An analytic estimation for the largest Lyapunov exponent of the Rössler chaotic system based on the synchronization method

In this paper, we propose an analytical approach to estimate the largest Lyapunov exponent (LLE) of a Rössler chaotic system, leveraging the synchronization method. This research focuses on establishing an analytical criterion for the synchronization of two identical Rössler chaotic systems through the linear coupling of state variables. This is crucial because the LLE of such systems can be estimated based on the critical coupling required for synchronization. Unlike previous studies, we first transform the synchronization error system between two identical Rössler chaotic systems into a set of Volterra integral equations by using the Laplace transform and convolution theorem. The critical coupling for synchronization is analytically derived using integral equation theory to solve the error system. As compared to the numerical results of the Rössler chaotic system's LLE, our analytical estimates demonstrate high accuracy. Our findings suggest that the challenge of estimating the Rössler chaotic system's LLE can be simplified to solving a cubic algebraic equation, offering a novel perspective on the analysis of how parameters influence the LLE's value in the Rössler chaotic system.