World Journal of Mechanics, 2012, 2, 334-338
doi:10.4236/wjm.2012.26039 Published Online December 2012 (http://www.SciRP.org/journal/wjm)
Stability Control of Stretch-Twist-Fold Flow by Using
Numerical Methods
Shahab Ud-Din Khan1, Yonglu Shu1, Salah Ud-Din Khan2
1Department of Mathematics and Statistics, Chongqing University, Chongqing, China
2Sustainable Energy Technologies Center, King Saud University, Riyadh, KSA
Received September 21, 2012; revised October 24, 2012; accepted November 5, 2012
ABSTRACT
In this study, the multistep method is applied to the STF system. This method has been tested on the STF system, which
is a three-dimensional system of ODE with quadratic nonlinearities. A computer based Matlab program has been de-
veloped in order to solve the STF system. Stable and unstable position of the system has been analyzed graphically and
finally a comparison as well as accuracy between two-step sizes with detail. Newton’s method has been applied to show
the best convergence of this system.
Keywords: STF System; Chaos; Modified Method; Fixed Point Iteration Method; Newton’s Method
1. Introduction
The distinction between slow and fast dynamos was first
drawn by Vainshtein & Zeldovich (1972) in this research;
we describe the stretch-twist-fold (STF) fast dynamo,
which is the archetype of the elementary models of the
process. Basically, stretch-twist-fold is applied in fluid
mechanics in aerospace. In space, any fluid can be D-
Tracked easily so a magnetic field is required to compel
the fluid to be in the same orbit and this method is called
STF system. In this paper, we will investigate the accu-
racy of numerical method. The Multistep method was
first introduced by Goldstine, Herman H. in the begin-
ning of 1977’s. This iterative method has proven rather
successful in dealing with various scientific problems
[1-4] since it provides analytical solutions, which is a
standard numerical method. This method has also been
applied to solve nonlinear systems of ordinary differen-
tial equations. For example, H. B. Keller [5] presented an
extensive comparative study on the accuracy of the
multistep method and C. Lubich [6] studied the effects of
time steps on the stiff problem. J. O. Fatokun and I. K. O.
Ajibola [7] studied multistep method for integrating or-
dinary differential equations on manifolds. Differential
equations are used to model problems in science and en-
gineering that involve the change of some variables with
respect to another. Most of their problems require the
solution to an initial-value problem that is the solution to
a differential equation that satisfies a given initial condi-
tion. In most real-life situations the differential equation
that models the problem is too complicated to solve ex-
actly and one of two approaches is taken to approximate
the solution. The first approach is to simplify the differ-
ential equation to one that can be solved exactly and then
use the solution of the simplified equation to approxi-
mate the solution to the original equation. The other ap-
proach, which we will examine in this paper, uses meth-
ods for approximating the solution of the original prob-
lem. This is the approach that is most commonly taken,
since the approximation methods give more accurate
results and realistic error information. The objective of
this research is to solve STF system and test nonlinear
behavior with different time steps. This modified method
is able to find a stable and unstable position of STF sys-
tem. This method can also give the exact values after
iteration results. Newton’s method is able to show the
best convergence than fixed point iteration method.
2. Stretch-Twist-Fold Flow (STF)
The STF flow is defined as




222
8,
11 33,
2,
xtz xy
ytxy zxz
ztx yzxy






 



(1)
where α = 0.1, β = 1 are positive real parameters and re-
lated to the ratios of intensities of the stretch, twist and
fold ingredients of the flow.
3. Description of Methods
The methods we consider in this section do not produce
a continuous approximation to the solution of the ini-
tial-value problem. Rather, approximations are found at

S. U. D. KHAN ET AL.
335
certain specified and often equally spaced points. Some
method of interpolation is used if intermediate values are
needed. We need some definitions and results from the
theory of ordinary differential equations before consid-
ering methods for approximating the solutions to ini-
tial-value problems.
Definition 3.1: A function

,
f
xy is said to satisfy a
Lipschtiz condition in the variable y on a set


,,Dxyaxb y
If a constant 0L exists with the property that


,,
f
xyf xyLyy





,,,
x
yxy D


This first part of this section is concerned with ap-
proximation the solution

y
x to a problem of the form

d,,for
d
y
f
xyax b
x
Subject to an initial conditions

0.
y
ay
Lemma 3.1: Suppose that


,
f
xy is continuous on D
if f satisfies a Lipschitz condition on D in the variable y,
Then the initial-value problem


