2 . dt C1
dv10 = g(v20 − v10 ) − i01 , dt
region |v| ≤ d ≤ 1. Then, two hyperchaotic circuits are run independently without a driving signal. In this case, the maximum real part of the eigenvalues of the variational system in this region, that is, = 1.71, is slightly smaller than that of the continuous synchronization (= 1.96). Thus, the selective synchronization is expected to have a smaller CLE. As we can see from Figs. 4 and 5, all of the CLEs become negative. This implies that the hyperchaotic circuits will be synchronized by using only one control variable. We note that the selective synchronization scheme gives the smallest CLE. 4
if |v20 − v10 | ≤ d ≤ 1, 1.71 ± 4.58i, 0.935 ± 0.851i.
1.71
if |v20 − v10 | ≥ d ≥ 1, −0.0951 ± 2.35i, −65.7,
0.751,
0.751.
However, the convergence time4 is not as quick as shown in Table 3. Note that the convergence time depends on the initial conditions, but the data in Table 3 serves as a reference of the performance. In our laboratory experiment, the synchronization of the hyperchaotic circuits is easily broken by small external noise, if the variational system has completely unstable regions. Thus, we synchronized the hyperchaotic circuits by transmitting the two driving signals via time-division-based impulsive synchronization [Itoh et al., 1997]. In terms of this interpretation, Pyragas’s conjecture may be
We set the convergence condition such that the difference e(t) between the same components is less than 10−12 .
Reconstruction and Synchronization of Hyperchaotic Circuits via One State Variable 2075 0.4
Maximum CLE
0.2
0
-0.2
-0.4
’Hyperchaotic circuit’
-0.6
’Hyperchaos generator’ -0.8
-1 0
1
2
3
4
5
d Fig. 4. Maximum CLE of the hyperchaotic circuit and the hyperchaos generator for selective synchronization. The symbol d indicates the threshold, that is, if |v| > d, then the driving signal is used for synchronizing the circuit. A selective synchronization for d = 0 is equivalent to a continuous synchronization.
0.3
’Hyperchaotic circuit’
0.2
’Hyperchaos generator’
Maximum CLE
0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 0
0.2
0.4
0.6
0.8
1
P/Q Fig. 5. Maximum CLE of the hyperchaotic circuit and the hyperchaos generator for impulsive synchronization. The symbols P and Q indicate the impulse width and the frame length, respectively. An impulsive synchronization for P/Q = 1 is equivalent to a continuous synchronization.
Table 3.
Convergence time of hyperchaotic dynamical systems.
Convergence Time Hyperchaotic circuit Hyperchaos generator
Continuous Synchronization 119 83.8
Selective Synchronization 149 66.5
Impulsive Synchronization (P = 0.05, Q = 0.03) 119 81.1
2076 M. Itoh & L. O. Chua
Table 4.
Hyperchaos generator. Eigenvalues of the Variational Systems
Systems
Maximum LE or CLE
Master circuit C1
dv1 = Gv1 + i1 , dt
C2
dv2 = −i1 − i2 , dt
L1
di1 = v2 − v1 , dt
L2
di2 = v2 − f (i2 ) , dt
f (i) = bi + 0.5(a − b)(|i + 1| − |i − 1|) .
0.223
Slave circuit A
if |i02 | ≤ d ≤ 1,
0 0 = v2 − f (i2 ) .
if |i02 | ≥ d ≥ 1,
dv 0 C2 2 = −i01 − i02 , dt L1 L2
di01 dt di02 dt
= v20 − v1 ,
65.7.
65.7
0.506 ± 3.74i.
−65.7,
−0.506
−0.506 ± 3.74i.
Slave circuit B 0 dv2 0 0 C2 = −i1 − i2 , dt 0 di L1 1 = v20 − v10 , dt 0 di2 0 0 L2 = v2 − f (i2 ) . dt C1
dv10 = Gv10 + i01 , dt
if |i02 | ≤ d ≤ 1, 65.3, 1.04,
65.3
1.10 ± 2.43i. if |i02 | ≥ d ≥ 1, −67.2, −2.49,
2.45
2.45 ± 1.80i.
i
L2
v=-f(i) + Chua’s v diode
C2
Fig. 6.
L1
i1
+
+
v1
v
2
A hyperchaos generator.
