Impulsive Synchronization Between Double-Scroll Circuits Ch.K. Volos, S.G. Stavrinides, I.M. Kyprianidis, I.N. Stouboulos, M. Ozer, and A.N. Anagnostopoulos
Abstract Two identical, double-scroll circuits, in their chaotic mode of operation, are unidirectionally connected via an externally triggered electronic switch. Thus, the case of impulsive synchronization is established. Their synchronization and its dependence on the switch on-off frequency and duty cycle is demonstrated.
1 Introduction Nowadays, wireless networks and mobile telephony has become part of everyday life. Serving multiple receivers by a single cell transmitter, without reducing connection speed or creating interference problems, has always been a major matter. Next to proper coding, the time sharing technique—dividing a certain time span into discrete timeslots—has been proposed as a suitable solution addressing this problem. On the other hand chaotic synchronization between transmitter and receiver has been successfully applied for ultra wideband and secure signal transmission [1, 2] and related phenomena have been extensively studied [3]. Reclaiming the advantages of those two approaches, impulsive synchronization arises as a synergy between them.
Ch.K. Volos Department of Mathematics and Engineering Studies, Hellenic Army Academy, Athens, Greece S.G. Stavrinides () Department of Electrical Engineering, Kavala Institute of Technology, Kavala, Greece e-mail:
[email protected] I.M. Kyprianidis I.N. Stouboulos A.N. Anagnostopoulos Physics Department, Aristotle University of Thessaloniki, 54124, Greece M. Ozer Physics Department, Kultur University (Atakoy Campus), Bakirkoy, Istanbul, Turkey S.G. Stavrinides et al. (eds.), Chaos and Complex Systems, DOI 10.1007/978-3-642-33914-1 65, © Springer-Verlag Berlin Heidelberg 2013
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The case of chaotic intermittent chaotic synchronization between identical, double-scroll circuits has been already studied in [4]. In the present work impulsive chaotic synchronization, in the case of unidirectional coupling, is experimentally demonstrated.
2 The Coupled Double-Scroll Circuit System The system presented in Fig. 1 consists of two identical, autonomous, nonlinear circuits, producing double-scroll chaotic attractors by properly adjusting their parameters [5]. An op-amp buffer (built around an LF411), a linear resistor RC and an electronic switch (CD4066), externally driven, provide the impulsive, unidirectional coupling, between the two nonlinear circuits. The coupled-circuit system equations are displayed in (1). In this set, the three equations on the left describe the driving circuit (transmitter), while the three equations on the right describe the driven circuit (receiver). dx1 dx2 D y1 D y2 dt dt dy1 dy D z1 2 D z2 C .y1 y2 / (1) dt dt dz2 dz1 D a.x1 C y1 C z1 / C b f .x1 / D a.x1 C y1 C z1 / C b f .x1 / dt dt State parameters x1;2 ; y1;2 , and z1;2 stand for normalized voltages at the outputs of the correspondingly numbered at each coupled circuit, operational amplifiers, (Fig. 1). Parameters ˛ and b are defined as follows: ˛ D .RC/1 b D .Rx C /1
(2)
The function f .x1;2 / in (1) is a saturation function (with saturation plateaus at ˙1 and an intermediate linear region slope k D R3 =R2 /. It stands for the output voltage at op-amp “5” (Fig. 1) and it is defined by the following expression:
f .x/ D
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R2 =R3 .V / R3 =R2 x; R2 =R3 .V / x R2 =R3 .V / : 1; x < R2 =R3 .V /
(3)
Coupling coefficient , is defined—in the case of discontinuous coupling—in (4): D
R=RC ; 0;
for t < Td for t > Td
(4)
Impulsive Synchronization Between Double-Scroll Circuits
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Fig. 1 The double-scroll coupled scheme Table 1 Circuit parameters
Element R R1 R2
Value 20 k 1 k 14:3 k
Element R3 RX RC
Value 28:6 k 10 k 10 k
The circuits’ parameter values were set at ˛ D 0:5, b D 1:0 and k D 2:0, ensuring chaotic operation of each double-scroll circuit. All the op-amps used, were LF411, while power supply voltages were set to ˙15 V. Capacitor C D 1nF and resistor values appear in Table 1.
