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International Journal of Bifurcation and Chaos, Vol. 16, No. 7 (2006) 1961–1976 c World Scientific Publishing Company
SYNCHRONIZING CHAOTIC ATTRACTORS OF CHUA’S CANONICAL CIRCUIT. THE CASE OF UNCERTAINTY IN CHAOS SYNCHRONIZATION
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I. M. KYPRIANIDIS, A. N. BOGIATZI, M. PAPADOPOULOU and I. N. STOUBOULOS 3rd Lab. of Physics, Physics Department, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece G. N. BOGIATZIS Informatics and Communications Department, Technological Educational Institute of Serres, Serres, Greece T. BOUNTIS Center of Research and Applications of Nonlinear Systems and Mathematics Department, University of Patras, Patras 26500, Greece Received November 15, 2004; Revised May 10, 2005 In this paper, we have studied the dynamics of two identical resistively coupled Chua’s canonical circuits and have found that it is strongly affected by initial conditions, coupling strength and the presence of coexisting attractors. Depending on the coupling variable, chaotic synchronization has been observed both numerically and experimentally. Anti-phase synchronization has also been studied numerically clarifying some aspects of uncertainty in chaos synchronization. Keywords: Chua’s canonical circuit; coupled systems; chaos synchronization.
1. Introduction Electric circuits have emerged as a simple yet powerful experimental and analytical tool in studying chaotic behavior in nonlinear dynamics. Most chaotic and bifurcation effects cited in the literature have been observed in electric circuits e.g. the period-doubling route to chaos [Linsay, 1981; Van Buskirk & Jeffries, 1985; Kennedy, 1993; Chua et al., 1993a; Murali & Lakshmanan, 1991], intermittency route to chaos [Jeffries & Perez, 1982; Huang & Kim, 1987; Richetti et al., 1986; Fukushima & Yamada, 1988; Chua & Lin, 1991], quasiperiodicity route to chaos [Matsumoto et al., 1987; Chua et al., 1986b; Petrani et al., 1994], crisis [Jeffries & Perez, 1983; Ikezi et al., 1983; Rollins & Hunt, 1984; Kyprianidis et al., 1995],
antimonotonicity [Kocarev et al., 1993; Kyprianidis et al., 2000] and hyperchaos [Matsumoto et al., 1986; Mitsubori & Saito, 1994; Kapitaniak & Chua, 1994; Kapitaniak et al., 1994c; Tamaˇseviˇcious et al., ˇ 1997; Tamaˇseviˇcious & Cenys, 1998; Kyprianidis & Stouboulos, 2003a]. It has been shown, that it is possible to construct a set of chaotic systems, so that their common signals will have identical or synchronized behavior [Pecora & Carroll, 1990; Chua et al., 1993b; Murali & Lakshmanan, 1993]. Generally, there are two methods of chaos synchronization available in the literature. In the first method due to Pecora and Carroll [1990], a stable subsystem of a chaotic system could be synchronized with a separate chaotic system under certain suitable conditions. The second method to achieve chaos
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synchronization between two identical nonlinear systems is due to the effect of resistive coupling without requiring to construct any stable subsystem [Murali & Lakshmanan, 1994; Kapitaniak et al., 1994b; Murali et al., 1995; Kyprianidis & Stouboulos, 2003a, 2003b]. For all these coupled systems there exists a critical value, or synchronization threshold, that corresponds to a blowout bifurcation [Ott & Sommerer, 1994]. Two interesting phenomena, namely, on–off intermittency and attractors with riddled basins of attraction are expected to take place in the vicinity of the bifurcation point. In contrast to the studies before, Zhong et al. [2001] investigated the uncertainty of chaos synchronization in the sense that coupled systems, which satisfy the standard synchronization criterion, do not necessarily operate in a synchronization regime. They proved that depending on the initial states of the coupled systems, distinct behaviors of synchronization, such as in-phase synchronization, anti-phase synchronization, oscillation-quenching, and attractor bubbling, may take place. Their study was focused on the dynamical behaviors of two resistively coupled identical Chua’s circuits. Chua’s circuit is a paradigm for chaos [Madan, 1993]. Among the members of Chua’s circuit family, the autonomous canonical Chua’s circuit introduced by Chua and Lin [1990] is of considerable importance. This is because it is capable of realizing the behavior of every member of the Chua’s circuit family [Chua & Lin, 1990, 1991; Kyprianidis et al., 1995]. It consists of two active elements, one linear negative conductor, and one nonlinear resistor with odd-symmetric piecewise linear υ–i characteristic. In this paper we have studied chaotic dynamics of two resistively, unidirectionally and bidirectionally coupled, identical canonical Chua’s circuits. We have focused on the cases of existence of synchronization and of uncertainty in chaos synchronization.
with two coupled subsystems u and v, the state equation of the system in the case of bidirectional coupling can be written as: du = f1 (u) + E(v − u) dt
(1)
dv = f2 (v) − E(v − u) dt where u, v ∈ Rn and E = diag[ε1 , . . . , εn ]T . In the case of unidirectional coupling, we have the following state equation du = f1 (u) dt
(2)
dv = f2 (v) − E(v − u) dt where u, v ∈ Rn and E = diag[ε1 , . . . , εn ]T . The synchronization problem is formulated in the following way: find E such that s = v(t) − u(t) → 0 as t → ∞. This means that in the case of bidirectional coupling the desired limit (synchronous) solution cannot be predetermined. In the case of unidirectional coupling, the submanifold v(t) tends to coincide with the submanifold u(t) of the system. In this study we have focused on the case that the two subsystems are identical, that is, f1 = f2 = f . We have studied the influence of initial conditions on chaos synchronization, especially the cases when the two subsystems have: (a) equal initial conditions with opposite signs, (b) arbitrary initial conditions, which create chaotic coexisting attractors. Zhong et al. [2001] have proved that, when two identical bidirectionally coupled systems with odd symmetry have equal initial conditions with opposite signs, i.e. u(t = 0) = −v(t = 0), and, if there is a constant K > 0 such, that f (q) ≤ KI < 0 uniformly for all q ∈ Rn , where I is the identity matrix, then u(t) = −v(t) for all t ≥ 0, independent of the value of E ≥ 0. They named this type of dynamical behavior as anti-phase synchronization.
2. Dynamics of Coupled Systems It is well known that two coupled nonlinear systems can exhibit a variety of distinct performances of dynamical behavior, among which chaotic synchronization is of special importance. To achieve synchronization between two systems, which are chaotic, the linear coupling technique is commonly used [Chua et al., 1993b; Kapitaniak et al., 1994b; Pecora & Carroll, 1991]. In particular, for a system
3. The Chua’s Canonical Circuit We have adopted Chua’s canonical circuit [Chua & Lin, 1990, 1991] as a paradigm to study the dynamical behaviors referred to in Sec. 2. Chua’s canonical circuit is shown in Fig. 1(a). It has a nonlinear resistor NR with an odd symmetric υ–i characteristic of type-N [Fig. 1(b)]. Gn is a negative linear conductance.