C1
-G
Reconstruction and Synchronization of Hyperchaotic Circuits via One State Variable 2077
correct. We guess that the selective synchronization is robust for external noise, since we shut off the driving signal with the strong desynchronization effect. Next, we consider the hyperchaos generator from [Saito, 1990], whose dynamics are given in Table 4. The circuit contains two capacitors Cj , two inductors Lj , a negative conductance −G, and an eventually strictly monotonically increasing Chua’s diode v = g(i) (see Fig. 6). The impedance Z(s) of the linear part of the circuit is given by Z(s) =
G1 (s) G2 (s)
the selective synchronization has an advantage of stable synchronization.
4. Chaotic Circuits In this section, we apply the synchronization schemes to low-dimensional chaotic circuits. We first consider the canonical Chua’s oscillator (see Fig. 7, Tables 6 and 7). The admittance Y (s) has the form Y (s) =
1
= sL2 +
1
sC2 + sL1 +
.
1
= sC1 +
(13)
sL1 + r +
1 sC1 − G
The denominator G2 (s) = sC2 (sL1 (sC1 − G) + 1) + sC1 − G has a root with positive real part. This implies that the slave circuits will not be synchronized by the driving signal i(t). Since the circuit consists of a ladder network and a Chua’s diode, all voltages and currents can be reconstructed from the linear functions of the derivative of v1 : dv1 i1 = C1 − Gv1 , dt d2 v1 dv1 v2 = L1 C1 2 − L1 G + v1 , dt dt d3 v1 d2 v1 i2 = L1 C1 C2 3 + L1 C2 G 2 dt dt dv1 + (C1 − C2 ) − Gv1 . dt
G1 (s) G2 (s) 1 sC2 − G
.
Since Y (s) is expanded into a continued fraction for ladder-networks, all voltages and currents can be reconstructed from the linear functions of the derivative of v2 (t): i1 = −C2
dv2 + Gv2 , dt
d2 v2 dv2 + (rC2 − LG) 2 dt dt + (1 − rG)v2 .
(16)
v1 = C2 L1
(14)
This chaotic attractor has two positive Lyapunov exponents. In order to synchronize the hyperchaotic circuit, the voltage across the negative conductance, that is, v2 is used as a driving signal. Then, the variational system becomes asymptotically stable in the regions where the v–i characteristic of Chua’s diode is strictly monotonically increasing. We found that the maximum real part of the eigenvalues for this region is slightly smaller than that of the continuous synchronization. Thus, the selective synchronization scheme is expected to have a smaller CLE. As can be seen from Figs. 4 and 5 and Table 3, the selective synchronization scheme gives the smallest Lyapunov exponent, and so this scheme is the most stable one. Furthermore, the circuits will be synchronized quickly by this scheme. Therefore, we conclude that
(15)
Furthermore, the denominator G2 (s) = C2 L1 s2 + (C2 r − L1 G)s + (1 − Gr) has a root with positive real part. This implies that the slave circuits will not be synchronized by the driving signal v1 (t) (the voltage across the Chua’s diode). Thus, the voltage across the negative conductance, that is, v2 , is used as a driving signal. As can be seen from Figs. 8 and 9, we found that the impulsive synchronization i1
L1
r
i + Chua’s v diode
C1
+
+
v1
v
2
i=-f(v) Fig. 7.
Canonical Chua’s oscillator.
R C2
2078 M. Itoh & L. O. Chua
Table 5.
Chaotic dynamical systems.
Dynamical Systems
Parameters
Driving Signal
Canonical Chua’s oscillator
C1
dv1 = i − f (v1 ) , dt
dv2 C2 = Gv2 − i1 , dt di1 L1 = −v1 + v2 − ri1 , dt f (v) = bv + 0.5(a − b)(|v + 1| − |v − 1|) .
C1 =
C2 = L1 =
G= r= a= b=
25 , 11 1.0, 1, 0.1, −1, 5. 1 , 8
v2
Coupled canonical Chua’s oscillator 25 C2 = , 11 1 C3 = , 5 20 L1 = , 23 20 L2 = , 17 G = 1, 2 r1 = , 23 8 r1 = , 85 a1 = −1, b1 = 5, a2 = −2.5, b2 = 10.
C1 =
C1
dv1 = i1 − f (v1 ) , dt
C2
dv2 = Gv2 − i1 − i2 , dt
L1
di2 = −v1 + v2 − r1 i1 , dt
di2 L2 = −v3 + v2 − r2 i2 , dt f (v) = b1 v + 0.5(a1 − b1 )(|v + 1| − |v − 1|) , g(v) = b2 v + 0.5(a2 − b2 )(|v + 1| − |v − 1|) .
