Torus Bifurcations in a Mechanical System

Abstract

To study the dynamics and bifurcations of periodic solutions and tori, we consider a self-excited as well as parametrically excited three-mass chain system (a Tondl model) in 1:2:3 resonance. For the analysis both averaging-normalization and numerical simulations are used. First, we consider the case with the upper and lower mass almost equal, but not necessarily in 1:2:3 resonance. Surprisingly, this case simplifies at first order to a system of two coupled oscillators and one uncoupled. A set of necessary and sufficient conditions is then given for the general system to be in 1:2:3 resonance; the conditions can be resolved analytically. Using averaging-normalization, we are able to locate different periodic solutions. A bifurcation diagram is produced for each of the resonances generated by the quasi-periodic solutions, revealing interesting dynamics like a stable 2-torus, torus doubling and in the neighborhood of a Hopf–Hopf bifurcation a stable 3-torus. These tori eventually break up, leading to strange attractors and chaos, in agreement with the Ruelle–Takens (Commun Math Phys 20:167–192, 1971) scenario. Comparing the results of averaging-normalization with the dynamics of the original system shows good agreement. The bifurcation diagram of the normal form shows a complex accumulation of period doublings.