Generic and degenerate fold-Hopf bifurcations in jerk systems: Reduction, dynamics, and applications

In this paper we investigate the occurrence of fold-Hopf bifurcations in a two-parameter family of jerk systems. Under suitable generic assumptions, we show that the system can be transformed into generic fold-Hopf jerk forms. More precisely, we consider the case where the parameters follow a saddle–node bifurcation curve and impose conditions ensuring that a pair of complex eigenvalues becomes purely imaginary. By means of Taylor expansions and smooth invertible transformations of the variables and parameters, the system is reduced to generic fold-Hopf jerk systems. We further discuss the possible types of zero-Hopf singularities arising in this setting. Using the first-order averaging theory, we establish the existence of periodic orbits bifurcating from the zero-Hopf singularity. Next, we observe that the simplest generic systems mentioned above exhibit a degenerate fold-Hopf bifurcation, and we analyze their dynamics. Finally, to illustrate the applicability of our results, we analyze the fold-Hopf bifurcation in a variant of the Rössler system expressed in an equivalent jerk form. Furthermore, we show that introducing a control term into the jerk formulation of a given system enables the corresponding original system to undergo a fold-Hopf bifurcation. In addition, we study the occurrence of the fold-Hopf bifurcation in a three-dimensional extension of the Liénard equation and we highlight the occurrence of chaotic behavior of a particular jerk system exhibiting a degenerate fold-Hopf bifurcation.