Dynamical Systems in Neuroscience:

Bifurcations 223• Andronov-Hopf bifurcation when a < b and a 2 − 2ab + 4I = 0,• Bogdanov-Takens bifurcation when a = b = 2 √ I.Use results of Ex. 15 to prove that the Andronov-Hopf bifurcation in the modelabove is always subcritical.14. Use (6.7) to prove that the relaxation oscillator˙v = f(v) − u˙u = µ(v − b)with an N-shaped fast nullcline u = f(v) undergoes Andronov-Hopf bifurcationwhen f ′ (b) = 0, i.e., at the knee. The bifurcation is supercritical when f ′′′ (b) < 0and subcritical when f ′′′ (b) > 0.15. Prove that the Andronov-Hopf bifurcation point in˙v = F (v) − u˙v = µ(bv − u)satisfies F ′ = µ and b > µ. Use (6.7) to show thata = {F ′′′ + (F ′′ ) 2 /(b − µ)}/16 .16. Prove that the Andronov-Hopf bifurcation point in˙v = F (v) − u˙u = µ(G(v) − u)satisfies F ′ = µ and G ′ > µ. Use (6.7) to show thata = {F ′′′ + F ′′ (F ′′ − G ′′ )/(G ′ − µ)}/16 .17. Prove that the Andronov-Hopf bifurcation point in˙v = F (v) − (v + 1)u˙u = µ(G(v) − u)satisfies F ′ = µ and G ′ > µ. Use (6.7) to show thata = {F ′′′ + µ − (F ′′ − µ)(1 + µ[G ′′ − F ′′ + 2µ]/ω 2 )}/16 .18. Use (6.7) to show that a two-dimensional relaxation oscillator˙v = F (v, u)˙u = µG(v, u)at an Andronov-Hopf bifurcation point hasa = 1 { [Fvv G u − F u G vvF vvv + F vv16F u G v− F vuF u]}+ O( √ µ) .