SIMULATION AND EXAMINATION OF A SIGNAL MASKING CHAOTIC COMMUNICATION SYSTEM, BASED ON THE DUFFING OSCILLATOR SIMULATION AND EXAMINATION OF A SIGNAL MASKING CHAOTIC COMMUNICATION SYSTEM, BASED ON THE DUFFING OSCILLATOR

The study in this paper is focused on the applications of the chaotic Duffing oscillator to signal masking communication scheme. Using the concept of synchronized chaotic systems the signal masking approach is demonstrated with the Duffing oscillator implemented in both the transmitter and receiver. The chaotic masking signal is added at the transmitter to the information signal, and at the receiver, the masking is regenerated and subtracted from the received signal. Chaotic behavior and chaotic masking scheme are designed and simulated using Simulink/Matlab program. All the results are used to illustrate the effectiveness and the applicability of the Duffing oscillator in signal masking communication.


Chaotic masking approach description
The chaotic signal masking approach is based on the identical synchronization, where the state of response system converges asymptotically to the state of the drive system [6].

Two continuous-time chaotical systems
x (1) and x (2) are said to synchronize identically if (3) for any combination of initial states [4,7].
In the equations (1) and (2) x, x ෂ are according the vectors of the state of the two systems, t -the time.
The system (1) is located in the transmitter and system (2) in the receiver (Fig. 1). If the same initial condition is chosen for the transmitter and the receiver, the both systems will evolve in a syn- chrony in the sense that, x (1) will continue being equal to x (2) for all t Ͼ 0.
The idea of a chaotic masking approach is that the information signal is masked by directly adding a chaotic signal at the transmitter. Because chaotic signals are typically broadband, noise like, and difficult to predict, they can be used for masking informationbearing waveforms.
At the receiver which is synchronized to the transmitter the chaotic component is subtracted from the received signal to recover the original transmitted message.
The transmitter state evolution is given by the chaotic dynamics A chaotic signal u(t) which is a function of the transmitter state x(t) is added to the information signal m(t).

The transmitted signal s(t) is governed by
The evolution of the receiver state x r (t) dynamics is given by the dynamics . (6) and v(t) is the chaotic signal which is a function of the receiver state x r (t).
, dt Then the combined signal (5), received by the receiver, can be used to regenerate the chaotic signal u(t) through chaos synchronization. If the synchronization is successful, the v(t) converge to u(t) after a short initial transient and the original message can be recovered by subtracting the reproduced chaotic signal v(t) from the received signal s(t): The masking scheme works only when the amplitudes of the information signals are much smaller than the masking chaotic signals The systems with chaotic masking works effectively in small disturbances in the channel (high ratio signal/noise).

Model construction
The Duffing oscillator is given by the second-order differential equation [4] with an external periodic drive term: , (9) or by an equivalent set of two first order non -autonomous equations , where a, A and ω are parameters.
The transmitter state variables are denoted by the vector d ϭ (x, y) (11) and the receiver variables by the vector Assume that the transmitter is described by the equation (10). A response system that will synchronize to the chaotic signals at the transmitter (10) is described by , where δ is a coupling parameter to couple the receiver with the transmitter.
This system is obtained from the transmitter equations by renaming variables (x, y) to (x r , y r ). The synchronization between the transmitter (10) and receiver (13) has been proved by analyzing the stability of the error signal. It is derived from the difference between the output of the receiver system and that received from the transmitter: The error signal is then used to modify the state of the receiver such that it can be synchronized with the transmitter. This signal is given by the system with .
The characteristic equation of system (15) is If a 1 Ͼ 0 and a 2 Ͼ 0, then e 1 ϭ e 2 ϭ 0 is a stable fixed point [3] and the receiver will eventually synchronize with the transmitter.

Simulation results
Chaotic signal masking communication scheme, based on the Duffing oscillator, is designed and simulated using Simulink/Matlab program. Figure 2 shows the simulation scheme. The information signal m(t) is a sinus wave of amplitude 1 volts and of 5 kHz frequency.
Transmitter and receiver systems are identical. In the present case a ϭ 0.2 is accepted.
As follows of (19) the transmitter and receiver are synchronized when δ Ն 5. In the following analysis the synchronization between the transmitter and the receiver is obtained with an absence of disturbance.
The information signal is added to the generated chaotic signal u(t), and the signal is fed into the receiver.
The chaotic signal v(t) is generated in receiver system and The examination is based on a simulation of the systems in the transmitter and the receiver unto different initial conditions and comparison of the driving signal u(t) with the response signal v(t).
The phase plane of the transmitter and receiver chaotic signals is shown in Figs. 3 and 4. Figure 5 shows the difference u(t) Ϫ v(t) of scope.
It is seen that after a transient period the error signal of the transmitter and receiver systems goes to zero, showing that the two systems are synchronized.
One can see from the figures that the synchronization is satisfactory.
When the frequency of the information signal is increased to 10 kHz , the error signal shown in Fig. 6 is much smaller indicating that the quality of the message signal recovery is greatly improved.
The simulation results show that, when frequency increasing, the two chaotic systems come faster in synchronisation and the oscillation signal (u-v) is smaller. This is confirmed by the graph of Fig. 7, which shows the error signal for f ϭ 15kHz.

Conclusions
In this paper, chaotic masking communication system of the Duffing oscillator is designed and simulated. The proposed simula-tion model of Simulink / Matlab allows to monitor the difference between chaotic signals in the transmitter and the receiver and the synchronization time of the two systems.
According to numerical simulations, by a choice of parameters the synchronization error states converge to zero and hence the synchronization between chaotic systems is achieved.
In this paper was investigated the dependence for error signal of the frequency, that with increasing frequency the chaotic systems enter more easily into synchronisation.
Simulation results are used to visualize and illustrate the effectiveness of Duffing chaotic system in signal masking.
The results of simulatition show that the Duffing chaotic system can be used for transmission of confidental information. Fig. 7 Error state of the transmitter system and receiver system for f ϭ 15kHz