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We perform a detailed study of the dynamics of a nonlinear, one-dimensional oscillator
driven by a periodic force under hysteretic damping, whose linear version was originally
proposed and analyzed by Bishop in [1]. We first add a small quadratic stiffness term in
the constitutive equation and construct the periodic solution of the problem by a systematic
perturbation method, neglecting transient terms as t → ∞. We then repeat the analysis
replacing the quadratic by a cubic term, which does not allow the solutions to escape
to infinity. In both cases, we examine the dependence of the amplitude of the periodic
solution on the different parameters of the model and discuss the differences with the
linear model. We point out certain undesirable features of the solutions, which have also
been alluded to in the literature for the linear Bishop’s model, but persist in the nonlinear
case as well. Finally, we discuss an alternative hysteretic damping oscillator model
first proposed by Reid [2], which appears to be free from these difficulties and exhibits
remarkably rich dynamical properties when extended in the nonlinear regime
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