Synchronization via multiplex pulse trains - Circuits and Systems I ...

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Synchronization via Multiplex Pulse Trains Hiroyuki Torikai, Member, IEEE, Toshimichi Saito, Member, IEEE, and Wolfgang Schwarz, Member, IEEE

Abstract— The master–slave synchronization of a complex chaotic system using multiplex pulse trains is considered. Each master outputs a chaotic pulse train of narrow pulses, the intervals of which are governed by a chaotic map. The slave system has the winner-take-all function. A simple realization of the systems is shown. We provide a theorem which guarantees the synchronization in the case that all masters and passive slaves have identical parameter values. Experiments confirm that the synchronized state changes intermittently and we clarify this mechanism. Furthermore, in order to obtain robust synchronization, a stabilization technique of the chaotic system is proposed and applied. The stabilization is guaranteed theoretically and some experimental results are given. Index Terms— Chaos, demultiplexing, multiplex communication, multiplexing, pulse train, synchronization, winner-take-all.

I. INTRODUCTION

R

ECENTLY, synchronization phenomena in chaotic systems have been treated intensively, e.g., chaos synchronization in master–slave systems and mutually coupled systems [1]–[10]. There are many studies on the application of such phenomena to communication systems (see [11] and references therein). In these systems the chaotic signals are treated as continuous- or discrete-time analog valued signals. Then the synchronized state can sometimes be broken by disturbances and/or distortions as discussed in [12]. We consider the chaos synchronization using multiplex pulse trains. In our system, the information for synchronization is contained in the time intervals and not in the signal value. Synchronization by such a pulse train is an interesting topic since the pulse trains can be transmitted via a noisy channel. Synthesis and analysis of such chaotic pulse-train generators (CPG’s) has been studied recently [9], [10], [13]–[19]. Also, the synchronization of multiple chaos by means of a multiplex signal is interesting from the theoretical viewpoint of chaos synchronization and might be applicable in multiplex communication schemes with chaos, as discussed in [20]–[22]. The objectives of this paper are the following. • Design of the System Structure: A master–slave system scheme is considered. In the master system, the multiplex signal is obtained by simply adding the pulse trains. The slave system has the winner-take-all function.

Manuscript received September 16, 1997; revised May 4, 1998. This paper was recommended by Associate Editor C. W. Wu. H. Torikai and T. Saito are with the Department of Electronics and Electrical Engineering, Hosei University, Tokyo, Japan (e-mail: [email protected] and [email protected]). W. Schwarz is with the faculty of Electronic Engineering, Technical University Dresden, Dresden, Germany (e-mail: [email protected]). Publisher Item Identifier S 1057-7122(99)07206-2.

• Analysis of the Synchronization Behavior: We prove a theorem which guarantees the synchronization between the master system and passive slave system where all masters and slaves have identical parameter values. We also analyze an implemented circuit. • Improvement of the Synchronization Performance: Stabilized periodic pulse trains are used in order to obtain robust synchronization. • Personalization of the Master–Slave Pairs: In order to personalize the masters and the slaves, different parameters are assigned to different master–slave pairs. Experiments confirm the personalization. The master is a CPG [13], [14] which is a simplified version of a forced relaxation oscillator discussed in [23] and [24]. The pulse trains are generated by one-dimensional (1-D) return maps which can easily be changed, thus providing the generation of a variety of pulse trains. Experiments confirm that chaos synchronization is intermittent. Intermittent synchronization has been studied recently [6]–[8] and we clarify this phenomenon for our structure. In order to obtain robust synchronization, a stabilization using a higher frequency external periodic signal is proposed and applied. This leads to a discretization of the return map [13], [14], and the resulting finite state system generates periodic signals only. Although some experimental results on taming chaos by periodic perturbations have been reported [10]–[26], we treat this from a more theoretical viewpoint. (Some comments on the words taming and stabilizing are given in Section III.) Finally, using a stabilization procedure, we confirm analytically and experimentally that the system shows robust synchronization.