C3
dv3 = i2 − g(v3 ) , dt
1 , 8
v2
Chua’s oscillator C1 = C1
dv1 = G(v2 − v1 ) − f (v1 ) , dt
dv2 C2 = G(v1 − v2 ) + i , dt di L = −v2 − ri , dt f (v) = bv + 0.5(a − b)(|v + 1| − |v − 1|) .
1 , 10
C2 = 1, L=
1 , 15
1 r= , 150 a = −1.27, b = −0.68.
G = 1,
v1
Reconstruction and Synchronization of Hyperchaotic Circuits via One State Variable 2079 Table 6.
Variational systems for the canonical Chua’s oscillator. Eigenvalues of the Variational Systems
Canonical Chua’s Circuit
Maximum LE or CLE
Master circuit C1
dv1 = i − f (v1 ) , dt
C2
dv2 = Gv2 − i1 , dt
di1 L1 = −v1 + v2 − ri1 , dt f (v) = bv + 0.5(a − b)(|v + 1| − |v − 1|) .
0.105
Slave circuit A
C1
dv10 dt
= i − f (v10 ) ,
if |v10 | ≤ d ≤ 1,
6.85, 1.05.
6.85
if |v10 | ≥ d ≥ 1,
di0 L1 1 = −v10 + v2 − ri01 . dt
−0.301
−39.8, −0.301.
Slave circuit B
if |v10 | ≤ d ≤ 1,
0 0 0 = −v1 + v2 − ri1 .
if |v10 | ≥ d ≥ 1,
dv 0 C1 1 = i0 − f (v10 ) , dt dv 0 C2 2 = Gv20 − i01 , dt L1
di01 dt
6.84, 1.34, −0.725.
−39.8,
6.84
−0.0693
−0.0693 ± 0.552i.
0.3 ’Canonical Chua’s oscillator’ ’Coupled canonical Chua’s oscillator’ 0.2
’Chua’s oscillator (v2-drive)’
Maximum CLE
0.1
0
-0.1
-0.2
-0.3
-0.4 0
0.5
1
1.5
2
d Fig. 8. Maximum CLE of chaotic circuits for selective synchronization. The symbol d indicates the threshold. A selective synchronization for d = 0 is equivalent to a continuous synchronization.
2080 M. Itoh & L. O. Chua 0.3
’Canonical Chua’s oscillator’ ’Coupled canonical Chua’s oscillator’
0.2
’Chua’s oscillator (v1-drive)’ ’Chua’s oscillator (v2-drive)’
Maximum CLE
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6 0
0.2
0.4
0.6
0.8
1
P/Q Fig. 9. Maximum CLE of the chaotic circuits for impulsive synchronization. The symbols P and Q indicate the impulse width and the frame length, respectively. An impulsive synchronization for P/Q = 1 is equivalent to a continuous synchronization.
is the most stable scheme. Furthermore, there is no great difference between continuous synchronization (d = 0 in Fig. 8) and selective synchronization (d 6= 0). Thus, there is no advantage for choosing selective synchronization. Note that the CLE is decreasing in the neighborhood of d = 1 in view of the following reason: in selective synchronization, the maximum real part of eigenvalues for the region |v1 | ≤ d ≤ 1 is equal to 6.85, which is close to that of the continuous synchronization (= 6.84). However, if the threshold d is increased from 1, then the maximum CLE for the region |v1 | ≤ d is decreased from 6.85 to 0.105 (= the maximum Lyapunov exponent of the canonical Chua’s oscillator). Thus, the CLE of the selective synchronization is expected to become slightly smaller than the continuous synchronization in the neighborhood of d = 1.
Y11 = sC1 +
Y12 = Y21
We also examined the performance of the synchronization scheme for the coupled canonical Chua’s oscillator (see Fig. 10, Tables 5 and 7). Since the circuit has two Chua’s diodes, the circuit equation can be described as a four-terminal network: "
and
#
"
Ia Y11 Y12 = Ib Y21 Y22
ia = −f (va ) , ib = −g(vb ) ,
#
Va , Vb
(17)
(18)
where Ia , Ib , Va and Vb are the Laplace transforms of the ia , ib , va and vb , respectively. The symbol Y = [yij ] is called as the admittance matrix, which is given by
1 , 1 1 sL1 + r1 + + sC2 − G sL2 + r2
−1 , (sL1 + r1 )(sL2 + r2 )(sC2 − G) + s(L1 + L2 ) + r1 + r2
−1 = , (sL1 + r1 )(sL2 + r2 )(sC2 − G) + s(L1 + L2 ) + r1 + r2
Y22 = sC2 +
#"
1 . 1 1 sL2 + r2 + + sC2 − G sL1 + r1
(19)
Reconstruction and Synchronization of Hyperchaotic Circuits via One State Variable 2081 i1
i
L1
C1
L2
r2
i
+
+
+ Chua’s v diode a
i2
r1
v
v1
2
C2
C3
v3
i b = -g ( v b )
i a = -f ( v a )
Fig. 10.
