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In this paper, a sliding mode projective synchronization strategy based on disturbance observer and fuzzy system is presented to implement projective synchronization of hyperjerk system with low time-varying disturbance and white noise. Theoretical analysis and numerical calculation show that the disturbance observer can approach the low time-varying disturbance very well. The application of disturbance observer reduces the chattering of the controller. Variable universe adaptive fuzzy control (VUAFC) method is utilized to further reduce the chattering phenomenon. The simulation results demonstrate the effectiveness of the proposed controller.

For nonsmooth Filippov systems, the stability of the system is assumed to be proved by nonsmooth Lyapunov functions, such as piecewise smooth Lyapunov functions. This extension was based on the Filippov solution and Clarke generalized gradient. However, it is difficult to estimate the gradient of a non-smooth Lyapunov function. In some cases, the nonsmooth system can be divided into continuous and discontinuous components. If the Lebesgue measure of the discontinuous components is zero, the smooth Lyapunov function can be utilized to prove the stability of the system owing to the inner product of the gradient of the Lyapunov function of the discontinuous components being zero. In this paper, we apply the smooth Lyapunov function to prove the stability of the nonsmooth ratio-dependent predator-prey system. In contrast to the existing literature, in this paper, although the system is divided into continuous and discontinuous components, the inner product of the gradient of the Lyapunov function of the discontinuous part is not zero but negative. In the proof of stability, the negative value condition is stricter than the zero-value condition. This proof method only needs to construct a smooth Lyapunov function, which is simpler than a non-smooth Lyapunov function or other methods.

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