<span>The main purpose of the paper is to projective synchronous chaotic oscillation in the real four-dimensional hyperchaotic model via designing many adaptive nonlinear controllers. Firstly, in view that there are many strategies in the design process of existing controllers, a nonlinear control strategy is considered as one of the important powerful tools for controlling the dynamical systems. The prominent advantage of the nonlinear controller lies in that it deals with known and unknown parameters. Then, the projective synchronize behavior of a four-dimensional hyperchaotic system is analyzed by using the Lyapunov stability theory and positive definite matrix, and the nonlinear control strategy is adopted to synchronize the hyperchaotic system. Finally, the effectiveness and robustness of the designed adaptive nonlinear controller are verified by simulation.</span>

Nonlinear control strategy was established to realize the Projective Synchronization (PS) and Hybrid Projective Synchronization (HPS) for 4-D hyperchaotic system at different scaling matrices. This strategy, which is able to achieve projective and hybrid projective synchronization by more precise and adaptable method to provide a novel control scheme. On First stage, three scaling matrices were given in order to achieving various projective synchronization phenomena. While the HPS was implemented at specific scaling matrix in the second stage. Ultimately, the precision of controllers were compared and analyzed theoretically and numerically. The long-range precision of the proposed controllers are confirmed by third stage.

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