Synchronizing Chaotic Attractors of Chua’s Canonical Circuit
(a) Fig. 1.
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(b)
(a) Chua’s canonical characteristic, (b) υ–i characteristic of the nonlinear resistor NR .
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The state equations of the circuit are 1 dυ C1 {iL − iN } = dt C1 dυ C2 1 = {−Gn υC2 − iL} dt C2
(3)
1 diL = {−υC1 + υC2 − RiL} dt L while the current iN resistor is
through the nonlinear
iN = g(υC1 ) = Gb υC1 + 0.5(Ga − Gb ) ×{|υC1 + Bp | − |υC1 − Bp |} (4) These state equations take the dimensionless form dx = α{z − h(x)} dτ dy = −βy − z dτ
(5)
dz = γ(y − x − z) dτ where h(x) = mb x + 0.5(ma − mb ){|x + 1| − |x − 1|} (6) is the dimensionless function of the current through the nonlinear resistor. The dimensionless variables and parameters of Eqs. (5) and (6) are defined as follows: υC2 RiL υC1 , y= , z= x= Bp Bp Bp α=
C2 , C1
β = RGn ,
ma = RGa ,
γ=
R2 C2 , L
(7)
mb = RGb
t RC2
We have chosen the following parameter values for the Chua canonical circuit throughout our study: L = 100 mH, R = 330 Ω, C1 = 33.0 nF, Gn = −0.5 mS, Ga = −0.105 mS, Gb = 4.70 mS, Bp = 0.68 V. C2 can take different values giving a double-scroll strange attractor or a spiral strange attractor. In Fig. 2, the bifurcation diagram υC1 versus C2 is shown for the chosen parameter values.
4. Linearly Coupled Chua’s Canonical Circuits Based on the Chua’s canonical circuit presented above, the state equation of the unidirectionallycoupled system (2), in the case of f1 = f2 , is written dx1 = α{z1 − h(x1 )} dτ dy 1 = −βy1 − z1 dτ
with the dimensionless time variable τ=
Fig. 2. Bifurcation diagram υC1 versus C2 of the Chua’s canonical circuit for the chosen parameter values.
(8)
dz 1 = γ(y1 − x1 − z1 ) dτ
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dx2 = α{z2 − h(x2 )} + εx (x1 − x2 ) dτ
h(x2 ) = mb x2 + 0.5(ma − mb ){|x2 + 1| − |x2 − 1|} (11)
dy2 = −βy 2 − z2 + εy (y1 − y2 ) dτ
and εj (j = x, y, z) are the coupling factors. The state equations of the bidirectionallycoupled system (1), in the case of f1 = f2 , on the other hand, take the form
dz 2 = γ(y2 − x2 − z2 ) + εz (z1 − z2 ) dτ
(9)
dx1 = α{z1 − h(x1 )} − εx (x1 − x2 ) dτ
where
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h(x1 ) = mb x1 + 0.5(ma − mb ){|x1 + 1| − |x1 − 1|} (10)
dy 1 = −βy 1 − z1 − εy (y1 − y2 ) dτ
(a)
(b)
(c)
(d)
Fig. 3. Experimental phase portraits υC2 versus υC1 : (a) the spiral attractor (C2 = 64.0 nF); horizontal axis 0.5 V/div, vertical axis 1.0 V/div, (b) the double-scroll attractor (C2 = 58.0 nF); horizontal axis 0.5 V/div, vertical axis 1.0 V/div, (c) the double-scroll attractor (C2 = 43.0 nF); horizontal axis 0.5 V/div, vertical axis 1.0 V/div, (d) period-1 limit cycle (C2 = 41.0 nF); horizontal axis 0.5 V/div, vertical axis 5.0 V/div.
Synchronizing Chaotic Attractors of Chua’s Canonical Circuit
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dz 1 = γ(y1 − x1 − z1 ) − εz (z1 − z2 ) dτ dx2 = α{z2 − h(x2 )} + εx (x1 − x2 ) dτ dy2 = −βy2 − z2 + εy (y1 − y2 ) dτ dz 2 = γ(y2 − x2 − z2 ) + εz (z1 − z2 ) dτ
(12)
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where h(x1 ) = mb x1 + 0.5(ma − mb ){|x1 + 1| − |x1 − 1|} (13) h(x2 ) = mb x2 + 0.5(ma − mb ){|x2 + 1| − |x2 − 1|} (14)
Fig. 4. The unidirectionally-coupled Chua’s canonical circuits (x-coupling).
and εj (j = x, y, z) are the coupling factors.
4.1. x-Coupled system Figure 4 shows the experimental set up of the unidirectionally-coupled system. By removing the voltage follower from the coupling branch, we obtain the bidirectionally x-coupled system. In this case, only the x state variable is coupled. This means that εy = εz = 0, while εx = αR/Rc . For C2 = 60.0 nF, the uncoupled circuits are in a chaotic state having a double scroll strange
attractor. In Fig. 5, the bifurcation diagrams y2 −y1 versus εx are shown for unidirectional and bidirectional coupling. Synchronization, y2 − y1 = 0 is observed in both cases. Clearly, in the second case, synchronization is established for a lower critical value of the coupling factor εx . When the strange attractor of the canonical Chua’s circuit is a spiral one, then coexisting attractors are observed, as shown in Fig. 6. We have
(a)
(b)
Fig. 5. Bifurcation diagrams y2 − y1 versus εx , for unidirectional and bidirectional coupling, in the case C2 = 60.0 nF. Synchronization is observed in both cases.
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(a)
(b) Fig. 6.
Coexisting spiral attractors for C2 = 63.0 nF.
studied the dynamics of such a coupled system in the following cases:
4.2. y-Coupled system
(a) the two sub-circuits have initial conditions belonging to the same basin of attraction, (b) the two sub-circuits have initial conditions belonging to different basin of attraction. In both cases, synchronization is not observed (Figs. 7 and 8) independently of the way of coupling (unidirectional or bidirectional).
Figure 9 shows the experimental set up of a unidirectionally-coupled system with y-coupling. By removing the voltage follower from the coupling branch, we obtain the bidirectionally y-coupled system. Since only the y state variable is coupled, this means that εx = εz = 0, while εy = R/Rc . For C2 = 60.0 nF, the uncoupled circuits are in a chaotic state having a double scroll strange
(a)
(b)
Fig. 7. Bifurcation diagrams y2 − y1 versus εx , for (a) unidirectional, and (b) bidirectional coupling, in the case C2 = 63.0 nF. Synchronization is not observed in any case. The initial conditions of each individual circuit belong to different basins of attraction.
