This work consists of three parts. These are linked by the common theme of nonlinear phenomena in solid state systems, but are otherwise independent and self contained
PART 1: In this part an experimental study is made of the interactions between spin wave modes excited in a sphere of yttrium iron garnet by pumping the Suhl subsidiary absorption at 9.2 GHz with the dc field parallel to [111]. The dynamical behavior of the magnetization is observed under high resolution by varying two control parameters, dc field (580 < H < 2100 Oe) and microwave pump power (1 < P < 200 mW). Within this parameter space quite varied behavior is found: (i) onset of the Suhl instability by excitation of a single spin wave mode with very narrow linewidth (< 0.5 G); (ii) when two or more modes are excited, interactions lead to auto-oscillations with a systematic dependence of frequency (10^4 to 10^6 Hz) on pump power, these oscillations displaying period-doubling to chaos; (iii) quasiperiodicity, locking, and chaos occur when three or more modes are excited; (iv) abrupt transition to wide band power spectra (i.e., turbulence), with hysteresis; (v) irregular relaxation oscillations and aperiodic spiking behavior. A theoretical model is developed from first principles, using the plane wave approximation and including anisotropy effects, obtaining the lowest order nonlinear interaction terms between the excited modes. Extension of this analysis to the true spherical spin-modes is discussed. Bifurcation behavior is examined, and dynamical behavior is numerically computed and compared to the experimental data, explaining a number of features. A theory is developed regarding the nature of the experimentally observed relaxation oscillations and spiking behavior based on the interaction of"weak" and "strong" modes, and this is demonstrated in the numerical simulations for two modes. Quasiperiodicity is shown to occur in the numerical study when at least 3 modes are excited with appropriate parameter values. A possible mechanism for generating microwave subharmonics at half of the pumping frequency is discussed.
PART 2: This is an experimental study of a forced symmetric oscillator containing a saturable inductor with magnetic hysteresis. It displays a Hopf bifurcation to quasiperiodicity, entrainment horns, and chaos. The bifurcations and hysteresis occurring near points of resonance (particularly "strong resonance") are studied in detail and it is shown how the observed behavior can be understood using Arnold's theory. Much of the behavior relating to the entrainment horns is explored: period doubling and symmetry breaking bifurcations; homoclinic bifurcations; and crises and other bifurcations taking place at the horn boundaries. Important features of the behavior related to symmetry properties of the oscillator are studied and explained through the concept of a half-cycle map. The system is shown to exhibit a Hopf bifurcation from a phase-locked state to periodic "islands" similar to those found in Hamiltonian systems. An initialization technique is used to observe the manifolds of saddle orbits and other hidden structure. An unusual differential equation model is developed which is irreversible and generates a noninvertible Poincare map of the plane. Noninvertibility of this planar map has important effects on the behavior observed. The Poincare map may also be approximated through experimental measurements, resulting in a planar map with parameter dependence. This model gives good correspondence with the system in a region of the parameter space.
PART 3: This part takes a new look at an old problem, namely the observed "noise rise"; in superconducting Josephson junction parametric amplifiers. By exploiting recent insights from dynamical systems theory, it is shown how the interplay of random noise and (nonchaotic) deterministic dynamics can result in a noise rise like that observed in experiments. This analysis leads to a universal first order equation which applies to all similar systems in the high-gain regime. several predictions are proposed which can be tested experimentally, including that a similar noise rise should occur in modulated semiconductor injection lasers. An analysis is also made of a previously unknown mode of operation - a "six-photon" mode associated with a symmetry breaking bifurcation - and its potential advantages over the previously studied three-photon and four-photon modes are discussed.