Skiadas C.H., Dimotikalis I. (editors) Chaotic Systems: Theory and Applications
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338
Chaos in a Fractional-Order Jerk Model
Using Tanh Nonlinearity
Banlue Srisuchinwong
Sirindhorn International Institute
of
Technology, Thammasat University
Pathum-Thani, 12000, Thailand
(e-mail:
banlue@siit.tu.ac.th)
Abstract:
Chaos in a fractional-order jerk model using hyperbolic tangent
nonlinearity
is
presented. A fractional integrator
in
the model
is
approximated by
linear transfer function approximation
in
the frequency domain. Resulting chaotic
attractors are demonstrated with the system orders
as
low
as
2.1.
Keywords: chaos, fractional order, jerk model, hyperbolic tangent nonlinearity.
1. Introduction
Since the discovery
of
the famous Lorenz chaotic attractor in 1963, chaos has
been extensively studied and found ubiquitous applications in scientific,
engineering and mathematical communities, for example, in astronomy [1],
galaxies [2], quantum chaos [3], circuits and systems [4], chaos-based
communications [5-9], medical science [10-12] and chaotic oscillators [13-
IS]. An integer-order chaotic system, according to the Poincare-Bendixon
theorm [19], must have a minimum order
of
3 for chaos to appear. For
example, the well-known Chua's circuit is based on three first-order ordinary
differential equations (ODEs), whilst Sprott [14] has alternatively proposed
chaotic oscillators based on a single third-order ODE called a "jerk" (time
derivative
of
acceleration) function
of
the form [14-17] :
d
3
x
+A
d
2
x +
dx
=G(x)
dt
3
dt
2
dt
(1)
or
(2)
where
'x',
'v'
and
'a'
may represent position, velocity and acceleration
of
an
object, respectively, and G(x) is a nonlinear component such
as
j
G
J
X)=IX
I
-2
;A=0.6
[14])
G(x)=
G
2
(x)=1.2s-4.5sgn(x)
;A=0.6
[14]
G
3
(x)=-1.2x+2sgn(x)
;A=0.6
[14]
G
4
(x)=-x+2tanh(x)
;A=0.19
[20]
(3)
Recently, fractional-order chaotic systems
of
orders less than 3 have been
demonstrated based on fractional calculus [21,22]. For example, fractional-
Chaos in a Fractional-Order Jerk Model Using
Tanh
Nonlinearity 339
order Chua's circuit
of
order as low as 2.7 can produce a chaotic attractor
[23], whilst chaos in a fractional-order jerk model [24]
of
orders as low
as
2.1
has been demonstrated. The fractional-order jerk model is
of
the
form:
where
da A
dv
d"'x
-G(
)
- +
-+---
x
dt
dt
dt'"
d"'x
-
-=v
dt'"
(4)
(5)
and m is a fractional number (0 < m
~
I). Equation (5) can be integrated
using a fractional integrator in the frequency domain. Although chaos in the
fractional-order jerk model (4) has been illustrated using the similar
nonlinear components GJ(x), G
2
(x) and G
3
(x)
of
(3), chaos in the fractional-
order jerk model (4) has never been demonstrated using the nonlinear
component
Gix)
=
-x
+ 2tanh(x) and A = 0.19
of(3).
In this paper, chaos
in
a fractional-order jerk model using the hyperbolic
tangent nonlinearity G(x)
= -x+2tanh(x) is presented. The fractional
integrator in the model is approximated by linear transfer function
approximation in the frequency domain. Resulting chaotic attractors are
demonstrated with the system orders as low as 2.1.
2. A Fractional Integrator
In the frequency domain
(s
= jro), a fractional integrator
of
order
'm'
can be
represented by a transfer function [22]:
I
F(s)
=-
sl1l
(6)
where the magnitude in the Bode plots
is
characterized by a slope
of
-20m
dB/decade. The transfer function (6) may be approximated by pole-zero pairs
(Pi,
Zi)
of
the form [22-24],
n
I+~
N-I(
)
I I
,
~
O
z,
-;;;
= (
s)'"
N ( s. )
1+-
n 1+ -
P
T
i=O P I
N-
I
n(s+z,)
G
'-
0
o N
n(
s+
p,)
i=O
where
PT
is the corner frequency,
Go
is a constant and N is given by
N = 1 + Integer
og
Po
log(ab)
[
I
(wmaX)J
(7)
(8)
340
B.
