Small-amplitude periodic solutions in the polynomial jerk equation of arbitrary degree

Abstract

A zero-Hopf singularity for a 3-dimensional differential system is a singularity for which the Jacobian matrix of the differential system evaluated at it has eigenvalues zero and $\pm$ω$i$ with ω ≠ 0. In this paper we investigate the periodic orbits that bifurcate from a zero-Hopf singularity of the $n$th-degree polynomial jerk equation $\stackrel{⃛}{x}-$ ϕ$(x,\dot{x},\ddot{x})=0$, where ϕ$(\ast ,\ast ,\ast )$ is an arbitrary $n$th-degree polynomial in three variables. We obtain sharp upper bounds on the maximum number of limit cycles that can emerge from such a zero-Hopf singularity using the averaging theory up to the second order. The result improves upon previous findings reported in the literature on zero-Hopf singularities and averaging theory. As an application we characterize small-amplitude periodic traveling waves in a class of generalized non-integrable Kawahara equations. This is accomplished by transforming the partial differential models into a five-dimensional dynamical system and subsequently analyzing a jerk system on a normally hyperbolic critical manifold, leveraging the averaging method and singular perturbation theory.