Transient Chaos in a Jerk System: Zero-Hopf Bifurcation and Fractional Order Dynamics

Transient chaos is a phenomenon in which chaotic dynamics persists for a finite time before transitioning to periodic or steady-state behavior. TS has profound implications across disciplines, from neuroscience to quantum physics and machine learning. Recent studies have highlighted its role in crisis-induced transitions, early-time entanglement growth in quantum system, and pathological neuronal activity. In this paper, we present a novel jerk model characterized by a single nonlinearity in the form of a Lambert function. This system exhibits a short-time chaotic transient followed by convergence to a regular regime. For the proposed system, the conditions under which it exhibits a zero-Hopf equilibrium at the origin are identified. Furthermore, it is shown that applying the first-order averaging theory leads to the emergence of a unique periodic solution bifurcating from this zero-Hopf equilibrium point. Finally, the study extends to fractional order cases both commensurate and incommensurate revealing that one equilibrium point exhibits nonchaotic behavior while another presents chaos. The Gru..nwald–Letnikov method is employed to compute Lyapunov exponents and visualize phase trajectories, confirming the complex fractional dynamics of the system.