Synchronizing Chaotic Attractors of Chua’s Canonical Circuit
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(a)
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(b)
Fig. 8. Bifurcation diagrams y2 − y1 versus εx , for (a) unidirectional, and (b) bidirectional coupling, in the case C2 = 70.0 nF. Synchronization is not observed in any case. The initial conditions of each individual circuit belong to the same basin of attraction.
(a)
Fig. 9. The unidirectionally-coupled Chua’s canonical circuits (y-coupling).
attractor. In Fig. 10, the bifurcation diagrams y2 − y1 versus εy are shown for unidirectional and bidirectional y-coupling. Synchronization, y2 − y1 = 0, is now observed in both cases. In the second case, however, synchronization is established for a lower critical value of the coupling factor εy . When the strange attractor of the canonical Chua’s circuit is a spiral one, synchronization is always observed even though the initial conditions of the y-coupled circuits belong to different basin of attraction (Fig. 11).
(b) Fig. 10. Bifurcation diagrams y2 − y1 versus εy , for (a) unidirectional, and (b) bidirectional y-coupling, in the case C2 = 60.0 nF. Synchronization is observed in both cases.
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(a)
(b)
Fig. 11. Bifurcation diagrams y2 − y1 versus εy , for (a) unidirectional and (b) bidirectional y-coupling, in the case C2 = 67.0 nF. Synchronization is observed in both cases. The initial conditions of each individual circuit belong to different basins of attraction.
5. Experimental Results In the following figures, (Figs. 12 and 13), we now exhibit some characteristic experimental phase space portraits of the coupled system for different values of the coupling factor. In Figs. 12(a) and 13(a) the system is out of synchronization,
while in Figs. 12(b) and 13(b) the system is in chaotic synchronization. The results of Fig. 13(b) demonstrate that the system can experimentally achieve synchronization, even at parameter values not predicted by the simulations. That the converse may also occur is discussed in the next section.
(a)
(b)
Fig. 12. Experimental phase portraits, υC11 versus υC12 , for the unidirectionally y-coupled system, when C2 = 60.0 nF; (a) εy = 0.0015, the system is out of synchronization, (b) εy = 0.04, chaotic synchronization.
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Synchronizing Chaotic Attractors of Chua’s Canonical Circuit
(a)
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(b)
Fig. 13. Experimental phase portraits, υC11 versus υC12 , for the unidirectionally y-coupled system, when C2 = 67.0 nF (spiral attractor); (a) εy = 0.003, the system is out of synchronization, (b) εy = 0.016, chaotic synchronization.
6. Uncertainty in Chaos Synchronization According to the bifurcation diagram of Fig. 5(a), in the case of unidirectional x-coupling for C2 = 60.0 nF, chaotic synchronization must be observed for εx > 0.315. The experimental realization of the coupled system revealed that a different situation can also be observed (Fig. 14), in which the system, for a sufficiently high value of εx , can be out of synchronization. To see how this can occur, let us examine more closely the experimental attractors of the driving (master) and the driven (slave) circuit shown here in Fig. 15. The attractor of the driving circuit is the well-known double scroll attractor [Fig. 15(a)], while the attractor of the driven circuit is an unknown chaotic attractor [Fig. 15(b)]. In the absence of coupling, for C2 = 60.0 nF, there are two coexisting attractors. The chaotic double scroll attractor (see also Fig. 15(a) and a limit-cycle of period-1 (Fig. 16)). The initial conditions of the chaotic double scroll attractor are x0 = −1.0, y0 = 0.40, and z0 = 0.50, while the initial conditions of the limit-cycle of period-1 are x0 = −1.0, y0 = 1.0, and z0 = 2.0. For εx = 1.20, the computer simulated phase portraits, x1 versus x2 of the x-coupled system, and
Fig. 14. Experimental phase portrait, υC11 versus υC12 , for the unidirectionally x-coupled system, when C2 = 60.0 nF. The coupling factor is εx = 1.20, and the system is out of synchronization. Horizontal axis 0.5 V/div, vertical axis 0.5 V/div.
y2 versus x2 of the driven system, are shown in Fig. 17. The driven circuit oscillates in the basin of attraction of the big limit cycle giving a new type of chaotic attractor, similar to the one observed experimentally in Fig. 15(b). In contrast to this property of the double scroll attractor, the chaotic spiral attractor does not
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(a)
(b)
Fig. 15. Experimental phase portrait, for the unidirectionally x-coupled system, when C2 = 60.0 nF. The coupling factor is εx = 1.20. (a) υC21 versus υC11 , horizontal axis 0.5 V/div, vertical axis 0.5 V/div; (b) υC22 versus υC12 , horizontal axis 5.0 V/div, vertical axis 0.5 V/div.
For C2 = 60.0 nF, each Chua’s canonical circuit of the system with the parameter values listed above operated in a double-scroll attractor before being coupled. We have chosen the following sets of initial conditions for the computer simulation: x1 (0) = −1.0, y1 (0) = 0.4, z1 (0) = 0.5, x2 (0) = 1.0, y2 (0) = −0.4, z2 (0) = −0.5.
6.1.1. x-Coupled system
Fig. 16. The coexisting attractors for C2 = 60.0 nF. The initial conditions are (a) x0 = −1.0, y0 = 0.40, z0 = 0.50, for the chaotic double scroll attractor, (b) x0 = −1.0, y0 = 1.0, z0 = 2.0, for the limit-cycle of period-1.
have a coexisting periodic attractor, so an uncertainty of chaos synchronization of this kind is not observed in that case.
6.1. Anti-phase synchronization As we have mentioned in Sec. 2, anti-phase synchronization can be observed, when two identical bidirectionally coupled systems with odd symmetry have equal initial conditions with opposite sign.
As the coupling factor εx is increased, starting from zero (the uncoupled case), a change of the type of the attractor of each subsystem is observed. The double-scroll attractor gives its place to a chaotic spiral attractor and then a reverse period doubling sequence is observed, while anti-phase synchronization is conserved. The reverse period doubling sequence as the coupling factor εx increases is shown in Fig. 20. For 0.215 ≤ εx ≤ 0.680, the oscillations in the two Chua’s canonical circuits were quenched and each circuit operated in a point attractor located at (0, 0). For εx ≥ 0.690 oscillations were present again and each Chua’s canonical circuit operated in a period-1 attractor with anti-phase.
6.1.2. y-Coupled system In the case of the y-coupled system, the reverse period-doubling sequence is also observed, as the
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Synchronizing Chaotic Attractors of Chua’s Canonical Circuit
(a)
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(b)
Fig. 17. The computer simulated phase portraits, (a) x1 versus x2 of the x-coupled system, and (b) y2 versus x2 of the driven system, for C2 = 60.0 nF and εx = 1.20.
