Bo Sang, Rizgar Salih, Ning Wang, Zero-Hopf bifurcations and chaos of quadratic jerk systems, Vol. 2020 (2020), Article ID 25, pp. 1-16

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DOI: 10.23952/jnfa.2020.25

Received February 29, 2020; Accepted June 10, 2020; Published June 25, 2020

Abstract. The purpose of this paper is to propose some coefficient conditions, characterize the stability of the periodic solutions bifurcated from zero-Hopf bifurcations of the general quadratic jerk system, and apply these  theoretical results to a special jerk system in order to predict chaos.  First, we characterize the zero-Hopf bifurcations of the general quadratic jerk system in $\mathbb{R}^3$. The coefficient conditions on the stability of the periodic solutions are obtained via the averaging theory of first order. Next, we apply the theoretical results to a two-parameter jerk system. Finally, special attention is paid to a jerk system with one non-negative parameter $\epsilon$ and one non-linearity. By studying the continuation of periodic solution initiating at the zero-Hopf bifurcation, we numerically find a sequence of period doubling bifurcations, which leads to the creation of chaotic attractor.