Catastrophic sliding bifurcations and onset of oscillations in a superconducting resonator

This paper presents a general analysis and a concrete example of the catastrophic case of a discontinuity-induced bifurcation in so-called Filippov nonsmooth dynamical systems. Such systems are characterized 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 dynamics including chaos. In contrast to previously analyzed cases, in this work a periodic orbit is assumed to graze the boundary of a repelling sliding region, resulting in its abrupt destruction without any precursive 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 modeled as discontinuous, and whose fast temperature versus slow current fluctuations are modeled by a slow-fast timescale separation in the dynamics. 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 experimental device.

Figure 1

Phase portraits for the four sliding bifurcations in three dimensions. The discontinuity manifold is represented by a horizontal plane in which the attracting sliding region is shaded.

Figure 2

A Filippov system, depicting the different types of dynamics that occur at a switching manifold: attracting or repelling sliding (double arrows show sliding vector field), and crossing. At the boundaries between sliding and crossing, trajectories graze the switching manifold.

Figure 3

A periodic orbit above the switching manifold $h=0$ is destroyed by a catastrophic grazing-sliding bifurcation. The local phase portrait is shown with the sliding region shaded.

Figure 4

(i) Image of the resonator device showing the NbN ring, and (ii) optical microscope (inverted color) image of the microbridge in the ring (provided courtesy of Eran Segev and Eyal Buks). (iii) Schematic model of the device showing the driving amplitude $b^{\text{in}}$ and frequency ${\omega }_{\text{p}}$, the response $B\left(t\right)e^{-\text{i}{\omega }_{\text{p}}t}$ of the resonator, and the temperature $T$ of the microbridge [16].

Figure 5

Vector fields $f_{\mathsf{N}1}$, $f_{\mathsf{N}2}$, $f_{\mathsf{S}1}$, and $f_{\mathsf{S}2}$ apply in regions bounded by the planar switching manifold $\Sigma$, and the parabolic switching manifolds ${\Sigma }_{\mathsf{N}}$ and ${\Sigma }_{\mathsf{S}}$ on which the sliding vector fields $F_{\mathsf{N}}$, $F_{\mathsf{S}}$ apply.

Figure 6

(Color online) Example of an orbit of the Filippov system. (i) In the three-dimensional phase space $\left(\text{Re}\beta ,\text{Im}\beta ,\Theta \right)$, orbits are attracted to and slide on the switching manifolds ${\Sigma }_{\mathsf{N}}$ and ${\Sigma }_{\mathsf{S}}$, (ii) projection onto the complex $\beta$ phase plane, showing intersections of the switching manifolds ${\Sigma }_{\mathsf{N},\mathsf{S}}$ with the switching manifold $\Sigma$. Fast dynamics are shown with double arrows, sliding orbits with single arrows.

Figure 7

(Color online) Geometry of the stripline resonator model. The switching manifold $\Sigma$ and slow manifolds ${\Sigma }_{\mathsf{N},\mathsf{S}}$ are shown. Arrows indicate flow directions in the original system (8), in which the flow changes radial direction through ${\Sigma }_{\mathsf{N},\mathsf{S}}$ at the bold curves R, which transect $\Sigma$ at the condition (32). Black dots are the equilibria (23). Inset: illustrations of (i) a periodic orbit, and (ii) catastrophic grazing-sliding bifurcation of a periodic orbit.

Figure 8

(Color online) Plot of $P\propto {\left|b^{\text{in}}\right|}^2$ against $w={\omega }_{\text{p}}/g$ as given by Eq. (33) for $\text{Re}{\Lambda }_{\mathsf{S}}=-0.1$, $\text{Re}{\Lambda }_{\mathsf{N}}=-1$, ${\omega }_{0\mathsf{S}}{\gamma }_1{\gamma }_{2\mathsf{S}}=1$, ${\omega }_{0\mathsf{N}}{\gamma }_1{\gamma }_{2\mathsf{N}}=0.1$, and ${\omega }_{0\mathsf{S}}/g=1$, where ${\omega }_{0\mathsf{N}}/g$ takes values: (i) 1, (ii) 1.1, and (iii) 1.5. $\mathsf{N}$ indicates the admissible equilibrium in existence above the red full parabola given by Eq. (24), $\mathsf{S}$ indicates the admissible equilibrium existing below the blue full parabola given by Eq. (25), elsewhere 0 denotes no equilibria. A periodic orbit exists in the shaded zone bounded by the bifurcation curves of catastrophic grazing-sliding in $\mathsf{N}$ (red/upper dotted curve) or $\mathsf{S}$ (blue/lower dotted curve).

Figure 9

(Color online) Simulations at the points (a)–(e) in Fig. 8(ii), showing the $-\frac{\pi }{4}<\text{arg}\beta <0$ quadrant of the complex $\beta$ plane. The circular arcs are the switching boundaries ${\left|\beta \right|}^2=1/{\sigma }_{\mathsf{N}}$ and ${\left|\beta \right|}^2=1/{\sigma }_{\mathsf{S}}$, the orbits belong to $\mathsf{N}$ (red) or $\mathsf{S}$ (blue) are labeled. Filled circles are admissible equilibria, empty circles are inadmissible. The cases are: (a) $w=1.22$ and $P=0.026$, showing an equilibrium in region $\mathsf{S}$; (b) $w=1.22$ and $P=0.039$, showing grazing in $\mathsf{S}$, a periodic orbit appears in a catastrophic grazing-sliding bifurcation; (c) $w=1.22$ and $P=0.079$, the periodic orbit in the astable zone; (d) $w=1.22$ and $P=0.101$, showing a periodic orbit homiclinic to the equilibrium of $\mathsf{N}$; and (e) $w=1.58$ and $P=0.165$, showing the codimension two scenario where the bifurcation curves meet in the bistable zone, implying catastrophic grazing-sliding simultaneously at both switching boundaries.

Figure 10

(Color online) A magnification of Fig. 8(iii) near resonance, showing the boundary of the $\mathsf{S}$ stability zone (full curve), the catastrophic grazing bifurcation curve (dotted curve). Shown for comparison is a bifurcation curve where periodic orbits vanish (dashed curve) in a smoothed version of the slow-fast system in Eq. (8).