HOMOCLINIC BIFURCATIONS AND CHAOS IN A TWO-DEGREE-OF-FREEDOM MAGNETO-MECHANICAL NONLINEAR SYSTEM

This Ph.D. thesis is primarily composed of three parts: (1) the theoretical development of homoclinic bifurcation theory for a multiple degree-of-freedom nonlinear system that possesses both homoclinic and heteroclinic orbits, and the application to a two-degree-of-freedom nonlinear oscillator, (2) extensive numerical simulations of the oscillator, and (3) experimental verification of the theoretical predictions.; The development of the theory is primarily based on the results obtained by Palmer. We have successfully applied the theory to the nonlinear oscillator. The homoclinic and heteroclinic orbits have been analytically computed, and the bifurcation integrals have been carried out. When the external forcing exceeds the bifurcation value, chaotic motion becomes possible even though the system is completely deterministic. A characteristic feature of this type of motion is that solutions with arbitrarily close initial conditions generally diverge after sufficient time, and the behavior appears unstable.; A variety of numerical techniques have been used to simulate the system. The Poincare map and Fast-Fourier-Transform (FFT) methods have been used to check the chaotic behavior of an individual motion, and the Lyapunov exponent technique has been used in a systematic way to check the behavior of motions in the driving frequency and amplitude plane (a positive Lyapunov exponent indicates that the motion is chaotic). Using these techniques the chaotic region in the parameter space has been found numerically. The numerical results show that homoclinic and heteroclinic bifurcations are necessary conditions for chaotic motion. It was also observed that the emergence of fractal basin boundaries coincides with the heteroclinic bifurcation criterion, or that absolute predictability of the final state might be impossible. In practice this has important implications in dynamic modelling, particularly if numerical simulations are to be attempted.; The main nonlinearity of our experimental system is created by the motion of a ferromagnetic ball in an non-uniform magnetic field. Using the FFT technique, transient chaotic motions of our experimental oscillator in the x-y plane were detected. This was done with the use of a dynamic signal analyzer, a digital oscilloscope, and supporting electronic devices. The experimental results closely match the theoretical prediction.; The technique developed in this dissertation can be applied to any situation where the unforced system possesses multiple equilibria separated by hyperbolic, or in certain cases, nonhyperbolic points. It is a new approach to a wide range of problems in which irregular oscillations are observed and have traditionally been ascribed to the influence of stochastic forces.