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This paper addresses a unified mathematical expression describing a class of chaotic and hyper-chaotic systems, for which the problem of synchronization between different chaotic and hyper-chaotic systems with uncertain parameters is studied. Based on the Lyapunov stability theory, a novel, simple and systemic adaptive synchronization controller is designated, the analytic expression of the controller and the adaptive laws of parameters are developed. Two different chaotic systems and a hyper-chaotic system are employed as the examples to show the effectiveness of the proposed adaptive synchronization scheme. Additionally, the robustness of the proposed method and the effect of noise are investigated by measuring the mean squared error of the systems.