292
7 Pulse Programming of Memristor Circuits
which is well defined under the next additional assumption (named Assumption 4,
or (A4) for short; see also Appendix 1).
(A4) Submatrix H 22 is nonsingular, i.e., det H 22 = 0.
It follows from (7.26b) that Y(t) − Y(t 0 ) = y(t) − y(t 0 ) with Y(t 0 ) =
−H
−1
22
H 21 x M (t 0 ) + M y ˙
y(t 0 )
. Hence, the change of variables (7.26) is also
equivalent to using the following compact expressions X(t) − x M (t 0 ) = x(t) and
Y(t) − Y(t 0 ) = y(t), that permits to identify the relationship with the circuit
variables given in (7.17).
By (7.26) and (7.20) we can derive the following form of the SEs in the (ϕ, q)domain such that manifolds and the dynamics of N with respect to manifolds can
be grasped:
M x ˙
X(t)
M y ˙
Y(t)
= −
H 11 H 12
H 21 H 22
X(t)
Y(t)
−
F(X(t))
0
−
u x (t)
u y (t)
+
k 0
0
(7.27)
where we have let
k 0 = S 22 x M (t 0 ) + F(x M (t 0 )) + M x ˙
x(t 0 ) − H 12 H
−1
22 M y ˙
y(t 0 )
(7.28)
and
S 22 = H/H 22 = H 11 − H 12 H
−1
22 H 21
is the Schur complement of H 22 in H [14].
Note that k 0 ∈ R n M depends upon the ICs (x M (t 0 ), ˙
x(t 0 ), ˙
y(t 0 )) for the state
variables in the (v, i)-domain (see (7.24)). The SEs (7.27) make clear that k 0 plays
a crucial role in the nonlinear dynamics and bifurcation phenomena of N. 6
In the following we investigate:
• the geometric properties of manifolds, i.e., how the state space in the (v, i)domain can be foliated into manifolds with specific geometric properties;
• the dynamic properties of manifolds, i.e., conditions under which the manifolds
are positively invariant for the dynamics of N in the (v, i)-domain described by
the SEs (7.23). In the latter case, the goal is to study how solutions of (7.23)
evolve in time through different manifolds.
6 Note that k 0 depends also on circuit parameters and memristor nonlinearities (through the vector
F(·) and the submatrices of H and M). In this chapter, the chief interest is on the dependency on ICs
in order to highlight the concept of bifurcation without parameters, whereas standard bifurcations
due to the change of circuit parameters and memristor nonlinearities are not considered.
7 Pulse Programming of Memristor Circuits
which is well defined under the next additional assumption (named Assumption 4,
or (A4) for short; see also Appendix 1).
(A4) Submatrix H 22 is nonsingular, i.e., det H 22 = 0.
It follows from (7.26b) that Y(t) − Y(t 0 ) = y(t) − y(t 0 ) with Y(t 0 ) =
−H
−1
22
H 21 x M (t 0 ) + M y ˙
y(t 0 )
. Hence, the change of variables (7.26) is also
equivalent to using the following compact expressions X(t) − x M (t 0 ) = x(t) and
Y(t) − Y(t 0 ) = y(t), that permits to identify the relationship with the circuit
variables given in (7.17).
By (7.26) and (7.20) we can derive the following form of the SEs in the (ϕ, q)domain such that manifolds and the dynamics of N with respect to manifolds can
be grasped:
M x ˙
X(t)
M y ˙
Y(t)
= −
H 11 H 12
H 21 H 22
X(t)
Y(t)
−
F(X(t))
0
−
u x (t)
u y (t)
+
k 0
0
(7.27)
where we have let
k 0 = S 22 x M (t 0 ) + F(x M (t 0 )) + M x ˙
x(t 0 ) − H 12 H
−1
22 M y ˙
y(t 0 )
(7.28)
and
S 22 = H/H 22 = H 11 − H 12 H
−1
22 H 21
is the Schur complement of H 22 in H [14].
Note that k 0 ∈ R n M depends upon the ICs (x M (t 0 ), ˙
x(t 0 ), ˙
y(t 0 )) for the state
variables in the (v, i)-domain (see (7.24)). The SEs (7.27) make clear that k 0 plays
a crucial role in the nonlinear dynamics and bifurcation phenomena of N. 6
In the following we investigate:
• the geometric properties of manifolds, i.e., how the state space in the (v, i)domain can be foliated into manifolds with specific geometric properties;
• the dynamic properties of manifolds, i.e., conditions under which the manifolds
are positively invariant for the dynamics of N in the (v, i)-domain described by
the SEs (7.23). In the latter case, the goal is to study how solutions of (7.23)
evolve in time through different manifolds.
6 Note that k 0 depends also on circuit parameters and memristor nonlinearities (through the vector
F(·) and the submatrices of H and M). In this chapter, the chief interest is on the dependency on ICs
in order to highlight the concept of bifurcation without parameters, whereas standard bifurcations
due to the change of circuit parameters and memristor nonlinearities are not considered.