ABSTRACT
In this work, the
existence of symmetric periodic solutions of the Sitnikov problem was studied.
Analytical solutions were obtained for the homogenous equation. The
complementary function thus obtained, confirmed the existence of periodic
solution. Further test for periodicity was carried out using the Bendixson
Criterion. Due to the high nonlinear nature of the equation, Runge-Kutta
fourth-order and Euler methods were again used to obtain an approximate
solution which was unbounded and were compared with the analytical solution for
the interval of eccentricities. Numerical simulation was obtained using MATCAD
which extend some results in literature.
TABLE OF CONTENTS
Title
Page i
Certification ii
Declaration iii
Dedication iv
Acknowledgements v
Table
of Contents vi
List
of Tables viii
List
of Figures ix
Abstract x
CHAPTER
1 (INTRODUCTION) 1
1.1 Background
to the Study 1
1.2
Statement
of the Problem 2
1.3
Aim
and Objectives of the Study 3
1.4
Significance
of the Study 3
1.5
Justification
of the study 4
1.6
Scope
and Limitations of the Study 4
1.7 Definition
of Terms 4
CHAPTER 2 (LITERATURE
REVIEW) 7
2.1
History of the Sitnikov 7
2.2
Methods of Solutions 8
2.3
Advantages of Sitnikov Problem 10
2.4
Areas of Application 10
2.5
Existence of 
Periodic Solution 10
2.6
Existence
of Second Order Nonlinear Differential Equation 11
2.7
Equation of Motion 12
2.8
Resonant and Non-Resonant Equations 12
2.9
Equations of Symmetries 13
2.10
Periodic Solutions 14
CHAPTER
3 (METHODOLOGY) 16
3.1 Periodic
Problems 16
3.2
Basic Properties of Euler Methods 18
3.3
Derivative of Euler’s Method 18
3.4
Runge-Kutta Methods 22
3.5
Existence of Periodic Solutions 25
CHAPTER
4 (RESULTS AND DISCUSSION) 28
4.1
Analytical Result 28
4.2
Periodic Solution 34
4.3
Application of Runge-Kutta Methods 36
4.4
Error Bound 41
4.5
Similarities of Analytical Approach and 4th Order R-K Method 42
4.6
Differences of Analytical Approach and 4th Order R-K Method 42
CHAPTER
5 (SUMMARY, CONCLUSION AND RECOMMENDATION) 53
5.1
Summary 53
5.2 Conclusion 53
5.3
Recommendations 54
References 56
LIST OF TABLES
4.1
(Euler method) 27
4.2 (Fourth-order Runge-Kutta method) 39
4.3 (Error Bound) 42
LIST OF FIGURES
CHAPTER 1
INTRODUCTION
1.1 BACKGROUND TO THE STUDY
Sitnikov
problem is a restricted three-body problem which allows oscillatory type of
motion, citing two primaries such as m1, m2 which are non-zero equal masses, moving around
each other of eccentricity e. The massless body m3 performs motion which is perpendicular to the
primary orbit plane through the barycentre of the primaries. Let 
be the position of the body z in a coordinate system, then the equation of
motion of the Sitnikov problem becomes
with
the initial conditions;
Where;
is the distance from the center of the orbit
to 
is acceleration. e is the eccentricity, 
covers the distance from the barycentre to the
mass and it is given by
It
can either be circular or an elliptic solution of the Kepler problem
with
eccentricity 
respectively. We have
the eccentricity anomaly u(t), which is a function
of time through Kepler equation
Few
works have been done by researchers on existence of symmetric periodic solution
in the Sitnikov problem. Such as Belbruno (1994) who worked on the families of
periodic orbits which bifurcate from the circular restricted three-body
problem, the analytical part was gotten by using elliptic functions. The
analytical expressions were for the solution of the circular Sitnikov problem. He
also analyzed the qualitative and quantitative behaviour of the periodic
function. Llibre and Ortega (2008) analytically made use of the global
continuation theorem to investigate the families of the elliptic Sitnikov
problem for non-necessarily small values of the eccentricity e, and they stated that some
periodic orbits for e = 0 can be continued in 
Rafael Ortega (2016)
studied Symmetric periodic solutions in the Sitnikov problem, using Shooting
method and Sturm oscillation theory. The results showed the existence of a
periodic solution with minimal period.
