Nonclassical responses of oscillators with hysteresis

The responses and codimension-one bifurcations in Masing-type and Bouc-Wen hysteretic oscillators are investigated. The pertinent state space is formulated for each system and the periodic orbits are sought as the fixed points of an appropriate Poincare map. The implemented path-following scheme is a pseudo-arclength algorithm based on arclength parameterization. The eigenvalues of the Jacobian of the map, calculated via a finite-difference scheme, are used to ascertain the stability and bifurcations of the periodic steady-state solutions. Frequency-response curves for various excitation levels are constructed considering representative hysteresis loop shapes generated with the two models in the primary and superharmonic frequency ranges. In addition to known behaviors, a rich class of solutions and bifurcations, mostly unexpected for hysteretic oscillators-including jump phenomena, symmetry-breaking, complete period-doubling cascades, fold, and secondary Hopf-is found. Complex (mode-locked) periodic and nonperiodic responses are also investigated thereby allowing to draw a more comprehensive picture of the dynamical behavior exhibited by these systems.