II. SYNCHRONIZATION VIA MULTIPLEX PULSE TRAINS A. The System 1) General Principle: We consider the synchronization behavior in the master–slave system depicted in Fig. 1. In the masters which have the discretemaster system, there are . The masters output pulse time states . These pulse trains are multiplexed by simply trains is sent to the slave system adding and the multiplex signal slaves and via a single line. In the slave system, there are the pulse distributor, which distributes incoming pulses to the slaves. The pulse distributor has the winner-take-all function. The th master outputs the pulse train , as shown in and Fig. 1(b), where the th pulse position is denoted by st pulse positions the pulse interval between the th and . The pulse intervals is denoted by

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(a)

Fig. 3. A direct realization of the master.

(b) Fig. 1. The master–slave system. (a) Whole scheme. (b) The output pulse trains yi of the masters and the multiplex signal p( ).

other one is a subsystem of the master, which cannot oscillate by itself. The states of the slaves are fed back to the pulse distributor, which is designed to realize the synchronization. 2) Master Structure: In order to realize the master, the structure in Fig. 3 could be used [15], [16]. In this figure, is a controlled analog delay. The value-time converter outputs successive instantaneous pulses. The intervals between them . At every pulse moment, the time-value converter are and gets the next pulse interval clocks the analog delay . Here we introduce a system structure (see Fig. 4) which directly generates the pulse train without using a time-value converter. This structure is a simplified version of a forced relaxation oscillator discussed in [23] and [24]. The basic unit (BU) consists of an integrator, an adder, and the controlled switches. The pulse generator (PG) consists of the BU, the comparator, and the pulse former. The masters’ state equations are given by for

Fig. 2. An example of the return map f . si = 0:3 and ri = 2.

is set to

(2)

if

if otherwise

from the th master are given by

where the is a linear scaling function and is the parameter vector of the th master and slave. The state is governed by the following return map:

where is the normalized time, is the continuous-time state of the th master, is the number of the masters and the is one system parameter. The signal slaves, and is an internal signal of the th BU which is given by (3)

(1) An example of the return map is shown in Fig. 2. According : to [27] and [28], the system (1) will generate chaos for . the masters generate chaotic pulse trains We consider two kinds of slave structures. One is the same structure as the master and can oscillate autonomously and the

is the second system parameter and is periodic where with period one and is called the base signal, defined as (4) The time-domain waveforms are shown in Fig. 5. The state increases and reaches of the th master starting from

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The pulse interval

is given by (see Fig. 5)

(8) Using the pulse position return map and the scaling , we as get the return map for the pulse intervals

Fig. 4. The CPG.

(9) and the pulse position Note that the pulse interval are equivalent via the scaling , hence, we analyze the . pulse train by using are given by the differThe absolute pulse positions ence equation (10) with the solution (11)

Fig. 5. Time-domain waveforms from the CPG.

the threshold value . At this moment the comparator switches, an instantaneous pulse is generated by the pulse . After this, the process former, and the state is set to is repeated. Recall that the th pulse position of the th master . Then the th master outputs a pulse train is denoted by such that if otherwise.

(5)

Here we analyze the pulse train by means of the pulse intervals, in which the information for synchronization is contained. The is determined by the value of the internal pulse interval which is periodic with period one. Hence, for signal convenience, we introduce the restricted time coordinate , as shown in Fig. 5. Then the pulse positions are governed by the following pulse position return map: (6) which is identical to the return map (1) by transforming the parameters as (7)

Since generates chaos, the masters in (2) generate chaotic . Hence, we call each master a CPG. pulse trains The graph of the return map is determined by the form . Hence, by adjusting the base signal, of the base signal the CPG can produce a variety of pulse trains. Return maps of the form (1) have been treated intensively (e.g., [29]). Then, using the substructures of the CPG, the master system and the slave system are realized, as shown in Fig. 6. The master system consists of PG’s (see Fig. 4), a common base signal generator (BSG), and an adder. The multiplex signal is obtained by simply adding the chaotic pulse trains (12) In our system this is realized by a multiple logical OR gate. An example of the chaotic pulse trains and the multiplex signal is shown in Fig. 1(b). 3) Slave Structure: In the slave system, the base signal , which is identical to the base reconstructor outputs [see Fig. 6(a)]. The pulse distributor is a dynamic signal winner-takes-all circuit which controls the switches depending on its inputs . The switch is closed if and only if the input is the maximum in . We consider the slave systems. First, the BU’s (see Fig. 4) are used as the slaves: the terminal (b) of the th BU is connected to the switch . Then the slave system is described

