Adaptive control and synchronization of complex dynamical systems with various types of delays and stochastic disturbances

Adaptive control and synchronization techniques have been extensively developed over the years due to their ability to operate without prior analytical knowledge of the system. However, their application in real-world systems necessitates the consideration of various factors such as delays and stochastic disturbances. Moreover, the conventional hypothesis of a global Lipschitzian condition is often overly restrictive for real-world systems. In this paper, we address these challenges by jointly incorporating these factors. We introduce a novel parameter θ in the algebraic power associated with the adaptive rule. This modification enhances the effectiveness of the adaptive scheme in more realistic systems. Additionally, we investigate how the power θ and the nonlinear degrees of vector fields affect the performance of the adaptive scheme. The adaptive scheme proves to be effective in controlling or synchronizing stochastic complex dynamics in ecosystems and biosystems, demonstrating its broad applicability in various fields.