In this study, we investigated the electroencephalogram (EEG) dynamics in normal and epileptic subjects using three newly defined quantifiers adapted from nonlinear dynamics and Hilbert transform scatter plots (HTSPs): dispersion entropy (DispEntropy), dispersion complexity (Disp Comp), and forbidden count (FC), hypothesizing that analysis of electroencephalogram (EEG) signals using nonlinear and deterministic chaos theory may provide clinicians with information for medical diagnosis and assessment of the applied therapy. DispEntropy evaluates irregularity of the EEG time series. DispComp and FC quantify degree of variability of the time series. Receiver operating characteristic (ROC) analysis reveals that all the three quantifiers can discriminate between seizure and non-seizure states with very high accuracy. The application of such a technique is justified by ascertaining the presence of nonlinearity in the EEG time series through the use of surrogate test. The false positive rejection of the null hypothesis is eliminated by employing Welch window before the computation of the Fourier transform and randomizing the phases, in the generation of the surrogate data. Paired t-test revealed significant differences between the measures of the original time series and those of their respective surrogated time series, indicating the presence of deterministic chaos in the original EEG time series.