A plasma double layer is created by an anode grid in the target chamber of the double plasma device. When the separation grid of the double plasma device is floating, very coherent plasma oscillations, triggered by the unstable double layer, are observed. The oscillations have a frequency close to 1 kHz and very high amplitude. The amplitude and frequency of these oscillations depend on the anode bias and on the discharge voltage of the plasma source. The anode characteristics exhibit hysteresis. When plasma oscillations are coupled to the external oscillator, by modulating the floating potential of the separation grid, synchronization and periodic pulling are observed. When external oscillations are tuned to the second, third or fourth harmonics, period doubling bifurcations are observed, when the external amplitude is increased. At synchronization with the third and fourth harmonics, oscillations become chaotic after the first period doubling bifurcation. The correlation dimension of the underlying attractor is found to be fractal between 2.3 and 2.8, with a positive largest Lyapunov exponent.