Figure 7
Bifurcation analysis of the A and R models.
(a) In addition to the previously discovered saddle-node (green line) and Hopf bifurcation curves (blue dashed lines) of the A model30, we discovered a limit cycle curve (brown line) and a homoclinic orbit (red dashed line). We also discovered a Bogdanov-Takens bifurcation point (BT) which is a bifurcation of co-dimension 2. (b–d) Phase portraits using parameter values indicated by p1, p2, and p3, respectively, in (a). Red circles indicate equilibrium points and for each panel, two representative phase portraits illustrate the stability of the equilibrium points and limit cycles. (e) Two-parameter bifurcation analysis of the R model on the parameters S and ΓE. (f) Two-parameter bifurcation analysis of the R model on the parameters S and ΓM. Abbreviations: GH = Generalized Hopf, LPC = Limit Point Cycle, CP = Cusp Point, BT = Bogdanov-Takens, IFS = Inclination-flip with respect to the stable manifold, LP = saddle-node, H = Hopf, HH = double Hopf, ZH = zero-Hopf. For the A model, the fixed parameter values were: ε = 0.1, κ = 5, Γ1′ = 1, while for the R model, the fixed model parameters are given in Table 1.
