ABSTRACT
This study focuses on the generation and evolution of chaotic motions in single and double-well Duffing oscillators under certain parametrical excitations. The Melnikov approach and Lyapunov exponent are proposed to calculate the threshold values for the chaotic motion in a Duffing system. The minimum and maximum values were obtained and the dynamical behaviors showed the intersections of manifold which was illustrated with MATHCAD software. Similar results obtained from the two methods show that the behavior of the perturbed Duffing oscillator is chaotic and highly unstable with repeated resonances of successively higher periods. As a result, the functions with symmetric wells were separated by the barrier at a point and the unperturbed system produces three equilibria points showing similar behaviors. Also, the method of the Lyapunov exponents narrow the range for the critical threshold values and detect mutations in the chaotic system. Numerical simulations showed that as the parameter was varied, repeated resonances of successively higher periods occurred and unstable chaotic motion was observed.
TABLE
OF CONTENTS
Title
page
i
Declaration ii
Certification 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
3
1.3 Aims and Objectives of the Study
4
1.4 Purpose of the Study
5
1.5 Significance of the Study
5
1.6 Justifications of the study
6
1.7 Scope and Limitations of the Study 7
1.8
Definition of Terms 7
CHAPTER 2 (LITERATURE
REVIEW) 9
2.1 History of Chaos 9
2.2 Review on variations of parameters in Duffing
Oscilltor
17
2.3
Review on excitation of parameters of Duffing oscillator
18
2.4
Chaos
20
2.5 Characteristics of chaos 20
2.6 Method of solutions for chaotic Duffing
Double and single-well oscillator 20
2.7 Advantages of chaos
21
2.8 Applications of chaos
22
2.9 Types of chaos
22
2.10
Double-well oscillators
23
2.11 Single-well oscillators 23
CHAPTER 3 (RESEARCH
METHODS)
25
3.1
Melnikov Method
25
3.2 Melnikov method for predicting chaos
25
3.3 Method of Lyapunov Exponent
28
3.4 Hamiltonian Equation
30
CHAPTER 4 (RESULTS) 32
4.1
The single-well Duffing Oscillator
32
4.2 Melnikov method for the perturbed single-well
Duffing oscillator
33
4.3 The Lyapunov exponent of a single-well
Duffing Oscillator 35
4.4 Numerical simulation of single-well well
Duffing oscillator 38
4.5 The double-well Duffing oscillator 44
4.6 Melnikov method for the perturbed double-well
Duffing oscillator
46
4.7 The Lyapunov exponent of a Double-well
Duffing oscillator 47
4.8 Numerical simulation of a double-well Duffing
oscillator 50
CHAPTER 5 (SUMMARY AND
CONCLUSIONS)
5.1 Summary 56
5.2 Conclusions 56
5.3 Recommendations 57
References
58
LIST
OF TABLES
1: Solution matrix table for Solution Functions
37
2: Solution matrix for the independent variable
values
51
LIST
OF FIGURES
4.1: Trajectory x(t) as a function of time
41
4.2: Velocity x(t) as a function of time 41
4.3: Phase portrait 42
4.4: Trajectory
x(t) as a function of time 43
4.5: Velocity x(t) as a function of time 43
4.6: Phase portrait 44
4.7: Trajectory
x(t) as a function of time 52
4.8: Velocity x(t) as a function of time 52
4.9: Phase
portrait 53
4.10:
Trajectory x(t) as a function of time 53
4.11: Velocity
x(t) as a function of time 54
4.12: Phase portrait 54
CHAPTER
INTRODUCTION
1.1
BACKGROUND
TO THE STUDY
The Duffing oscillator or Duffing equation named after
George Duffing is a second-order nonlinear differential equation which is used to model
some driven and damped oscillators. It has been used widely in engineering,
economics, physics, and various other physical phenomena. Given its chaotic
nature and the characteristic of oscillation, many scientists are motivated by
the Duffing equation which is nonlinear differential equation due to its nature
to repeat like dynamics in our natural world (Eze et al., 2019). The equation has become a common example of
nonlinear oscillations in textbooks and research article. The equation is of
the form;
The equation is a easy model that can show different
types of oscillation such as chaos and limit cycles. The terms associated with
this system represent:
δ controls the amount of damping,
α
controls the linear stiffness,
β
controls the amount of non-linearity in the
restoring force; if β = 0,
the Duffing equation or oscillator explains a damped and driven simple harmonic
oscillator. 
is the amplitude of the external force; if γ = 0,
the system is without driving force and 
is the angular frequency of the periodic
force.
