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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. 


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



5.1  Summary                                                                                                                         56

5.2  Conclusions                                                                                                                    56

5.3  Recommendations                                                                                                          57      

References                                                                                                                            58                                                                                                                 









 1:  Solution matrix table for Solution Functions                                                           37

 2:  Solution matrix for the independent variable values                                               51    










 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










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.


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.


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.


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.


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:


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.           



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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