This document discusses chaotic systems and synchronization. It describes the tent map, which is a mathematical function that exhibits chaotic behavior. An electronic circuit implementation of the tent map is also presented, using operational amplifiers, diodes, and resistors. The document explains how varying parameters in the circuit allows replicating the theoretical behavior of the tent map. Synchronization between two chaotic systems is achieved when their outputs become identical after a number of iterations when exchanging information.

2.  CHAOTIC SYSTEM
 TENT MAP
 SYNCHRONIZATION
 HARDWARE IMPLEMENTATION
 ELECTRONIC IMPLEMENTATION OF TENT MAP
 SCHEMATIC DIA. OF TENT MAP ELECTRONIC CIRCUIT
 COMPONENTS
 SOFTWARE VRIFICATION CIRCUIT
 SYCHRONIZATION IMPLEMENTATION
 REFERENCE

3.  Chaotic systems are unstable since they tend not to resist
any outside disturbances but instead react in significant
ways.
 Chaotic systems are common in Nature.

4.  Many natural phenomena can also be characterized as
being chaotic. They can be found in meteorology , solar
system, heart and brain of living organisms and so on.
 It is difficult for two chaotic systems to synchronize with
each other, but if the two systems exchange information in
just the right way, they can synchronize

5.  The tent map with parameter μ is the real-valued
function fμ defined by:
 In particular, iterating a point x0 in [0, 1] gives rise to a
sequence :
where μ is a positive real constant.

6.  This is called ‘tent map’ due to tent like shape of the
graph of fμ.

7. Synchronization between two system means feeding both the
transmitter and the receiver with the same carrier signal for
Demodulating the modulated signal accurately.
Need for Synchronization:
 Synchronization is one of the most important
requirements for designing a Digital Communication
System

8.  Here we present one of the simplest electronic
implementation of the tent map, which at same time is a
good engineering model of the corresponding
mathematical system.
 Through the variation of tent map control parameter we
are able to reproduce the theoretical diagram with high
accuracy.

9. There are various processes for electrical implementation of
tent map-
1. Using analog multipliers
2. Using digital signal processor
3. New approach [1]

12. The above diagram consists of-
 five operational amplifiers(from U1 to U5),
 four diodes (D1 - D4),
 thirteen resistors (from R1 to R13
 a dc voltage source (Vdc).
 variable resisors R5 & R6(5kῼ)
 variable resistors R11(100kῼ) & R12(10kῼ)

15. When y(n)=x(n) after around few iterations (say 5), f1=f2. [5]

17.  A very simple tent map electronic circuit has been presented
here and its implementation using only analog components as
operational amplifiers, diodes, and resistors was also
provided.
 The circuit that replicates the whole known range of
behaviors of the tent map has been determined.
 The employed techniques are simple and the approach can be
extended to other types of chaotic maps such.
 Such circuit realizations have many potential applications, for
example: random number generation, frequency hopping,
ranging, and spread-spectrum communications.
 Finally, we notice that this design can be manufactured in just
one chip because the final electronic circuit contains only
semiconductors and passive components.

18. 1. Campos-Cantón, I., E. Campos-Cantón, J. S. Murguıa, and H. C. Rosu. "A
Simple Electronic Circuit Realization of the Tent Map." Chaos, Solitons
& Fractals 42 (2009): 12-16.
2. May, Robert M. 1976. "Simple mathematical models with very
complicated dynamics." Nature 261(5560):459-467.
3. Gleick, James (1987). Chaos: Making a New Science. Penguin Books.
4. Strogatz, S. H. Nonlinear Dynamics and Chaos Perseus Books 1994. ISBN
0-201-54344-3
5. Hardware Implementation of Tent Map, Debanjana Datta, Bidisha De
and Subhajit Paul, Submitted in ICETIC 2015
6.https://www.google.co.in/search?q=block+diagram+of+digital+communi
cation+system&biw=1024&bih=677&source=lnms&tbm=isch&sa=X&ved=
0ahUKEwjczNvIpZ_JAhVRTo4KHe1ZAQIQ_AUIBigB#imgrc=oFOLMrjZFT2Hh
M%3A
7.https://www.google.co.in/search?q=synchronisation+in+digital+communi
cation&biw=1024&bih=677&source=lnms&tbm=isch&sa=X&ved=0ahUKEw
jWnLqrs5_JAhWCSo4KHcAoAmoQ_AUICCgD&dpr=1#imgrc=ty0UO-
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