Skiadas C.H., Dimotikalis I. (editors) Chaotic Systems: Theory and Applications
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8
M. M.
Alomari
and
1.
G.
Zhu
References
[1]
E.
H.
Abed and
P. P.
Varaiya, "Nonlinear Oscillations
in
Power Systems", Int.
J.
Elecr. Power and Energy System, Vol.
6,
pp. 37-43, Jan., 1984.
[2]
W.
Zhu, R.R. Mohler,
R.
Spce,
W.A
Mittelstadt and
D.
Maratukulman, "Hopf
Bifurcation in a SMIB Power System with SSR," IEEE Trans. on Power Systems,
Vol.1l, No.3, pp. 1579-1584, 1996.
[3]
M.
A Tomim, A
C.
Zambroni de Souza,
P.
P. Carvalho Mendes, and
G.
Lambert-
Torres, "Identification of Hopf Bifurcation in
Power Systems Susceptible to
Subsynchronous
Resonance", IEEE Bologna Power Tech Conference, June 23-26,
2003, Bologna, Italy.
[4]
A.H Nayfeh,
AM.
Harb, and C-M.Chin,
AM.
Hamdan and
L.
Mili, Application
of
Bifurcation Theory
to
Subsynchronous Resonance
In
Power Systems,
InU.Bifurcation and Chaos,YoI.8,No.l,pp.157-l72, 1998
[5]
Kundur,
P.
1994, Power System Stability
and
Control, Electric Power Research
Institute, McGraw-Hill, New York.
[6]
Harly,
R.
G., Limebeer,
D.
J., and Chirricozzi,
E.
, 1980, "Comparative study
of
the
saturation methods in synchronous machine
models," IEEE Proceedings, Vol. 127,
Part
B.
[7]
De Mello,
F.
P., and Hannet,
L.
N.
"Representation
of
Saturation
in
Synchronous
Machines," IEEE Transactions on Power Systems
PWRS-l,
1986.
[8]
AM.
Harb,
L.
Mili,
AH
Nayfeh and C-M. Chin, On the Effect
of
the Machine
Saturation on SSR
in
Power Systems, Electrical Machines and Power Systems,
Vo1.20, pp.
10
19-1035, 2000.
[9]
AH.
Nayfeh and
B.
Balachandran, Applied Nonlinear Dynamics, Wiley, New York,
1995.
[10] IEEE SSR Working Group, "Second Benchmark Model for Computer Simulation
of
Sub synchronous Resonance," IEEE Trans. On Power Apparatus and Systems.
Vol.
PAS- 104, No.5, 1057-\064, May 1985.
[II]
Majdi
M.
Alomari and Benedykt S. Rodanski, "The Effects
of
Machine Components
on Bifurcation and Chaos
as
Applied to Multimachine System", CHAOS2008,
Chaotic Modeling and Simulation International Conference, Chania, Crete, Greece,
3-6 June
2008.
[12]
S.
H.
Minnich,
R.
P.
Schulz,
D.
H.
Baker,
D.
K. Sharma,
R.
G.
Farmer, and
J.
H.
Fish,
"Saturation Functions for Synchronous Generators from Finite Elements", IEEE
Trans. Energy Conversion, vol. EC-2, December 1987.
Chaos
In
Multiplicative
Systems
Dorota
Aniszewska
and
Marek
Rybaczuk
Institute
of
Materials
Science
and
Applied
Mechanics
Wroclaw
University
of
Technology
50-370 Wroclaw, Smoluchowskiego 25,
Poland
(e-mail:
dorota.
aniszewska@pwr.wroc.pl,
marek.rybaczuk@pwr.wroc.pl)
9
Abstract:
The
goal
of
the
paper
is
chaos
exa
mination
in
multiplicative
systems.
The
paper
collects
results
of
numerical
simulations
as well as
presentation
of
meth-
ods
applicable
in
the
case
of
multiplicative
systems.
Chaos
examination
concerns
one-dimensional
multiplicative
version
of
logistic
equation
and
multi-dimensional
nonlinear
system
described
with
multiplicative
derivatives.
