The nonlinear saturation of the r-mode instability and its effects on the spin evolution of Low Mass X-ray Binaries (LMXBs) are modeled using the triplet of modes at the lowest parametric instability threshold. We solve numerically the coupled equations for the three mode amplitudes in conjunction with the spin and temperature evolution equations. We observe that very quickly the mode amplitudes settle into quasi-stationary states. Once these states are reached, the mode amplitudes can be found algebraically and the system of equations is reduced from eight to two equations: spin and temperature evolution. Eventually, the system may reach thermal equilibrium and either (1) undergo a cyclic evolution with a frequency change of at most 10%, (2) evolve toward a full equilibrium state in which the accretion torque balances the gravitational radiation emission, or (3) enter a thermogravitational runaway on a very long timescale of about $10^6$ years. Alternatively, a faster thermal runaway (timescale of about 100 years) may occur. The sources of damping considered are shear viscosity, hyperon bulk viscosity and boundary layer viscosity. We vary proprieties of the star such as the hyperon superfluid transition temperature T_c, the fraction of the star that is above the threshold for direct URCA reactions, and slippage factor, and map the different scenarios we obtain to ranges of these parameters. For all our bound evolutions the r-mode amplitude remains small $sim 10^{-5}$. The spin frequency is limited by boundary layer viscosity to $ u_{max} sim 800 Hz [S_{ns}/(M_{1.4} R_6)]^{4/11} T_8^{-2/11}$. We find that for $ u > 700$ Hz the r-mode instability would be active for about 1 in 1000 LMXBs and that only the gravitational waves from LMXBs in the local group of galaxies could be detected by advanced LIGO interferometers.
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