Dynamical Systems in Neuroscience:

204 BifurcationseigenvaluesHopffoldeigenvaluesfold-Hopf bifurcationBogdanov-Takens bifurcationFigure 6.37: Two ways an equilibrium can undergo a saddle-node (fold) and anAndronov-Hopf bifurcations simultaneously.• (Fold-Hopf) The Jacobian matrix at the equilibrium has a pair of pure imaginarycomplex-conjugate eigenvalues (Andronov-Hopf bifurcation) and one zeroeigenvalue (saddle-node bifurcation). In this case the two bifurcations occur indifferent subspaces.• (Bogdanov-Takens) The Jacobian matrix has two zero eigenvalues.case the two bifurcations occur in the same subspace.In thisThe fold-Hopf bifurcations occurs in systems having dimension 3 and up, while theBogdanov-Takens bifurcation can occur in two-dimensional systems. Both bifurcationshave codimension-2; that is, they require 2 bifurcation parameters. Notice thatfold-Hopf bifurcation has 3 eigenvalues with zero real part, whereas Bogdanov-Takensbifurcation has only 2 zero eigenvalues. This bifurcation can on the one hand be viewedas a saddle-node bifurcation in which another (negative) eigenvalue gets arbitrary closeto zero, and on the other hand as an Andronov-Hopf bifurcation in which imaginarypart of the complex-conjugate eigenvalues goes to zero.The Jacobian matrix of an equilibrium at the Bogdanov-Takens bifurcation satisfiestwo conditions: det L = 0 (saddle-node bifurcation) and tr L = 0 (Andronov-Hopfbifurcation). For example, it can have the formL =( 0 10 0). (6.10)Because of these two conditions, the codimension of this bifurcation is 2. There arealso certain non-degeneracy and transversality conditions (see Kuznetsov 1995). Thecorresponding topological normal form,˙u = v ,˙v = a + bu + u 2 + σuv ,(6.11)