Dynamical Systems in Neuroscience:

Solutions to Exercises, Chap. 3 409(b) The equation 0 = x − x 3 has three solutions: x = ±1 and x = 0, hence there are threeequilibria in the system (b). The eigenvalues are the derivatives at each equilibrium, λ =(x − x 3 ) ′ = 1 − 3x 2 . The equilibria x = ±1 are stable because λ = 1 − 3(±1) 2 = −2 < 0.The equilibrium x = 0 is unstable because λ = 1 > 0. The same conclusions also followfrom the geometrical analysis in Fig. 10.3.1.51x2 x -110.5xx-x 30.50x0x-0.5-0.5-1-2 -1 0 1 2a-1-2 -1 0 1 2bFigure 10.3: Phase portraits of the systems (a) ẋ = −1 + x 2 , (b) ẋ = x − x 3 .4. The equilibrium x = 0 is stable in all three cases.5. See Fig. 10.4. Topologically equivalent systems are in (a), (b), and (c). In (d) there aredifferent numbers of equilibria; no stretching or shrinking of the rubber phase line can producenew equilibria. In (e) the right equilibrium is unstable in ˙V = F 1 (V ), but stable in ˙V = F 2 (V );no stretching or shrinking can change the stability of an equilibrium. In (f) the flow betweenthe two equilibria is directed rightward in ˙V = F 1 (V ) and leftward in ˙V = F 2 (V ); no stretchingor shrinking can change the direction of the flow.6. (Saddle-node (fold) bifurcation in ẋ = a + x 2 ) The equation 0 = a + x 2 has no real solutionswhen a > 0, and two solutions x = ± √ |a| when a ≤ 0. Hence there are two branches ofequilibria, depicted in Fig. 10.5. The eigenvalues are7.λ = (a + x 2 ) ′ = 2x = ±2 √ |a| .The lower branch − √ |a| is stable (λ < 0), and the upper branch + √ |a| is unstable (λ > 0).They meet at the saddle-node (fold) bifurcation point a = 0.(a) x = −1 at a = 1, (b) x = −1/2 at a = 1/4 (c) x = 1/2 at a = 1/4(d) x = ±1/ √ 3 at a = ±2/(3 √ 3) (e) x = ±1 at a = ∓2 (f) x = −1 at a = 18. (Pitchfork bifurcation in ẋ = bx − x 3 ) The equation 0 = bx − x 3 has one solution x = 0 whenb ≤ 0, and three solutions x = 0, x = ± √ b when b > 0. Hence there is only one branch ofequilibria for b < 0 and three branches for b > 0 of the pitchfork curve depicted in Fig. 10.6.The eigenvalues areλ = (bx − x 3 ) ′ = b − 3x 2 .The branch x = 0 exists for any b and its eigenvalue is λ = b. Thus, it is stable for b < 0 andunstable for b > 0. The two branches x = ± √ b exist only for b > 0, but they are always stablebecause λ = b−3(± √ b) 2 = −2b < 0. We see that the branch x = 0 loses stability when b passesthe pitchfork bifurcation value b = 0, at which point a pair of new stable branches bifurcates(hence the name bifurcation). In other words, the stable branch x = 0 divides (bifurcates) intotwo stable branches when b passes 0.