EMERGENT BEHAVIOUR IN NON-LINEAR OSCILLATIOR SYSTEMS

Non-linear oscillators have seen intense research interest over recent decades.Throughout this time they have served as representative models to help peopleunderstand theories in physics more simply and intuitively in areas as diverse asquantum mechanics and mechanical friction. They also continue to be of interestin their own right, for the unexpected dynamics which may emerge in even thesimplest non-linear oscillator models. In this thesis a series of increasingly complexnon-linear oscillator systems are studied, all in the presence of a horizontaldriving force, and the nature of the dynamics is uncovered in each case, revealingbehaviour such as sub-harmonic resonances through to energy localisation.Firstly, beginning with the simplest system, a single damped-shaken non-linearpendulum system is studied. Bi-stability is found to be supported over a criticalrange of driving frequencies, and the development of chaos at large drivingamplitude is uncovered. Period-1 and multiple-period oscillatory solution familiesare traced numerically and their basins of attraction are shown. In addition,rotating solutions are found and investigated.Then, the complexity of the system is increased by effectively splitting the singlependulum into two masses, which are coupled through impacts. Regular modesare observed in the system along with impact-specific dynamics and rotations.Quasi-periodic dynamics is shown to be a dominant feature of the system, andin order to return to periodic motion, stronger damping and forcing are considered.The solution families are then followed numerically, and the existence of a symmetry-breaking bifurcation is uncovered, and the dependence of the systemfamilies on physical characteristics such as mass size and hardness is explored.Lastly, a coupled damped-shaken pendulum chain is investigated. Period-1 oscillatorysolutions are found and compared with the results of the single pendulumsystem. Modulation instability is analysed analytically and numerically, andenergy-localised breather solutions are found. These solutions and their stabilityare analysed and a new type of mixed-frequency breather is introduced, and itsappearance in the dynamics is examined. Finally further directions are brieflyconsidered, such as the effect of mass defects on the system dynamics