11.6 Chaotic Circuit with Memcapacitor and Memristor
413
As predicted by the theoretical results in the previous section, the SEs have an
order reduction n vi − n ϕq = 2 in passing from the (v, i)-domain to the (ϕ, q)domain, that is exactly the number of mem-elements in Fig. 11.7.
Let (t) = (σ MC (t), q M (t), q MC (t), i L (t), v C (t)) T be the vector of state
variables in the (v, i)-domain, then (11.27) gives
K() =
−Gf MC (σ MC ) + q MC + Cv C − q M
Li L + f M (q M ) − RCv C + f MC (σ MC ) + Rq M
.
Moreover, by (11.32), K((t; t 0 , 0 )) along a solution (t; t 0 , 0 ) of the SEs in the
(v, i)-domain (see (11.29)) is
K((t; t 0 , 0 )) = K( 0 ) +
q a (t; t 0 )
ϕ e (t; t 0 )
(11.36)
for any t ≥ t 0 , where 0 = (σ MC 0 , q M 0 , q MC 0 , i L 0 , v C 0 ) T are the ICs for the state
variables in the (v, i)-domain.
Let us focus first on the case in which the mem-circuit in Fig. 11.7 has zero input
sources, i.e., a(t) = 0 and e(t) = 0, thus q a (t; t 0 ) = 0 and ϕ e (t; t 0 ) = 0, t ≥
t 0 . In this case, due to Theorem 11.1, there exist ∞ 2 three-dimensional manifolds
M( ¯
K) = { ∈ R 5 : K() = ¯
K}, for any ¯
K ∈ R 2 , that are positively invariant for
the mem-circuit dynamics in the (v, i)-domain. On each manifold, due to (11.30),
the reduced third-order nonlinear dynamics can be rewritten as
⎛
⎝
˙
X(t)
L ˙
Y (t)
C ˙
Z(t)
⎞
⎠ =
⎛
⎝
(−G +
1
R )f MC (X(t)) +
Z(t)
R + K 1 ( 0 )
−Z(t) + f M (Y (t)) + K 2 ( 0 )
1
R f MC (X(t)) + Y (t) −
Z(t)
R
⎞
⎠
(11.37)
where Z(t) is in (11.30) and K( 0 ) = (K 1 ( 0 ), K 2 ( 0 )) with
K 1 ( 0 ) = −Gf MC (σ MC 0 ) + q MC 0 + Cv C 0 − q M 0
K 2 ( 0 ) = Li L 0 + f M (q M 0 ) − RCv C 0 + f MC (σ MC 0 ) + Rq M 0 .
The ICs in the (v, i)-domain act as constant inputs in the SEs in the (ϕ, q)domain via the terms K 1 ( 0 ) and K 2 ( 0 ). It turns out that there coexist ∞ 2 different
reduced-order dynamics (one for each manifold M(K 1 ( 0 ), K 2 ( 0 )).
Suppose that the circuit parameters are G = 2.1, R = 0.2083, C = 5.8823, L =
0.136 and let f MC (σ MC ) = 0.7σ MC + 0.5σ 2
MC and f M (q M ) = 0.01q M − 0.05q 2
M
(cf. [24]). Figure 11.8 reports the bifurcation diagram for (11.37), as a function
of K 1 and K 2 /L, numerically obtained by means of the software MatCont [25].
The black (resp., blue) solid curve corresponds to points where MatCont detects a
supercritical (resp., subcritical) Hopf bifurcation. The black dashed line represents
points of the parameter-space where a period-doubling bifurcation occurs. Finally,
413
As predicted by the theoretical results in the previous section, the SEs have an
order reduction n vi − n ϕq = 2 in passing from the (v, i)-domain to the (ϕ, q)domain, that is exactly the number of mem-elements in Fig. 11.7.
Let (t) = (σ MC (t), q M (t), q MC (t), i L (t), v C (t)) T be the vector of state
variables in the (v, i)-domain, then (11.27) gives
K() =
−Gf MC (σ MC ) + q MC + Cv C − q M
Li L + f M (q M ) − RCv C + f MC (σ MC ) + Rq M
.
Moreover, by (11.32), K((t; t 0 , 0 )) along a solution (t; t 0 , 0 ) of the SEs in the
(v, i)-domain (see (11.29)) is
K((t; t 0 , 0 )) = K( 0 ) +
q a (t; t 0 )
ϕ e (t; t 0 )
(11.36)
for any t ≥ t 0 , where 0 = (σ MC 0 , q M 0 , q MC 0 , i L 0 , v C 0 ) T are the ICs for the state
variables in the (v, i)-domain.
Let us focus first on the case in which the mem-circuit in Fig. 11.7 has zero input
sources, i.e., a(t) = 0 and e(t) = 0, thus q a (t; t 0 ) = 0 and ϕ e (t; t 0 ) = 0, t ≥
t 0 . In this case, due to Theorem 11.1, there exist ∞ 2 three-dimensional manifolds
M( ¯
K) = { ∈ R 5 : K() = ¯
K}, for any ¯
K ∈ R 2 , that are positively invariant for
the mem-circuit dynamics in the (v, i)-domain. On each manifold, due to (11.30),
the reduced third-order nonlinear dynamics can be rewritten as
⎛
⎝
˙
X(t)
L ˙
Y (t)
C ˙
Z(t)
⎞
⎠ =
⎛
⎝
(−G +
1
R )f MC (X(t)) +
Z(t)
R + K 1 ( 0 )
−Z(t) + f M (Y (t)) + K 2 ( 0 )
1
R f MC (X(t)) + Y (t) −
Z(t)
R
⎞
⎠
(11.37)
where Z(t) is in (11.30) and K( 0 ) = (K 1 ( 0 ), K 2 ( 0 )) with
K 1 ( 0 ) = −Gf MC (σ MC 0 ) + q MC 0 + Cv C 0 − q M 0
K 2 ( 0 ) = Li L 0 + f M (q M 0 ) − RCv C 0 + f MC (σ MC 0 ) + Rq M 0 .
The ICs in the (v, i)-domain act as constant inputs in the SEs in the (ϕ, q)domain via the terms K 1 ( 0 ) and K 2 ( 0 ). It turns out that there coexist ∞ 2 different
reduced-order dynamics (one for each manifold M(K 1 ( 0 ), K 2 ( 0 )).
Suppose that the circuit parameters are G = 2.1, R = 0.2083, C = 5.8823, L =
0.136 and let f MC (σ MC ) = 0.7σ MC + 0.5σ 2
MC and f M (q M ) = 0.01q M − 0.05q 2
M
(cf. [24]). Figure 11.8 reports the bifurcation diagram for (11.37), as a function
of K 1 and K 2 /L, numerically obtained by means of the software MatCont [25].
The black (resp., blue) solid curve corresponds to points where MatCont detects a
supercritical (resp., subcritical) Hopf bifurcation. The black dashed line represents
points of the parameter-space where a period-doubling bifurcation occurs. Finally,