In this paper, we study a three-dimensional jerk system \( \dddot{x} + a\,x + \dot{x} + b\,\ddot{x} + x\,\dot{x} - d\,\ddot{x}^{\,2} = 0\) , with \(a,b,d \ge 0\) , introduced by Li et al. [1] in the context of hidden chaotic dynamics. While previous studies focused on a Hopf bifurcation at \(a=b\) and \(d=1\) , we show that this system undergoes a zero-Hopf bifurcation at the origin when \(a=b=0\) , where the linearization has a simple zero eigenvalue and a pair of purely imaginary eigenvalues. By applying second-order averaging, we prove the existence and orbital stability of a small-amplitude periodic orbit that bifurcates from the zero-Hopf equilibrium under small parameter perturbations. In contrast to the general classification of zero-Hopf bifurcations in quadratic polynomial jerk systems by Llibre and Makhlouf [2], the system considered here is structurally minimal, containing a single nonlinear term and lying outside their framework. Our results therefore provide a complementary contribution by identifying the simplest quadratic–cubic jerk system in which zero-Hopf dynamics arise.
