Bifurcation analysis of a two-DoF mechanical system subject to digital position control. Part I: theoretical investigation

This paper analyzes the double Neimark–Sacker bifurcation occurring in a two-DoF system, subject to PD digital position control. In the model the control force is considered piecewise constant. Introducing a nonlinearity related to the saturation of the control force, the bifurcations occurring in the system are analyzed. The system is generally losing stability through Neimark–Sacker bifurcations, with relatively simple dynamics. However, the interaction of two different Neimark–Sacker bifurcations steers the system to much more complicated behavior. Our analysis is carried out using the method proposed by Kuznetsov and Meijer. It consists of reducing the dynamics of the nonlinear map to its local center manifold, eliminating the non-internally resonant nonlinear terms and transforming the nonlinear map to an amplitude map, that describes the local dynamics of the system. The analysis of this amplitude map allows us to define regions, in the space of the control gains, with a close interaction of the two bifurcations, which generates unstable quasiperiodic motion on a 3-torus, coexisting with two stable 2-torus quasiperiodic motions. Other regions in the space of the control gains show the coexistence of 2-torus quasiperiodic solutions, one stable and the other unstable. All the results described in this work are analytical and obtained in closed form, numerical simulations illustrate and confirm the analytical results.