One model which attempts to explain the reversal of the Earth's magnetic field is the Rikitake system (Rikitake, 1958). This system describes the currents of two coupled dynamo disks (Ito, 1980; McMillen, 1999). The governing equations are:
Here a and μ are parameters which we will assume to be nonnegative. Synchronization
of chaotic systems has in recent years become an area of active research. Different
approaches are proposed and being pursued. Among themes the technique of Pecora-Carroll
(Pecora and Carroll, 1990), who show that, when a state variable from a chaotically
evolving system is transmitted as an input to a replica of part of the original
system, the replica subsystem (receiver) sometimes synchronizes to the original
system (sender). They and others suggest that this phenomenon of chaos synchronism
may serve as the basis for new ways to achieve secure communication (Carroll
and Pecora, 1993), (Cuomo and Oppenheim, 1993). This study focuses on the identical
synchronization using Pecora-Carroll technique to Rikitake system for secure
Simulink and Pspice simulations of Rikitake attractor: Using Matlab
Simulink modeling of Rikitake Attractor in Fig. 1., Poincare
maps in the xy, xz and yz planes are attained in Fig. 2.
Figure 3 shows the circuit schematic for implementing the
Rikitake Eq. (1). We use TL084 opamps, the Analog Devices AD633JN
multipliers, appropriate valued resistors and capacitors for Pspice simulations.
The circuit is supplied ±12 V power supplies. Acceptable inputs to the
AD633 multiplier IC are -10 to +10 V.
||Matlab-Simulink model of Rikitake attractor
||Phase portraits of the Rikitake attractor when μ = 2,
a = 5, x0 = 0, y0 = 0.1 and z0 = 0
||Circuit scheme of Rikitake Attractor
||The Pspice simulation results of the Rikitake attractor circuit.
(a) x, y phase portrait (b) x, z phase portrait (c) y, z phase portrait
||Lyapunov exponents of the Rikitake system (1) for merely parameters
μ = 2, a = 5
||Block diagram of a cascaded synchronization system
||Simulink Modeling of Pecora-Carroll Synchronization of Rikitake
||Simulink outputs of Pecora-Carroll Synchronization of Rikitake
Attractor a) Drive system X signal b) Response system Xr signal c) Synchronization
between X and Xr
The resistors R1-R7,
are all shown with nominal values in Fig. 3. Figure
4 shows Pspice simulation results of this circuit. Pspice and Matlab-Simulink
simulations (Fig. 2) give the same conclusions.
Figure 5 shows the calculus of the Lyapunov Exponents performed
the Rikitake system in the case of μ = 2, a = 5. It is worth noting that
only one positive LE is present.
||Pspice Circuit of Pecora-Carroll Synchronization of Rikitake
Synchronization of the Rikitake system: Synchronization between chaotic
systems has received considerable attention and led to communication applications.
||Pspice simulation outputs of Pecora-Carroll Synchronization
of Rikitake Attractor Circuit a) Drive system X signal b) Response system
Xr signal b) Synchronization between X and Xr
There are two major methods for coupling and synchronizing identical chaotic
systems, the cascading method and the one-way coupling method. With these methods,
a message signal sent by a transmitter system can be reproduced at a receiver
under the influence of a single chaotic signal through synchronization. This
paper presents the study of numerical simulation of chaos synchronization for
chaotic Rikitake System. The method of synchronization is Pecora-Carroll method;
drive subsystem and response subsystem were constructed. Figure
6 shows block diagram of a cascaded synchronization system. Figure
7 pointed out simulation modeling of P-C Synchronization of Rikitake Attractor.
There are two major methods in chaos synchronization of coupled identical systems; the cascading method and the one way coupling method. The idea of the methods is to reproduce al1 the signals at the receiver under the influence of a single chaotic signal from the driver. Therefore, chaos synchronization provides potential applications to communications and signal processing. However, to build secure communications system, some other important factors, need to be considered. Simulations of synchronization of Rikitake Attractor are presented as shown in Fig. 8.
Figure 9 shows the circuit schematic for implementing the
Rikitake attractos Pecora-Carroll Synchronization. We use TL081 opamps,
the Analog Devices AD633JN multipliers, appropriate valued resistors and capacitors
for Pspice simulations. The circuits are supplied ±12 V power supplies.
Figure 10 shows Pspice simulation results of this circuit.
Pspice and Matlab-Simulink simulations (Fig. 8) give the same
In this study, we have studied applications of chaos synchronization. Method is the one-way coupling method, the idea is using a parameter to couple two identical chaotic systems and makes them synchronizing. But this coupling is not mutual. The behavior of the response system depends on the behavior of the drive system but not invertible. We have demonstrated in simulation that chaos can be synchronized and applied to secure communication schemes. This paper focuses on the identical synchronization for secure communications. Chaos synchronization was investigated using Matlab-Simulink and Pspice programmes. Related Fig. 2, 4, 8 and 10) show that Pspice and Matlab-Simulink simulations give the same conclusions.