Srisuchinwong
(9)
(10)
b =
10(yIlOm)
(11)
where y is the discrepancy (dB) between the actual and the approximated
lines, and
IDmax
is the frequency bandwidth
of
the system. Such pole-zero
pairs are found
as
-
10[yIlO(l-m)]
Zo
-
Po
Zi
= apo(ab)i
Pi
=
Po
(ab/
(12)
(13)
(14)
Table I summarizes a list
of
integer-order approximation
of
fractional
integrators
of
order
'm'
using y = 2 dB,
~nax
= 100,
PT
= 0.01.
Table!. List
of
integer order approximation
of
fractional integers
of
order
'm'
using y = 2 dB,
IDmax
= 100,
PT
= 0.01, Y = 2 dB.
_'"
1584.8932(s+0.1668)(s+27.83)
SOl
-
(s
+ O.l)(s + 16.68)(s + 2783)
_
79.4328(s
+
0.05623)(s
+ I)(s + 17.78)
?2 =
(s
+0.03162)(s
+0.5623)(s
+ lO)(s + 177.8)
1 _ 39.8107(s
+0.0416)(s
+0.3728)(s
+ 3.34)(s + 29.94)
S03
= (s + 0.02154)(05 + 0.1931)(s + 1.73)(s + 15.51)(s + 138.9)
1 _ 35.4813(s +
0.03831)(s + 0.261)(s +
1.778)(05
+ 12.12)(s + 82.54)
~
=
(05
+0.01778)(s
+0.1212)(s
+
0.8254)(s
+ 5.623)(s + 38.31)(s + 261)
_ 15.8489(s +
0.03981)(05 + 0.2512)(s + 1.585)(s + 63.1)
?s
=
(s
+ 0.01585)(s +
0.1)(05
+ 0.631)(s + 3.981)(s + 25.12)(s + 158.5)
I _ 1O.7978(s
+0.04642)(s
+0.3162)(s
+ 2.154)(s + 14.68)(s + 100)
S06
=
(s
+ 0.0\468)(05 + O.I)(s + 0.6813)(s +
4.642)(s
+
31.62)(s
+ 215.4)
I _
9.3633(s
+0.06449)(s
+0.578)(s
+ 5.179)(s +
46.42)(s
+ 416)
? = (s + 0.01389)(s
+0.1245)(s
+ 1.116)(s + lO)(s + 89.62)(s + 803.1)
1 _
5.3088(s+0.1334)(s+2.371)(s+42.17)(s+749.9)
?s
= (s +0.01334)(05 + 0.2371)(s + 4.217)(s +
74.99)(s
+ 1334)
I _
2.2675(s + 1.292)(s + 215.4)
~
= (s +
0.01292)(s
+ 2.154)(s + 359.4)
Chaos in a Fractional-Order Jerk Model Using
Tanh
Nonlinearity
341
3. Fractional-Order Jerk Model using Tanh Nonlinearity
Figure 1 shows a structural implementation
of
the fractional-order jerk model
described in (4) and
(S) using G(x) =
-x
+ 2tanh(x) and K = A = 0.19. For
simplicity, let
X = position
'x',
Y = velocity
'v'
and Z = acceleration
'a'
of
the jerk model.
In
Fig.
1,
the pole-zero block diagram represents the
fractional integrator described in
(7) and may be approximated by the pole-
zero pairs shown in Table 1 where y
= 2 dB.
~
________
~L
__________
~Z
Y
~----~----~y
x
X
x
Scope
x
Trigonometric
Gain
3
Function
Figure
1.
A fractional-order jerk model using Tanh nonlinearity.