Table 7.
Coupled canonical Chua’s oscillator.
Variational systems for the coupled canonical Chua’s oscillator.
Coupled Canonical Chua’s Oscillator C1
+ v b Chua’s diode
+ R
Eigenvalues of the Variational Systems
dv1 = i1 − f (v1 ) , dt
dv2 = Gv2 − i1 − i2 , dt di2 L1 = −v1 + v2 − r1 i1 , dt
C2
dv3 = i2 − g(v3 ) , dt
0.0843
di2 L2 = −v3 + v2 − r2 i2 , dt f (v) = b1 v + 0.5(a1 − b1 )(|v + 1| − |v − 1|) , g(v) = b2 v + 0.5(a2 − b2 )(|v + 1| − |v − 1|) .
C3
Maximum LE or CLE
Slave circuit A 0 di2 0 0 L1 = −v1 + v2 − r1 i1 , dt dv 0 C3 3 = i02 − g(v30 ) , dt 0 di2 0 0 L2 = −v3 + v2 − r2 i2 . dt dv 0 C1 1 = i01 − f (v10 ) , dt
if |v10 | ≤ d ≤ 1 and |v30 | ≤ d ≤ 1 , 6.64, 1.25,
12.2
12.2, 0.28. if |v10 | ≥ d ≥ 1 and |v30 | ≥ d ≥ 1, −39.8, −0.347,
−0.153
−49.9, −0.153.
Slave circuit B C1
dv10 = i01 − f (v10 ) , dt
C2
dv20 = Gv20 − i01 − i02 , dt
di02 = −v10 + v20 − r1 i01 , dt 0 dv3 0 0 C3 = i2 − g(v3 ), dt 0 di2 0 0 0 L2 = −v3 + v2 − r2 i2 , . dt L1
if |v10 | ≤ d ≤ 1 and |v30 | ≤ d ≤ 1, 12.2, 6.66, 0.712,
12.2
0.0617 ± 0.863i. if |v10 | ≥ d ≥ 1 and |v30 | ≥ d ≥ 1, −49.9, −39.8, −0.243, 0.0912 ± 0.868i.
0.0912
2082 M. Itoh & L. O. Chua
R
i
+ Chua’s diode v
C1
+
+
v1
v2
L
C2
r i
i=-f(v) Fig. 11.
Table 8.
Chua’s oscillator.
Variational systems for Chua’s oscillator.
Chua’s Oscillator (v1 -drive)
Eigenvalues of the Variational Systems
Maximum LE or CLE
Master circuit C1
dv1 = G(v2 − v1 ) − f (v1 ) , dt
C2
dv2 = G(v1 − v2 ) + i , dt
di L = −v2 − ri , dt f (v) = bv + 0.5(a − b)(|v + 1| − |v − 1|) .
0.294
Slave circuit A dv2 = G(v1 − v20 ) + i0 , dt di L = −v20 − ri0 . dt
C2
−0.55 ± 3.85i
−0.55
Slave circuit B dv10 = G(v20 − v10 ) − f (v10 ) , dt 0 dv2 C2 = G(v10 − v20 ) + i0 , dt 0 di 0 0 L = −v2 − ri . dt
C1
The elements Yij are easily obtained by using the classical circuit theory or the Laplace transform of the circuit equation in Table 5 (under zero initial conditions). In this case, the currents and volt-
if |v10 | ≤ d ≤ 1, 3.86, −1.13 ± 3.09.
3.86
if |v10 | ≥ d ≥ 1,
0.17
−4.64, 0.17 ± 3.19i.
ages cannot be described as linear functions of the derivatives of vj (t) or ij (t), since the linear part of the circuit is described as a four-terminal network. The performance for each scheme is similar to that
Reconstruction and Synchronization of Hyperchaotic Circuits via One State Variable 2083 Table 9.
Variational systems for Chua’s oscillator.