Chaotic
motion in nonlinear systems has become a popular research over the years. Many
investigations have been carried out on the different nonlinear chaotic systems
to understand the various behavior of these systems (Ueda1979). Duffing
oscillator is one of the three fundamental forced oscillators viz; Duffing, Van
Der Pol and Rayleigh. These oscillators
have been extensively examined and its characteristics embedded in the physical
systems can be realized from these three systems (Chang ,2017). Amongst them,
the Duffing oscillator is the most useful nonlinear dynamical systems which is
considered as a model for various physical and engineering problems such as
dynamics of a bucked elastic beam, particles in double and single well (Sang
and Kim, 2000). The Duffing type of nonlinear system is well noted for the
occurrence of chaos behavior over the years (Zhang et al., 2016). As a model, Zeeman (2000), opined that the equation
involves an electro-magnetized vibrating beam analysed as exhibiting cusp
catastrophic behavior parameter values. According to Thompson and Stewart
(2002), Duffing oscillator or equation is a typical example of dynamical system
that has chaotic nature.
Several methods for evaluating the occurrence of
chaotic behavior has existed in the literature like the Melnikov approach which
is an analytical tool to giving the measure for the occurrence of chaotic
behavior which was presented by Melnikov in (1963). The main idea of his
approach is to obtain the distance between unstable and stable manifolds in the
Poincare section. If the unstable and stable manifolds cross once, they will
intersect countless times. Thus according to Holmes (1979),this criterion was
analyzed as the qualitative methods for bifurcation of complex systems. In his
work, he considered systems whose dynamical behaviors may be represented by an
autonomous ordinary differential equation with parameters.
Further, the investigation carried out by Ueda in
(1979) was centered on the chaotic phenomena in Duffing equation. In his
research, changes of attractors was seen using the numerical simulation under
various parameters. Hence, the chaotic nature in Duffing equation under
harmonic excitation is called a common dynamical phenomenon. However, Ott,
(1993) defined and analyzed chaotic behaviors and its bifurcation. The crisis
has been widely used in double well Duffing equation under harmonic parametric
excitation to carry out the transformations between different dynamical
behaviors. Also (2002), he points that dynamics on attractor can be chaotic if there is an exponential
sensitivity to initial conditions. Thus, describing chaos as the dynamics of
attractors, for most cases involving differential equation, chaos commonly
occur.
The Lyapunov exponent is also an interesting
quantities to dynamical systems in that it provides a quantitative amount of
the convergence or divergence of nearby trajectories or initial points for a
dynamical system. Therefore, it can be used to study the stability of limits set
and to check the presence of chaotic attractors ( Wolf 1986)). For instance, by
computing Lyapunov exponents, one may construct phase diagrams with the regions
showing the characterized periodic
motions (negative exponent) or aperiodic behaviors (positive exponent). In
other words, Lyapunov exponents can be used to discriminate regions
characterized by chaotic and non chaotic behaviors. ( Zeni and Gallas 1995).
1.2
STATEMENT
OF THE PROBLEM
In this research, we seek the chaotic motion of the
nonlinear differential equation in single and double-well Duffing’s oscillator.
The work provides an appropriate solution of the Duffing equation considering
different damping effect and with different initial conditions.
The single-well Duffing equation under parametrical excitation is shown below;
The double-well Duffing system under a harmonic
parametrical excitation is given as the following:
where a, b and c are positive coefficients, 
and 𝛺 are amplitude and
frequency of excitation and 0 < ɛ < 1.
By denoting x = y,
equation (1.2) can be re-written as the following;
x = y
hence, the Duffing system (1.3) is symmetric
to the origin (0,0)
However, if α and β are varied and γ and ω are excited in the above equation, the
behavior may not be sensitive to initial or boundary condition, most previous
work direct on the use of perturbation. However, the present study
investigates Duffing equation using
Melnikov method and the Lyapunov exponent for the nonlinear system and to verify if their
behaviors will be chaotic or not.
1.3 AIM
AND OBJECTIVES OF THE STUDY
The aim of this research work is to study chaos in
Double and Single well Duffing oscillator under certain parametric variations
and excitations. The sub-objectives are;
i.
To detect the chaotic phenomena
for the nonlinear differential equation.
ii.
To gain insight into the
evolution of chaotic dynamical behaviors.
iii.
