Dynamical Systems in Neuroscience:

Solutions to Exercises, Chap. 4 417abcdFigure 10.16: Approximate directions of the vector in each region between the nullclines.If ∆ < 0 (left half-plane in Fig. 4.15), then the eigenvalues have opposite signs. Indeed,√τ2− 4∆ > √ τ 2 = |τ| ,whenceτ + √ τ 2 − 4∆ > 0 and τ − √ τ 2 − 4∆ < 0 .The equilibrium is a saddle in this case. Now consider the case ∆ > 0. When τ 2 < 4∆ (insidethe parabola in Fig. 4.15), the eigenvalues are complex-conjugate, hence the equilibrium is afocus. It is stable (unstable) when τ < 0 (τ > 0). When τ 2 > 4∆ (outside the parabola inFig. 4.15), the eigenvalues are real. They both are negative (positive) when τ < 0 (τ > 0).5. (van der Pol oscillator) The nullclines of the van der Pol oscillator,y = x − x 3 /3 (x-nullcline) ,x = 0 (y-nullcline) ,are depicted in Fig. 10.22. There is a unique equilibrium (0, 0). The Jacobian matrix at theequilibrium has the form( )1 −1L =.b 0Since tr L = 1 > 0 and det L = b > 0, the equilibrium is always an unstable focus.6. (Bonhoeffer–van der Pol oscillator) The nullclines of the Bonhoeffer–van der Pol oscillator withc = 0 have the formy = x − x 3 /3 (x-nullcline) ,x = a (y-nullcline) ,