Analytical modeling of bifurcations, chaos and multifractals in nonlinear dynamics

Several analytical models of bifurcations, chaos and multifractals are proposed in this thesis.;To study a continuous system, the Duffing oscillator is employed. The Chirikov overlap criterion and the renormalization group technique are used to independently derive the chaotic condition near a subharmonic resonant orbit of the undamped Duffing oscillator. To analyze the stability and bifurcation of periodic solutions of the damped Duffing oscillator with strong nonlinearities, an improved harmonic balance method is proposed. The physical system studied is the buckling of a nonlinear rod and for this structure, four types of Duffing oscillator are identified. Chaos in the weakly damped Duffing oscillator is studied using the Melnikov method.;A new method based on the incremental energy approach is developed to model stochastic layers near the homoclinic and the heteroclinic orbits, and also, resonant layers in the vicinity of the resonant orbit. In the case of the stochastic layers, the outer and inner strengths for the Duffing and forced planar-pendulum oscillators are obtained, and for the resonant layers, the appearance, disappearance and accumulated disappearance strengths for the Duffing oscillator are determined. Employing a Naive discretization of the differential equation of motion of the Duffing oscillator and the subsequent application of the cubic renormalization on its discrete mapping, the universal character of the oscillator is studied. The jump phenomenon and the strange attractor are clearly seen in the cascades of bifurcations.;To examine a discontinuous system, the impact oscillator is used. Based on the differential equation of motion of a ball bouncing on a massive vibrating table, the stability and bifurcation conditions are derived. Analyzing the mappings of the motion, three types of stable motion and two types of unstable motion are found. From the Poincare mappings of the unstable period-1 motion, the two saddles are shown to possess identical Smale horseshoe structures. However, this is not necessarily true for the higher periodic solutions. Another example of a discontinuous system is that of a horizontal impact pair. A theory for a system with discontinuities and applied to the impact analysis of a horizontal impact pair is developed. Mappings for four switch planes are defined and from these, five impact motions are derived. One finding is that period doubling bifurcations cannot occur for equispaced impacts of the Model I motion.;A highly accurate method for the analysis of period doubling bifurcations in 1-D iterative maps is proposed. The technique consists of constructing similar structures of the period doubling solutions and applying a renormalization procedure to evaluate the appropriate length scaling factors. The weight parameter function, several generalized fractal dimensions, the scaling index and the fractal spectrum functions are derived.;To develop a theory for multifractals in chaotic dynamics, the m-D horseshoe map is adopted. The results for 1-D uniform and nonuniform Cantor sets are first derived, and then extended to handle 2-D uniform and nonuniform Smale horseshoes. Fractal characteristics for the invariant sets generated via the Cantor sets and Smale horseshoes can be easily determined using this new theory.;One of the key features of the models developed is that bifurcations and the onset of chaos can be theoretically predicted by employing computed instead of prescribed input parameters in numerical simulations. This ability is beneficial as it can significantly reduce the amount of numerical experimentations. (Abstract shortened by UMI.)...