For a coupled nonlinear oscillator system with diffusion and gradient couplings, spatial Fourier transformation is performed and the dynamic equations of various space modes are derived. By calculating the Lyapunov exponents of the transverse modes, one can determine the stable region of the synchronous chaos on the plane of coupling parameters. On the boundary of the stable region, a couple of conjugate transverse modes destabilize, and a Hopf bifurcation takes place. Numerical simulations are carried out for the coupled Lorenz oscillator system. An electronic circuit is designed for simulating the bifurcation in the system. Results from the simulations show that the frequency created by the Hopf bifurcation is equal to the oscillation frequency of the destabilized transverse modes.