A new three-dimensional (3D) chaotic system is proposed with four nonlinear terms which include two quadratic terms. To analyze the dynamical properties of the new system, mathematical tools such as Lyapunov exponents (LEs), Kaplan–York dimensions, observability constants, and bifurcation diagram have been exploited. The results of these calculations verify the specific features of the new system and further determine the effect of different system parameters on its dynamics. The proposed system has been experimentally implemented as an analog circuit which practically confirms its predicted chaotic behavior. Moreover, the problem of master–slave synchronization of the proposed chaotic system is considered. To solve this problem, we propose a new method for designing a nonfragile Takagi–Sugeno (T–S) fuzzy static output feedback synchronizing controller for a general chaotic T–S system and applied the method to the proposed system. Some practical advantages are achieved employing the new nonlinear controller as well as using system output data instead of the full-state data and considering gain variations because of the uncertainty in values of practical components used in implementation the controller. Then, the designed controller has been realized using analog devices to synchronize two circuits with the proposed chaotic dynamics. Experimental results show that the proposed nonfragile controller successfully synchronizes the chaotic circuits even with inexact analog devices.