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(a)

(b) Fig. 6. Realization of the master–stave system. (a) Whole scheme. (b) The slave generator.

is described by

by is set to

if

is set to

and

if

and

or otherwise,

otherwise (14)

(13) is the state of the th BU and is where the internal signal of the th BU. According to this algorithm, the input pulse train of the th BU will be if

and

otherwise. Since this slave system cannot oscillate without the signal , it is called the passive slave system. Second, we consider a slave system which consists of slave generators [see Fig. 6(b)]. The terminal ( ) of the th slave generator is connected to the switch . Then the slave system

is a small value. The input pulse train of the th where slave generator will be if

and

or otherwise. Since this system can oscillate by itself, we call it the active slave system. From active slave systems a better synchronization performance can be expected than from passive ones. 4) Implementation: Fig. 7(a) shows an implementation example of the BSG and the CPG. MF is the mono-stable flip flop which outputs an instantaneous pulse if its input changes from ) to high ( ). In this example, the current sources are low (

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is given by (15) . In the where is the period of the base signal and correspond to CPG, the capacitor voltage and the output , the BU the state and the output , respectively. If in (2). If reaches zero, the MF outputs an realizes is closed instantaneously, instantaneous pulse, the switch is set to the internal signal and consequently, . Then the CPG is described by (2)–(4) with the normalization

(16)

(a)

Fig. 7(b) shows measured waveforms from the implemented circuit. In Fig. 8(a) an implementation example of the master system and the slave system is depicted. In the master system, the multiplexing is realized by the logical OR gate which outputs . In the slave system, the base signal the multiplex signal which reconstructor (BSR) [see Fig. 8(b)] produces using the periodic clock pulse train . The repeats slave generator is shown in Fig. 8(c). Using the transformation , its circuit equation is transformed to (16) and the slave generator in (14). The passive slave system (13) is realized using the BU’s, and the active slave system (14) is realized using the slave generators, as described in point 3). The dynamic winner-take-all circuit consists of the maximum value selector, the comparators, and the sample and hold at circuits (S&H’s). The S&H’s sample the slaves’ states every incoming pulse moment and hold them until the end of each pulse. All outputs of the OP-amps are compared with where is the lower bound of . If is the maximum, the th AND gate is opened and thus the th BU (or slave generator) gets the actual pulse for synchronization. In in Fig. 6(a). this way, the AND gates realize the switches By means of the S&H’s, the maximum value selector is decoupled from the BU’s (or slave generators) and racing is avoided. repeats . In the Note that we assumed that implementation circuit, this can be achieved by the clock pulse which is sent to the slave system via an additional train line [see Fig. 8(a)]. A discussion on the clock pulse train is given in Section IV.

(b) Fig. 7. An implementation example of the BSG and the CPG and an experimental result. (a) Implementation example. (b) Measured time-domain : 3:3 [nF], I =: 11 [A], V =: 1:9 [V], C =: 3:3 [nF], waveforms. CB B B I1 =: 11 [A], VA1 =: 2:0 [V]. (Corresponding to 1 = 1 and a1 = 1.)

=

realized by operational transconductance amplifiers (OTA’s) where is and the output current is is the input voltage. the transconductance of the OTA and Here we assume that is always applied to the terminal (C) of the BSG, that is, the MF is driven only by the comparator. This terminal is used in the next section. Then the base signal

B. Behavior Analysis 1) Passive Slave with Identical Parameter Values: First, we give a theoretical result on synchronization between a master system (2) and a passive slave system (13) (see Fig. 6), where all slaves and masters have identical parameters, i.e., and for all . Thus, the internal signals in both for all . systems are identical, i.e., We assume that the multiplexed pulse trains are disjoint, i.e., for (17)

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(a)

(b)

(c) Fig. 8. An implementation example of the master-slave system. (a) Whole scheme (master system and passive/active slave system). (b) The base signal reconstructor. (c) The slave generator.