The
classical
Lorenz
system
transformed
into
multiplicative
version was chosen for
analysis
of
stability
and
chaotic
behaviour.
Keywords:
multiplicative
calculus, logistic
equation,
the
Lorenz
system,
Runge-
Kutta
method,
Lyapunov
ex
ponent.
1
Introduction
Main
subj
ect
of
our
work is
examination
if chaos
may
occur
in process
of
defects evo
lution
in
material.
Model of
fractal
defects
growth
has
been pre-
sented
in
Rybaczuk
and
Stoppel[4].
It
is
based
on
single
fractal
approximation
and
links
the
energy
uniformly
distributed
over fractal, its
measure
VD
and
energy dens
ity
a(D), which is essential
material
characteristics
depending
on
fractal
dim
ension:
[ = a(D)vD.
(1)
Appearance
of
fractals requires
chang
es in
the
form
of
physical equations.
Mathematical
theory
of
dynamical
syst
ems describing physical
proc
esses
must
pay
attention
to
physical
quantities.
In
model
(1) shift
of
fractal
dimension
means
shift
of
physical dimension. Classic
derivativ
e
based
on
diff
ere
ntial
quotient
containing
a(D
+
L1D)
- a(D)
has
no physical sense.
The
labile
physical dimension of a(D) is
the
esse
ntial
reason for employing multiplica-
tive calculus
to
formulate
dynamical
system
of
defects growth.
This
paper
collects results of chaos
examination
for
exemplary
multiplica-
tive systems.
It
is
expect
ed
that
pr
es
ented
methods
are
applicable
to
a wide
variety
of
multiplicative
dynamical
systems
including
dynamical
models
of
fractal defects evolution.
10
D.
Aniszewska
and
M.
Rybaczuk
2
Multiplicative
calculus
Multiplicative
calculus
has
been
presented
in
Volterra[5].
The
differential
multiplicative
calculus
has
been
restored
in
Rybaczuk[3]'
Rybaczuk
and
Stop-
pel[4]. Definition
of
the
multiplicative
derivative
offunction
f (x)
with
respect
to
x follows as:
1
7r
f (x) = lim { f (x
(1
+
E))
} ,
7rX
£->0
f(x)
,
(2)
where
shift
of
independent
variable
x is
multiplicative
x
--)
x(l
+
E),
there-
fore x E
lR.+
and
f(x)
E
lR.+.
In
Rybaczuk[3]
the
generalized
methods
to
examine
equations
comprising
quantities
with
labile
physical
dimensions
are
presented.
All
notions
and
theorems
of
the
classic differential calculus
have
their
multiplicative
counterparts.
To
assist
further
applications
we
present
some
of
them.
If
all
subsequent
multiplicative
derivatives
exist,
multiplicative
derivative
of
sum
and
product
of
functions
f(x)
and
g(x)
equals
respectively:
7rf(x)+g(x)
f(x)
g(x)
7r
f(x)
f(xJ+g(x)
7rg(x)
f(xJ+g(x)
7rX 7rX
7rX
(3)
7rf(x)g(x) 7rf(x)
7rg(x)
----
(4)
7rX
7rX
7rX
The
formula
for a
complex
function
is
more
complicated:
7rf(g(x))
I
~
f(cc)
In
~
f(g)
7r
f
(g)
n ----;ex 7r g ( X )
----:;cg
(5)
The
function
f(x)
can
have
extrema
in
such
points
for which:
7rf(x) = 1
7rX
(6)
3
Lyapunov
stability
and
Lyapunov
exponent
Lyapunov
type
stability
for
system
of
multiplicative
differential
equations
is
introduced
in
Aniszewska
and
Rybaczuk[2]
in
details.
In
this
paper
it
is
summarized
briefly.
Equilibria
of
system
of
autonomous
multiplicative
differential
equations
~
=
fj
(x!,
...
, x
n
),
where
j = 1,
...
, n
are
obtained
from:
(7)
S
. 1
fi
d·
((0)
(0)
(0))
d b d b
olutlOn c ose
to
xe
pomt
Xo
=
Xl
,x
2
,
...