Fig. 18. Phase portrait y2 versus y1 in the case of anti-phase synchronization (y-coupled system, εy = 0.001) and the computational environment.
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(a1 )
(a2 )
(b1 )
(b2 )
(c1 )
(c2 )
Fig. 19. y2 versus x2 and y2 versus y1 phase portraits, in the case of anti-phase synchronization of the bidirectionally x-coupled identical Chua’s canonical circuits with opposite identical initial conditions, for C2 = 60.0 nF: (a) εx = 0.010. Double-scroll attractor. (b–d) εx = 0.020, 0.130, 0.142. Spiral attractors. (e) εx = 0.146. Period-4 attractor. (f) εx = 0.150. Period-2 attractor. (g) εx = 0.180. Period-1 attractor.
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Synchronizing Chaotic Attractors of Chua’s Canonical Circuit
(d1 ) (d2 )
(e1 ) (e2 )
(f1 )
(f2 )
Fig. 19.
(Continued )
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(g1 )
(g2 ) Fig. 19.
(Continued )
7. Concluding Remarks
Fig. 20. Bifurcation diagram x2 versus εx of the bidirectionally x-coupled identical Chua’s canonical circuits with opposite identical initial conditions.
coupling factor εy is increased. For 0.030 ≤ εy ≤ 0.390 the oscillations in the two Chua’s canonical circuits were quenched and each circuit operated in a stable point attractor. The coordinates of the point attractor are dependent on the value of εy . For εy ≥ 0.40, the point attractor is located at (0, 0, 0). In the case of y-coupled identical Chua’s canonical circuits with opposite identical initial conditions, the transition of the system from the point attractor to the period-1 attractor was not observed.
In the present work, using two identical Chua’s canonical circuits resistively coupled, we have studied the process of synchronization, both in unidirectional and bidirectional coupling forms, using computer simulation and experiment. The results have shown that the dynamics of the coupled system is strongly affected by the initial states and the coupling strength. Furthermore, although the experimental results generally agree with the predictions of the simulations, there are some differences, which demonstrate, that the actual electrical circuits may include additional effects, which are not present in our systems of differential equations. Our observations suggest that the synchronization state of the y-coupled system is the most robust. When each individual circuit operates in a double-scroll or a spiral chaotic attractor, chaotic synchronization is always observed, when the coupling strength exceeds a critical value. In the case of the x-coupled system, however, when each individual circuit operates in a spiral attractor mode, chaotic synchronization is not observed. On the other hand, computer simulations show that when the circuits display double-scroll attractor dynamics, chaotic synchronization can be observed. Generally, the convergence of the system to the state of synchronization is very slow. Our experimental studies have also demonstrated that the state of synchronization is very sensitive to the unavoidable mismatches of the components of the
Synchronizing Chaotic Attractors of Chua’s Canonical Circuit
coupled circuits, leading to an experimental uncertainty of chaotic synchronization. The study of anti-phase synchronization by computer simulation verified the reverse period-doubling scenario and oscillation-quenching observed by Zhong et al. [2001]. A new feature we noted, in the case of the x-coupled system, is the period-1 oscillation, which become possible after the state of oscillation-quenching, as the coupling factor is further increased.
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References Chua, L. O., Komuro, M. & Matsumoto, T. [1986a] “The double-scroll family,” IEEE Trans. Circuits Syst. CAS-33, 1073–1118. Chua, L. O., Yao, Y. & Yang, Q. [1986b] “Devil’s staircase route to chaos in a nonlinear circuit,” Int. J. Circuit Th. Appl. 14, 315–329. Chua, L. O. & Lin, G. N. [1990] “Canonical realization of Chua’s circuit family,” IEEE Trans. Circuits Syst. CAS-37, 885–902. Chua, L. O. & Lin, G. N. [1991] “Intermittency in a piecewise-linear circuit,” IEEE Trans. Circuits Syst. CAS-38, 510–520. Chua, L. O., Wu, C. W., Huang, A. & Zhong, G.-Q. [1993a] “A universal circuit for studying and generating chaos,” IEEE Trans. Circuits Syst.-I 40, 732– 761. Chua, L. O., Itoh, M., Kocarev, Lj. & Eckert, K. [1993b] “Chaos synchronization in Chua’s circuit,” J. Circuits Syst. Comput. 3, 93–108. Fukushima, K. & Yamada, T. [1988] “Type-III intermittency in a coupled nonlinear LCR circuit,” J. Phys. Soc. Japan 57, 4055–4062. Huang, J. Y. & Kim, J. J. [1987] “Type-II intermittency in a coupled nonlinear oscillator: Experimental observation,” Phys. Rev. A 36, 1495–1497. Ikezi, H., de Grassie, J. S. & Jensen, T. H. [1983] “Observation of multiple-valued attractors and crises in a driven nonlinear circuit,” Phys. Rev. A 28, 1207– 1209. Jeffries, C. & Perez, J. [1982] “Observation of a Pomeau– Manneville intermittent route to chaos in a nonlinear oscillator,” Phys. Rev. A 26, 2117–2122. Jeffries, C. & Perez, J. [1983] “Direct observation of crises of the chaotic attractor in a nonlinear oscillator,” Phys. Rev. A 27, 601–603. Kapitaniak, T. & Chua, L. O. [1994a] “Hyperchaotic attractor of unidirectionally-coupled Chua’s circuit,” Int. J. Bifurcation and Chaos 4, 477– 482. Kapitaniak, T., Chua, L. O. & Zhong, G.-Q. [1994b] “Experimental synchronization of chaos using continuous control,” Int. J. Bifurcation and Chaos 4, 483– 488.