0
d,,for
d
y
f
xyax b
x
ya y





has a unique solution

y
x for axb.
The methods of Euler and Runge-kutta are called one-
step methods because the approximation for the mesh
point 1
i
x involves information from only one of the
previous mesh points i
x
although these methods can
use functional evaluation information at points between
i
x
and 1
i
x, they do not retain that information for
direct use in future approximations. All the information
used by these methods is obtained within the subinterval
over which the solution is being approximated. Since the
approximate solution is available at each of the mesh
points 01
,, ,
i
x
xx before the approximation at 1
i
x

is obtained and because the error

11ii
yyx

 tends
to increase with I, it seems reasonable to develop meth-
ods that these more accurate previous data when ap-
proximation the solution at 1
i
x

.
Methods using the approximation at more than one
previous mesh point to determine the approximation at
the next point are called multistep methods.
Definition 3.2: An m-step multistep method for solv-
ing the initial-value problem (3.1) is one whose differ-
ence equation for finding the approximation 1i
y

.
At the mesh point 1
i
x can be represented by the
following equation,
where p is an integer greater than 0

 
1011111 11
011
11
1
01
,
,,
iimimii
iimim im
mm
ji jij
jj
yayayay hbfxy
bfxyb fxy
ayh bf

 




 


 



(A)
When b–1 = 0 then the method is called explicit or
open. Since Equation (A) then gives 1i
y explicitly in
terms of previously determine values. When 10b

then the method is called implicit or closed. Since 1i
y

occurs on both sides of Equation (A) and is specified
only implicitly.
To begin the derivation of the multistep methods, note
that the solution to the initial-value problem (3.1), if in-
tegrated over the interval


1
,
ii
x
x has the property that
  

1
1,d
i
i
x
ii
x
yxyxf xyxx

  (B)
Since we cannot integrate
 
,
f
xy x without know-
ing


y
x the solution to the problem, we instead inte-
grate an interpolating


Lx to
 
,
f
xy x that is de-
termined by some of the previously obtained data points






00 11
,,,,,,
ii
x
yxy xy Equation (B) becomes
 
1d i
i
x
ii
x
y
xy Lxx



3.1. Modified Method
Use the modified APC method to solve STF system. This
method is derived from ABF-Explicit m-step technique
and AM-Implicit m-step technique. The simulation done
of this paper is for the time range


0, 1t with two
time steps 0.01t

 and 0.001t

.
Represented formula:


111223344nn
y
yhbgbgbgbg
 


145516273nn
y
yhbgbgbgbg
 
where,
1234
567
55, 59,37, 9
19,5, 1.
bb b b
bbb



12132
4351
,,,
,.
nn n
nn
g
fgf gf
gfgy


 

3.2. Unstable Position
When α = 1, β = 0.1 and h = 0.01 then we can determine
the unstable position of the system that is shown in Ta-
ble 1 and easily analyzed by the Figure 1.
3.3. Stable Position
When α = 0.1, β = 1 and h = 0.001 then we can determine
the stable position of the system that is shown in Tabl e 2
and easily analyzed by the Figure 2.

S. U. D. KHAN ET AL.
336
Table 1. X, Y, Z-Direction for β = 0.1.
T ∆x ∆y ∆z
0 0 0 0
0.1 0.0533 0.5102 0.0081
0.2 0.0801 0.8065 0.0210
0.3 0.0947 0.8007 0.0386
0.4 0.1035 0.7961 0.0605
0.5 0.1097 0.7909 0.0863
0.6 0.1147 0.7854 0.1155
0.7 0.1191 0.7796 0.1473
0.8 0.1233 0.7734 0.1811
0.9 0.1275 0.7670 0.2158
1 0.1319 0.7602 0.2507
00.1 0.20.3 0.40.5 0.6 0.70.8 0.91
-0. 8
-0. 6
-0. 4
-0. 2
0
0. 2
0. 4
0. 6
0. 8
1
t
f(t)
Figure 1. The unstable position of the system when α = 1, β
= 0.1 and h = 0.01.
Table 2. X, Y, Z-Direction for β = 1.
T ∆x ∆y ∆z
0 0 0 0
0.1 0.0051 0.0508 0.0001
0.2 0.0100 0.1013 0.0004
0.3 0.0146 0.1511 0.0004
0.4 0.0190 0.1998 0.0006
0.5 0.0232 0.2472 0.0009
0.6 0.0271 0.2929 0.0012
0.7 0.0309 0.3367 0.0016
0.8 0.0345 0.3785 0.0021
0.9 0.0379 0.4180 0.0025
1 0.0411 0.4551 0.0031
00.1 0.20.3 0.40.5 0.6 0.70.8 0.91
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
t
f(t)
Figure 2. The stable position of the system when α = 0.1, β =
1 and h = 0.001.
4. Fixed Points for Function of Several
Variables
In this section, we will discuss about fixed point iteration
method and Newton’s method.
A system of nonlinear equations has the form




