Dynamical Systems in Neuroscience:

Bifurcations 219Figure 6.50: The founder of Russian school of nonlineardynamics, Alexander Aleksandovich Andronov (1901-1952) in 1950 (picture provided by M.I. Rabinovich).xx=bstablestable0unstablebunstableFigure 6.51: Transcritical bifurcation in ẋ = x(b − x).bifurcation in Fig. 6.36 could result in the appearance of a stable equilibrium x = 0 ifb decreases past 0. Our classification of bifurcations into subcritical and supercriticalis consistent with the following widely accepted rule: Let the bifurcation parameterchange in the direction leading to the increase in a number of objects (equilibria, limitcycles). The bifurcation is supercritical if stable objects appear, subcritical if unstableobjects appear, and transcritical, such as in Fig. 6.51, if equal numbers of stableand unstable objects appear or disappear. The condition for supercritical (subcritical)Andronov-Hopf bifurcation, Eq. (6.7), is taken from Guckenheimer and Holmes (1983).Delayed loss of stability was first described by Shishkova (1973), and then studied indetail by Nejshtadt (1985), though many find his paper difficult to read. An alternativedescription is given by Arnold et al. (1994) and Baer et al. (1989).Canard (French duck) solutions were reported by Benoit et al. (1981). Due to therecent political climate in the USA, some refer to “French ducks” as “freedom ducks”,probably to emphasize that “French = freedom”. Canards in R 3 were studied by Benoit(1984), Samborskij (1985, in R n ), and recently by Szmolyan and Wechselberger (2001,2004) and Wechselberger (2005).