New Insights on a Memristor-Based Chaotic System: Jacobi Stability, Integrability and Dynamics at Infinity

This paper explores the complex dynamics of a memristor-based dynamical system, which includes a third-order voltage-controlled generalized memristor and represents a chaotic memristive circuit. Through three complementary analytical frameworks, the fundamental insights into memristor-based chaotic systems are provided: (1) Using the Kosambi–Cartan–Chern (KCC) framework, the Jacobi stability of the system’s unique equilibrium point is explored. The results show that it exhibits Jacobi instability because there exists at least one positive eigenvalue of the deviation curvature tensor. (2) By investigating the Galois groups of the normal variational equations along a particular straight-line solution, it is demonstrated that the system is neither rationally integrable nor possesses a rational first integral for nearly all parameter values. (3) For investigating the convergence properties of the system’s orbits, the Poincaré compactification method is applied to describe its dynamical behavior at infinity, revealing two families of nonisolated equilibria and heteroclinic connections that bound the system’s asymptotic evolution. Together, these results establish a comprehensive analytical framework for understanding memristor-based chaos. The above findings offer some new geometric and algebraic insights into its intricate dynamics.