_{1}

^{*}

For the model of a Closed Phase Locked Loop (CPLL) communication System consists of both the transmission and receiver ends. This model is considered to be in a multi-order intermittent chaotic state. The chaotic signals are then synchronized along side with our system. This chaotic synchronization will be demonstrated and furthermore, a modulation will be formed to examine the system if it will perfectly reconstruct or not. Finally we will demonstrate the synchronization conditions of the system.

A chaotic system has various properties one of which we will consider in our paper here is the property of uncorrelated trajectory that forms by the exponential divergence of the initial conditions, which varies with time. It is not straightforward to demonstrate a pair of chaotic systems that get synchronized together perfectly.

In previous work done by Pecora and Carroll [

Our model’s state variables are time variant composed of a set of pair differential equations for each identical system that is in a chaotic state.

The eminent theory of synchronization contains in the deterministic concept of chaotic systems. With time variation, the deterministic system of the transition of the state variables is connected by a given set of differential equations. In our case, two identical chaotic systems are assumed to be simultaneously active. Hence, the differential equations will govern the state variables of each system. Moreover, the set of the initial conditions is reflected by the divergence of the trajectories that were initially stated.

In this paper, the use of a closed phase locked loop CPLL system is studied. This system consists of a transmission and receiver that will be forced by a common chaotic signal. Furthermore, the system representing the mathematical model was derived of the differential equation and then was simulated. These results will be developed through the synchronization of the multi-order intermittent CPLL model [

Chaos in closed phase locked loops has been researched by many researchers in various institutions around the world for at past few years. Closed phase locked loops similar to many chaotic systems that are close to Chua’s circuit series [

The CPLL that was considered as a chaos generator for this systems has a response that is undesirable for many of the typical communication systems. In one case, the CPLL is used to demodulate an FM signals, as well as the output of the system may become chaotic for particular loop parameters. This behavior drives the CPLL to reach to unlock its state [

Both, Endo and Watada [

Finally, both Harb & Harb [

Next we superimpose the control signal into the CPLL system, and then integrate it into the new system form; this will enable it to obtain the simulation results as illustrated in

The closed phase locked loop model is sketched in the block diagram shown in

We have a chaotic 1-CPLL that will drive the complete system to be at state of out of lock. Thus, this will convert the receiver chaotic signal of 0-CPLL to produce anther chaotic situation. Meanwhile, 2-CPPL will in turn becomes an attractor of chaos, and the common chaotic signal c(t) will synchronize the complete system. We note that the initial conditions and parameters are unchanged and are identical throughout the numerical solution [

We then modulate the information signal m(t) in the transmission end of the system with a chaotic signal. Furthermore, we then synchronize the receiver end of the system along with the transmission end of the system.

We initially start by driving the differential equations of the system. This is done by taking the receiving end containing 0-CPLL and 1-CPLL to a chaotic state as represented in Equation (1),

ϕ ⃛ o + a ϕ ⃛ o + b cos ( ϕ o ) ϕ ¨ o + c ϕ ˙ o + d cos ( ϕ o ) ϕ ˙ o − b sin ( ϕ o ) ϕ ˙ o 2 + sin ( ϕ o ) = δ (1)

Next, we take the differential equation representing 1-CPLL is represented by Equation (2),

ϕ ⃛ 1 + a ϕ ¨ 1 + b cos ( ϕ 1 ) ϕ ¨ 1 + c ϕ ˙ 1 + d cos ( ϕ 1 ) ϕ ˙ 1 − b cos ( ϕ o ) ϕ ˙ o − d cos ( ϕ o ) ϕ ˙ o + b sin ( ϕ o ) ϕ ˙ o 2 − b sin ( ϕ 1 ) ϕ ˙ 1 2 + sin ( ϕ 1 ) − sin ( ϕ o ) = 0 (2)

And for the transmission side 2-CPLL we have Equation (3),

ϕ ⃛ 2 + a ϕ ¨ 2 + b cos ( ϕ 2 ) ϕ ¨ 2 + c ϕ ˙ 1 + d cos ( ϕ 2 ) ϕ ˙ 2 − b cos ( ϕ o ) ϕ ˙ o − d cos ( ϕ o ) ϕ ˙ o + b sin ( ϕ o ) ϕ ˙ o 2 − b sin ( ϕ 2 ) ϕ ˙ 2 2 + sin ( ϕ 2 ) − sin ( ϕ o ) = 0 (3)

We will introduce a set of variables to develop our model into a state variable set of space equations,

x 1 = ϕ o , x 2 = ϕ ˙ o , x 3 = ϕ ¨ o , x 4 = ϕ 1 , x 5 = ϕ ˙ 1 , x 6 = ϕ ¨ 1 , x 7 = ϕ 2 , x 8 = ϕ ˙ 2 , x 9 = ϕ ¨ 2

Hence, our system’s state space equations becomes,

x ˙ 1 = x 2 x ˙ 2 = x 3 x ˙ 3 = δ − a x 3 − b cos ( x 1 ) x 3 − c x 2 − d cos ( x 1 ) x 2 + b sin ( x 1 ) x 2 2 − sin ( x 1 ) x ˙ 4 = x 5 x ˙ 5 = x 6 x ˙ 6 = − a x 6 − b cos ( x 4 ) x 6 − c x 5 − d cos ( x 4 ) x 5 + b cos ( x 1 ) x 3 + d cos ( x 1 ) x 2 − b sin ( x 1 ) x 2 2 + b sin ( x 4 ) x 5 2 + sin ( x 1 ) − sin ( x 4 ) x ˙ 7 = x 8 x ˙ 8 = x 9 x ˙ 9 = − a x 9 − b cos ( x 7 ) x 9 − c x 8 − d cos ( x 7 ) x 8 + b cos ( x 1 ) x 3 + d cos ( x 1 ) x 2 − b sin ( x 1 ) x 2 2 + b sin ( x 7 ) x 8 2 + sin ( x 1 ) − sin ( x 7 ) (4)

We have run the simulation in order to verify the synchronization within the communication system that is based on our CPLL model. These simulations resulted as expected that the synchronization depended on the initial conditions along with the parameters of our system.

We selected the phase errors Ф_{1} and Ф_{2} to be as the transmission end and receiver end respectively. Furthermore, we introduce an information sinusoidal signal and denote it by m(t) to be multiplied by the transmission phase error Ф_{1}. As a result,

As a result of a mismatch in the initial conditions for both ends of the transmission and receiver ends would indicate a non-synchronized signal as it is apparent in

In general, we have proven that the chaotic system depends implicitly to the origination of a matched initial conditions and matched parameters of the complete system in study.

Our simulation and analysis indicate that the two chaotic outputs of identical

CPLLs, V_{co} would result in a common chaotic input synchronized sinusoidal signal. The initial conditions along with the original parameters matching as well as with both the transmission and receiver ends will result into a one to one synchronization system. It has also been proven that a small mismatch in the either the initial conditions or the original parameters would result in an impossible recovery of the information sinusoidal signal transmitted and received.

Shariff, S.M. (2018) Multi-Order Intermittent Chaotic Synchronization of Closed Phase Locked Loop. International Journal of Modern Nonlinear Theory and Application, 7, 48-55. https://doi.org/10.4236/ijmnta.2018.72004