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Kapitaniak, T., Chua, L. O. & Zhong, G. O. [1994c] “Experimental hyperchaos in coupled Chua’s circuits,” IEEE Trans. Circuits Syst.-I 41, 499–503. Kennedy, M. P. [1993] “Three steps to chaos,” IEEE Trans. Circuits Syst.-I 40, 640–674. Kocarev, Lj., Halle, K. S., Eckert, K. & Chua, L. O. [1993] “Experimental observation of antimonotonicity in Chua’s circuit,” in Chua’s Circuit: A Paradigm for Chaos, ed. Madan, R. N. (World Scientific, Singapore), pp. 137–144. Kyprianidis, I. M., Petrani, M. L., Kalomiros, J. A. & Anagnostopoulos, A. N. [1995] “Crisis-induced intermittency in a third-order electrical circuit,” Phys. Rev. E 52, 2268–2273. Kyprianidis, I. M., Stouboulos, I. N., Haralabidis, P. & Bountis, T. [2000] “Antimonotonicity and chaotic dynamics in a fourth-order autonomous nonlinear electric circuit,” Int. J. Bifurcation and Chaos 10, 1903–1915. Kyprianidis, I. M. & Stouboulos, I. N. [2003a] “Synchronization of two resistively coupled nonautonomous and hyperchaotic oscillators,” Chaos Solit. Fract. 17, 317–325. Kyprianidis, I. M. & Stouboulos, I. N. [2003b] “Chaotic synchronization of three coupled oscillators with ring connection,” Chaos Solit. Fract. 17, 327–336. Linsay, P. S. [1981] “Period doubling and chaotic behavior in a driven anharmonic oscillator,” Phys. Rev. Lett. 47, 1349–1351. Madan, R. N. (ed.) [1993] Chua’s Circuit: A Paradigm for Chaos (World Scientific, Singapore). Matsumoto, T., Chua, L. O. & Kobayashi, K. [1986] “Hyperchaos: Laboratory experiment and numerical confirmation,” IEEE Trans. Circuits Syst. CAS-33, 1143–1147. Matsumoto, T., Chua, L. O. & Tokunaga, R. [1987] “Chaos via torus breakdown,” IEEE Trans. Circuits Syst. CAS-34, 240–253. Mitsubori, K. & Saito, T. [1994] “A four-dimensional plus hysteresis chaos generator,” IEEE Trans. Circuits Syst.-I CAS-41, 782–789. Murali, K. & Lakshmanan, M. [1991] “Bifurcation and chaos of the sinusoidally driven Chua’s circuit,” Int. J. Bifurcation and Chaos 1, 369–383. Murali, K. & Lakshmanan, M. [1993] “Synchronizing chaos in driven Chua’s circuit,” Int. J. Bifurcation and Chaos 3, 1057–1066. Murali, K. & Lakshmanan, M. [1994] “Drive-response scenario of chaos synchronization in identical nonlinear systems,” Phys. Rev. E 49, 4882–4885. Murali, K., Lakshmanan, M. & Chua, L. O. [1995] “Controlling and synchronization of chaos in the simplest dissipative nonautonomous circuit,” Int. J. Bifurcation and Chaos 5, 563–571. Ott, E. & Sommerer, J. C. [1994] “Blowout bifurcation: The occurrence of riddled basins and on-off intermittency,” Phys. Lett. A 188, 39–47.
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Pecora, L. M. & Carroll, T. L. [1990] “Synchronization in chaotic systems,” Phys. Rev. Lett. 64, 821– 824. Pecora, L. M. & Carroll, T. L. [1991] “Driving systems with chaotic signals,” Phys. Rev. A 44, 2374–2383. Petrani, M. L., Karakotsou, Ch., Kyprianidis, I. M. & Anagnostopoulos, A. N. [1994] “Characterization of the attractor governing the neon bulb RC relaxation oscillator,” Phys. Rev. E 49, 5863–5866. Richetti, P., Argoul, F. & Arneodo, A. [1986] “Type-II intermittency in a periodically driven nonlinear oscillator,” Phys. Rev. A 34, 726–729. Rollins, R. W. & Hunt, E. R. [1984] “Intermittent transient chaos at interior crisis in the diode resonator,” Phys. Rev. A 29, 3327–3334.
ˇ Tamaˇseviˇcious, A., Cenys, A., Mykolaitis, G., Namajunas, A. & Lindberg, E. [1997a] “Hyperchaotic oscillator with gyrators,” Electron. Lett. 33, 542–544. ˇ Tamaˇseviˇcious, A., Cenys, A., Mykolaitis, G., Namajunas, A. & Lindberg, E. [1997b] “Synchronization of hyperchaotic oscillators,” Electron. Lett. 33, 2205–2206. ˇ Tamaˇseviˇcious, A. & Cenys, A. [1998] “Hyperchaos in dynamical systems with a monoactive degree of freedom,” Chaos Solit. Fract. 9, 115–119. Van Buskirk, R. & Jeffries, C. [1985] “Observation of chaotic dynamics of coupled nonlinear oscillators,” Phys. Rev. A 31, 3332–3357. Zhong, G.-Q., Man, K.-F. & Ko, K.-T. [2001] “Uncertainty in chaos synchronization,” Int. J. Bifurcation and Chaos 11, 1723–1735.
SYNCHRONIZING CHAOTIC ATTRACTORS OF CHUA’S CANONICAL CIRCUIT. THE CASE OF UNCERTAINTY IN CHAOS SYNCHRONIZATION
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I. M. KYPRIANIDIS, A. N. BOGIATZI, M. PAPADOPOULOU and I. N. STOUBOULOS 3rd Lab. of Physics, Physics Department, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece G. N. BOGIATZIS Informatics and Communications Department, Technological Educational Institute of Serres, Serres, Greece T. BOUNTIS Center of Research and Applications of Nonlinear Systems and Mathematics Department, University of Patras, Patras 26500, Greece Received November 15, 2004; Revised May 10, 2005 In this paper, we have studied the dynamics of two identical resistively coupled Chua’s canonical circuits and have found that it is strongly affected by initial conditions, coupling strength and the presence of coexisting attractors. Depending on the coupling variable, chaotic synchronization has been observed both numerically and experimentally. Anti-phase synchronization has also been studied numerically clarifying some aspects of uncertainty in chaos synchronization. Keywords: Chua’s canonical circuit; coupled systems; chaos synchronization.
