Synchronization of chaos in a SAB based chaotic electronic oscillator Tanmoy Banerjee, Biswajit Karmakar, Debabrata Biswas and B. C. Sarkar Department of Physics, Burdwan University, Burdwan 713 104, West Bengal, India. (email:
[email protected])
Abstract In the present paper we have explored the synchronization of chaotic oscillations in two-coupled Single Amplifier Biquad (SAB) based autonomous chaotic electronic oscillators. The synchronization scheme we have employed is the unidirectional feedback synchronization. The mathematical model of the proposed feedbackcoupled circuit is a set of eight first order coupled nonlinear differential equations involving voltages and currents. The behavior of the coupled circuit has been investigated through numerical simulation and SPICE based circuit simulator and it has been found that for certain system parameter zone, two chaotic circuits synchronize with each other.
1. Introduction The field of chaotic synchronization has grown considerably since last decade [1]. Researchers have realized that chaotic systems can be synchronized even though these systems defy synchronization due to their
high
sensitivity to initial conditions. There are several types of possible synchronization schemes e.g. feedback synchronization, phase synchronization, complete synchronization, generalized synchronization, delay synchronization, anticipatory synchronization, impulsive synchronization, etc. From application point of view Feedback synchronization is the simplest one, which is achieved by one way coupling of two identical chaotic systems. By one-way coupling we mean that the behavior of second chaotic system (slave or response) is dependent on the behavior of the first identical system (master or drive) but the converse is not true. The one way coupling method is also called as feedback method as in this we choose a drive variable from the drive system and feedback control is applied to the response system. Under suitable conditions, as time elapses, the amount of feedback decreases and soon both the drive and response systems achieve complete synchronization by following the same trajectory and afterward the feedback amount becomes zero and the identical synchronization persists. The importance of chaotic synchronization has been identified in the form of chaos based electronic communication system, which is believed to be secured and interference free. In the present paper we have explored the synchronization of chaos in a SAB based autonomous chaotic electronic oscillator proposed recently [2]. The unidirectional feedback synchronization is considered here. To realize the chaotic oscillator we have converted the SAB into a sinusoidal oscillator using proper positive feedback and then it has been modified for generating chaos using suitable passive nonlinearity and a storage element in the form of an inductor. Two such identical electronic oscillators have been coupled by a unity gain non-inverting buffer and a variable resistance which acts as a coupling parameter. One of the oscillators acts as Drive oscillator (D) and the other one, which is driven by the drive, acts as Response oscillator (R).
2. Circuit description with Feedback scheme Figure 1 shows the circuit diagram of unidirectional coupled chaotic SAB oscillator. Each unit (i.e. drive and response) comprises of a chaotic SAB oscillator proposed recently. In this circuit a parallel Diode Inductor (DL)
1
arrangement has been introduced between the V1 terminal (through a resistance R1) and ground. The diode switches on and off according the voltage developed across the inductor. This inductor voltage appears across diode parasitic transit capacitance CD and let that voltage be VCD. The circuit behavior can be modeled by a set of four first order coupled autonomous differential equations in voltages (V0, V1 and VCD) and inductor current IL. Unity gain noninverting buffer
C
C
R2 V1
+
C
V0
R2
V1 C
+
Ra R1
D
+ Vcd -
Ra
R1
L
Rb
D
+ Vcd -
L
V0
Rb
2 (k 1) (k 1) dV0 C dt R V0 R V1 R V CD 2 1 1 dV1 (2k 1) k k C dt ( k 1) R V0 R V1 R VCD 2 1 1 (1) dI L VCD dt L dV C D CD 1 V1 I L 1 VCD I D dt R1 R1 The nonlinear diode current ID has been modeled as a piece-wise linear function of voltage such that
Drive
Response
ID = 1 / RD (VCD-V γ), if VCD ≥V γ; 0 if VCD
