A model of a reaction system consisting of two parallel, isothermal autocatalytic reactions in a CSTR has been examined. It is shown that self-sustained chaotic behavior can occur in this system. The region of chaos is entered and exited according to period-doubling and halving sequences, with both sets of bifurcations giving rise to Feigenbaum's number. Power spectral density calculations show that the nature of the chaotic behavior depends quite strongly on the parameter values. From a calculation of the Lyapunov exponents it is found that the Lyapunov dimension of the strange attractor is only slightly greater than that of a two-periodic torus.