1. Introduction Electric circuits have emerged as a simple yet powerful experimental and analytical tool in studying chaotic behavior in nonlinear dynamics. Most chaotic and bifurcation effects cited in the literature have been observed in electric circuits e.g. the period-doubling route to chaos [Linsay, 1981; Van Buskirk & Jeffries, 1985; Kennedy, 1993; Chua et al., 1993a; Murali & Lakshmanan, 1991], intermittency route to chaos [Jeffries & Perez, 1982; Huang & Kim, 1987; Richetti et al., 1986; Fukushima & Yamada, 1988; Chua & Lin, 1991], quasiperiodicity route to chaos [Matsumoto et al., 1987; Chua et al., 1986b; Petrani et al., 1994], crisis [Jeffries & Perez, 1983; Ikezi et al., 1983; Rollins & Hunt, 1984; Kyprianidis et al., 1995],
antimonotonicity [Kocarev et al., 1993; Kyprianidis et al., 2000] and hyperchaos [Matsumoto et al., 1986; Mitsubori & Saito, 1994; Kapitaniak & Chua, 1994; Kapitaniak et al., 1994c; Tamaˇseviˇcious et al., ˇ 1997; Tamaˇseviˇcious & Cenys, 1998; Kyprianidis & Stouboulos, 2003a]. It has been shown, that it is possible to construct a set of chaotic systems, so that their common signals will have identical or synchronized behavior [Pecora & Carroll, 1990; Chua et al., 1993b; Murali & Lakshmanan, 1993]. Generally, there are two methods of chaos synchronization available in the literature. In the first method due to Pecora and Carroll [1990], a stable subsystem of a chaotic system could be synchronized with a separate chaotic system under certain suitable conditions. The second method to achieve chaos
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synchronization between two identical nonlinear systems is due to the effect of resistive coupling without requiring to construct any stable subsystem [Murali & Lakshmanan, 1994; Kapitaniak et al., 1994b; Murali et al., 1995; Kyprianidis & Stouboulos, 2003a, 2003b]. For all these coupled systems there exists a critical value, or synchronization threshold, that corresponds to a blowout bifurcation [Ott & Sommerer, 1994]. Two interesting phenomena, namely, on–off intermittency and attractors with riddled basins of attraction are expected to take place in the vicinity of the bifurcation point. In contrast to the studies before, Zhong et al. [2001] investigated the uncertainty of chaos synchronization in the sense that coupled systems, which satisfy the standard synchronization criterion, do not necessarily operate in a synchronization regime. They proved that depending on the initial states of the coupled systems, distinct behaviors of synchronization, such as in-phase synchronization, anti-phase synchronization, oscillation-quenching, and attractor bubbling, may take place. Their study was focused on the dynamical behaviors of two resistively coupled identical Chua’s circuits. Chua’s circuit is a paradigm for chaos [Madan, 1993]. Among the members of Chua’s circuit family, the autonomous canonical Chua’s circuit introduced by Chua and Lin [1990] is of considerable importance. This is because it is capable of realizing the behavior of every member of the Chua’s circuit family [Chua & Lin, 1990, 1991; Kyprianidis et al., 1995]. It consists of two active elements, one linear negative conductor, and one nonlinear resistor with odd-symmetric piecewise linear υ–i characteristic. In this paper we have studied chaotic dynamics of two resistively, unidirectionally and bidirectionally coupled, identical canonical Chua’s circuits. We have focused on the cases of existence of synchronization and of uncertainty in chaos synchronization.
with two coupled subsystems u and v, the state equation of the system in the case of bidirectional coupling can be written as: du = f1 (u) + E(v − u) dt
(1)
dv = f2 (v) − E(v − u) dt where u, v ∈ Rn and E = diag[ε1 , . . . , εn ]T . In the case of unidirectional coupling, we have the following state equation du = f1 (u) dt
(2)
dv = f2 (v) − E(v − u) dt where u, v ∈ Rn and E = diag[ε1 , . . . , εn ]T . The synchronization problem is formulated in the following way: find E such that s = v(t) − u(t) → 0 as t → ∞. This means that in the case of bidirectional coupling the desired limit (synchronous) solution cannot be predetermined. In the case of unidirectional coupling, the submanifold v(t) tends to coincide with the submanifold u(t) of the system. In this study we have focused on the case that the two subsystems are identical, that is, f1 = f2 = f . We have studied the influence of initial conditions on chaos synchronization, especially the cases when the two subsystems have: (a) equal initial conditions with opposite signs, (b) arbitrary initial conditions, which create chaotic coexisting attractors. Zhong et al. [2001] have proved that, when two identical bidirectionally coupled systems with odd symmetry have equal initial conditions with opposite signs, i.e. u(t = 0) = −v(t = 0), and, if there is a constant K > 0 such, that f (q) ≤ KI < 0 uniformly for all q ∈ Rn , where I is the identity matrix, then u(t) = −v(t) for all t ≥ 0, independent of the value of E ≥ 0. They named this type of dynamical behavior as anti-phase synchronization.
2. Dynamics of Coupled Systems It is well known that two coupled nonlinear systems can exhibit a variety of distinct performances of dynamical behavior, among which chaotic synchronization is of special importance. To achieve synchronization between two systems, which are chaotic, the linear coupling technique is commonly used [Chua et al., 1993b; Kapitaniak et al., 1994b; Pecora & Carroll, 1991]. In particular, for a system
3. The Chua’s Canonical Circuit We have adopted Chua’s canonical circuit [Chua & Lin, 1990, 1991] as a paradigm to study the dynamical behaviors referred to in Sec. 2. Chua’s canonical circuit is shown in Fig. 1(a). It has a nonlinear resistor NR with an odd symmetric υ–i characteristic of type-N [Fig. 1(b)]. Gn is a negative linear conductance.
Synchronizing Chaotic Attractors of Chua’s Canonical Circuit
(a) Fig. 1.
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(b)
(a) Chua’s canonical characteristic, (b) υ–i characteristic of the nonlinear resistor NR .
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The state equations of the circuit are 1 dυ C1 {iL − iN } = dt C1 dυ C2 1 = {−Gn υC2 − iL} dt C2
(3)
1 diL = {−υC1 + υC2 − RiL} dt L while the current iN resistor is
through the nonlinear
iN = g(υC1 ) = Gb υC1 + 0.5(Ga − Gb ) ×{|υC1 + Bp | − |υC1 − Bp |} (4) These state equations take the dimensionless form dx = α{z − h(x)} dτ dy = −βy − z dτ
(5)
dz = γ(y − x − z) dτ where h(x) = mb x + 0.5(ma − mb ){|x + 1| − |x − 1|} (6) is the dimensionless function of the current through the nonlinear resistor. The dimensionless variables and parameters of Eqs. (5) and (6) are defined as follows: υC2 RiL υC1 , y= , z= x= Bp Bp Bp α=
C2 , C1
β = RGn ,
ma = RGa ,
γ=
R2 C2 , L
(7)
mb = RGb
t RC2
We have chosen the following parameter values for the Chua canonical circuit throughout our study: L = 100 mH, R = 330 Ω, C1 = 33.0 nF, Gn = −0.5 mS, Ga = −0.105 mS, Gb = 4.70 mS, Bp = 0.68 V. C2 can take different values giving a double-scroll strange attractor or a spiral strange attractor. In Fig. 2, the bifurcation diagram υC1 versus C2 is shown for the chosen parameter values.
4. Linearly Coupled Chua’s Canonical Circuits Based on the Chua’s canonical circuit presented above, the state equation of the unidirectionallycoupled system (2), in the case of f1 = f2 , is written dx1 = α{z1 − h(x1 )} dτ dy 1 = −βy1 − z1 dτ
with the dimensionless time variable τ=
Fig. 2. Bifurcation diagram υC1 versus C2 of the Chua’s canonical circuit for the chosen parameter values.
