## 1 Introduction
[PERSON] and [PERSON] [38] firstly discussed the GFPS conditions of two different uncertain hyperchaotic systems when the scaling function factors were periodic functions and polynomial functions. However, only a few theoretical results about GFPS between two different uncertain chaotic systems have been obtained [41; 42; 43; 44; 45]. Furthermore, the aforementioned controllers all are designed through the construction of proper Lyapunov functions. The added controllers therefore are often very specific and may be too difficult to realize physically. In this paper, an effective approach is proposed to design an adaptive controller to realize GFPS between two different uncertain chaotic systems. Moreover, the unknown parameters in the coupled system can be estimated by given parameter update laws. The key to realizing GFPS between two chaotic systems is the determination of the stability of the synchronization error system. In previous works, this problem usually was solved by using the Lyapunov method. However, there are two main difficulties in this method: the first one is that it is challenging to construct a proper Lyapunov function; the second is that the GFPS conditions derived using the Lyapunov method are usually sufficient and highly conserved. The controllers and parameter update laws designed based on such conditions are often very specific. In fact, GFPS can be viewed as a unique case of GS, which can be investigated using the auxiliary system approach [46]. To avoid the construction of Lyapunov functions, the approach presented in this paper converts the differential equations describing the synchronization error system between the auxiliary system and the response system into a series of Volterra integral equations by using the Laplace transform method. Then, the dynamical behavior of the error system near its origin can be analyzed by employing the successive approximation method [47]. In the theory of integral equations, the significance of the Laplace transform method is that the GFPS condition can be obtained without the construction of Lyapunov functions, even if the scaling function factors are periodic functions, polynomial functions or any other complex functions. Compared to previous works, our approach allows the easier design of controllers and simpler parameter update laws and removes some restrictions on the uncertain parameters. The feasibility and effectiveness of the approach are illustrated by several numerical simulations. The rest of the paper is organized as follows. In Section 2, the GFPS scheme for two different uncertain chaotic systems is introduced. In Section 3, the GFPS condition is investigated using the Laplace transform method combined with the successive approximation method in the theory of Volterra integral equations. A numerical example is provided in Section 4 to demonstrate the effectiveness of our GFPS scheme. In Section 5, simple parameter update laws are designed according to the GFPS condition derived in Section 3 to estimate the uncertain parameters in chaotic systems, the validity of which are illustrated by numerical simulations. Finally, conclusions are drawn in Section 6.