410
11 Nonlinear Dynamics of Circuits with Mem-Elements
solution (t; t 0 , 0 ) is such that
(t; t 0 , 0 ) ∈ M(K((t; t 0 , 0 )))
where
K((t; t 0 , 0 )) = K( 0 ) + H 12 H
−1
22 U 2 (t; t 0 ) − U 1 (t; t 0 )
= K( 0 ) + H 12 H
−1
22 (B 21 ϕ e (t; t 0 ) + B 22 q a (t; t 0 ))
−(B 11 ϕ e (t; t 0 ) + B 12 q a (t; t 0 )).
(11.29)
Proof See Appendix 1.
This theorem permits to find at any instant t ≥ t 0 the manifold M(·) where the
solution of the SEs in the (ϕ, q)-domain belongs to. Such a manifold M(·) varies
according to the variations of K((t; t 0 , 0 )) induced by the input sources ϕ e (t; t 0 )
and q a (t; t 0 ) (cf. (11.29)).
11.5.2 Coexisting Dynamics and Bifurcations Without
Parameters
Suppose that a mem-circuit in LME has no input sources, thus ϕ e (t; t 0 ) = 0
and q a (t; t 0 ) = 0 (or equivalently e(t) = 0, a(t) = 0) for any t ≥ t 0 . Then,
Theorem 11.1 can be simplified as follows.
Corollary 11.1 If the assumptions of Theorem 11.1 hold, then
(t; t 0 , 0 ) ∈ M(K( 0 ))
for any t ≥ t 0 . Moreover, the nonlinear dynamics is described for t ≥ t 0 by the
following reduced-order SEs:
M
⎛
⎝
˙
X(t)
˙
Y(t)
˙
Z(t)
⎞
⎠ = −H
⎛
⎝
F a (X(t))
Y(t)
Z(t)
⎞
⎠ +
⎛
⎝
0
F b (Y(t))
0
⎞
⎠ +
⎛
⎝
K a ( 0 )
K b ( 0 )
0
⎞
⎠
where
Z(t) = z(t) − H
−1
22 H 21
F a (X 0 )
Y 0
− H
−1
22 M 22 R 0 .
(11.30)
Proof See Appendix 2.