CHAPTER ONE
1.0 Â Â Â Â INTRODUCTION
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             Most real life problems are nonlinear in nature, this has made the study of nonlinear systems which are very complex an important area of study and research. The Duffing oscillator with viscous damping is one of such important nonlinear system which can be generally described by the following equation of motion:
(1)
where
.
For the un-damped system, the total potential energy is given by:
, then we have a Hamiltonian system where H is the Hamiltonian. In this case, one can equate H to a constant and integrate for the displacement .
Where we have positive damping ( , we have the important property, .
If we set  in (1), we obtain,
(2)
where,
The system (2), describes the motion of the viscously damped cubic Duffing oscillator which can be used to model conservative double well oscillators which can occur in magneto-elastic mechanical systems [20]. A good and illustrating example of such system was described in [30]. The cubic Duffing equation can as well be used to model the nonlinear spring-mass system (hardening and softening) [2], [9], as well as the motion of a classical particle in a double well potential [20], [25]. The system (2) with forcing was proposed by Correig in [6] as a model of microseism time series and have been used in [22] to model the prediction of earthquake occurrence.
Generally, the viscously damped and forced cubic-quintic Duffing oscillator with random noise which can be obtained by setting , in (1) is given by the equation
(3)
where, ,
is the damping co-efficient.
is the proper or resonant frequency.
are the co-efficient of nonlinearity.
is the random noise.
Equation (3) with  was used in [8] to model the transverse oscillations of a nonlinear beam.
We can as well write (3) as a system in the form,
(4)
where,
is a tri-stable potential or a triple well potential.
Setting  in equation (4), then we get
(5)
where  implies as well from (4),
.                                                        (6)
The stability matrix of the system (4) is given by,
(7)
where the eigenvalues must satisfy the equation det
(8)
1.1Â PURPOSE OF STUDY
This study is aimed at first, observing the effect of the added quintic nonlinear term to the damped cubic Duffing oscillator, secondly, to show that we can only obtain centres and saddles for any arbitrary set of parameters in the case of an un-damped cubic-quintic Duffing oscillator, thirdly, to observe the effect of positive/negative damping coefficient to the Duffing oscillator and finally to observe the effect strong nonlinearity to the damped Duffing oscillator.
1.2Â Â DEFINITION OF ESSENTIAL TERMS
         Positive Damping
This is understood as the basic damping in which case, the damping acts opposite the direction of the velocity of a given system thereby bringing the system to equilibrium at a given time. This is very feasible in all physical systems.
Negative Damping
In this case we mean to represent a situation where the damping acts in the same direction as the velocity. This results in oscillations with increasing or growing amplitude. Physically, this is very possible, many physical systems behave this way, as was noted in [34] where catastrophic failures recorded for a large, long stroke and high speed extrusion press was attributed to negative damping. In [35], it was also noted that systems such as the laser driven pendulum studied in [36] and systems close to thermodynamic equilibrium exhibit such damping, [37] investigated the asymptotic behavior of the solutions for nonlinear wave equations of Kirchhoff type with positive/negative damping, it was also remarked in [38] that negative damping is responsible for LCOs (limit cycle oscillations) arising in nonlinear aero-elastic systems. Motivated by the fact that the drop in steady-state wind turbine rotor thrust with wind speed above rated would lead to negative damping of the barge-pitch mode and contribute to the large system-pitch motions, [39] investigated the influence of control on the pitch damping of a floating wind turbine and interesting results were obtained. Though practical measures are taken to prevent/reduce negative damping in most physical systems, its occurrence persist after a given period of time. Systems like the nose wheel shimmy of an airplane exhibits negative damping while the airplane is in motion as it eventually comes in contact with an external object (say a pebble on the runway ). We must note that it is usually very hard to construct simple systems that exhibit negative damping. Obviously negative damping is no abstract or mathematical idealization as it is frequently obtainable and observable in most physical systems.
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