Nonlinear resonance and devil's staircase in a forced planer system containing a piecewise linear hysteresis. (June 2018)

• Record Type: Journal Article
• Authors: Miino, Yuu; Ueta, Tetsushi; Kawakami, Hiroshi
• Is Part Of: Chaos, solitons and fractals. Volume 111(2018)
• Page Start: 75
• Page End: 85
• Publication Date: 2018-06
• Subjects: Bifurcation analysis -- Hybrid system -- Nonlinear resonance -- Duffing equation -- Devil's staircase -- Chaos; Chaotic behavior in systems -- Periodicals; Solitons -- Periodicals; Fractals -- Periodicals; Chaotic behavior in systems; Fractals; Solitons; Periodicals; 003.7
• DOI: 10.1016/j.chaos.2018.04.007
• Languages: English
• ISSNs: 0960-0779

Abstract

Highlights: We describe a planer system including a hysteresis by a hybrid system. This system exhibits a bifurcation which is often observed in discrete systems. We solve bifurcation sets in B − B 0 parameter plane by using the continuation. A devil's staircase is found even though the system is continuous-time. The grazing bifurcations as typical phenomena in hybrid systems are observed. Abstract: The Duffing equation describes a periodically forced oscillator model with a nonlinear elasticity. 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 hysteresis because of difficulties in their mathematical treatment. In this paper, we investigate a forced planer system obtained by replacing a cubic term in the Duffing equation with a hysteresis function. For simplicity, we approximate the hysteresis to a piecewise linear function. Since the solutions are expressed by combinations of some dynamical systems and switching conditions, a finite-state machine is derived from the hybrid system approach, and then bifurcation theory can be applied to it. We topologically classify periodic solutions and compute local and grazing bifurcation sets accurately. In comparison with the Duffing equation, we discuss the effects caused by the hysteresis, such as the devil's staircase in resonant solutions.