tokens: on the low branch. Mass is being fed into the annulus at a greater rate than it is able to process it and so mass accumulates and Σ increases. Eventually, the annulus reaches the low-temperature bifurcation point and leaps to the high hot state. The annulus, now above the Σ-nullcline, begins losing mass because its accretion rate exceeds the supply rate. Once the high-temperature bifurcation point is reached, the annulus drops to the low-temperature state and the process is repeated. The red curve in Fig. 4 (a) corresponds to a numerical simulation of equations ( 15 ) and ( 17 ) and illustrates this basic behaviour quite clearly. In Fig. 4 (b), we plot the evolution of Σ and T c as a function of time. Here, one can observe the two time-scales at work, especially in the (blue) T c curve. In between outbursts there is little change in temperature, but a build up in mass (seen in the red curve) occurring on the slow viscous time-scale t Σ . Interleaving these phases are the abrupt bursts occurring on the short thermal time-scale t th . 3.2.2 Z Cam type simulations Z Cam variables intermittently behave in a similar manner to U Gem variables but occasionally remain on the high-temperature branch in a quasi-steady state for months to years before dropping to quiescence and undergoing a fresh batch of outbursts. This is attributed to a shift in the Σ-nullcline upwards (due to a change in mass transfer, δ ≠ 0), so that the intersection is beyond the saddle node. Of course, the system must be close to criticality if this is to be possible. To demonstrate this in our model, and δ are chosen such that the fixed point oscillates between the stable high-temperature branch and the unstable branch. When the fixed point is on the upper branch the system is stable and will remain in a standstill. A decrease in pushes the fixed point on to the unstable branch and the system undergoes periodic orbits resulting in trains of outbursts. In Fig. 5 , we show an example of standstill–outburst-train cycles. Figure 5. Time evolution of the surface density and central temperature with and b 0 = 0 with mass transfer variation. The variation amplitude is δ = 0.1 and t I = 5 × 10 6 s. The system clearly switches between outburst trains and standstills. 3.3 Results: stochastic With the basics established, we include a fluctuating viscosity (β ≠ 0). The fluctuation time-scale is taken to be 10 t orb , which is motivated by MRI shearing box simulations (Hawley et al. 1995 ). The amplitude is set at b 0 = 0.1. First, we consider the case where there is no mass transfer variability (δ = 0). To obtain a fixed point on the high-temperature branch close to the bifurcation point we choose . A deterministic system with this configuration would remain in the high-temperature state indefinitely. A stochastic system however will see the fixed point jostle about as the two nullclines fluctuate with α, sometimes sitting on the upper branch, sometimes dropping to the middle branch
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