THREE PROBLEMS IN NONLINEAR DYNAMICS WITH 2:1 PARAMETRIC EXCITATION

Abstract

Parametric excitation is epitomized by the Mathieu equation, x''+(d + e cos t)x = 0, which involves the characteristic feature of 2:1 resonance. This thesis investigates three generalizations of the Mathieu equation:

1) the effect of combining 2:1 and 1:1 parametric drivers:

x''+(d + e cos t + e cos wt)x = 0

2) the effect of combining parametric excitation near a Hopf bifurcation:

x''+ (d + e cos t) x + e Ax' + e(b_1 x^3 + b_2 x'^2 x + b_3 x x'^2 + b_4 x^3) = 0

3) the effect of combining delay with cubic nonlinearity:

x''+(d + e cos t)x + eg x^3 = e b x(t-T)

Chapter 3 examines the first of these systems in the neighborhood of 2:1:1 resonance. The method of multiple time scales is used including terms of O(e^2) with three time scales. By comparing our results with those of a previous work on 2:2:1 resonance, we are able to approximate scaling factors which determine the size of the instability regions as we move from one resonance to another in the d-w plane.

Chapter 4 treats the second system which involves the parametric excitation of a Hopf bifurcation. The slow flow obtained from a perturbation method is investigated analytically and numerically. A wide variety of bifurcations are observed, including pitchforks, saddle-nodes, Hopfs, limit cycle folds, symmetry-breaking, homoclinic and heteroclinic bifurcations. Approximate analytic expressions for bifurcation curves are obtained using a variety of methods, including normal forms. We show that for large positive damping, the origin is stable, whereas for large negative damping, a quasiperiodic behavior occurs. These two steady states are connected by a complicated series of bifurcations which occur as the damping is varied.

Chapter 5 examines the third system listed. Three different types of phenomenon are combined in this system: 2:1 parametric excitation, cubic nonlinearity, and delay. The method of averaging is used to obtain a slow flow which is analyzed for stability and bifurcations. We show that certain combinations of the delay parameters b and T cause the 2:1 instability region in the d-e plane to become significantly smaller, and in some cases to disappear. We also show that the delay term behaves like effective damping, adding dissipation to a conservative system.