Dynamical Systems in Neuroscience:

202 Bifurcationsc 1c 2xFigure 6.34: Cusp surface.in particular, c 1 = c 2 = 0 at the cusp point. The cusp bifurcation is supercritical whena < 0 and subcritical otherwise. It is explained by the shape of the surfacec 1 + c 2 x + ax 3 = 0depicted in Fig. 6.34.Let us treat c 1 and c 2 as two independent parameters, and check that there aresaddle-node bifurcations in any neighborhood of the cusp point. The bifurcation setsof the topological normal form can easily be found. Differentiating c 1 + c 2 x + ax 3 withrespect to x gives c 2 +3ax 2 . Equating both of these expressions to zero and eliminatingx gives the saddle-node bifurcation curvesc 1 = ± √ 2 ( c2) 3/2,a 3depicted at the bottom of Fig. 6.34.Since c 1 = c 1 (b) and c 2 = c 2 (b), varying the bifurcation parameter b results in apath on the (c 1 , c 2 )-plane. Depending on the shape and location of this path, one canget many 1-dimensional bifurcation diagrams. A summary of some special cases isdepicted in Fig. 6.35 showing that there can be many interesting dynamical regimes inthe vicinity of a cusp bifurcation point.An important special case is when c 1 = 0 and c 2 (b) = b, so that the topologicalnormal form isẋ = bx + ax 3 .This form corresponds to a pitchfork bifurcation, whose diagram is depicted in Fig. 6.36(see also the bottom bifurcation diagram in Fig. 6.35). This bifurcation has an infinitecodimension unless one considers dynamical systems with symmetry, e.g., ẋ = f(x, b)with f(−x, b) = −f(x, b) for all x and b.6.3.3 Bogdanov-TakensCan an equilibrium undergo Andronov-Hopf and saddle-node bifurcations simultaneously?There are two possibilities, illustrated in Fig. 6.37: