Hopf Bifurcation of a Novel Chaotic System Without Linear Terms

Chaotic systems characterized by exponential nonlinearities without linear terms pose significant challenges in analyzing stability and bifurcation, with applications spanning optics, neuroscience, control theory and quantum physics. This paper studies the Hopf bifurcation in these systems. We introduce a novel chaotic system governed by exponential functions, aiming to examine its dynamic behavior, which includes the coexistence of attractors, the bifurcation structure, local stability and Hopf bifurcation. Our methodology employs the Routh–Hurwitz criteria for assessing local stability and calculates the first Lyapunov coefficient to discern the existence and nature of limit cycles emerging from the Hopf bifurcation. Furthermore, we obtained the coexistence of chaotic attractors under certain initial conditions. Despite the intricate behavior induced by exponential nonlinearities and the system’s reliance on the Lambert W function, the analysis verifies the presence of limit cycles, underscoring the system’s complex dynamic behavior.