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Academic Paper
Yuu Miino, Tetsushi Ueta, Hiroshi Kawakami, Nonlinear resonance and devil's staircase in a forced planer system containing a piecewise linear hysteresis, Elsevier, Chaos, Solitons & Fractals, in-press, Apr. 2018. Abstract: The Duffing equation describes a periodically forced oscillator model with a nonlinearelasticity. In its circuitry, a saturable-iron core often exhibits a hysteresis. However,a few studies about the Duffing equation has discussed the effects of the hysteresisbecause of difficulties in their mathematical treatment. In this paper, we investigatea forced planer system demonstrating the Duffing equation containing the hysteresis.For simplicity, we approximate the hysteresis to a piecewise linear function. Sincethe solutions are expressed by combinations of some dynamical systems and switchingconditions, a finite-state machine is derived from the hybrid system approach, and thenbifurcation theory can be applied to it. We topologically classify periodic solutions andcompute local and grazing bifurcation sets accurately. In comparison with the Duffingequation, we discuss the effects caused by the hysteresis, such as the devils staircase inresonant solutions.

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