Bifurcation Structure for Modulated Vibration of Strings subjected to harmonic boundary excitations

In the studies of nonlinear dynamics, phase plan plot is a most commonly used tool for solution characteristics interpretation. Phase plan plot provides an adequate representation of the dynamic characteristics of single solution, but it does not provide information on the interrelation between neighboring solutions; therefore, the evolution of solutions is studied by examining fragmented information of many individual responses. When the solution space becomes complicated, accurate information of the interrelation between responses is essential for an overall comprehension of the characteristics of the solutions. Noticing the characterizing amplitude variation of the limit cycles for typical modulated responses, bifurcation structure was proposed and developed to examine the overall characteristics of the modulated solutions of nonlinear string vibration. The bifurcation structure depicts all solutions in one plot by recording the upper and lower limits of the modulated vibration limit cycles. All kinds of bifurcations and interrelations between typical solutions can be qualitatively and quantitatively identified using the bifurcation structure. In this study, bifurcations and solution interrelations identified and revealed by bifurcation structures include forward and reverse Hopf bifurcations through period-doubling, appearance of isolated solution branch, solution branch transitions between the Hopf branch and isolated branch, appearance of chaotic attractors, chaotic attractor transitions between the Rossler and Lorenz types, and attractor disappearance by boundary crisis.