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

Abstract

Abstract This article focuses on investigating local bifurcations in a special type of chaotic jerk system. It examines the occurrence and non-occurrence of saddle-node, transcritical, zero-Hopf, Hopf, and pitchfork bifurcations at the origin. The parameters that result in a zero-Hopf equilibrium point at the origin are characterized for the proposed system. Additionally, a demonstration is provided to show that the utilization of the first-order averaging theory leads to the emergence of a single periodic solution branching out from the zero-Hopf equilibrium located at the origin. Furthermore, the focus quantities method is applied to explore the periodicity of the cubic part of the system. This method helps 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 are found. Under specific conditions, it is shown that three periodic solutions can bifurcate from the origin of the system. Finally, the study also examines the chaotic attractors of the system.