1.2 STATEMENT OF THE PROBLEM
We
consider the equation of Sitnikov problem of the form
The
questions that naturally arise are;
I.
Does the solution exist in the interval [0.1]?
II.
Is the solution periodic?
It
is our duty in this work to answer the above questions.
1.3 AIM
AND OBJECTIVES OF THE STUDY
This
work studies the existence of symmetric periodic solutions in the Sitnikov
problem. The sub-objectives are:
I.
To investigate the Existence of symmetric periodic solutions in [0,1].
II.
To test for the periodicity of the solutions.
III.
To obtain results for all values of eccentricity.
1.11
SIGNIFICANCE
OF THE STUDY
Sitnikov
type has been found to be significant in some areas of study and applications
such as Physics, Engineering, Chemistry, Economics, Agriculture and Biological
phenomena. In Physics it can be considered in a spacecraft and relevant celestial
bodies, example is the Earth and the Moon. This happens when we consider a free
return trajectory around the moon. In Chemistry it is significant under
Spectroscopy, Nuclear magnetic Resonance, (Streitwieser 1976). It also plays a
great role in mathematical biology and population dynamics, also in
mathematical economics, where they consider systems often subject to seasonal
variations.
1.4 JUSTIFICATION
OF THE STUDY
This
work is justified by studying Ortega (2016), where existence of a periodic
solution with minimal period were obtained using Shooting method and Sturm
oscillation theory. Ortega and Llibre (2008), where families of symmetric
periodic orbits for all 
were studied using global
continuation method.
Unlike
Ortega (2016), Ortega and Llibre (2008), this work proposes both analytical and
numerical approaches using Euler and fourth-order Runge-Kutta methods. Further
tests for the periodicity was also proposed.
1.5 SCOPE
AND LIMITATIONS OF THE STUDY
The
scope of the study is on a second order nonlinear differential equation of the
Sitnikov type, it also covers the movement of the three restricted body
systems, the periodicity of the solutions of the Sitnikov problem, the nature
of the eccentricity of the orbit, and the rate of change of displacement
between the bodies. It is limited to only second order nonlinear differential
equation and not higher order differential equation.
1.6 DEFINITION
OF TERMS
1.7.1 Periodic Solution: Periodic
solutions of equations are solutions that describe regularly repeating
processes. That is, a solution z(t) is said to be periodic if
Where
is the period.
1.7.2 (Runge-Kutta Method): Let 
real numbers and let 
An
s-state Runge-Kutta method is given by
1.7.3 (Euler Method): Euler
method is based on approximating a graph of solution y(t) with a sequence of tangent line approximation
computed sequentially.
1.7.4 (Step Size): A
step size (nesh size) is a decomposition of the interval
the
spacing between the points is called the step size.
1.7.5 (Phase plane analysis):
This is a technique which is often very useful in order to analyse the phase
plane behaviour of a two dimensional autonomous system.
1.7.6 (Step Point): This
is the partition of the interval 
on which the solution is desired into finite
number of subintervals by the points
1.7.7 (Euler and Runge-Kutta Fourth-order
Methods): The Euler and Runge-Kutta fourth-order methods are numerical
techniques use to solve an ordinary differential equation of the form;
1.7.8 (Symmetry):
A geometric shape or object is symmetric if it can be divided into two or more
identical pieces that are arranged in an organized fashion.
1.7.9 (Resonance): this
is the tendency of a system to vibrate with increasing amplitudes at some
frequencies of excitation.
1.7.10 (Eccentricity): This
is the deviation of a curve or orbit from circularity. Or a measure of the
extent of deviation from circularity.
Eccentricity
at different values, has the following shapes:
1.7.11 (Orbit): This
is a curved path of a celestial object or spacecraft round a star, planet, or
moon, especially a periodic elliptical revolution.
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