Dynamical Systems in Neuroscience:

178 Bifurcationsrstable limit cycleunstable limitcyclerstableequilibriumunstable equilibriumstable equilibriumunstableequilibriumstable limit cyclecunstable limitcyclecSupercritical (a0)Figure 6.9: Andronov-Hopf bifurcation: A stable equilibrium becomes unstable insystem (6.8, 6.9).• (Transversality) Let c(b) ± iω(b) denote the complex-conjugate eigenvalues of theJacobian matrix of (6.4) for b near 0, with c(0) = 0 and ω(0) = ω The real part,c(b), must be non-degenerate with respect to b, that is, c ′ (0) ≠ 0.The Andronov-Hopf bifurcation has codimension one, since only one condition involvesstrict equality (tr L = 0), and the other two involve inequalities (“≠”).The sign of a determines the type of the Andronov-Hopf bifurcation, depicted inFig. 6.9:• Supercritical Andronov-Hopf bifurcation occurs when a < 0. It corresponds to astable limit cycle appearing from a stable equilibrium.• Subcritical Andronov-Hopf bifurcation occurs when a > 0. It corresponds to anunstable limit cycle shrinking to a stable equilibrium.Finding a in applications could be challenging. A few useful examples are consideredin Exercises 14-18.Any system undergoing an Andronov-Hopf bifurcation can be reduced by a changeof variables to the topological normal form (see also Ex. 4)ṙ = c(b)r + ar 3 , (6.8)˙ϕ = ω(b) + dr 2 , (6.9)where r ≥ 0 is the amplitude (radius), and ϕ is the phase (angle) of oscillation, as inFig. 6.10, and a, b, c(b), and ω(b) as above.