A necessary condition for that is that the initial states in the , , master system are disjoint: . Further, we assume that the initial states of the BU’s satisfy (18)

That is, each BU has different initial state which is larger . In order to analyze the behavior of the system under than the above assumptions, we give some definitions. of the th BU is set Definition 1: Suppose that the state , where . Then to the internal signal at . The the th BU is said to accept the pulse train

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Fig. 10. Time-domain waveforms from the passive slave system.

Fig. 9. Time-domain waveforms from the master system and the passive slave system.

th BU is said to trace for if for . for and the Suppose that the th BU traces th BU accepts at . Then the th BU is said to be by the th BU at . intercepted Fig. 9 shows time-domain waveforms from both the master . At system and the slave system for the second BU accepts the pulse train and traces for . At the first BU accepts although the second BU traced . Then the second BU is by the first BU. The second BU accepts at intercepted and traces for . Also, the first BU for . traces Definition 2: The master system and the slave system are if there exist a said to be synchronized for for all during unique BU which traces . . In Fig. 9 the synchronization is achieved for , the slave states are set to Note that after only when they are equal to zero. This property can be used as a criterion for synchronization. be the maximum (latest) pulse position Definition 3: Let and let be the maximum in such that . pulse position in

In Fig. 9 these are given by , and . for Definition 4: If a BU traces some pulse train , the BU is said to be a tracing basic unit (TBU) for , otherwise said to be a nontracing basic unit . If a pulse train is traced by some (NBU) for TBU, the pulse train is said to be a traced pulse train (TPT), otherwise it is said to be a nontraced pulse train (NPT). In Fig. 9, the second BU is a TBU and correspondingly, is TPT for . The first BU is an NBU for and it intercepts and changes to a TBU at . Consequently, the second BU becomes an NBU . The pulse train is an NPT during for . Then we have the main theorem for any . Theorem: Suppose that the PG’s in the master system and the BU’s in the passive slave system have identical parameters, and for all and (17) and (18) are i.e., at , satisfied. If the th BU accepts some pulse train for . This statement holds for all then it traces and, thus, the synchronization is achieved between the master system and the passive slave system for . Proof: We first show that the number of TBU’s and TPT’s are equal, that is, TBU’s and TPT’s correspond uniquely. Assume (for contradiction) that the number of TBU’s is larger than that of the TPT’s and, thus, there exists at least one TPT which is traced by more than one TBU. In this case, from the initial states of those TBU’s, either at least two are . This contradicts (18). identical or at least one is less than Also, it follows from (17) that the number of the TBU’s must not be less than that of the TPT’s. for Next we show that a BU traces some state if it accepts at . We focus on the state of the th BU, which is described by the bold trajectory in Fig. 10. Let . Now us suppose that the th BU accepts at for contradiction assume that (19) where

is some integer and that according to the algorithm

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(a)

(b)

(c)

(d)

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=

3 with Fig. 11. Experimental results from the master–slave systems for N identical values of Ii (i = 1; 2; 3) and VAj (j = 1; 2; 3). The parameters are the same as Fig. 7(b). (a) States of the PG’s and the multiplex signal in the master system. (b) Synchronization in the passive slave system. (c) Break of synchronization in the same passive slave system as Fig. 11(b). (d) : Time-domain waveforms from the active slave system. V = 0:3 [V].

(13), the th BU accepts at . At this changes from TPT to NPT and then there must moment, exist at least one NBU since the number of the NPT’s is equal to that of the NBU’s. For such an NBU two cases are possible: ; (A) the BU was an NBU during (B) the BU changed from a TBU to an NBU at some and was an NBU moment where is some during integer. An example of the trajectories for the two cases are shown in since Fig. 10. The NBU (A) satisfies and it does not decrease until . since The NBU (B) satisfies and it does not decrease until . Obviously, both cases contradict (19). Q.E.D. According to this theorem, the synchronization between the whole master system and the whole passive slave system is if noise does not exist. achieved after a transition period Note that, which specific slave is synchronized with which master depends upon the initial conditions. Fig. 11 shows measured time-domain waveforms from an implemented circuit. In Fig. 11(a) the states in the master system and the multiplexed pulse trains are shown. Fig. 11(b) shows waveforms from the passive slave system. It can be seen that the slave system is synchronized to the master system. Fig. 11(c) also shows the snapshot at another moment where synchronization is broken. As shown in these figures, the passive slave system is synchronized intermittently. The reasons for this imperfect synchronization include the following. may overlap or may be located very R1) The pulses in close to each other because of the mixing property of chaos [27]. Then the dynamic winner-take-all circuit

Fig. 12. Time-domain waveforms from the active slave system. The broken lines are the trajectories [(x01 ), (x02 ), and (x03 )] from the passive slave system.

cannot distinguish these pulses and the synchronization is broken. This will be discussed in the next point 2). R2) One pulse may be located very close to because of the ergodic property of chaos may be set to [27]. Then while its master state (synchronized state) is set to and vice versa. This also breaks synchronization. We have analyzed this phenomenon in [8]. Referring to [8], [27], and [28] in the return map , the pulse must be located very close to at least one positions discontinuous point. Since such pulse positions are mapped to or , synchronization is broken. very close to and 2) Active Slave System with Identical Values of : Next, we consider the active slave system (14). The timedomain waveforms from the master system and the active slave system are shown in Fig. 12. In this figure, the synchronization . Note that the pulses are very close is achieved at . If this occurs in the to each other at may overshoot zero, the pulse train passive slave system, may be intercepted by the first BU, and the synchronized state may be changed (the broken lines in Fig. 12). In the is set to due to the threshold active slave system, . Hence, the synchronized state can be held as long is the maximum at . Fig. 11(c) shows the as : experimental results for the active slave system for the system exhibits synchronization recovery performance. 3) Different Parameter Values: In order to personalize the slaves and the masters, different parameter values are assigned for to different master–slave pairs, i.e., . Fig. 13(a) and (b) shows the experimental results , where for the passive and active slave systems for

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(a) (a)

(b)

(b)

(c) Fig. 14. The principle of stabilization: (a) An example of f , unstable periodic points with period three, and corresponding orbits. r = 2, s = 1. (b) An example of h.  = 2; = 1=7; = =2. (c) The return map g .

(c)

to ) are assigned. In Fig. 13(d), the synchronized state is due to R1), but is recovered immediately broken at since the active slave has its own threshold (see Fig. 12). The personalization is observed intermittently. In both cases, the active system exhibits a more robust synchronization than the passive one. More detailed analysis will be done in the near future. (d) Fig. 13. Experimental results from the master–slave systems with different 3. The left snapshots are the time-domain parameter values for N waveforms and the right ones are the (v1 vi0 ); i = 1; 2; 3; planes. (a) : Intermittent synchronization from the passive slave system (VA1 = 2:0 [V], : : VA2 = 1:2 [V], VA3 = 2:6 [V]). The other parameters are the same as Fig. : in 7(b). (b) Intermittent synchronization from the active slave system (V = 0:3 [V]). The other parameters are the same as in Fig. 13(a). (c) Intermittent : : synchronization from the passive slave system (I1 = 11 [A], I2 = 17 : [A], I3 = 7:0 [A]). The other parameters are the same as in Fig. 7(b). : (d) Intermittent synchronization from the active slave system (V = 0:3 [V]). The other parameters are the same as in (c).

