Processes 2019, 7,20
the considered system using the bifurcation theory. From the 2D and 3D projection in Figures 4–6 one
might get the impression that the limit cycle starting from the second Andronov-Hopf bifurcation is
not completed, but this limit cycle terminates at the unstable equilibrium branch, cf. the projection
on the (G Ca 2+ , x, V)–space in Figure 8a. In Figure 8b we show for comparison the corresponding
bifurcation diagram with ˜
τ f = 0.8 · τ f and ˜
τ x = 0.8 · τ x instead of with τ f and τ x , cf. Figure 5b. Here,
one sees that the behaviour is different compared to the standard setting, while in the discussion of the
GSPT this change has no influence. Thus, it is important to use both approaches for the investigation
of such phenomena.
(a)3D projection on the (G Ca 2+ , x, V)–space.
0.0305
0.031
0.0315
0.032
0.0325
bifurcation parameter G Ca
-80
-70
-60
-50
-40
-30
-20
-10
0
10
voltage V (mV)
(b)Zoom: Bifurcation diagram of system (1) with τ d = 20 ms,
˜
τ f = 0.8 · τ f and ˜
τx = 0.8 · τx.
Figure 8. In (a) a different point of view of Figure 6b is given to illustrate that the limit cycle branch
terminates at the unstable equilibrium branch, while in (b) the corresponding bifurcation diagram with
˜
τ f = 0.8 · τ f and ˜
τ x = 0.8 · τ x is stated.
Finally, if we consider the bifurcation diagram of (1)–(3) with τ d = 40 ms and C m = 2
μF
m 2 instead of
τ d = 20 ms and C m = 1
μF
m 2 , we see again the importance to consider all these parameters, cf. Figure 9.
0.0305 0.0306 0.0307 0.0308 0.0309 0.031 0.0311 0.0312
bifurcation parameter G Ca
-80
-70
-60
-50
-40
-30
-20
-10
0
voltage V (mV)
supercritical Andronov-Hopf bifurcation
period doubling bifurcation
limit point of cycles
Figure 9. Zoom of bifurcation diagram: C m = 2
μF
m 2 and τ d = 40 ms.
In Figures 9 and 10a we see that the system (1) may exhibit different type of MMOs and critical
transient regions depending on the choice of the system parameters. Even more, it also shows that
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