(8)
dz 1 = γ(y1 − x1 − z1 ) dτ
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dx2 = α{z2 − h(x2 )} + εx (x1 − x2 ) dτ
h(x2 ) = mb x2 + 0.5(ma − mb ){|x2 + 1| − |x2 − 1|} (11)
dy2 = −βy 2 − z2 + εy (y1 − y2 ) dτ
and εj (j = x, y, z) are the coupling factors. The state equations of the bidirectionallycoupled system (1), in the case of f1 = f2 , on the other hand, take the form
dz 2 = γ(y2 − x2 − z2 ) + εz (z1 − z2 ) dτ
(9)
dx1 = α{z1 − h(x1 )} − εx (x1 − x2 ) dτ
where
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h(x1 ) = mb x1 + 0.5(ma − mb ){|x1 + 1| − |x1 − 1|} (10)
dy 1 = −βy 1 − z1 − εy (y1 − y2 ) dτ
(a)
(b)
(c)
(d)
Fig. 3. Experimental phase portraits υC2 versus υC1 : (a) the spiral attractor (C2 = 64.0 nF); horizontal axis 0.5 V/div, vertical axis 1.0 V/div, (b) the double-scroll attractor (C2 = 58.0 nF); horizontal axis 0.5 V/div, vertical axis 1.0 V/div, (c) the double-scroll attractor (C2 = 43.0 nF); horizontal axis 0.5 V/div, vertical axis 1.0 V/div, (d) period-1 limit cycle (C2 = 41.0 nF); horizontal axis 0.5 V/div, vertical axis 5.0 V/div.
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dz 1 = γ(y1 − x1 − z1 ) − εz (z1 − z2 ) dτ dx2 = α{z2 − h(x2 )} + εx (x1 − x2 ) dτ dy2 = −βy2 − z2 + εy (y1 − y2 ) dτ dz 2 = γ(y2 − x2 − z2 ) + εz (z1 − z2 ) dτ
(12)
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where h(x1 ) = mb x1 + 0.5(ma − mb ){|x1 + 1| − |x1 − 1|} (13) h(x2 ) = mb x2 + 0.5(ma − mb ){|x2 + 1| − |x2 − 1|} (14)
Fig. 4. The unidirectionally-coupled Chua’s canonical circuits (x-coupling).
and εj (j = x, y, z) are the coupling factors.
4.1. x-Coupled system Figure 4 shows the experimental set up of the unidirectionally-coupled system. By removing the voltage follower from the coupling branch, we obtain the bidirectionally x-coupled system. In this case, only the x state variable is coupled. This means that εy = εz = 0, while εx = αR/Rc . For C2 = 60.0 nF, the uncoupled circuits are in a chaotic state having a double scroll strange
attractor. In Fig. 5, the bifurcation diagrams y2 −y1 versus εx are shown for unidirectional and bidirectional coupling. Synchronization, y2 − y1 = 0 is observed in both cases. Clearly, in the second case, synchronization is established for a lower critical value of the coupling factor εx . When the strange attractor of the canonical Chua’s circuit is a spiral one, then coexisting attractors are observed, as shown in Fig. 6. We have
(a)
(b)
Fig. 5. Bifurcation diagrams y2 − y1 versus εx , for unidirectional and bidirectional coupling, in the case C2 = 60.0 nF. Synchronization is observed in both cases.
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(a)
(b) Fig. 6.
Coexisting spiral attractors for C2 = 63.0 nF.
studied the dynamics of such a coupled system in the following cases:
4.2. y-Coupled system
(a) the two sub-circuits have initial conditions belonging to the same basin of attraction, (b) the two sub-circuits have initial conditions belonging to different basin of attraction. In both cases, synchronization is not observed (Figs. 7 and 8) independently of the way of coupling (unidirectional or bidirectional).
Figure 9 shows the experimental set up of a unidirectionally-coupled system with y-coupling. By removing the voltage follower from the coupling branch, we obtain the bidirectionally y-coupled system. Since only the y state variable is coupled, this means that εx = εz = 0, while εy = R/Rc . For C2 = 60.0 nF, the uncoupled circuits are in a chaotic state having a double scroll strange
(a)
(b)
Fig. 7. Bifurcation diagrams y2 − y1 versus εx , for (a) unidirectional, and (b) bidirectional coupling, in the case C2 = 63.0 nF. Synchronization is not observed in any case. The initial conditions of each individual circuit belong to different basins of attraction.
Synchronizing Chaotic Attractors of Chua’s Canonical Circuit
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(a)
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(b)
Fig. 8. Bifurcation diagrams y2 − y1 versus εx , for (a) unidirectional, and (b) bidirectional coupling, in the case C2 = 70.0 nF. Synchronization is not observed in any case. The initial conditions of each individual circuit belong to the same basin of attraction.
(a)
Fig. 9. The unidirectionally-coupled Chua’s canonical circuits (y-coupling).
attractor. In Fig. 10, the bifurcation diagrams y2 − y1 versus εy are shown for unidirectional and bidirectional y-coupling. Synchronization, y2 − y1 = 0, is now observed in both cases. In the second case, however, synchronization is established for a lower critical value of the coupling factor εy . When the strange attractor of the canonical Chua’s circuit is a spiral one, synchronization is always observed even though the initial conditions of the y-coupled circuits belong to different basin of attraction (Fig. 11).
(b) Fig. 10. Bifurcation diagrams y2 − y1 versus εy , for (a) unidirectional, and (b) bidirectional y-coupling, in the case C2 = 60.0 nF. Synchronization is observed in both cases.
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(a)
(b)
Fig. 11. Bifurcation diagrams y2 − y1 versus εy , for (a) unidirectional and (b) bidirectional y-coupling, in the case C2 = 67.0 nF. Synchronization is observed in both cases. The initial conditions of each individual circuit belong to different basins of attraction.
5. Experimental Results In the following figures, (Figs. 12 and 13), we now exhibit some characteristic experimental phase space portraits of the coupled system for different values of the coupling factor. In Figs. 12(a) and 13(a) the system is out of synchronization,
while in Figs. 12(b) and 13(b) the system is in chaotic synchronization. The results of Fig. 13(b) demonstrate that the system can experimentally achieve synchronization, even at parameter values not predicted by the simulations. That the converse may also occur is discussed in the next section.
(a)
(b)
Fig. 12. Experimental phase portraits, υC11 versus υC12 , for the unidirectionally y-coupled system, when C2 = 60.0 nF; (a) εy = 0.0015, the system is out of synchronization, (b) εy = 0.04, chaotic synchronization.
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(b)
Fig. 13. Experimental phase portraits, υC11 versus υC12 , for the unidirectionally y-coupled system, when C2 = 67.0 nF (spiral attractor); (a) εy = 0.003, the system is out of synchronization, (b) εy = 0.016, chaotic synchronization.
