This chapter introduces a new two-dimensional (2D) fractional sigmoidal-based quadratic memristive map. The map’s behaviors and complexity are explored using bifurcation charts, phase portraits, and analyzing the maximum Lyapunov exponent (MLE). Additionally, we demonstrate chaos and evaluate its complexity in the fractional map using the sample entropy approach. The results revealed that the fractional sigmoidal-based quadratic memristive map exhibits diverse dynamical proprieties, including symmetry, asymmetry, chaos, and hidden chaos. Finally, the presented map is stabilized and synchronized by means of 2D nonlinear controllers.