4. Numerical Results
The fractional-order jerk model shown in Fig. 1
is
simulated and the order m
is
reduced from 1
to
0.1. This result in a total system order reduced from 3
to
2.1. For m =
1,
Figs 2 and 3 show the resulting chaotic attractor in XY and
XZ axes, respectively. For m = 0.9, Figs 4 and S show the trajectories in XY
and XZ axes using
Go
= 3 and y = 2 dB. Note that
Go
is
simply a scaling
factor and therefore may not be the same
as
Go
shown in Table
1.
It
can be
observed that the double-scroll-like attractor
is
preserved and
is
not
completely destroyed by order reduction. For m
=
O.S,
Figs 6 and 7 display
the trajectories in
XY and XZ axes using
Go
= 30 and y = 2 dB.
By using Table 1 where the discrepancy y
= 2 dB, the chaotic attractor can be
observed for m
= 0.9
to
0.2. For m = 0.1, the chaotic attractors can be
observed using A
= 0.02 and the discrepancy y = 3 dB, i.e.
S01.14(s +0.6811)
}T=
(s+O.3162)(s+681.1)
(15)
In
(IS),
Go
=
SO
1.14. Figures 8 and 9 show the trajectories in XY and XZ
axes using m = 0.1,
Go
= IS00. Figures
10
and
11
show the trajectories in XY
and XZ axes using m = 0.1,
Go
= 2000. Figures
12
and
13
show the
trajectories in
XY and XZ axes using m = 0.1,
Go
= 3000.
342
B.
Srisuchinwong
Figure
2.
Trajectorie in
XY
axes
for
111
=
1.
:r
I
I
1f
f/)
i
'x of
'P
I
>:
1
1
-
~~
--
-
-
~
2
---
0
~--~
2
-
~
J
4
.~
X-axi
s
Figure 4.
Tra
je
ctor
ie
in XY axes
for
111
= 0.9,
Go
= 3, Y = 2 dB.
Figure 6.
Traje
ctor
ie
in
XY
ax
es
for
ill
= 0.5,
Go
= 30, Y = 2 dB.
06
1
0.6
1'
OAf
0.21
"
~
0
>-
-0.2
·0.4
·0.6
I
-
0
8
L
--
-
-
-~
~~
--
~
-3 -2
-1
0
X-a
xis
Figure
8.
Trajectorie in
XY
axes,
for
ill
= 0.1,
Go
= 1500, Y = 3 dB,
A
= 0.
02
.
U>
.~
0
N
-1
-2
-~4
-2
0 2 4
X-axis
Figure 3. Trajectories in
XZ
axes
for
ill
=
1.
3
2
'"
'
iii
0
N
·1
-2
-2
0 2 4
X-axis
Figure 5. Trajectories in
XZ
axes
for
ill
= 0.9,
Go
=
3,
Y = 2 dB.
I
I
-1
0
3
X-axis
Figure 7. Trajectories in
XZ
axes
for
ill
= 0.5,
Go
= 30, Y = 2 dB.
0.8
0.6
0.4
0.2
-0.2
-0.4
-0.6
._
~_~_
.
__ ,
__
~
__
i
-2
-1
0 1 2 3
X-axis
Figure 9. Trajectories in
XZ
axes,
for
ill
= 0.1,
Go
= 1500, Y
==
3 dB,
A=0.02.
Chaos
ill
a Fractional-Order Jerk Model
Us
ing
Tanh
Nonlinearity
343
-O:~
.
5
'
2
-
-1
~.
S
~
-
I
-~S
-
I
~
~1.5
-~
2
2.5
X-axis
Figure
10.
Trajectorie in
XY
axe
s,
for M = 0
.1
,
Go
= 2000, Y = 3 dB,
A=0.02.
-():~.5
-=2
'''--
-::ts
--:-1
-:O
.-s
-o~
-
-t5
-
~I--
t5-
--
2 2.5
X-axis
Figure 12. Trajectorie in XY axes,
for
M =
0.1
,
Go
= 3000, Y = 3 dB,
A=
0.02.
5. Conclusions
·2 -1
.5
-1
-0
.5 0 0.5 1
1.
5 2 2.5
X-axis
Figure
II.
Tr~jectorie
in XY axe
s,
for m = 0.1,
Go
= 2000, Y = 3
dB
,
A=O
.02.
