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

Cristian Lăzureanu and Jinyoung Cho

Chaos, Solitons & Fractals, 2026, vol. 208, issue P1

Abstract: 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.

Keywords: Fold bifurcation; Fold-Hopf bifurcation; Jerk system; Stability (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:208:y:2026:i:p1:s0960077926002390

DOI: 10.1016/j.chaos.2026.118098

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