This paper presents a general analysis and a concrete example of the catastrophic case of a discontinuity-induced bifurcation in so-called Filippov nonsmooth dynam- ical systems. Such systems are characterised by discontinuous jumps in the right- hand-sides of differential equations across a phase space boundary and are often used as physical models of stick-slip motion and relay control. Sliding bifurcations of periodic orbits have recently been shown to underlie the onset of complex dy- namics including chaos. In contrast to previously analysed cases, in this work a periodic orbit is assumed to graze the boundary of a repelling sliding region, re- sulting in its abrupt destruction without any pre-cursive change in its stability or period. Necessary conditions for the occurrence of such catastrophic grazing-sliding bifurcations are derived. The analysis is illustrated in a piecewise-smooth model of a stripline resonator, where it can account for the abrupt onset of self-modulating current fluctuations. The resonator device is based around a ring of NbN containing a microbridge bottleneck, whose switching between normal and super conducting states can be modelled as discontinuous, and whose fast temperature versus slow current fluctuations are modeled by a slow-fast timescale separation in the dynam- ics. By approximating the slow component as Filippov sliding, explicit conditions are derived for catastrophic grazing-sliding bifurcations, which can be traced out as parameters vary. The results are shown to agree well with simulations of the slow- fast model and to offer a simple explanation of one of the key features of this novel experimental device.

Catastrophic Sliding Bifurcations and onset of oscillations in a superconducting resonator

DI BERNARDO, MARIO;
2010

Abstract

This paper presents a general analysis and a concrete example of the catastrophic case of a discontinuity-induced bifurcation in so-called Filippov nonsmooth dynam- ical systems. Such systems are characterised by discontinuous jumps in the right- hand-sides of differential equations across a phase space boundary and are often used as physical models of stick-slip motion and relay control. Sliding bifurcations of periodic orbits have recently been shown to underlie the onset of complex dy- namics including chaos. In contrast to previously analysed cases, in this work a periodic orbit is assumed to graze the boundary of a repelling sliding region, re- sulting in its abrupt destruction without any pre-cursive change in its stability or period. Necessary conditions for the occurrence of such catastrophic grazing-sliding bifurcations are derived. The analysis is illustrated in a piecewise-smooth model of a stripline resonator, where it can account for the abrupt onset of self-modulating current fluctuations. The resonator device is based around a ring of NbN containing a microbridge bottleneck, whose switching between normal and super conducting states can be modelled as discontinuous, and whose fast temperature versus slow current fluctuations are modeled by a slow-fast timescale separation in the dynam- ics. By approximating the slow component as Filippov sliding, explicit conditions are derived for catastrophic grazing-sliding bifurcations, which can be traced out as parameters vary. The results are shown to agree well with simulations of the slow- fast model and to offer a simple explanation of one of the key features of this novel experimental device.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11588/368861
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