Figure
13
. Trajectorie in XY axes,
for M
= 0.1,
Go
= 3000, y = 3
dB
,
A = 0.02.
In this paper, chaos in a fractional-order jerk model using the hyperbolic
tangent nonlinearity has been presented. A fractional integrator in the model
has been approximated by linear transfer function approximation in the
frequency domain. Resulting chaotic attractors have been observed with the
system orders
as
low
as
2.
1.
Acknowledgements
This work
is
supported by Telecommunications Research and Industrial
Development Institute (TRJDI
),
National Telecommunications Conunission
(NTC), Thailand.
References
[1]
G.
Contopoulos, Order
and.
Chaos
in
Dynamical Astronomy, New York:
Springer, 2004.
[2]
G.
Contopoulos, and N. Voglis,
Eds_
Galaxies and Chaos, Lecture Notes
in
Physics, Germany: Springer, Germany, 2003.
l3]
H,
-J. Stockmann, Quantum Chaos : an introduction, UK: Cambridge University
Press, 2000.
344
B.
Srisuchinwong
[4]
G. Chen and T. Ueta, Chaos
in
Circuits and Systems, Singapore: World
Scientific,
2002.
[5]
F.
C. M. Lau, and
C.
K.
Tse, Chaos-Based Digital Communication Systems,
Germany: Springer, 2003.
[6]
S.
MandaI, and S. Banerjee, Analysis and CMOS Implementation
of
a Chaos-
Based Communication System,
IEEE Trans. Circuits and
Systems,S
1 (9): 1708-
1722,2004.
[7]
G.
Kolumban, M. P. Kennedy and L. O. Chua, The Role
of
Synchronization
in
Digital Communications Using Chaos - Part I: Fundamentals
of
Digital
Communications,
IEEE Trans. Circuits and Systems, 44(10):
927-936
, 1997.
[8]
G.
Kolumban, M. P. Kennedy and L.
O.
Chua, The Role
of
Synchronization in
Digital Communications Using Chaos - Part II: Chaotic Modulation and Chaotic
Synchronization,
IEEE Trans. Circuits and Systems, 45(11): 1129-1140, 1998.
[9]
P. Stavroulakis, Chaos Applications
in
Telecommunications, New York: Taylor
& Francis, 2006.
[10]
H.
Adeli,
S.
Ghosh-Dastidar,
N.
Dadmehr, A Wavelet-Chaos Methodology for
Analysis
of
EEGs and EEG Subbands to Detect Seizure and Epilepsy, IEEE
Trans Biomedical Engineering,
54(2):205-21
I,
2007.
[11]
S.
Ghosh-Dastidar,
H.
Adeli, N. Dadmehr, Mixed-Band Wavelet-Chaos-Neural
Network Methodology for Epilepsy and Epileptic Seizure Detection,
IEEE
Trans. Biomedical Engineering,
54(9):1545-1551, 2007.
[12] A. Dalgleish, The relevance
of
non-linear mathematics (chaos theory) to the
treatment
of
cancer, the role
of
the immune response and the potential for
vaccines,
Q.
1.
Med
.,
92:347-359,1999.
[13] M. P. Kennedy, Robust
Op Amp Realization
of
Chua
's
Circuit, Frequenz, 46(3-
4):66-80, 1992.
[14] J.
C. Sprott, A New Class
of
Chaotic Circuit, Physics Letters A, 266:19-23, 2000.
[IS]
B. Srisuchinwong, and
C.
-H.
Liou, High-Frequency Implementation
of
Sprott's
Chaotic Oscillators
Using Current-Feedback Op Amps, Proceedings
of
International Symposium on Signals Circuits and Systems - ISSCS 2007, 1:97-
99,2007.
[16] B. Srisuchinwong,
C.
-H.
Liou, and T. Klongkumnuankan, Prediction
of
Dominant Frequencies
of
CFOA-Based Sprott's Sinusoidal and Chaotic
Oscillators, Proceedings
of
CHAOS
2008,
Chaotic Modeling and Simulation
International Conference,
3-6 June, Crete, Greece, 1-8, 2008.
