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

In this paper, we study a three-dimensional jerk system x + a\,x + ẋ + b\,ẍ + x\,ẋ - d\,ẍ\,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.