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31I15 IJAET0715576 v6 iss3 1299to1312.pdf


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International Journal of Advances in Engineering & Technology, July 2013.
©IJAET
ISSN: 22311963
The present study utilised the popular constant operation time step fourth and fifth order Runge-Kutta
schemes to simulate equation (1) in the first order rate equations (2) and (3). The respective details of
each scheme are provided in equations (4) to (8) and (9) to (15) substituting y  1 ,2 , x  t and
constant time step h.
Fourth-Order Runge-Kutta Scheme

h
K1  2( K2  K3 )  K 4 
6
K1  f ( xi , yi )
Kh
h
K 2  f ( xi  , yi  1 )
2
2
K2h
h
K3  f ( xi  , yi 
)
2
2
K4  f ( xi  h, yi  K3h)
yi 1  yi 

(4)
(5)
(6)
(7)
(8)

Fifth-Order Runge-Kutta Method

h
7 K1  32K3  12K4  32K5  7 K6 
90
f ( xi , yi )
Kh
h
f ( xi  , yi  1 )
2
2
(3K1  K 2 )h
h
f ( xi  , yi 
)
4
16
Kh
h
f ( xi  , yi  3 )
2
2
(3K 2  6 K3  9 K 4 )h
3h
f ( xi  , yi 
)
4
16
( K  4 K 2  6 K3  12 K 4  8K5 )h
f ( xi  h, yi  1
)
7

yi 1  yi 

K1 
K2 

K3 
K4 

K5 
K6 

(9)
(10)
(11)
(12)
(13)
(14)
(15

Solutions Schemes
The under-listed four distinct solution schemes were implemented in the present study.
 RK41-Constant single simulation time step fourth order Runge-Kutta scheme.
 RK42-Constant double simulation time step fourth order Runge-Kutta scheme.
 RK51-Constant single simulation time step fifth order Runge-Kutta scheme.
 RK52-Constant double simulation time step fifth order Runge-Kutta scheme.
Study Parameters
In tune with literature research interest this study focuses on the parameter plane defined by

2.0  q  4.0 and 0.9  g  1.5 , fixed drive frequency D 

h

2
, and fixed simulation time step
3

TD
2
for TD 
. The initial conditions for all studied cases is (0, 0) and the simulation was
500
D

executed for 2010-excitation periods including 10-periods of transient and 2000-periods of steady
solutions.
The associated novel attractor of Poincare solutions were investigated for their space filling ability
using fractal disk dimension characterisation, see Salau and Ajide (2012). Ten (10) systematic
observation scales of disk size variation and quantity of disks required for complete covering of the
attractor were made in five (5) different trials per observation scale using random number generation
seed value of 9876. This process enables the determination of ‘optimum’ number of disks required at
specified observation scale.

1302

Vol. 6, Issue 3, pp. 1299-1312