[17] B. Srisuchinwong,
C.
-H.
Liou, and
T.
Klongkumnuankan, Prediction
of
Dominant Frequencies
of
CFOA-Based Sprott's Sinusoidal and Chaotic
Oscillators, in Topics on Chaotic
Systems:
Selected Papers from CHAOS 2008
International Conference, c.
H.
Skiadas, Ed. Greece: World Scientific, (in
press).
[18] B. Srisuchinwong, and
W.
San-Urn, A Chua's Chaotic Oscillator Based on a
Coarsely Cubic-Like
CMOS Resistor, Proceedings
of
Asia-Pacific Conference
on Communications
-APCC
2007,2007,47-49,2007.
[19] M. W. Hirsch, and S. Smale, Differential Equations: Dynamical Systems and
Linear Algebra,
New York: Academic Press, 1974, Chapter 1 I, 238-54.
[20]
J.
C. Sprott, Simple chaotic systems and circuits, Am.
1.
Phys. 68(8):758-763,
2000.
[21] K. B. Oldham and J. Spanier, The Fractional Calculus, New York: Dover
Publications, 1974.
Chaos in a Fractional-Order Jerk Model Using
Tanh
Nonlinearity
345
[22] A. Charef,
H.
H. Sun, Y.
Y.
Tsao, and
B.
Onaral, Fractal System as Represented
by
Singularity Function, IEEE Trans. Automatic Control, 37(9): 1465-1470,
1992.
[23] T. T. Hartley,
C.
F. Lorenzo, and
H.
K. Qammer, Chaos
in
a Fractional Order
Chua's System IEEE Trans. Circuits and Systems, 42(8):485-490,1995.
[24]
W.
M.
Ahmad, and
1.
C.
Sprott, Chaos
in
Fractional-Order Autonomous
Nonlinear Systems,
Chaos, Solitons & Fractals, 16:339-351, 2003.
346
Nonlinear
Systems
in
Brunovsky
Canonical
Form: A
Novel
Neuro-Fuzzy
Algorithm
for
Direct
Adaptive
Regulation
with
Robustness
Analysis
Dimitris C.
Theodoridisl,
Yiannis
Boutalis
1
,2,
and
Manolis A.
Christodoulou
3
1
Department
of
Electrical
and
Computer
Engineering
Democritus
University
of
Thrace
67100
Xanthi,
Greece
(e-mail:
dtheodo@ee.duth.gr
)
2
Department
of
Electrical,
Electronic
and
Communication
Engineering
Chair
of
Automatic
Control
University
of
Erlangen-Nuremberg
91058
Erlangen,
Germany
(e-mail:
ybout@ee.duth.gr)
3
Department
of
Electronic
and
Computer
Engineering
Technical
University
of
Crete
73100
Chania,
Crete,
Greece
(e-mail:
manolis@ece.tuc.gr
)
Abstract:
The
direct
adaptive
regulation
of
unknown
nonlinear
dynamical
sys-
tems
in
Brunovsky
form
with
modeling
error
effects, is
considered
in
this
chapter.
The
method
is
based
on
a
new
Neuro-Fuzzy
Dynamical
System
definition,
which
uses
the
concept
of
Fuzzy
Adaptive
Systems
(FAS)
operating
in
conjunction
with
High
Order
Neural
Network
Functions
(HONNF's).
Since
the
plant
is
considered
unknown,
we
propose
its
approximation
by
a special
form
of
a
Brunovsky
type
fuzzy
dynamical
system
(FDS)
assuming
also
the
existence
of
disturbance
expressed
as
modeling
error
terms
depending
on
both
input
and
system
states.
The
develop-
ment
is
combined
with
a
sensitivity
analysis
of
the
closed
loop
in
the
presence
of
modeling
imperfections
and
provides
a
comprehensive
and
rigorous
analysis
of
the
stability
properties
of
the
closed loop
system.
Simulations
illustrate
the
potency
of
the
method
and
its
applicability
is
tested
on
well
known
benchmarks,
where
it
is
shown
that
our
approach
is
superior
to
the
case
of
simple
Recurrent
High
Order
Neural
Networks
(RHONN's).
