A novel approach to synchronizing a biological snap oscillator within a fixed time and expanding the method to various chaotic systems.

The purpose of this research is to address the problem of synchronization between two identical biological hyperchaotic snap oscillator systems with unknown parameters at a fixed time, as well as to design a new theory for synchronizing two different chaotic systems. Firstly, we designed an adaptive controller using Lyapunov stability and fixed-time stability theory to achieve synchronization between two identical biological hyperchaotic snap oscillator in fixed time under unknown parameters. Among the results obtained, we found that the synchronization time for system states occurred in a very short time compared to results in previous research. Secondly, we extended and generalized synchronization between non-identical chaotic systems by creating a theory for achieving synchronization in which adaptive control and fixed-time stability are combined. Through the new theory, the unknown parameters in chaotic systems were estimated, and the settling time was determined, which turned out to be very short and independent of the initial conditions of the system states. One of what makes the scheme important is that it can be easily applied to most chaotic and even hyperchaotic systems in the shortest possible time. Finally, we gave examples of different systems to test the feasibility and effectiveness of the scheme that was designed. [ABSTRACT FROM AUTHOR]