Using the disc instability model for dwarf novae and soft X-ray transients, we investigate the stability of accretion discs in long-period binary systems. We simulate outbursts due to this thermal-viscous instability for two symbiotic systems, RS~Ophiuchi and Z~Andromedae. The outburst properties deduced from our simulations suggest that, although the recurrent nova events observed in RS~Oph are due to a thermonuclear runaway at the white dwarf surface, these runaways are triggered by accretion disc instabilities. In quiescence, the disc builds up its mass and it is only during the disc-instability outburst that mass is accreted on to the white dwarf at rates comparable to or larger than the mass-transfer rate. For a mass-transfer rate in the range $10^{-8}$ to $10^{-7}~{rm M}_{odot}$ yr$^{-1}$, the accretion rate and the mass accreted are sufficient to lead to a thermonuclear runaway during one of a series of a few dwarf nova outbursts, barely visible in the optical, but easily detectable in X-rays. In the case of Z~And, persistent irradiation of the disc by the very hot white-dwarf surface strongly modifies the dwarf-nova outburst properties, making them significant only for very high mass-transfer rates, of the order of $10^{-6}~{rm M}_{odot}$ yr$^{-1}$, much higher than the expected secular mean in this system. It is thus likely that the so-called `combination nova outburst observed in years 2000 to 2002 was triggered not by a dwarf-nova instability but by a mass-transfer enhancement from the giant companion, leading to an increase in nuclear burning at the accreting white-dwarf surface.
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