Appendix 1: Bifurcations Without Parameters of Equilibrium Points
265
Appendix 1: Bifurcations Without Parameters of Equilibrium
Points
We want to verify that the bifurcation of EPs observed in the M − C studied
in Sect. 6.1.2 exactly corresponds to one type of bifurcations without parameters
introduced theoretically in [6].
Suppose the memristor has a CR q M = −ϕ M +
1
3 ϕ 3
M . In the (v, i)-domain the
M − C circuit satisfies
˙
ϕ M (t) = v C (t)
C ˙
v C (t) = (1 − ϕ
2
M (t))v C (t).
There exists a manifold (a line) of EPs coinciding with the ϕ M axis. By letting
Y = ϕ M and Z = v C , assuming C = 1 F, and omitting dependence on t we can
rewrite the system as
˙
Y = F
Y (Y, Z) = Z
˙
Z = F
Z (Y, Z) = (1 − Y
2 )Z
which is in the form of system (10) in [6]. Let F = (F Y , F Z ). We have F(Y, 0) = 0
for any Y ∈ R, i.e., the Y -axis is a line of EPs.
To find the EPs that are candidates for a bifurcation without parameter we impose
condition (cf. (11) in [6])
∂ Z F
Z (Y, 0) = 0.
This yields
∂ Z F
Z (Y, 0) = 1 − Y
2
= 0
hence the EPs candidate for a bifurcation without parameter are Y e = ±1.
Let us consider Y e = −1 (a similar analysis holds for Y e = 1). The change of
variables y = Y − Y e = Y + 1 and z = Z yields
˙
y = f
y (y, z) = z
˙
z = f
z (y, z) = (2y − y
2 )z.
Let f = (f y , f z ). We have f(y, 0) = 0 for y ∈ R, hence the y-axis is a line of EPs.
Moreover, we have
∂ z f
z (0, 0) = 0
∂ yz f
z (0, 0) = 2 = 0