Zero-Hopf dynamics in a quadratic–cubic jerk system: a complement to quadratic classification

Abstract In this paper, we study a three-dimensional jerk system x + a\, x + ẋ + b\, x + x\, ẋ - d\, x^\, 2 = 0 x ⃛ + a x + x ˙ + b x ¨ + x x ˙ - d x ¨ 2 = 0, with a, b, d 0 a, b, d ≥ 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 a = b and d=1 d = 1, we show that this system undergoes a zero-Hopf bifurcation at the origin when a=b=0 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.