The equations of motion of the symmetrical dc superconducting quantum interference device (SQUID) are numerically investigated with a 16-term Taylor-expansion routine. The coherent phase or single-junction state is found to be unstable in large regions of the phase space including areas of interest in magnetometer design. The stability problem is discussed analytically in terms of the Hill equation. Feigenbaum sequences to chaos are found at borders of stability of various types of solution. Some of these, at finite external fluxes, occur at quite physical values of SQUID parameters.