Poincarè Map-Based Continuation of Periodic Orbits in Dynamic Discontinuous and Hysteretic Systems

A numerical algorithm is proposed to compute variation of periodic solutions and their codimension-one bifurcations in discontinuous and hysteretic systems. For general nonsmooth systems such as those exhibiting hysteresis, the nondifferentiable nature of the vector field makes the Poincaré map method one of the viable numerical strategy for continuation and stability analysis. Here the Jacobian of the map is evaluated via a finite-difference approach. The continuation scheme is based on arclength parameterization. The eigenvalues of the Jacobian of the map - Floquet multipliers - are computed to ascertain the stability of the periodic orbits and the associated bifurcations. The procedure is used to investigate the response of a class of one-dof systems with different representative restoring forces. The objective of the investigation is twofold: (i) to show the effectiveness of the procedure in dealing with various typical bifurcation scenarios of nonlinear dynamic systems and (ii) to investigate more in-depth some peculiar characteristics of softening-type oscillators having multi-linear and hysteretic restoring forces. Specifically, a bilinear system with either a sharp or a smooth transition in the force-displacement curve, Masing-type, and Bouc-Wen hysteretic systems are analyzed. A rich class of solutions and bifurcations - including jump phenomena, pitchfork, and period-doubling - are captured effectively by the procedure. Therefore, the implemented numerical strategy proves to be a powerful tool for analyzing the bifurcation behavior of general hysteretic systems shedding light onto some nonthoroughly explored nonlinear phenomena in these systems.