, ;Z;n is escri e y
small
multiplicative
shift
Ej
(t):
(8)
Chaos
in
Multiplicative Systems
11
Behaviour
of
derivative
7rE;i
t
)
for long
time
determines
if
equilibrium
Xo is
stable.
Calculation
of
multiplicative
derivatives
gives following result:
(9)
where
EjO)
is
determined
by
initial
conditions.
Stability
of
fixed
point
Xo de-
pends
on
eigenvalues
of
matrix
[In
7r
~~:o)].
For
complex
eigenvalue
stability
depends
on
their
real
part.
If
it
is
negative,
for long
time
Ej
(t)
---+
0
and
Xo
is
stable.
If
it
is positive, value
of
E j (t) increases
according
to
power
function
of
time
and
Xo is
unstable.
Divergence
or
convergence
of
trajectories
with
close
starting
points
pro-
ceeds
according
to
power
of
time
Ej
:::::;
EjO)t>',
therefore
Lyapunov
exponents
for
multiplicative
dynamical
system
can
be
defined as:
\ . 1
Ej
Aj
=
11m
-1
- In
-(0)
t~oo
n t
E.
J
and
if
at
least
one
of
them
is positive,
trajectory
behaviour
is chaotic.
4
Multiplicative
logistic
equation
X2n
=
rx
1
-
x
(10)
The
simplest
systems
employed
to
chaos
examination
are
one-dimensional
ones.
Numerical
simulations
and
calculation
of
Lyapunov
exponent
have
been
performed
for
multiplicative
version
of
logistic
equation,
which
can
have
a few
forms.
One
of
them
follows as:
(11)
where
n is a
step
number.
In case
of
multiplicative
system,
shift
of
step
must
be
multiplicative: n
---+
n(l
+
.:1n),
where
.:1n
= 1 for
discrete
system.
There-
fore following
numbers
of
steps
are
calculated
as
2n
equals
1,2,4,8,16,
....
This
version
of
equation
is
analysed
in
details
in
Aniszewska
and
Rybaczuk[2].
In
this
paper
we
present
another
multiplicative
version
of
the
classical logistic
equation,
which
has
the
form:
l-x
X2n
=
rx
(12.)
and
for
this
equation
chaos
appears
for
certain
range
of
r
parameter
values.
Phase
diagrams
and
step
series for various r values
are
presented
in
Figures
1
and
2.
For
r = 2
there
are
only
two
solutions
and
for r = 4
chaotic
behaviour
appears.
Bifurcation
diagram
is
shown
in
Figure
3(a).
Calculation
of
Lyapunov
exponent
is
based
on
two
trajectories
started
from
points
with
distance
determined
by
small
multiplicative
shift
EO.
When
[2
D.
Alliszewska alld
M.
Rybaczuk
,
if-
i
05t
oc--~-·
-~
o 0.5
---
....
~.--.-."-,----.--,-------.-
.
.)
1.5
2 2.5
x"
( a) (b)
Fig.
1.
Phase
diagrams
and step series of the multiplicative logistic equation :C2n =
r::rl-~"
fix T =
2.
(
a)
(b)
Fig.
2.
Phase diagrarns and step series of the multiplicative logistic equation :(;2", =
rxl~'J.'
for T =
,1.
in
each
k
step
multiplicative shift
tk
bet"veen
points
is calculated.
then
for
many
steps
,YO
obtain
value
of
Lyapunov
exponent:
N
\
1.
"1
tk
A=
----
~
n--
InN
.
to'
k=.l
(13)
where
N is
number
of
steps.
Relationship
between
Lyapullov
exponent
and
T
parameter
is
presented
in
Figure
3(b).
Slighly
different version
of
multiplicative logistic
equation
has
the
form
:r2n
=
(rx)
1-",.
Its
behaviour
its similar,
but
chaos
appears
for
other
range
of
T'
values.
Bifurcation
diagram
for
this
version is
presented
in
Figure
'l(a)
and
relationship
between Lyapul10v
exponent
and
T'
parameter
is shown in
Figure
4(b).
Chaos ill Multiplicative Systems
13
10
15
20
(a)
(b)
Fig.
3.
