Non-chaotic behaviour for a class of quadratic jerk equations

It is shown that a class constituted by 27 different types of non-linear third-order differential equations of the form x⃛=j(x,x˙,x¨), where j   is a quadratic polynomial with only one or two terms, and for which ∂j(x,y,z)/∂z∂j(x,y,z)/∂z is not a constant function of time, does not exhibit chaos. The three-dimensional dynamical systems associated to these equations are not necessarily dissipative everywhere nor conservative everywhere in the corresponding phase spaces. Our results include and improve some recent results obtained by Yang and Chen who only considered the case where j was a homogeneous quadratic polynomial with two terms.