Communication through chaotic modeling of languages
Murilo S. Baptista,
1,2,*
Epaminondas Rosa, Jr.,
3
and Celso Grebogi
1,2,4
1
Institute for Plasma Research, University of Maryland, College Park, Maryland 20742
2
Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742
3
Nonlinear Dynamics Laboratory, Department of Physics, University of Miami, Coral Gables, Florida 33146
4
Department of Mathematics, University of Maryland, College Park, Maryland 20742
Received 2 March 1999
We propose a communication technique that uses modeling of language in the encoding-decoding process of
message transmission. A temporal partition time-delay coarse graining of the phase space based on the symbol
sequence statistics is introduced with little if any intervention required for the targeting of the trajectory.
Message transmission is performed by means of codeword, i.e., specific targeting instructions are sent to the
receiver rather than the explicit message. This approach yields i error correction availability for transmission
in the presence of noise or dropouts, ii transmission in a compressed format, iii a high level of security
against undesirable detection, and iv language recognition.
PACS numbers: 05.45.Vx, 05.45.Gg
I. INTRODUCTION
Recent developments in communicating with chaos 1–4
have produced a wealth of potential practical applications
including synchronization 5–7, encoding-decoding tech-
niques 1–4,8–11, noise filtering 12, and signal masking
and recovery 13,14. This is so because chaotic systems
have peculiar properties that make them natural candidates to
play a significant role in nonlinear communication systems.
One of these properties, the sensitivity of the dynamics to
small perturbations, is useful for targeting the trajectory in
phase space to specific regions to which particular symbols
have been assigned. This targeting feasibility provides cha-
otic systems with a natural type of dynamics to be used in
communication. The symbol sequence to be followed by the
chaotic trajectory corresponds then to the information to be
transmitted 1–4,8,9,11,15. Indeed, the ergodicity or the
eventual visit of the trajectory to all partitions without any
targeting or control of chaotic systems has been used re-
cently 14 in a chaotic communication scheme.
Symbolization of a chaotic trajectory can be useful for
extracting relevant information about the system under con-
sideration. Correlation function computing 16,17, param-
eter estimation 18, and data compression 19 are examples
of symbolic dynamics 20 application toward a better under-
standing of the system dynamics. Also, different signals gen-
erated by the same dynamics can be identified with the help
of the conditional entropy 21 obtained from the symbolic
dynamics of the chaotic process. Of course, the symbolic
sequence generated by a chaotic trajectory depends on how
the phase space is partitioned. It also depends on the time
delay interval sampling rate for symbol sequence construc-
tion, which has been used to measure correlation lengths
from given symbolic sequences 19. Much emphasis has
been placed on the characterization of the complexity of
symbol sequences based on patterns and transmission rules
estimated from symbolic time series 22.
In this work we present a language approach for a chaotic
communication system. The text message to be transmitted is
generated by a chaotic process that respects the grammar of a
language. Symbols are assigned to judiciously chosen re-
gions of phase space, and the chaotic trajectory is controlled
to visit these regions generating a symbol sequence that cor-
responds to the desired message. The message itself is not
transmitted. Rather, what is transmitted is a set of instruc-
tions, the codeword, that enables the receiver to decode the
message. A temporal partition is introduced as a time-delay
coarse graining 19 of the phase space. The phase space is
divided into a number of cells to which different symbols are
assigned 16,17. As the chaotic trajectory visits these re-
gions, symbols are generated, producing a symbol sequence
that corresponds to a message to be transmitted. The parti-
tions are chosen in such a way that the message is consistent
with the grammar of a language. For the purpose of illustra-
tion we use an artificial language created as an approxima-
tion to a real language in terms of statistical structure. We
assume a communication system consisting basically of a
transmitter where the message is encoded, a communication
channel that carries the message from one place to another,
and the receiver where the message is decoded. Transmitter
and receiver have complete knowledge about the dynamical
system being used. The procedure involves a minimum of
information transmission, is secure against unwanted detec-
tion, and is robust against noise and dropouts.
This paper is organized as follows. In Sec. II we introduce
concepts and definitions related to languages, paying special
attention to their statistical structure. In Sec. III, we show
how this statistical structure is used in the construction of the
dynamical model process. Section IV details how the com-
munication system is built based on language modeling, and
a technique for optimal transmission of information is pre-
sented in Sec. V. In Sec. VI, we introduce a language recog-
nition scheme and explain how this proposed communication
system is secure against undesired decoding. Section VII
proposes an error correcting code that is able to recover in-
formation when the transmission is corrupted by noise or lost
*Permanent address: Instituto de Fı ´sica, Universidade de Sa ˜o
Paulo, C.P. 66318, 05315-970 Sa ˜o Paulo, SP, Brazil.
PHYSICAL REVIEW E APRIL 2000 VOLUME 61, NUMBER 4
PRE 61 1063-651X/2000/614/359011/$15.00 3590 © 2000 The American Physical Society