112
212
12
,,, 0
,,, 0
,,, 0
n
n
mn
fxx x
fxxx
fxx x











Here each function fi can be thought of as mapping a
vector

T
12
,,,
n
x
xx xof n-dimensional space Rn into
the real line R.
The system of n nonlinear equations in n unknowns
can alternatively be represented by defining a function f,
mapping Rn into Rn by

T
12
,,, .
n
fff f
Then we have


0fx
In an iterative process for solving an equation


0fx

was developed by transforming the equation
into one of the form


x
gx. The function g is defined
to have fixed points precisely at solutions to the original
equation. A similar procedure will be investigated for
function from Rn to Rn.
Definition 4.1: A function g from n DR into Rn
has a fixed point at
x
D


if

.
g
xx


Consider the STF system (see Figure 3):
3
1
2
22
13 13
2
1
31
2
8
11
13
1
22
x
xx
x
xxx
x
x
xx
x









 






(C)

S. U. D. KHAN ET AL.
337
(a) x-y-z
(b) x-y
(c) y-z
Figure 3. Portrait and x, y, z direction of STF system.
To approximate the fixed point
x
 we choose

T
00.3,0.2, 0.1x
the sequence of vectors generated
by
 
13
1
2
22
13 13
1
2
11
31
2
8
11
13
1
22
k
k
k
kk
kk
k
k
kk
k
x
xx
x
xxx
x
x
xx
x





















If k = 1, 2, 3, 4 then



T
1
T
2
T
3
0.00625,0.82664,0.22500
0.00340,0.99136,0.003503
0.00004,0.99998,0.00187
x
x
x





13
0.62664,0.221497,8.6 10
kk
xx


Now we have tested this system in Newton’s method
and comparison with fixed point iteration results.
Newton’s method for systems, like the one-dimen-
sional Newton’s method, a fixed point iteration based on
a linearization of


.
f
x If :n
f
RR then the Tay-
lor series for


f
x has the form





 

kkk
f
xfx Jxxx Ex 
Newton’s method is derived just as it was for the
one-dimensional case: neglecting the remainder term, we
have




kkk
f
xfx Jxxx 
And setting


0fx

gives what we hope is an im-
proved estimate





0
kk
fx Jxxx


If




det 0,
k
Jx

the iteration
 
1
1,0,1,2, kk kk
xxJxfxk


 
 
This is Newton’s method for systems.
Where,

1
k
Jx




 is the inverse of

J
x.
In practice it is preferable to solve


kk
J
xx




k
f
x for k
x

and then add this quantity to k
x
we
have
1kkk
x
xx



where
1kk k
x
xx



By system (C), the Jacobi matrix

J
x for this sys-
tem is

S. U. D. KHAN ET AL.
338
Table 3. Convergence rate of STF.
k 1
k
x
2
k
x
3
k
x
1kk
x
x

0 0.3 0.2 –0.1
1 0.007523 0.00430 0.00005 2.9 × 10–1
2 0.962467 0.10136 0.10002 9.5 × 10–1
3 0.338002 0.00450 0.00258 6.2 × 10–1

21
13 31
231 2
880.1
226 2
0.1 22
xx
J
xxx xx
xxxx



 




The results are given in Table 3.
According to previous examples, we can easily ana-
lyze that Newton’s method is more accurate than fixed
point iteration method.
5. Conclusion
In this paper, MATLAB programming has been used to
solve the STF system with variable time steps (∆t = 0.01,
0.001). We have obtained good results by using two
methods applied to the STF system concerning the sys-
tem is stable and unstable state. The modified method
was computed by developing simple algorithm without
perturbation techniques i.e. linearization or discretization.
In all the considered cases, it has been proved that the
modified multistep method appears to be the best method
to approximate this solution based on its accuracy and
Newton’s method is a good example to solve root finding
problem in STF system. Newton’s method is able to
show the best convergence than fixed point iteration
method.
6. Acknowledgements
The first author is very thankful to all of his co-authors
and especially to Professor Shu Yonglu for advising and
giving me the opportunity to conduct this research and
also very much thankful to Sustainable Energy Tech-
nologies Centre, King Saud University for funding the
research.
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