7.8 Further Examples
307
7.7.3 Memristor Star-CNNs
This short section discusses an application of the pulse programming of nonlinear
dynamics on manifolds in large memristor circuits with a neural architecture.
Consider a neural network with a star topology and a (possibly) large number
N of cells, where each cell is represented by an element D ϕ with a capacitor and
a flux-controlled memristor in parallel. Neural networks with a star topology are
also referred to as Star-CNNs [17, 18]. Hereinafter, the studied Star-CNNs with
memristors are named Memristor Star-CNNs (MS-CNNs).
MS-CNNs fall into the class of memristor circuits in Sect. 7.5.3. To simplify
the notation, time is omitted, we let t 0 = 0 and denote ϕ
(i)
γ M = ϕ i , q
(i)
γ M = q i ,
ϕ
(i)
M (0) = ϕ M 0 i , and G =
N
j =1 G j (see Fig. 7.12). The vectors of fluxes ϕ =
(ϕ 1 , . . . , ϕ N ) T , charges q = (q 1 , . . . , q N ) T , ICs ϕ M 0 = (ϕ M 0 1 , . . . , ϕ M 0 N ) T and
sources ϕ e = (ϕ e 1 , . . . , ϕ e N ) T permit to write via Millmann’s theorem the SEs of
MS-CNNs in the (ϕ, q)-domain in the compact form (see also (7.37))
M ˙
ϕ = −Hϕ − F(ϕ + ϕ M 0 ) + k 0 − u
(7.49)
k 0 = F(ϕ M 0 ) + M ˙
ϕ M 0 + Hϕ M 0
(7.50)
ϕ(0) = ϕ M 0
(7.51)
where u = Hϕ e + B 12 q a , the entry (i, j ) of B 11 (for 1 ≤ i ≤ N and
1 ≤ j ≤ N ) is [B 11 ] ij = −G i G j /G, H = diag(G 1 , . . . , G N ) + B 11 , B 12 =
(−G 1 /G, . . . , −G N /G) T and M = diag(C 1 , . . . , C N ).
If u is constant, the number and stability properties of EPs in the MS-CNNs
described by (7.51) depend on k 0 and u. Clearly, bifurcations without parameters of
such equilibria can be induced by applying pulses with finite time duration by means
of ϕ e and q a , which cause a change of k 0 into k 0 + k u with k u given in (7.43).
7.8 Further Examples
In the previous examples we considered memristor circuits that belong to the class
LM and satisfy assumptions (A1)–(A3) guaranteeing the existence of the hybrid
representation and the SE description. Of course, (A1)–(A3) are only sufficient
conditions for the existence of the hybrid representation. It can be seen that if
those conditions fail, but the hybrid representation exists, then we can still use
the developed procedure for writing the SEs and studying the invariant manifolds.
In this section, we provide some examples of memristor circuits containing active
elements as controlled sources and thus not belonging to the class LM. Due to the
presence of active elements, it is not easy to give conditions that a priori guarantee
the existence of the hybrid representation. Nevertheless, in the examples we directly
307
7.7.3 Memristor Star-CNNs
This short section discusses an application of the pulse programming of nonlinear
dynamics on manifolds in large memristor circuits with a neural architecture.
Consider a neural network with a star topology and a (possibly) large number
N of cells, where each cell is represented by an element D ϕ with a capacitor and
a flux-controlled memristor in parallel. Neural networks with a star topology are
also referred to as Star-CNNs [17, 18]. Hereinafter, the studied Star-CNNs with
memristors are named Memristor Star-CNNs (MS-CNNs).
MS-CNNs fall into the class of memristor circuits in Sect. 7.5.3. To simplify
the notation, time is omitted, we let t 0 = 0 and denote ϕ
(i)
γ M = ϕ i , q
(i)
γ M = q i ,
ϕ
(i)
M (0) = ϕ M 0 i , and G =
N
j =1 G j (see Fig. 7.12). The vectors of fluxes ϕ =
(ϕ 1 , . . . , ϕ N ) T , charges q = (q 1 , . . . , q N ) T , ICs ϕ M 0 = (ϕ M 0 1 , . . . , ϕ M 0 N ) T and
sources ϕ e = (ϕ e 1 , . . . , ϕ e N ) T permit to write via Millmann’s theorem the SEs of
MS-CNNs in the (ϕ, q)-domain in the compact form (see also (7.37))
M ˙
ϕ = −Hϕ − F(ϕ + ϕ M 0 ) + k 0 − u
(7.49)
k 0 = F(ϕ M 0 ) + M ˙
ϕ M 0 + Hϕ M 0
(7.50)
ϕ(0) = ϕ M 0
(7.51)
where u = Hϕ e + B 12 q a , the entry (i, j ) of B 11 (for 1 ≤ i ≤ N and
1 ≤ j ≤ N ) is [B 11 ] ij = −G i G j /G, H = diag(G 1 , . . . , G N ) + B 11 , B 12 =
(−G 1 /G, . . . , −G N /G) T and M = diag(C 1 , . . . , C N ).
If u is constant, the number and stability properties of EPs in the MS-CNNs
described by (7.51) depend on k 0 and u. Clearly, bifurcations without parameters of
such equilibria can be induced by applying pulses with finite time duration by means
of ϕ e and q a , which cause a change of k 0 into k 0 + k u with k u given in (7.43).
7.8 Further Examples
In the previous examples we considered memristor circuits that belong to the class
LM and satisfy assumptions (A1)–(A3) guaranteeing the existence of the hybrid
representation and the SE description. Of course, (A1)–(A3) are only sufficient
conditions for the existence of the hybrid representation. It can be seen that if
those conditions fail, but the hybrid representation exists, then we can still use
the developed procedure for writing the SEs and studying the invariant manifolds.
In this section, we provide some examples of memristor circuits containing active
elements as controlled sources and thus not belonging to the class LM. Due to the
presence of active elements, it is not easy to give conditions that a priori guarantee
the existence of the hybrid representation. Nevertheless, in the examples we directly