=

III. SYNCHRONIZATION VIA MULTIPLEX STABILIZED PULSE TRAINS

0

the different (corresponding to ) are assigned. In both cases, the first BU (slave) is synchronized, not perfectly, but intermittently with the first PG (master) and not synchronized with the other PG’s, that is, the first PG has its own BU. In synchronize for Fig. 13(b), and and the synchronized state is broken at due to R1). The personalization is observed intermittently. Fig. 13(c) and (d) shows the experimental results for the passive and active , where the different (corresponding slave systems for

A. Stabilization of Chaos by an External Periodic Signal Here we use a stabilization procedure for the chaotic system (2) by an external periodic signal. The principle of the stabilization technique is first explained by means of the return map. 1) Principle: In order to stabilize the periodic orbits of the return map in (1), we discretize the state of and obtain a new return map (see Fig. 14) (20) where

is given by

(21)

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the CPG is tamed by the external signal and we give some theoretical results of the basin of attraction for the stabilized periodic orbits. Then, by setting the initial state appropriately, we can get any desired stabilized periodic orbit. Hence, we use the term stabilization in this paper rather than taming. 3) Example: We consider the important case of in (1) with (26)

Fig. 15. Time-domain waveforms from the CPG with the external periodic signal ci ( ).

[see Fig. 14(b)]. The slope of is now zero almost everywhere and the variable is restricted to discrete values. Then all are periodic and stable. However, it orbits generated by is complicated to determine the periodic solution of (20) for arbitrary parameter values. 2) Realization: The above principle is realized by adding a periodic sawtooth wave (external signal) (22) of the th master in (2). Here , , to the internal signal and control the amplitude, frequency, and the phase of the , respectively. An example of is shown in signal Fig. 15. Then the internal signal of the th master is given by (23) is the base signal and is an offset. In order to where get the stabilized periodic pulse trains, we set

Then the return map is a Bernoulli map. We cite some basic definitions and results for periodic points and periodic orbits (see, for example, [31] and [32]). • A point is said to be a periodic point of with period if where denotes the -fold composition is of . A sequence said to be a periodic orbit. A periodic orbit is said to where and be stable if unstable otherwise. for all except for the discontinuous • Since points, all periodic orbits generated by the system (1) are unstable. An example of unstable periodic points and corresponding unstable periodic orbits is shown in Fig. 14(a). are given by • All unstable periodic points with period [32] (27) determines In the return map , an unstable periodic point an unstable periodic orbit. Therefore, a pulse train, generated by the CPG, started from , is to be periodic and we refer to it as an unstable periodic pulse train (UPP). A UPP started from is denoted by . From simple algebra it follows that all UPP’s with period , started from different periodic points, are disjoint (have no overlapping), i.e., for (28) In order to stabilize the unstable periodic orbits of with in (21) with period , we use the function

(24) (29) must always be negative in order to The internal signal guarantee oscillation, hence, must satisfy for all . As a result, instead of the chaotic map (6), we get the new return map

Then, according to the principle from point 1), the system is governed by the return map

(25)

(30)

which is identical to the system (20) via the transformation (7). Since the system (20) can generate only periodic orbits, the masters can output only periodic pulse trains. Some experimental results on taming chaos by periodic perturbations have been reported [10]–[26]. The taming of chaos implies changing a chaotic orbit to a periodic one by changing the stability of a chaotic system locally. On the other hand, the stabilization of chaos implies achieving a desired stabilized periodic orbit by some procedure. In this paper,

. Equation (30) Fig. 14(c) shows an example of for under implies that all unstable periodic orbits with period the parameter (26) can be stabilized by applying the function with the parameter values in (29). with the Using this result, the external periodic signal parameters (31)

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is added to the internal signal in order to obtain stabilized periodic pulse trains. Under these parameter values, the pulse positions are governed by the return map (30). Therefore, all UPP’s with period embedded in the chaotic pulse trains can because is be stabilized, with the exception of located at the edge point. determines the first pulse position The initial state . Hence, an initial state gives a stabilized UPP where (32) (a)

of initial states and the stabilized UPP’s The intervals have a one-to-one correspondence since the UPP’s in (28) are disjoint. 4) Implementation: Fig. 16(a) shows an implementation example of the external periodic signal generator (EPG). The circuit is similar to the BSG in Fig. 7(a). The mono-stable which flip-flop MF2 outputs a periodic pulse train synchronizes the BCG. The phase difference can be adjusted . Then the base signal and the by are given by external signal for the th master

(b)

(33) defined in (1) where is the period of with the function and the period of the base signal is the external signal Int . The external periodic given by signal is applied to the PG’s (see Fig. 6) in the master system, as shown in Fig. 15(b). Then the master system is described by (2) and (23) with the transformation (16) and (c)

Fig. 16(c) shows waveforms from an experimental circuit realization.

