Chaotic Communication
Communication with Chaotic Dynamical
Systems
Mattan Erez
December 2000
Chaotic Communication
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Not an oxymoron
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Chaos is deterministic
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Two chaotic systems can be synchronized
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Chaos can be controlled
Communicating with chaos
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Use chaotic instead of periodic waveforms
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Control chaotic behavior to encode information
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Outline
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What is chaos
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Synchronizing chaos
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Using chaotic waveforms
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Controlling chaos
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Information encoding within chaos
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Capacity
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Summary: Why (or why not) use chaos?
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References and links
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What is Chaos?
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Non-linear dynamical system
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Deterministic
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Sensitive to initial conditions
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x(t )  et x(0) ( - Lyapunov exponent)
Dense
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Infinite number of trajectories in finite region of phase space
perfect knowledge of present
imperfect knowledge of present
perfect prediction of future (practically) no prediction of future
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Continuous Time Systems
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Described by differential equations
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dimension  3 for chaotic behavior
Example: Lorenz System
 x   ( y  x)

 y  rx  y  xz
 z  xy  bz

, r, and b are parameters
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Useful Concepts
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Attractor: set of orbits to which the system
approaches from any initial state (within the
attractor basin)
Poincare` Surface of Section
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Discrete Time Systems
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Described by a mapping function

Can be one-dimensional

Logistic Map
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x(n)  x(n)(1  x(n))
Bernoulli Shift
1
xn 1  2 xn mod 1 0  x  1

0.5
Tent Map
1
time
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Chaos Synchronization
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Non-trivial problem
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sensitivity to initial conditions + density
initial state never accurate in a real system
trivial if dealing with finite precision simulations
Chaotic Synchronization
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Couple transmitter and receiver by a drive signal
Build receiver system with two parts
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(Pecora and Carrol Feb. 1990)
response system and regenerated signal
Response system is stable (negative Lyapunov exp.)
Converges towards variables of the drive system
Can synchronize in presence of noise and
parameter differences
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Example - Lorenz System
 x   ( y  x)

 y  rx  y  xz
 z  xy  bz

X
Y
Z
x(t)
s(t)
n(t)
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 xr   ( yr  xr )

 y r  rx  yr  xzr
 z  xy  bz
r
r
 r
Xr
Yr
Zr
xr(t)
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Chaotic Waveforms in Comm.
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Chaotic signals are a-periodic
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Spread spectrum communication
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Instead of binary spreading sequences
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Directly as a wideband waveform
Code-division techniques
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Replaces binary codes
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Chaotic Masking
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Mask message with noise-like signal
Amplitude of information must be small
X
Y
Z
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x(t)
m(t)
s(t)
n(t)
Xr
Yr
Zr
xr(t) - +
mr(t)
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Dynamic Feedback Modulation
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Mask message with chaotic signal
Removes restriction on small message amp.
Care must be taken to preserve chaos
X
Y
Z
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x(t)
m(t)
s(t)
n(t)
Xr
Yr
Zr
xr(t) - +
mr(t)
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Chaos Shift Keying
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Modulate the system parameters with the
message
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Similar concept to FSK but for a different parameter
Suitable mostly for digital communication
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Shift to a different attractor based on information symbol
m(t)

X
Y
Z
x(t)
s(t)
n(t)
Xr
Yr
Zr
xr(t) - +
detector
mr(t)
Also DCSK to simplify detection
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Problems in Conventional CDMA
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Binary m-sequences
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Binary gold sequences
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good cross-correlation
acceptable auto-correlation
few codes
Binary random maps
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good auto-correlation
bad cross-correlation
few codes
good auto-correlation
good cross-correlation
many codes
very large maps (storage)
Very long and complex (re)synchronization
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Chaotic Sequences for CDMA
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Simple description of chaotic systems
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Very large number of codes
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mostly based on numerical results
“Checbyshev sequences” yield 15% more users
Fast synchronization
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a-periodic with a flat (or tailored) spectrum
Good auto/cross correlation
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many useful maps
many initial states (sensitivity to initial conditions)
Good spectral properties
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one dimensional maps
If based on self-sync chaotic systems
Low probability of intercept
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chaotic sequence are real-valued and not binary
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Chaos in Ultra WB - CPPM
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Impulse communication
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uses PN sequences and PPM
PN spectrum has spectral peaks
Chaotic Pulse Position Modulation
t (n)  F (t (n  1)  tinforamation )
001101
t0 = 0
t1 = t
t(0)