Chua’s Oscillator (v2 -drive)
Eigenvalues of the Variational Systems
Master circuit C1
dv1 = G(v2 − v1 ) − f (v1 ) , dt
C2
dv2 = G(v1 − v2 ) + i , dt
Maximum LE or CLE
di L = −v2 − ri , dt f (v) = bv + 0.5(a − b)(|v + 1| − |v − 1|) .
0.294
Slave circuit A if |v10 | ≤ d ≤ 1,
dv 0 C1 1 = G(v2 − v10 ) − f (v10 ) , dt di0 L = −v2 − ri0 , dt
−0.1, 2.7. if |v10 | ≥ d ≥ 1, −0.1, −3.2.
2.7 −0.1
Slave circuit B if |v10 | ≤ d ≤ 1,
dv 0 C1 1 = G(v20 − v10 ) − f (v10 ) , dt 0 dv2 0 0 0 C2 = G(v1 − v2 ) + i , dt 0 di 0 0 L = −v2 − ri . dt
3.86, if |v10 | ≥ d ≥ 1, −4.64,
G1 (s) G2 (s)
1
= sC1 +
1
R+ sC2 +
.
(20)
1 sL + r
Since Y (s) is expanded into the ladder form, all voltages and currents can be reconstructed from the linear functions of the derivative of i(t): dv2 − ri , dt d2 i di v1 = −C2 LR 2 − (C2 rR + L) dt dt −(R + r)i.
0.17
0.17 ± 3.19i.
of canonical Chua’s oscillator. In this circuit, the selective synchronization is quicker than the other schemes (see Table 10). We also add the examples of Chua’s oscillator for reference (see Fig. 11, Tables 5 and 8). The admittance Y (s) has the form Y (s) =
3.86
−1.13 ± 3.09i,
v2 = −L
(21)
Furthermore, all roots of G2 (s) = C2 LRs2 + (C2 rR + L)s + (R + r) have negative real parts. This implies that the slave circuits will be synchronized by the driving signal v1 (t) (the voltage across the Chua’s diode). In this oscillator, the continuous synchronization is more stable than the impulsive synchronization (see Fig. 9). This is because the variational system is completely stable, and so the driving signal provides a strong synchronization effect to the slave system continuously. This situation is different from the canonical Chua’s oscillator and the coupled canonical Chua’s oscillator. We also studied the case where the voltage v2 (t) is used as the driving signal (Table 9). In this case, the selective synchronization is more effective than the continuous synchronization (see Fig. 8). This is due to the following reason: The variational system without driving signal is not stable, but the two systems converge quickly to the space spanned by the eigenvectors of the two eigenvalues: −1.13 ± 3.09i. Finally, we note that the impulsive synchronization
2084 M. Itoh & L. O. Chua Table 10.
Convergence time of chaotic dynamical systems.
Continuous Synchronization
Synchronization Schemes Canonical Chua’s oscillator
Selective Synchronization
94.6
97.4
92.3
Coupled circuit
312
184
312
Chua’s oscillator (v1 -drive) Chua’s oscillator (v2 -drive)
68.0 271
— 202
70.5 181
R1
ia
+ Chua’s diode va
C1
+
+
v1
v2
R2 L1 C2
R3 L2
r
r
i1
i2
C4
i a = -f ( v a )
5. Miscellaneous Hyperchaotic Systems It is well known that the coupled Chua’s circuits in Fig. 12 have two positive Lyapunov exponents [Brucoli et al., 1999]. The circuit consists of two Chua’s diodes and linear elements. Thus, the dynamic can be described as Ib
#
"
=
+
+
+
v4
v3
v b Chua’s diode
C3
Coupled Chua’s oscillators.
is not so stable (as we can see from Fig. 9), but the convergence is quick. From computer studies, we conclude that the performance of the synchronization schemes depends on the chaotic circuits.
Ia
ib
i b = -f ( v b ) Fig. 12.
