For three three-dimensional chaotic systems (Sprott NE1, NE8, and NE9) with only linear and quadratic terms and one parameter, but without equilibria, we consider the second order asymptotic approximations in the case that the parameter is small and near the origin of phase-space. The calculation leads to the existence and approximation of periodic solutions with neutral stability for systems NE1, NE9, and asymptotic stability for system NE8. Extending to a larger neighborhood in phase-space, we find a new type of relaxation oscillations with pulse behavior that can be understood by identifying hidden canards. The relaxation dynamics coexists with invariant tori and chaos in the systems.