t(1)
t(2) t(3) t(4)
Circuit implementation
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simple tent map and time-voltage-time converters
extremely fast synchronization (4 bits)
Low power
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Controlling Chaos
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Chaotic attractor (usually) consists of infinite
number of unstable periodic orbits
Small perturbation of accessible system
param forces the system from one orbit to a
more desirable one (Ott, Grebogi, and Yorke - Mar. 1990)
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the effect of the control is not immediate
each intersection of the phase-space coordinate eith
the surface of section a control signal is given
the exact control is pre-determined to shift the orbit to
the desired one, such that a future intersection will
occur at the desired point
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Encoding in Chaos
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Use symbolic dynamics to associate
information with the chaotic phase-space
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phase space is partitioned into r regions
each region is assigned a unique symbol
the symbol sequences formed by the trajectories of
the system are its symbolic dynamics
Identify the grammar of the chaotic system
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the set of possible symbol sequences (constraint)
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depends on the system and symbol partition
Exercise chaos control to encode the
information within the allowed grammar
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Example - Double Scroll System
0
1
1
0
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Symbolic Dynamics Transmission
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Use previous regions for two symbols
Build coding function - r(x)
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Build an inverse coding function s(r)
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for each intersection point (region) - record the
following n-bit sequence
define a region as the mean state-space point
corresponding to the n-bit sequence r.
Build a control function d(r)
 small perturbations: p = d(r)x
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Transmission (2)
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Encode user information to fit the grammar
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use a constrained-code based on the grammar
for the experimental setup demonstrated, the
constraint is a RLL constraint
Transmit the message
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load the n-bit sequence of r(x0) into a shift register
shift out the MSB and shift in the first message bit (LSB)
the SR now holds the word r1’ with the desired information bit
the next intersection occurs at x1=s(r1) of the original system
at that point we apply the control pulse to correct the
trajectory: p=d(r1)(x1-s(r1’))
repeat
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Receiver
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Threshold to detect 0 and 1
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decode the constrained-code
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Capacity of Chaotic Transmission
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The capacity of the system is its topological
capacity

define a partition and assign symbols w
count the number of n-symbol sequences the system
can then produce N(w,n)

H top  sup lim

w

n 
N ( w, n )
n
Additional restrictions on the code (for
noise resistance) decrease capacity
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Noise Resistance
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Force forbidden sequences to form a
“noise-gap”
0
1

In the example system - translates into
stricter RLL constraint
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Capacity vs. Noise Gap
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Devil’s staircase structure
1
.5 .5+e
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Summary
synchronization
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Chaos in spreadspectrum (and CDMA)
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spectral properties
synchronization can be
fast and simple
compact and efficient
representation
good multi-user
performance
worse single-user
performance
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control
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Direct encoding in chaos
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neat idea
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simple circuits?
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low power?
loss of synchronization
mismatched parameters
low power circuits
enhanced security
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LPI + numerous codes
(can be done with pseudo-chaos)
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References and Links
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http://rfic.ucsd.edu/chaos
Communication based on synchronizing chaos
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L. Pecora and T. Carroll, “Synchronization in Chaotic Systems,” Physical Review Letters,Vol. 64, No. 8, Feb.
19th, 1990
L. Pecora and T. Carroll, “Driving Systems with Chaotic Signals,” Physical Review A, Vol. 44, No. 4, Aug.
15th, 1991
K. Cuomo and A. Oppenheim, “Circuit Implementation of Synchronized Chaos with Application to
Communication,” Physical Review Letters, Vol. 71, No. 1, July 5th, 1993
G. Heidari-Bateni and C. McGillem, “A Chaotic Direct-Sequence Spread-Spectrum Communication System,”
IEEE Transactions on Communications, Vol. 42, No. 2/3/4, Feb./Mar./Apr. 1994
G. Mazzini, G. Setti, and R. Rovatti, “Chaotic Complex Spreading Sequences for Asynchronous DS-CDMAPart I: System Modeling and Results,” IEEE Transactions on Circuits and Systems-I, Vol. 44, No. 10, Oct.
1997
Communication based on controlling chaos
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E. Ott, C. Grebogi, and J. Yorke, “Controlling Chaos,” Physical Review Letters, Vol. 64, No. 11, Mar. 12th,
1990
S. Hayes, C. Grebogi, and E. Ott, “Communicating with Chaos,” Physical Review Letters, Vol. 70, No. 20,
May 17th, 1993
S. Hayes, C. Grebogi, E. Ott, and A. Mark, “Experimental Control of Chaos for Communication,” Physical
Review Letters, Vol. 73, No. 13, Sep. 26th, 1994
E. Bollt, Y-C Lai, and C. Grebogi, “Coding, Channel Capacity, and Noise Resistance in Communicating with
Chaos,” Physical Review Letters, Vol. 79, No. 19, Nov. 10th, 1997
J. Jacobs, E. Ott, and B. Hunt, “Calculating Topological Entropy for Transient Chaos with an Application to
Communicating with Chaos,” Physical Review E, Vol. 57, No. 6, June 1998.
I. Marino, E. Rosa, and C. Grebogi, “Exploiting the Natural Redundancy of Chaotic Signals in
Communication Systems,” Physical Review Letters, Vol 85, No. 12, Sep. 18th, 2000.
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