"
Impulsive Synchronization (P = 0.05, Q = 0.03)
Y11
Y12
Y21
Y22
#"
Va Vb
#
,
(22)
and ia = −f (va ) , ib = −f (vb ) ,
forms of ia , ib , va and vb , respectively. Since the circuit is a four-terminal network and Yij 6= 0, the currents and voltages of this circuit are not reconstructed from a linear combination of the derivatives of one state variable. Furthermore, even if we choose any variable as a driving signal, at least one eigenvalue of the difference system has a positive real part. Thus, the circuits cannot be synchronized by transmitting only one state variable. However, the circuits can be synchronized if we transmit the two driving signals via timedivision-based impulsive synchronization [Brucoli et al., 1999]. Next, we study the R¨ossler hyperchaos oscillator [R¨ossler, 1979]
(23)
where the symbol Y = [Yij ] is the admittance matrix, and Ia , Ib , Va and Vb are the Laplace trans-
dx dy = −y − z, = x + 0.25y + w , dt dt dz dw = 3 + xz, = −0.5z + 0.05w . dt dt
(24)
In this oscillator, the state variables are reconstructed from w as
Reconstruction and Synchronization of Hyperchaotic Circuits via One State Variable 2085
z = −2
dw d2 w − 0.1 +3 2 dt x = dt , dw 2 − 0.1w dt 2
dw + 0.1w , dt
d2 w d2 w dw d3 w d3 w dw + 0.2 2 − 0.2 3 w + 0.01 2 w 4 3 dw dt dt dt dt y=2 . − 0.1w − dt dt 2 dt dw 2 − 0.1w dt
−
d2 w −4 dt2
!2
dw − 0.01 dt
However, two state variables are needed to synchronize the R¨ossler hyperchaotic oscillators.5
6. Conclusions We have shown that some hyperchaotic circuits can be synchronized by using only one state variable. In particular, we presented two counter-examples to Pyragas’s conjecture. We also examined the performance of the continuous, impulsive and selective synchronization schemes.
Acknowledgments This research was partially supported by Fukuoka Institute of Technology, and by ONR grant N0001401-1-0741.
References Brucoli, M., Cafagna, D. & Camimeo, L. [1999] “Synchronization method of hyperchaotic circuits based on a scalar transmitted signal,” Proc. European Conf. Circuit Theory and Design, Stresa, pp. 365–368. Itoh, M., Wu, C. W. & Chua, L. O. [1997] “Communication systems via chaotic signals from a reconstruction viewpoint,” Int. J. Bifurcation and Chaos 7(2), 275–286. Itoh, M., Yang, T. & Chua, L. O. [1999] “Experimental study of impulsive synchronization of chaotic and hyperchaotic circuits,” Int. J. Bifurcation and Chaos 9(7), 1393–1424. Itoh, M., Tauchi, T., Yang, T. & Chua, L. O. [2000] “Synchronization of chaotic and hyperchaotic
5
2
−6
dw 2 − 0.1w dt
2
d2 w dw + 0.3 2 dt dt
(25)
.
systems,” Proc. 2000 IEEE Int. Conf. Industrial Electronics, Control, and Instrumentation, Nagoya, pp. 2407–2412. Itoh, M., Yang, T. & Chua, L. O. [2001] “Conditions for impulsive synchronization of chaotic and hyperchaotic systems,” Int. J. Bifurcation and Chaos 11(2), 551–560. Kennedy, M. P. [1992] “Robust OP-AMP implementation of Chua’s circuit,”Frequenz 46(3–4), 66–80. Matsumoto, T., Chua, L. O. & Kobayashi, K. [1986] “Hyperchaos: Laboratory experiment and numerical confirmation,” IEEE Trans. Circuits Syst. 33(11), 1143–1147. Panas, A. I., Yang, T. & Chua, L. O. [1998] “Experimental results of impulsive synchronization between two Chua’s circuits,” Int. J. Bifurcation and Chaos 8(3), 639–644. Pecora, L. & Carroll, T. [1990] “Synchronization in chaotic systems,” Phys. Rev. Lett. 64(8), 821–824. Peng, J. H., Ding, E. J., Ding, M. & Yang, W. [1996] “Synchronizing hyperchaos with a scalar transmitted signal,” Phys. Rev. Lett. 76(6), 904–907. Pyragas, K. [1993] “Predictable chaos in slightly perturbed unpredictable chaotic systems,” Phys. Lett. A181, 203–210. R¨ ossler, O. E. [1979] “An equation for hyperchaos,” Phys. Lett. A71(2 & 3), 155–157. Saito, T. [1990] “An approach toward higher dimensional hysteresis chaos generator,” IEEE Trans. Circuits Syst. 37(3), 399–409. Takens, F. [1981] Detecting Strange Attractors in Turbulence, Lecture Notes in Mathematics, Vol. 898 (Springer-Verlag, Berlin), pp. 366–381.
It is possible to synchronize two R¨ ossler hyperchaotic oscillators by using one scalar signal [Peng et al., 1996].
