EXISTENCE OF SYMMETRIC PERIODIC SOLUTIONS IN THE SITNIKOV PROBLEM

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 4^th Order R-K Method                                         42

4.6 Differences of Analytical Approach and 4^th 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 m_1, 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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