Degenerate Fold–Hopf Bifurcations in a Rössler-Type System

Abstract

We study the Hopf and the fold–Hopf bifurcations of the Rössler-type differential system [Formula: see text] with [Formula: see text]. 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 [Formula: see text] when [Formula: see text]. This Hopf bifurcation becomes a fold–Hopf bifurcation when [Formula: see text].

Theoretical conditions for the coexistence of viral strains with differences in phenotypic traits: a bifurcation analysis

“…The normal form equations for degenerate ZH bifurcations are not well known and previous studies have focused on different cases [73,74]. Tigan et al [75] showed that a degenerate ZH bifurcation with G 011 (0) ¼ 0 occurs in a Rö ssler-type system and used averaging theory to detect periodic orbits, leaving the structure of global bifurcations unresolved. We believe, as far as we know, that this is the first time a curve of TPO bifurcations has been observed to emanate from a degenerate ZH bifurcation of this type.…”