Summary: We expose the chaotic attractors of a time-reversed nonlinear system, further implement its behavior on an electronic circuit, and apply the pragmatical asymptotically stability theory to strictly prove that the adaptive synchronization of given master and slave systems with uncertain parameters can be achieved. In this paper, the variety chaotic motions of time-reversed Lorentz system are investigated through Lyapunov exponents, phase portraits, and bifurcation diagrams. For further applying the complex signal in secure communication and file encryption, we construct the circuit to show the similar chaotic signal of time-reversed Lorentz system. In addition, pragmatical asymptotically stability theorem and an assumption of equal probability for ergodic initial conditions [Z. M. Ge et al., “Pragmatical asymptotical stability theorem with application to satellite system,” Japanese J. Appl. Phys. 38, No. 10, 6178–6179 (1999); Z. M. Ge and J. K. Yu, “Pragmatical asymptotical stability theorems on partial region and for partial variables with applications to gyroscopic systems,” J. Mech. 16, No. 4, 179–187 (2000); Y. Matsushima, Differentiable manifolds. New York: Marcel Dekker (1972; Zbl 0233.58001)] are proposed to strictly prove that adaptive control can be accomplished successfully. The current scheme of adaptive control-by traditional Lyapunov stability theorem and Barbalat lemma, which are used to prove the error vector-approaches zero, as time approaches infinity. However, the core question – why the estimated or given parameters also approach to the uncertain parameters – remains without answer. By the new stability theory, those estimated parameters can be proved approaching the uncertain values strictly, and the simulation results are shown in this paper.