6. Uncertainty in Chaos Synchronization According to the bifurcation diagram of Fig. 5(a), in the case of unidirectional x-coupling for C2 = 60.0 nF, chaotic synchronization must be observed for εx > 0.315. The experimental realization of the coupled system revealed that a different situation can also be observed (Fig. 14), in which the system, for a sufficiently high value of εx , can be out of synchronization. To see how this can occur, let us examine more closely the experimental attractors of the driving (master) and the driven (slave) circuit shown here in Fig. 15. The attractor of the driving circuit is the well-known double scroll attractor [Fig. 15(a)], while the attractor of the driven circuit is an unknown chaotic attractor [Fig. 15(b)]. In the absence of coupling, for C2 = 60.0 nF, there are two coexisting attractors. The chaotic double scroll attractor (see also Fig. 15(a) and a limit-cycle of period-1 (Fig. 16)). The initial conditions of the chaotic double scroll attractor are x0 = −1.0, y0 = 0.40, and z0 = 0.50, while the initial conditions of the limit-cycle of period-1 are x0 = −1.0, y0 = 1.0, and z0 = 2.0. For εx = 1.20, the computer simulated phase portraits, x1 versus x2 of the x-coupled system, and
Fig. 14. Experimental phase portrait, υC11 versus υC12 , for the unidirectionally x-coupled system, when C2 = 60.0 nF. The coupling factor is εx = 1.20, and the system is out of synchronization. Horizontal axis 0.5 V/div, vertical axis 0.5 V/div.
y2 versus x2 of the driven system, are shown in Fig. 17. The driven circuit oscillates in the basin of attraction of the big limit cycle giving a new type of chaotic attractor, similar to the one observed experimentally in Fig. 15(b). In contrast to this property of the double scroll attractor, the chaotic spiral attractor does not
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(a)
(b)
Fig. 15. Experimental phase portrait, for the unidirectionally x-coupled system, when C2 = 60.0 nF. The coupling factor is εx = 1.20. (a) υC21 versus υC11 , horizontal axis 0.5 V/div, vertical axis 0.5 V/div; (b) υC22 versus υC12 , horizontal axis 5.0 V/div, vertical axis 0.5 V/div.
For C2 = 60.0 nF, each Chua’s canonical circuit of the system with the parameter values listed above operated in a double-scroll attractor before being coupled. We have chosen the following sets of initial conditions for the computer simulation: x1 (0) = −1.0, y1 (0) = 0.4, z1 (0) = 0.5, x2 (0) = 1.0, y2 (0) = −0.4, z2 (0) = −0.5.
6.1.1. x-Coupled system
Fig. 16. The coexisting attractors for C2 = 60.0 nF. The initial conditions are (a) x0 = −1.0, y0 = 0.40, z0 = 0.50, for the chaotic double scroll attractor, (b) x0 = −1.0, y0 = 1.0, z0 = 2.0, for the limit-cycle of period-1.
have a coexisting periodic attractor, so an uncertainty of chaos synchronization of this kind is not observed in that case.
6.1. Anti-phase synchronization As we have mentioned in Sec. 2, anti-phase synchronization can be observed, when two identical bidirectionally coupled systems with odd symmetry have equal initial conditions with opposite sign.
As the coupling factor εx is increased, starting from zero (the uncoupled case), a change of the type of the attractor of each subsystem is observed. The double-scroll attractor gives its place to a chaotic spiral attractor and then a reverse period doubling sequence is observed, while anti-phase synchronization is conserved. The reverse period doubling sequence as the coupling factor εx increases is shown in Fig. 20. For 0.215 ≤ εx ≤ 0.680, the oscillations in the two Chua’s canonical circuits were quenched and each circuit operated in a point attractor located at (0, 0). For εx ≥ 0.690 oscillations were present again and each Chua’s canonical circuit operated in a period-1 attractor with anti-phase.
6.1.2. y-Coupled system In the case of the y-coupled system, the reverse period-doubling sequence is also observed, as the
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(b)
Fig. 17. The computer simulated phase portraits, (a) x1 versus x2 of the x-coupled system, and (b) y2 versus x2 of the driven system, for C2 = 60.0 nF and εx = 1.20.
Fig. 18. Phase portrait y2 versus y1 in the case of anti-phase synchronization (y-coupled system, εy = 0.001) and the computational environment.
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(a1 )
(a2 )
(b1 )
(b2 )
(c1 )
(c2 )
Fig. 19. y2 versus x2 and y2 versus y1 phase portraits, in the case of anti-phase synchronization of the bidirectionally x-coupled identical Chua’s canonical circuits with opposite identical initial conditions, for C2 = 60.0 nF: (a) εx = 0.010. Double-scroll attractor. (b–d) εx = 0.020, 0.130, 0.142. Spiral attractors. (e) εx = 0.146. Period-4 attractor. (f) εx = 0.150. Period-2 attractor. (g) εx = 0.180. Period-1 attractor.
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Synchronizing Chaotic Attractors of Chua’s Canonical Circuit
(d1 ) (d2 )
(e1 ) (e2 )
(f1 )
(f2 )
Fig. 19.
(Continued )
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(g1 )
(g2 ) Fig. 19.
(Continued )
7. Concluding Remarks
Fig. 20. Bifurcation diagram x2 versus εx of the bidirectionally x-coupled identical Chua’s canonical circuits with opposite identical initial conditions.
coupling factor εy is increased. For 0.030 ≤ εy ≤ 0.390 the oscillations in the two Chua’s canonical circuits were quenched and each circuit operated in a stable point attractor. The coordinates of the point attractor are dependent on the value of εy . For εy ≥ 0.40, the point attractor is located at (0, 0, 0). In the case of y-coupled identical Chua’s canonical circuits with opposite identical initial conditions, the transition of the system from the point attractor to the period-1 attractor was not observed.
In the present work, using two identical Chua’s canonical circuits resistively coupled, we have studied the process of synchronization, both in unidirectional and bidirectional coupling forms, using computer simulation and experiment. The results have shown that the dynamics of the coupled system is strongly affected by the initial states and the coupling strength. Furthermore, although the experimental results generally agree with the predictions of the simulations, there are some differences, which demonstrate, that the actual electrical circuits may include additional effects, which are not present in our systems of differential equations. Our observations suggest that the synchronization state of the y-coupled system is the most robust. When each individual circuit operates in a double-scroll or a spiral chaotic attractor, chaotic synchronization is always observed, when the coupling strength exceeds a critical value. In the case of the x-coupled system, however, when each individual circuit operates in a spiral attractor mode, chaotic synchronization is not observed. On the other hand, computer simulations show that when the circuits display double-scroll attractor dynamics, chaotic synchronization can be observed. Generally, the convergence of the system to the state of synchronization is very slow. Our experimental studies have also demonstrated that the state of synchronization is very sensitive to the unavoidable mismatches of the components of the
Synchronizing Chaotic Attractors of Chua’s Canonical Circuit
coupled circuits, leading to an experimental uncertainty of chaotic synchronization. The study of anti-phase synchronization by computer simulation verified the reverse period-doubling scenario and oscillation-quenching observed by Zhong et al. [2001]. A new feature we noted, in the case of the x-coupled system, is the period-1 oscillation, which become possible after the state of oscillation-quenching, as the coupling factor is further increased.
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