Three problems in nonlinear dynamics with 2:1 parametric excitation
Parametric excitation is epitomized by the Mathieu equation, x¨ + (δ + ϵ 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¨ + (δ + ϵ cos t + ϵ cos ω t)x = 0; (2) the effect of combining parametric excitation near a Hopf bifurcation: x¨ + (δ + ϵ cos t)x + ϵAx˙ + ϵ(β1x3 + β 2x2x˙ + β 3xx˙2 + β4 x˙3) = 0; (3) the effect of combining delay with cubic nonlinearity: x¨ + (δ + ϵ cos t)x + ϵγx3 = ϵβx(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(ϵ2) 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 δ-ω 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 β and T cause the 2:1 instability region in the δ-ϵ 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.