The electroencephalogram signal, as a non-stationary signal, provides us with important information about various mental and biological dimensions, including an individual's gender. The use of quantitative measures of the signal's nonlinear dynamics is a method that can effectively extract information from brain signals. This research has investigated the ability of nonlinear dynamics of electroencephalogram signals as a biomarker for gender classification. This method, using parameters extracted from the Poincare plot, such as SD1, SD2 and some other methods to measure compactness and asymmetry in Poincare point distribution, to quantify complex neural patterns, has created a significant improvement in the accuracy of classifying signals based on gender. The gender-based classification accuracy using the proposed method reached 89%. After separating the frequency bands, it was observed that this effect has the greatest impact in the 10 to 15 Hz frequency range. The results show that the indices extracted from the Poincaré curve, compared to other complexity measurement criteria (such as various types of entropy, Lyapunov exponent, Hurst exponent, and fractal dimension), increase the classification precision in most frequency ranges by a statistically significant amount (p-value < 01/0). Significant differences between male and female genders were mainly observed in the parietal and frontal regions of the brain, which is consistent with previous studies on structural and functional differences between male and female brains in these regions. The findings of this research confirm that the nonlinear dynamics of brain signals can be used as reliable biomarkers for gender differentiation. This study is considered an important step towards developing more accurate methods for identifying gender differences in brain activity.