Dynamical Systems in Neuroscience:

218 Bifurcationsbifurcationssaddle-nodesaddle-node on invariant cicleAndronov-Hopfsaddle homoclinic orbitfold limit cyclesaddle-node homoclinic orbitBogdanov-TakensBautinflipalternative namesfold, limit point, saddle-node off limit cycleSNIC, saddle-node on limit cycle (SNLC), circle,saddle-node homoclinic, saddle-node central homoclinic,saddle-node infinite period (SNIPer), homoclinicHopf, Poincare-Andronov-Hopfhomoclinic, saddle-loop, saddle separatrix loop, Andronov-Leontovichsaddle-node of limit cycles, double limit cycle, fold cycle,saddle-node (fold) of periodicssaddle-node noncentral homoclinic, saddle-node separatrix-loopTakens-Bogdanov, double-zerodegenerate Hopf, generalized Hopfperiod doublingFigure 6.49: Popular alternative names to some of the bifurcations considered in thischapter.science. More bifurcation theory, including bifurcations in mappings x n+1 = f(x n , b),can be found in the excellent book “Elements of Applied Bifurcation Theory” by YuriKuznetsov (1995, new edition 2004), which, however, might be a bit technical fora non-mathematician. Some of the bifurcations considered in this chapter, such asthe blue-sky catastrophe, are classified as “exotic” by Kuznetsov (1995), though thecatastrophe was recently found in a model of a leech heart interneuron (Shilnikov andCymbalyuk 2005).There is no unified naming scheme for the bifurcations, mostly because they werediscovered and rediscovered independently in many fields and in many countries. Forexample, the Andronov-Hopf bifurcation was known to Poincare, so some scientistsrefer to it as Poincare-Andronov-Hopf bifurcation. Many refer to it as just Hopf bifurcationdue to the fault of no lesser men than the famous Russian mathematicianVladimir Igorevich Arnold and famous French mathematician Rene Thom. Accordingto Arnold’s own accounts, he was visited by Thom in the 1960s. While discussing variousbifurcations, Arnold put too much emphasis on the “recent” Hopf (1942) paper.As a result of Arnold’s misattribution, Thom popularized the bifurcation as being Hopfbifurcation. In Fig. 6.49 we provide some common alternative names to the bifurcationsconsidered in this chapter. The complete list of names of known bifurcations is toolong, and it resembles the list of faculty members of the department of RadioPhysics(Gorky State University, now Nizhnii Novgorod, Russia) founded by A.A. Andronovin 1945.The division of bifurcations into subcritical and supercritical ones may be confusingfor a novice. For example, some scientists erroneously think that supercritical bifurcationsresult in appearance of attractors (stable equilibria, limit cycles, etc.) andsubcritical bifurcations result in their disappearance. Let us emphasize here that theappearance or disappearance of an equilibrium or a limit cycle depends on the directionof change of the bifurcation parameter. For example, the subcritical pitchfork