3 Experimental The electronic switch, inserted in the coupling between the identical chaotic circuits, determines the value of coupling coefficient according equation (4). Switching frequency f and duty cycle Td served as the system’s control parameters. It should
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Fig. 2 Synchronization ensembles in the case of 50 % switch duty-cycle, when switch frequency is (a) f D 100 Hz, (b) f D 1:3 kHz, (c) f D 3 kHz and (d) f D 10 kHz
be mentioned that their values ranged from 10 Hz up to 3.5 kHz and from 30 % up to 70 %, respectively. A four channel digital oscilloscope (Yokogawa DL9140) was used to register voltage signals x1 (yellow), x2 (green), in both sub-circuits, as well as, the switch control pulse (purple) and the difference signal [x1 .t/ x2 .t/] (red), that depicted the system synchronization (all four signals are mentioned as they appear from top to bottom in the figures that follow). In order to perform the related statistics, the difference signal [x1 .t/x2 .t/] was registered by a second digital oscilloscope (Tektronix TDS1002B) that was further connected to a laptop, facilitating the software needed for the evaluation. It should be mentioned that during measurements the oscilloscope was controlled by this laptop. For this reason a custom-made acquisition setup [6], based on NI’s LabView, was created. We have chosen to vary frequency f for successive constant values of duty cycle Td . In any case both circuits remained chaotic, exhibiting a double scroll attractor. As expected in similar cases of impulsive synchronization, the circuits are synchronized as long as the switch is on and desynchronized when the switch goes off. But this is not always the case. For the most usual case of a 50 % duty cycle, synchronization and desynchronization, corresponding to on and off states of the switch, are observed in the range from 10 Hz to 2 kHz—Fig. 2a, b. In the range from 2 to 15 kHz the circuits’ dynamics
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Fig. 3 Synchronization ensembles in the case of 30 % switch duty-cycle, when switch frequency is (a) f D 100 Hz, (b) f D 3 kHz
Fig. 4 Synchronization ensembles in the case of 70 % switch duty-cycle, when switch frequency is (a) f D 100 Hz, (b) f D 3 kHz
lead to synchronization–desynchronization transitions not corresponding to the to on and off states of the switch—Fig. 2c—due to the appearance of a kind of slow transition decay. For higher frequencies, this decay prevails and the circuits remain synchronized independently from the switch state—Fig. 2d. The influence of switch duty cycle becomes critical at higher frequencies and for low duty cycles. For low duty cycle, as in case of 30 % and for low frequencies, the circuits are synchronized as long as the switch is on and desynchronized when the switch goes off (impulsive)—Fig. 3a. But for higher frequencies the circuits remain desynchronized independently from the switch state—Fig. 3b. For the case of high duty cycle such as 70 %, the circuits are impulsively synchronized, according to the switch on-off frequency, as long as the switch frequency is low—Fig. 4a. For higher frequencies they remain almost fully synchronized regardless of the switch state—Fig. 4b.
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4 Results For the purpose of technical applications, like those mentioned in the introduction, only the low and high frequency .f / cases are of technical importance. On the other hand the influence of switch duty cycle becomes critical at higher frequencies and for low duty cycles. For low duty cycle as in case of 30 % and for low frequencies the circuits are synchronized as long as the switch is on and desynchronized when the switch goes off (impulsive). But for higher frequencies the circuits remain desynchronized independently from the switch state. For the case of high duty cycle such as 70 %, the circuits are impulsively synchronized, according to the switch on-off frequency, as long as the switch frequency is low. Finally, for higher frequencies they remain almost fully synchronized regardless of the switch state.
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