APPROXIMATIONS IN SPROTT CIRCUITS
Liljana Stefanovska
1
,
Sanja Glamo
č
anin
2
1
Faculty of Tecnology and Metallurgy, Ss Cyril and Methodius Univ., Skopje, R. of Macedonia
liljana@ian.tmf.ukim.edu.mk
2
Faculty of Electrical Engineering.,
Ss Cyril and Methodius Univ., Skopje, R. of Macedonia
sanjag@etf.ukim.edu.mk
Abstract
–
Some approximation methods are
suggested to study quasiperiodic regimes in a class
of Sprott “jerk” electric chaotic circuits. The idea
is to replace piecewise linear characteristics of
nonlinear components with piecewise polynomial
ones. The methods proposed are chosen to fit two
proposals: a) to be the simplest possible; b) to have
one free parameter that regulates approximation
cleseness.
These are happening to be fulfilled by
cubic Hermite splines.
Several examples of chao-
tic circuits with approximated characteristics are
given.
Key Words
– Sprot “jerk” circuit,
Hermite appro-
ximations, chaos
1.
INTRODUCTION
Simple electric circuits are ideal tools for studying
features of chaos. According to Poincaré-Bendixon
theorem, autonomous first-order ODE with continu-
ous coefficients can not have bounded chaotic
solution unless it is at least three dimensional. The
classic example was the famous Lorenz model of
atmospheric heath conduction from 1963
()
,
(),
,
xA
y
x
yB
z
x
y
zx
y
C
z
=−
=−−
±
±
±
where A, B and C are real parameters.
Later, many
other 3D systems were found and in 1976 Rössler
came with even simpler 3D system than Lorenz’s. In
1997 Linz showed that Lorenz and Rössler system
can be transformed in the following form
(, , )
x
Jxxx
=
±±±
±± ±
,
(1)
where J is called
jerk function
upon the mechanical
term “jerk” for time derivative of acceleration. In this
way, one can consider DE (1) instead of 3D system in
order to stydy different aspects of chaos.
In 1996,
Gottlieb rised an interesting question: What is the
simplest jerk function that gives chaos? In [7] Julian
Sprott from Univ. of Wisconsin made experiments
with a simple jerk function
2
A
x x x
=
−+−
±±
±
,
showing that system exhibits a period-doubling rute
to chaos when
A
runs
between 2.017 and 2.082. In
further series of papers (see [6]-[9] ), Sprott replaced
linear term
x
in
J
with several
piecewise linear fun-
ctions, like
| |, max( ,0), sgn( ),
( )
sgn[max( ,0)]
xx
x
H
x
x
=
(2)
(
H
is Heaviside step function). Figure 1, reproduced
from [8], shows the way of modeling such functions
by electronic components.