To use Melnikov method
and Lyapunov exponent to provide solution to the problem and thereby simulating
the chaotic process.
iv.
To obtain the numerical
simulation using MATCAD software.
1.4 PURPOSE OF THE STUDY
In general, the Duffing equation or oscillator does
not have an exact symbolic solution. However, it should be noted that in many
works where chaos was present in a particular system, the analysis usually ends
up with a formula that established a border where bifurcation can occur. Further
analysis of chaos appearance, condition that would allow establishing links
between physical properties of double and single well has not been performed so
far. Zhang et al., (2016) opined that the double-well Duffing system with
parametrical excitation will keep on switching between chaotic, saddle and
permanent chaos by bifurcation and crisis, once the system becomes chaotic in
the sense of Smale horseshoe. Therefore, this research work presents the
analytical approaches to finding chaotic motion in double and single-well
Duffin system with the familiar concept of oscillators, exploring the two
control parameters of the system namely ,the amplitude 
and the angular frequency of the driven force as well as variation of
resonance and nonlinear coefficient of the cubic restoring force.
1.5 SIGNIFICANCE OF THE STUDY
Chaos is an interdisciplinary theory
stating that within the apparent randomness of chaotic complex systems, there
are underlying patterns, repetition, fractals etc on programming at the initial
point known as sensitive dependence on initial condition. Chaotic motion are
neither steady nor periodic, they appear to be complex. Some areas that will
benefit from this research work includes; mathematics, geology, computer
science, biology, robotics etc. Chaos theory formed the basis for such fields
of study as complex dynamical systems.
Computer science: The
chaos theory is not new to computer science since it has been for so many years
in cryptography and symmetric key, which relies on diffusion and confusion. It
can be used in DNA computing for efficient way to encrypt images and other
information.
Robotics: Robotics is an
area that has benefited from the chaos theory. Using the chaos theory developed
by Wang et al., (2012), a predictive
model was built to check trial and error type of refirnment, which aids
interaction with there environment.
Biology: Its application
to biology can be found in ecological systems such as hydrology and
cardiotocography. Dinggi et al.,
(2011) observed that the chaos metaphor can be used in verbal theory which
provides helpful insight to describing the complexity of small work groups that
go beyond the metaphor itself.
1.6 JUSTIFICATION OF THE STUDY
The Duffing oscillator is a well known model for
non-linear oscillator governed by the following dimensionless second order
differential equation;
The right hand side of eqn (1.4) represents the driving force at
a time t with amplitude f, and
angular frequency ω.
The system (1.2) is a generalization of the classic Duffing oscillator equation
and can be considered in two main physical situation wherein the dimensionless
potential:
1.7 SCOPE AND LIMITATIONS OF THE STUDY
This research work has been focused on Duffing oscillator with
external force which is nonlinear differential equation. However, it is limited
to only second order differential equation and not higher order differential
equation.
1.8 DEFINITIONS OF TERMS
Chaos: Chaos
in a common usage means a total disorder, but in a dynamical system, it denotes
a behavior that is extremely sensitive to initial conditions.
Duffing Equation: This
is a non-linear second-order differential equation used to model certain damped
and driven oscillators.
Perturbation:
A perturbation is a small change in the movement, especially an unusual change
or simply a disturbance of motion or state of equilibrium.
Oscillation:
A singe wing (as of an oscillating body) from one extreme limit to the order.
System:
A system is a set of interacting or inter-dependent of components parts forming
a complex or intricate whole.
Dynamics:
Is the study of how systems change in time.
Dynamical System:
A dynamical system is a system whose state evolves with time over a state space
according to a fixed rule or it can define as a system in which a function
describes the time dependence of a point in a geometrical space.
Parameter:
This is an arbitrary constant whose vale characterizes a member of a system.
Manifold:
Is a collection of points forming certain kind of set such as those of
topologically closed surface or an analogue of this in three or more dimension.
Bifurcation:
Is the change in the qualitative or topological structure of a given system.
Homoclinic Orbit:
This is a trajectory of a flow of dynamical systems which joins a saddle
equilibrium points to itself. More precisely, a homoclinic orbit lies in the
intersection of the stable and unstable manifold of an equilibrium.
Trajectories:
This is a solution curves of a differential equation.
Stable Manifold:
Is a set of points along the flow that are sent towards the fixed point
Unstable Manifold:
The unstable manifold is the set of points along the flow that are sent away
from the fixed point.
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