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Optical bistability and oscillating phases exist in a Sagnac interferometer and a single ring resonator made of a χ^{(3)} nonlinear medium where the refractive indices are modulated by the light intensity due to the Kerr nonlinearity. An array of coupled nonlinear ring resonators behave similarly but with more complexity due to the presence of the additional couplings. Here we theoretically demonstrate the bifurcation of edge modes, which leads to optical bistability in the Su-Schrieffer-Heeger lattice with the Kerr nonlinearity. We also demonstrate periodic and chaotic switching behaviors in an oscillating phase resulting from the coupling between the edge mode and bulk modes with different chiralities, i.e., clockwise and counterclockwise circulations.