Fig. 16. An implementation example of the stabilization procedure and an experimental result. (a) The EPG and the BSG. (b) The stabilization procedure. : 150 [mV], V =: 0:0 [V], (c) Measured time-domain waveform. VD G ki =: 2:0. The other parameters are the same as in Fig. 7(b). (Corresponding to  = 2, = 1=7, and = =2.)

=

B. Synchronization Behavior In order to confirm the Theorem in Section II-B, we consider first the passive slave system (13) where each PG and BU (see Fig. 6) have the identical parameter values (31). In disjoint stabilized UPP’s with period , we order to get . Under these conditions the pulse trains assume starting from different periodic points are disjoint. Then, by of the PG’s choosing the initial states in (32), disjoint stabilized UPP’s from different intervals will be obtained. Since the stabilized UPP’s are the solutions of the system (2) as well, the theorem holds, and robuster synchronization can be achieved. Fig. 17(a) shows measured , . waveforms from the implemented circuit for The upper-left and the lower-left diagrams show time-domain waveforms from the master system and the passive slave

system, respectively. The right diagrams show planes. The response speed of the winner-take-all circuit defines the allowable minimum pulse interval of the multiplex signal. This determines an upper limit for the frequency of the external . periodic signal Next we consider the case where the PG’s have different parameter values. From the theorem, the following condition guarantees the synchronization. The parameters and the initial state in the master system are such that the disjoint stabilized UPP’s. masters output

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Fig. 18. Generalized scheme of the stabilization. TABLE I STABILIZATION SCHEME Sb

Sc

d(  )

pulse-train y ( )

open

open

0a

periodic

closed

open

0 a + b( )

chaotic

closed

closed

0 a + b( ) + c( )

stabilized periodic

(a)

master–slave pairs can be achieved. The theoretical analysis of the behavior for the case where the above condition is not satisfied will be done in future. C. Generalized Stabilization Scheme

(b) Fig. 17. Experimental results from the master–slave systems with the stabilized periodic pulse trains. (a) Time-domain waveforms from the master system (the upper left) and the passive slave system (the lower left) and (v1 vi0 ); i = 1; 2; 3 planes (the right) with identical parameter values. The parameters are the same as in Figs. 11 and 16. (b) Time-domain waveforms from the master system (the upper left) and the passive slave system (the lower left) and (v1 vi0 ); i = 1; 2; 3 planes (the right) with different parameter values. The parameters are the same as in Figs. 13(a) and 16.

0

0

The initial states

of the BU’s satisfy if

where and

, . implies that both

Roughly speaking, the condition and are closely located. We provide experimental results which satisfy this condition in Fig. 17(b). It can be seen that the th BU is synchronized with the th PG for all , that is, the personalization of the

A generalized scheme of the stabilization procedure considered in this paper is depicted in Fig. 18. The periodic signal generator 1 (corresponding to PG in Fig. 4) has an output and an input which is determined by the switches and . The periodic signal generator 2 (corresponding to the , and the periodic signal generator BSG in Fig. 4) outputs [corresponding to 3 outputs a higher frequency signal . Here, and the external periodic signal in (22)] than are synchronized. In dependence from the switches the following cases, shown in Table I, can be distinguished. is open and is closed, i.e., , the If will be periodic or chaotic depending on the pulse train . Note that we treated the case where parameter values of [in (4)] and [in (2)], hence the system generates chaos only. We clarified theoretically that the frequency of the external determines the properties (e.g., period) of periodic signal the stabilized periodic orbits. The stabilizing chaos technique by external periodic signals is simpler than the ones described in [33] and can be applied up to higher frequencies since it does not require state feedback [26]. Our results contribute to the theoretical analysis of both stabilization and bifurcation phenomena in periodically perturbed chaotic systems [25], [26]. IV. CONCLUSIONS We have considered the master-slave synchronization via multiplex pulse trains. Two slave structures, a passive one

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TABLE II SYNCHRONIZATION PHENOMENA

