Dynamical Systems in Neuroscience:

220 BifurcationsExercises1. (Transcritical bifurcation) Justify the bifurcation diagram shown in Fig. 6.51.2. Show that the non-degeneracy and transversality conditions are necessary forthe saddle-node bifurcation. That is, present a system that does not exhibitsaddle-node bifurcation, but satisfies(a) the non-hyperbolicity and non-degeneracy conditions, or(b) the non-hyperbolicity and transversality conditions.3. Consider the model˙V = c(b − b sn ) + a(V − V sn ) 2 ,with positive a and c, and b > b sn . Show that the sojourn time in a boundedneighborhood of the point V = V sn scales asT =π√ac(b − bsn )when b is near b sn . (Hint: Find the solution that starts at −∞ and terminatesat +∞.)4. Show that the two-dimensional systemthe complex-valued systemand the polar-coordinate systemare equivalent.˙u = c(b)u − ω(b)v + (au − dv)(u 2 + v 2 ) , (6.14)˙v = ω(b)u + c(b)v + (du + av)(u 2 + v 2 ) , (6.15)ż = (c(b) + iω(b))z + (a + id)z|z| 2 ,ṙ = c(b)r + ar 3˙ϕ = ω(b) + dr 2 ,5. Show that the non-degeneracy and transversality conditions are necessary forthe Andronov-Hopf bifurcation. That is, present a system that does not exhibitAndronov-Hopf bifurcation, but satisfies(a) the non-hyperbolicity and non-degeneracy conditions, or(b) the non-hyperbolicity and transversality conditions.6. Show that the system (6.14, 6.15) with c(b) = b, ω(b) = 1, a ≠ 0 and d = 0exhibits Andronov-Hopf bifurcation. Check all three conditions.