Bifurcat.ion
diagram
for
the
multiplicative logi:>tic
equation
.T2n
and
relationship between
Lyapunov
exponent
and
r
parameter.
(a)
(b)
Fig.
4.
Bifurcation dia.gram for
the
multiplicative logistic equa.tion 3':2", = (r:r)
[-a'
and
relationship between
Lyapunov
exponent
and
r
parameter.
5
Multiplicative
Lorenz
system
Main
goal
of
this
paper
is chaos
examination
in
system
described
with
multi-
plicative differential
equations.
The
Lorenz
system
containing
three
nonlinear
equations
with
three
positive
real
parameters:
d:1:
--
=
(JY
-
0';1:
elt
.
dy
--- = r:D -
'!/
-
.TZ
rlt .
dz
-1
=
;r:y
-Ire
d
(14)
14
D.
Aniszewska and
M.
Rybaczuk
was chosen as
an
exemplary
chaotic
dynamical
system
and
it
has
been
trans-
formed
into
multiplicative
dynamical
system:
addition
was
replaced
by
multi-
plication
and
multiplication
by
raising
to
suitable
power.
The
multiplicative
Lorenz
system
equals:
(15)
Detailed
analysis
of
Lyapunov
type
stability
for
this
system
was
presented
in
Aniszewska
and
Rybaczuk[2].
It
provide
the
same
results
as for classical
Lorenz
system.
The
solutions
of
nonlinear
multiplicative
dynamical
systems
can
be
calculated
using
numerical
methods
therefore
multiplicative
Runge-
Kutta
methods
presented
in
[1]
was
employed
to
perform
numerical
simula-
tions.
0
,=
-----
'000
<10"
<10
'
o 0
(a) r = 10
(b) r = 28
50
15
rf
40
10
30
N
3'
0
10
10
LN(Y)
o 0
LN(X)
(c)r=10
(d)r=28
Fig.
5.
Trajectories
of
the
multiplicative
Lorenz
system
(15) for
CJ
= 10, b =
~
and
various
r values in
linear
and
logarithmic
scale.
Chaos in Multiplicative Systems
15
Numerical
solutions
of
the
multiplicative
Lorenz
system
(15) revealed
that
chaos
appears
for T > 24.84 for a = 10
and
b =
~.
Trajectory
spirals
out
two
equilibria
and
creates
butterfly
shape
attractor,
which
is visible
in
logarithmic
scale.
In
case
of
multi-dimensional
dynamical
systems
calculating
Lyapunov
ex-
ponents
is
more
complicated,
because
local
behaviour
of
close
trajectories
may
vary
with
the
direction.
For
calculation
of
the
largest
Lyapunov
expo-
nent
of
the
multiplicative
Lorenz
system
we
employed
multiplicative
version
of
method
introduced
in
[6].
For
classical
dynamical
system
the
largest
Lya-
punov
exponent
equals:
. 1 L(t)
A =
lun
-In-(-),
t-Hx)
t L 0 .
(16)
where
L( t) is
linear
extent
of
ellipsoid
created
in
space
by
two
trajectories.
For
multiplicative
dynamical
systems
the
largest
Lyapunov
exponent
should
be
calculated
as:
. 1 L(t)
A =
lun
-1
- In
-(
-)
,
t~oo
n t L 0 .
because
time
shift
is
multiplicative.
(17)
Results
of
the
largest
Lyapunov
exponent
computation
for
the
multiplica-
tive
Lorenz
system
are
given in
Table
1.
11-0.93
1-0.59
1-
0
.34
1-
0
.
12
1
0
.82
1
0
.91
Table
1.
The
largest
Lyapunov
exponent
of
the
multiplicative
Lorenz
system
for
IJ
= 10, b =
~,
40 000
steps,
starting
time
0.01
and
time
step
0.01.
6
Discussion
The
results
presented
in
the
paper
proof
that
all classical
conditions
concern-
ing
chaotic
behavior
can
be
extended
onto
multiplicative
systems.
The
paper
presents
exemplary
multiplicative
systems
with
chaotic
behaviour.
