By Ivana Suchy
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Above the bifurcation point a > 0, orbits diverge outwards from the ﬁxed point repellor to this attractor. Below the bifurcation point a < 0, the limit cycle repellor (dashed blue curve in Fig. 11b) separates the basin boundaries of the ﬁxed point attractor at the origin and the large-amplitude limit cycle attractor. In summary, Hopf bifurcations are of very high importance for an understanding of neural activity as they explain the onset and nature of oscillatory behaviour. Supercritical Hopf bifurcations lead to the appearance of small amplitude periodic oscillations.
An example, when a chaotic attractor collides with its fractal basin boundary was termed a crisis bifurcation when it was ﬁrst described (Celso et al. 1982). More speciﬁcally, a boundary crisis occurs when the collision is between an attractor and a boundary, whereas an interior crisis results from the collision between an attractor and a saddle. The former results in the sudden loss of an attractor and its basin, whereas the latter typically leads to the sudden increase in the extent of the attractor.
However, setting r > 0 but small has the eﬀect of introducing the third variable into the dynamics. 8 0 200 400 600 800 1000 x Fig. 21. Phase portrait and time series for the Fitz-Hugh-Nagumo model following a Hopf bifurcation. 1 as per Izhikevich (2005). Note the qualitative similarity to Fig. 19(e), (f) 36 Michael Breakspear and Viktor K Jirsa (b) y y (a) time x Fig. 22. Phase portrait and time series for the Hindmarsh-Rose model. 60; r = 0 equations as an additive term – the same as I although negative in size.