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  In our project we have studied the communication with chaotic system.

The Chaotic behavior has been observed in the laboratory in a variety of 

systems such as electrical circuits, lasers, oscillating chemical reactions,

fluid dynamics, and mechanical and magneto-mechanical devices.  veryday

 phenomena are showing e!amples of chaotic systems such as weather and

climate. There is some controversy over the e!istence of chaotic dynamics in

the  plate tectonics  and in economics. "e investigated some important

 properties of chaotic systems using the logistic map and its bifurcation

diagram. #lso we show the universality found in $$the route to chaos$$. "e

discussed analogue chaotic communication system and synchroni%ation.

Chaos is the study of comple! nonlinear dynamic system .The name &chaos

theory& comes from the fact that the theory describes a system is apparently

disordered. The logistic map is a  polynomial mapping, often cited as an

archetypal e!ample of how comple!, chaotic behavior can arise from very

simple non-linear   dynamical e'uations.The logistic model was originallyintroduced as a demographic  model  by (ierre )ran*ois +erhulst/.

0ifurcation theory is the study of how and when such bifurcations. #

 bifurcation occurs when a small smooth change made to the parameter 

values 1the bifurcation parameters2 of a system. 3uch a change causes a

sudden $'ualitative$ or topological change in the system$s long-term dynamical

 behavior. 0ifurcations occur in both continuous systems and discrete

systems. The analog signal is continuous in time and in amplitude. There has

 been considerable interest in the possibility of e!ploiting chaos in wide band

communication system. Two types of analog chaotic systems have been

investigated synchroni%ed and unsynchroni%ed chaotic systems. The general

approach to synchroni%ation is to ta4e two or more identical chaotic systems

and couple them together in such a way that the chaotic behavior of all the

systems is the same.

"e have considered some chaotic systems. )irst we have discussed

Chua chaotic system. The criteria for choosing Chua5s circuit is its

simplicity. It e!hibits a variety of chaotic phenomena simpler than by other 

comple! circuits, which ma4es it a popular circuit. #lso we have showed the

Chua5s generator circuit diagram, its component, and how does the circuitwor4 #lso we have demonstrated the simulation of Chua chaotic generator.

3econd, we have discussed 7oren% chaotic generator, its e'uations,

7oren% attractor, and simulation of 7oren% chaotic generator. The 7oren%

e'uations represent  the convective motion of fluid cell  which is warmed

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http://en.wikipedia.org/wiki/Electrical_circuitshttp://en.wikipedia.org/wiki/Lasershttp://en.wikipedia.org/wiki/Chemical_reactionshttp://en.wikipedia.org/wiki/Fluid_dynamicshttp://en.wikipedia.org/wiki/Plate_tectonicshttp://en.wikipedia.org/wiki/Polynomialhttp://en.wikipedia.org/wiki/Chaos_theoryhttp://en.wikipedia.org/wiki/Non-linearhttp://en.wikipedia.org/wiki/Demographyhttp://en.wikipedia.org/wiki/Model_(abstract)http://en.wikipedia.org/wiki/Pierre_Fran%C3%A7ois_Verhulsthttp://en.wikipedia.org/wiki/Topologicalhttp://planetmath.org/encyclopedia/RepresentableFunctor.htmlhttp://planetmath.org/encyclopedia/CellAttachment.htmlhttp://en.wikipedia.org/wiki/Lasershttp://en.wikipedia.org/wiki/Chemical_reactionshttp://en.wikipedia.org/wiki/Fluid_dynamicshttp://en.wikipedia.org/wiki/Plate_tectonicshttp://en.wikipedia.org/wiki/Polynomialhttp://en.wikipedia.org/wiki/Chaos_theoryhttp://en.wikipedia.org/wiki/Non-linearhttp://en.wikipedia.org/wiki/Demographyhttp://en.wikipedia.org/wiki/Model_(abstract)http://en.wikipedia.org/wiki/Pierre_Fran%C3%A7ois_Verhulsthttp://en.wikipedia.org/wiki/Topologicalhttp://planetmath.org/encyclopedia/RepresentableFunctor.htmlhttp://planetmath.org/encyclopedia/CellAttachment.htmlhttp://en.wikipedia.org/wiki/Electrical_circuits

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from below and cooled from above. The same system can also apply to

dynamos. It has been showed that the 7oren% #ttractor is a solution to a set

of differential e'uations originally developed to model small scale

atmospheric behavior. The plotted results of 7oren% attractor show a

 butterfly attractor that is unusual visuali%ations of deterministic chaos.

  Third, we have investigated the Chen$s chaotic generator. The Chen5s

attractor is recently observed and reported. In contrast with the 7oren%

 butterfly attractor8 Chen5s attractor is topologically more comple! but

without changing the smooth 'uadratic functions. The circuit

implementation of Chen5s attractor and the synchroni%ation of two Chen5s

systems have been discussed. The mathematical model of Chen5s system and

an electronic circuitry reali%ation for the system and simulation of Chen

chaotic generator are also elaborated, implemented and fabricated.

  )ourth, :ssler chaotic generator has been mentioned. :ssler$s system

is probably the simplest ;-< ordinary differential e'uations that have

'uadratic nonlinearity and e!hibits chaotic behavior. These differential

e'uations define a continuous-time dynamical system that e!hibits chaotic

dynamic associated with the fractal properties of the attractor. The :ssler 

attractor is a system of three non-linear ordinary differential e'uations.

  In our project we have considered, e!perimentally, the Chen$s chaoticgenerator. "e have discussed the mathematical model of Chen5s system and

the circuitry implementation of Chen5s attractor. #lso we have calculated

and designed the values of parameters and elements used in the circuit.

)inally we have illustrated the e!perimental setup, implementation, and

results of the Chen$s chaotic generator.

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