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By solving the recursion relation of a reaction-diffusion equation on a lattice, we find two distinct routes to turbulence, both of which reproduce commonly observed phenomena: the Feigeribaum route, with period-doubling frequencies; and a much more general route with noncommensurate frequencies and frequency entrainment, and locking. Intermittency and large-scale aperiodic spatial patterns, also observed in physical systems, are reproduced in this new route. The fractal dimension has been estimated to be about 2.6 in the oscillatory instability and about 6.0 in the turbulent regime.