Degenerate Fold–Hopf Bifurcations in a Rössler-Type System

G. Tigan, J. Llibre, L. Ciurdariu
2017 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering  
We study the Hopf and the fold-Hopf bifurcations of the Rössler-type differential systeṁ x = −y − z,ẏ = x + ay,ż = −cz + byz, with b = 0. We show that the classical Hopf bifurcation cannot be applied to this system for detecting the fold-Hopf bifurcation, which here is studied using the averaging theory. Our results show that a Hopf bifurcation takes place at the equilibrium (−ac/b, c/b, −c/b) when c = a < 0. This Hopf bifurcation becomes a fold-Hopf bifurcation when c = a = 0.
doi:10.1142/s0218127417500687 fatcat:cujnkabjzndp5kpnqupayiqe3q