We obtained an EEG from a dog in a deep sleep and a wakefulness state. The change of the spectral distribution, the standard deviation and the Poincare section constructed from the delay map were found between the two states. The many couplings between neurons in the wakefulness state cause various frequency modes. Some of the coupling are weakened in the deep sleep state, so the fundamental mode and its first harmonics are excited. The sequential decrease of the a-value from the wakefulness to the deep sleep state shows that entropy increases in the deep sleep state. The lower EEG activity in the deep sleep state can be observed from the value of SD which is smaller than the value in the wakefulness state. In the deep sleep state the delay map for a shape is similar to a quadratic shape, and the shape remains unchanged during the deep sleep state. However, in the wakefulness state, although the quadratic shape remains unchanged, it is rotated or translated in a random fashion.