A didactical note on the advantage of using two parameters in Hopf bifurcation studies

Figures & data

Figure 1. Bifurcation diagram in the (α₁,α₂)-plane showing a transcritical bifurcation curve (red) and Hopf bifurcation curves (blue) for kπ<ω<(k+1) π with k=0, 1, 2. The numbers of eigenvalues in the right half of the complex plane are indicated in the boxes. Each number (box) corresponds to a region bounded by the bifurcation curves.

Figure 2. Critical values of the parameters δ, 2 e^−γ, and β₀ and the Hopf period T for n=3. The transition from a stable steady state to an unstable steady state happens when we pass from below the curve to above the curve or when β₀ increases from below the critical value to above . For ω=0 we have .

Figure 3. Figure 1 repeated with a superimposed curve parametrized by δ in the (α₁,α₂)-plane corresponding to δ∈(0, 3), β₀=1.77, γ=0.2 and n=12.