(non-self-oscillating) and an active one (self-oscillating), have been proposed. The obtained results can be summarized in Table II. The following are some future problems: 1) generalization of the theorem for different parameter values assignment and multiplexing the clock pulse train into the multiplex signal; 2) more detailed analysis of the synchronization behavior and its break; 3) analysis of the chaotic pulse train for a wider parameter range; 4) generalization and detailed analysis of the proposed stabilization technique. Some application aspects such as the chaotic pulse-train generator and the stabilization procedure changing local stability of a system by the external periodic signal, may be developed into an information coding procedure, as discussed in [34]–[38]. It has been suggested that (chaotic) pulses play important roles in neural networks [39]–[41] and the intermittent synchronization of chaos is important in memory search dynamics [6], [42]. Our results may contribute to such chaos-based information processing and transmission schemes.

ACKNOWLEDGMENT The authors wish to thank the following for their valuable discussions and helpful comments: N. Suda, A. L. Fradkov, M. Hasler, M. J. Ogorzalek, M. P. Kennedy, T. Shimizu, T. Kohda, T. Ushio, K. Jin’no, and K. Mitsubori. REFERENCES [1] L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett., vol. 64, pp. 821–824, 1990. [2] C. W. Wu and L. O. Chua, “Synchronization in an array of linearly coupled dynamical systems,” IEEE Trans. Circuits Syst. I, vol. 42, pp. 430–447, Aug. 1995. [3] N. F. Rul’kov, A. R. Volkovskii, A. Rodriguez-Lozano, E. Del Rio, and M. G. Valarde, “Mutual synchronization of chaotic self-oscillators with dissipative coupling,” Int. J. Bifurcation Chaos, vol. 2, pp. 669–676, 1992. [4] T. Ushio, “Control of chaotic synchronization in composite systems with application to secure communication systems,” IEEE Trans. Circuits Syst. I, vol. 43, pp. 500–504, 1996. [5] H. Torikai and T. Saito, “Occasional linear connection for synchronization of chaos,” IEEE Trans. Circuits Syst. I, vol. 43, pp. 374–385, May 1996. [6] K. Kaneko, “Clustering, coding, switching, hierarchical ordering, and control in a network of chaotic elements,” Physica, vol. D41, pp. 137–172, 1990. [7] N. Platt, E. A. Spiegel, and C. Tresser, “On-off intermittency: A mechanism for bursting,” Phys. Rev. Lett., vol. 70, no. 3, pp. 279–282, 1993.

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Hiroyuki Torikai (S’95–M’99) received the B.E., M.E., and Ph.D. degrees in electrical engineering from Hosei University, Tokyo, Japan, in 1995, 1997, and 1999, respectively. He is currently an Assistant at the Department of Electronics and Electrical Engineering, Hosei University. His research interests are in chaotic circuits and neural networks.

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Toshimichi Saito (M’88) received the B.E., M.E, and Ph.D. degrees in electrical engineering, from Keio University, Yokohama, Japan, in 1980, 1982, and 1985, respectively. He is currently a Professor with the Department of Electronics and Electrical Engineering, Hosei University, Tokyo, Japan. His current research interests include chaos and bifurcation, artificial neural networks, digital communication, power electronics, and fuzzy systems.

Wolfgang Schwarz (M’92) received the Dipl.-Ing., Dr.-Ing., and Dr.-Ing. Habil. degrees from the Technical University, Dresden, Germany, in 1965, 1969, and 1976, respectively. From 1965 to 1969, he worked as a Lecturer at the Technical University, Dresden, and in 1969, he became an Associate Professor of Control Engineering at the University of Technology, Mittweida, Germany. From 1974 to 1978, he worked as an RD-Engineer of machine-tool control systems in the industry. In 1976, he became a full Professor of Information Technology at the University of Technology, Mittweida, and, since 1983, he has been a full Professor at the Technical University, Dresden. His research interest is in the field of nonlinear dynamic systems and circuits and analog circuit design. He teaches fundamentals of electrical engineering and electronic circuits at the Faculty of Electronic Engineering, Technical University, Dresden.