Bifurcation analysis with chaotic attractor for a special case of jerk system

Abstract

This paper is focused on investigating the local bifurcations of a special type of chaotic jerk system. The occurrence and non-occurrence of saddle-node, transcritical and pitchfork bifurcations at the origin, as well as the Hopf and Zero-Hopf bifurcations when n = 3, are examined. For the proposed system, the parameters that lead to a zero-Hopf equilibrium point at the origin are characterized. Additionally, it has been demonstrated that employing the first-order averaging theory results in the emergence of a single periodic solution branching out from the zero-Hopf equi-librium located at the origin. Furthermore, the focus quantities method is utilized to explore the periodicity of the system. This method helps to determine the number of periodic solutions that can emerge from the Hopf point. Due to the computational load for computing singular quantities, only three singular quantities were found. Under specific conditions, it is shown that three periodic solutions can emerge from the origin of the system. Finally, the chaotic attractors of the system are also studied.