7.5 Analysis of Manifolds
293
7.5.1 Geometric Properties
Using the expression of k 0 in (7.28), let us introduce:
• the function K(·) : R (n M +n C +n L ) → R n M of the state vector w = (x M , ˙
x, ˙
y) in
the (v, i)-domain (omitting dependence on t)
K(x M , ˙
x, ˙
y) = S 22 x M + F(x M ) + M x ˙
x − H 12 H
−1
22 M y ˙
y
(7.29)
• for any vector k ∈ R n M , the level set M(k) ⊂ R (n M +n C +n L ) of K(x M , ˙
x, ˙
y),
which is defined as
M(k) = {w = (x M , ˙
x, ˙
y) ∈ R
n M +n C +n L : K(w) = k}.
(7.30)
The next result illustrates some main geometric properties of sets M(k).
Theorem 7.3 If (A1)–(A4) are satisfied by N, then the following geometric properties hold:
1. for any k ∈ R n M , M(k) defines a nonempty, nonplanar, (n C + n L )-dimensional
manifold in the state space in the (v, i)-domain;
2. for any k 1 = k 2 ∈ R n M , we have M(k 2 ) = M(k 1 )+M −1
x (k 2 −k 1 ), i.e., M(k 2 )
is a rigid translation of M(k 1 ), and conversely;
3. there are ∞ n M nonintersecting manifolds, obtained by varying k in R n M , which
span the whole (n M + n C + n L )-dimensional state space in the (v, i)-domain.
Proof See Appendix 2.
The result confirms in particular the geometric properties of manifolds seen in
specific low-order circuits with one memristor in Chap. 6.
7.5.2 Dynamic Properties
The dynamic properties of manifolds can be inferred from the time evolution of the
solutions of the SEs (7.23) through the (n M + n C + n L )-dimensional state space in
the (v, i)-domain. Given w 0 ∈ R (n M +n C +n L ) , let w(t; t 0 , w 0 ) = (x M (t), ˙
x(t), ˙
y(t))
be the solution with ICs w 0 at t = t 0 of the SEs (7.23) in the (v, i)-domain.
Using (7.29) and (7.30), if we let
k(t; t 0 , w 0 ) = K(w(t; t 0 , w 0 )), ∀t ≥ t 0
(7.31)
then we have
w(t; t 0 , w 0 ) ∈ M(k(t; t 0 , w 0 )), ∀t ≥ t 0 .
(7.32)
293
7.5.1 Geometric Properties
Using the expression of k 0 in (7.28), let us introduce:
• the function K(·) : R (n M +n C +n L ) → R n M of the state vector w = (x M , ˙
x, ˙
y) in
the (v, i)-domain (omitting dependence on t)
K(x M , ˙
x, ˙
y) = S 22 x M + F(x M ) + M x ˙
x − H 12 H
−1
22 M y ˙
y
(7.29)
• for any vector k ∈ R n M , the level set M(k) ⊂ R (n M +n C +n L ) of K(x M , ˙
x, ˙
y),
which is defined as
M(k) = {w = (x M , ˙
x, ˙
y) ∈ R
n M +n C +n L : K(w) = k}.
(7.30)
The next result illustrates some main geometric properties of sets M(k).
Theorem 7.3 If (A1)–(A4) are satisfied by N, then the following geometric properties hold:
1. for any k ∈ R n M , M(k) defines a nonempty, nonplanar, (n C + n L )-dimensional
manifold in the state space in the (v, i)-domain;
2. for any k 1 = k 2 ∈ R n M , we have M(k 2 ) = M(k 1 )+M −1
x (k 2 −k 1 ), i.e., M(k 2 )
is a rigid translation of M(k 1 ), and conversely;
3. there are ∞ n M nonintersecting manifolds, obtained by varying k in R n M , which
span the whole (n M + n C + n L )-dimensional state space in the (v, i)-domain.
Proof See Appendix 2.
The result confirms in particular the geometric properties of manifolds seen in
specific low-order circuits with one memristor in Chap. 6.
7.5.2 Dynamic Properties
The dynamic properties of manifolds can be inferred from the time evolution of the
solutions of the SEs (7.23) through the (n M + n C + n L )-dimensional state space in
the (v, i)-domain. Given w 0 ∈ R (n M +n C +n L ) , let w(t; t 0 , w 0 ) = (x M (t), ˙
x(t), ˙
y(t))
be the solution with ICs w 0 at t = t 0 of the SEs (7.23) in the (v, i)-domain.
Using (7.29) and (7.30), if we let
k(t; t 0 , w 0 ) = K(w(t; t 0 , w 0 )), ∀t ≥ t 0
(7.31)
then we have
w(t; t 0 , w 0 ) ∈ M(k(t; t 0 , w 0 )), ∀t ≥ t 0 .
(7.32)