Similarly
like
in
case
of
dynamical
system
described
with
classical derivatives,
non-
linearity
and
three
indepenent
variables
are
necessary
for
chaos
occuring
in
systems
described
with
multiplicative
derivatives.
References
l.D.Aniszewska.
Multiplicative
runge-kutta
methods.
Nonlinear
Dynamics,
50:265
- 272, 2007.
16
D.
Aniszewska and
M.
Rybaczuk
2.D.Aniszewska
and
M.
Rybac
zuk.
Lyapunov
type
stability
and
lyapunov
expone
nt
for
exemplary
multiplic
ative
dynamical
syste
ms. Nonlinear
Dynamic
s,
54:345
~
354, 2008.
3.
M.Rybaczuk,
A.
Kedzia
,
and
W.
Zielinski.
The
conc
ept
of
physical
and
fracta
l
dimension
II.
the
diff
erentia
l calculus
in
dimensional
spaces. Chaos, Solitons
E'3
Fractals,
12:2537~2552,
200l.
4.M.Rybaczuk
and
P.
Stopp
e
l.
The
fractal
growth
of
fatigue
defects
in
m
ateria
ls.
International
Journal
of
Fracture,
103:71
~
94
,
2000.
5.V.
Volterra
and
B.Hostinsky.
Herman,
Paris,
1938.
6.A. Wolf,
J.B.
Swift, H.L. Swinney,
and
J.A.
Vastano
.
Determining
lyapunov
exponents
from
a
tim
e series. Physica
16
D,
pages
285
~
317, 1985.
Deterministic Chaos Machine:
Experimental vs. Numerical Investigation
Jan Awrejcewicz and Grzegorz Kudra
Department
of
Automatics and Biomechanics
Technical University
of
Lodz , Poland
(e-mail:
awrejcew@p.lodz.pl)
17
Abstract: Detenninistic chaos machine consisting
of
the plane triple pendulum
as
well
as
of
driving and measurement subsystems is presented and studied. The pendulum-driving subsystem
consists
of
two engines
of
slow alternating currents and optoelectronic commutation.
In
addition, a
mathematical model
of
the experimental rig is derived
as
a system
of
three second order strongly
nonlinear
ODEs. Mathematical modeling includes details, taking into account some characteristic
features (for example, real characteristics
of
joints built by the use
of
roller bearings)
as
well
as
some
imperfections (asymmetry
of
the forcing)
of
the real system. Parameters
of
the model are obtained
by a combination
of
the estimation from experimental data and direct measurements
of
the system's
geometric and physical parameters. A few versions
of
the model
of
resistance in the joints are tested
in the identification process. Good agreement between both numerical simulation results and
experimental measurements have been obtained and presented.
Keywords: triple pendulum, experiment, chaos, mathematical modeling, parameter estimation.
1.
Introduction
Development
of
mathematics, mechanics and related numerical methods allow
more exactly model the real dynamic phenomena that are exhibited by various
physical objects. Following historical overview
of
the natural sciences
mentioned, a significant role is played by a physical pendulum, which is the
very useful mechanism used in the design
of
various real processes.
As a single or a double pendulum are often studied experimentally [5,6],
a triple physical pendulum is rarely presented in literature from a point
of
view
of
real experimental object. For example, in the work
[7]
the triple pendulum
excited by horizontal harmonic motion
of
the pendulum frame is presented and a
few examples
of
chaotic attractors are reported. Experimental rigs
of
any
pendulums are still
of
interest
of
many researchers dealing with dynamics
of
continuous multi degrees-of-freedom mechanical systems. The model having
such a properties has been analyzed in work [3].
It
consists
of
a chain
of
N
identical pendulums coupled by dumped elastic joints subject to vertical
sinusoidal forcing on its base. Lately, also the monograph on the pendulum has
been published [4]. This is a large study on this simple system also from the
historical point
of
view.
In February,
2005, in the Department
of
Automatics and Biomechanics,
the experimental rig
of
triple physical pendulum was finished and activated.
This stand has been constructed and built in order to investigate experimentally
various phenomena
of
nonlinear dynamics, including regular and chaotic
motions, bifurcations, coexisting attractors, etc. In order to have more deep