Keywords:
direct
control,
Neuro-Fuzzy
Algorithm,
Brunovsky
form,
robustness
analysis.
1
Introduction
Adaptive
control
theory
has
been
an
active
area
of
research over
the
past
years [2,5-7,9,11,17,18,20,25,27].
The
identification
procedure
is
an
essential
part
in
any
control procedure.
In
the
Neuro
or
N euro -
Fuzzy
adaptive
Nonlinear Systems
in
Brunovsky Canonical
Form:
A Neuro-Fuzzy Algorithm 347
control two
main
approaches
are
followed. In
the
indirect
adaptive
control
schemes, first
the
dynamics
of
the
system
are
identified
and
then
a control
input
is
generated
according
to
the
certainty
equivalence principle. In
the
direct
adaptive
control schemes [5,9,18]
the
controller is
directly
estimated
and
the
control
input
is
generated
to
guarantee
stability
without
knowledge
of
the
system
dynamics. Also,
many
researchers focus
on
robust
adaptive
control
that
guarantees
signal
boundness
in
the
presence
of
modeling
errors
and
bounded
disturbances
[2,5-7,9,11,18,25,27].
It
has
been
established
that
neural
networks
and
fuzzy inference
systems
are
universal
approximators
[8,19,26]'i.e.,
they
can
approximate
any
nonlinear
function
to
any
prescribed
accuracy
provided
that
sufficient
hidden
neurons
and
training
data
or
fuzzy rules
are
available. Recently,
the
combination
of
these
two different technologies
has
given rise
to
fuzzy
neural
or
Neuro-
Fuzzy approaches,
that
are
intended
to
capture
the
advantages
of
both
fuzzy
logic
and
neural
networks.
Numerous
works have shown
the
viability
of
this
approach
for
system
modeling [3,10,13,14,16].
The
neural
and
fuzzy
approaches
are
most
of
the
time
equivalent, differ-
ing between each
other
mainly
in
the
structure
of
the
approximator
chosen.
Indeed, in
order
to
bridge
the
gap
between
the
neural
and
fuzzy
approaches
several researchers
introduce
adaptive
schemes using a class
of
parameterized
functions
that
include
both
neural
networks
and
fuzzy
systems
[3,10,14,16].
Recently
[4,12],
HONNF's
have
been
proposed
for
the
identification
of
nonlinear
dynamical
systems
of
the
form
i;
=
f(x,
u),
approximated
by a
Fuzzy
Dynamical
System.
This
approximation
depends
on
the
fact
that
fuzzy
rules could
be
identified
with
the
help of
HONNF's.
The
same
rationale
has
been
employed in [1,23,24]' where a Neuro-Fuzzy
approach
for
the
indirect
and
direct
control
of
square
unknown
systems
has
been
introduced
assuming
only
parameter
uncertainty
.
In
this
chapter
HONNF's
are
also used for
the
Neuro-Fuzzy
direct
control
of
nonlinear
dynamical
systems
in
Brunovsky
canonical form
with
modeling
errors.
In
the
proposed
approach
the
underlying
fuzzy model is
of
Mamdani
type
[15].
The
structure
identification
of
the
underlying fuzzy
system
is
made
off-line
based
either
on
human
expertise
or
on
gathered
data.
However,
the
required a-priori
information
obtained
by
linguistic
information
or
data
is
very
limited.
The
only required
information
is
an
estimate
of
the
centers
of
the
output
fuzzy
membership
functions.
Information
on
the
input
vari-
able
membership
functions
and
on
the
underlying
fuzzy rules is
not
necessary
because
this
is
automatically
estimated
by
the
HONNF's.
This
way
the
pro-
posed
method
is less
vulnerable
to
initial design
assumptions.
The
parameter
identification is
then
easily
addressed
by
HONNF's,
based
on
the
linguistic
information
regarding
the
structural
identification
of
the
output
part
and
from
the
numerical
data
obtained
from
the
actual
system
to
be
modeled.
We consider
that
the
nonlinear
system
can
be
expressed in
Brunovsky
canonical form. We also consider
that
its
unknown
nonlinearities could
be