8.2 Memristor-Based Chaotic Circuit (MCC)
325
y(t) = ϕ C 2 (t; t 0 ) + ϕ L (t 0 )
(8.6b)
z(t) = −Rq L (t; t 0 ) + ϕ L (t 0 ) − ϕ M (t 0 ) + Rq C 2 (t 0 ).
(8.6c)
The following SEs in adimensional form for t ≥ t 0 are obtained
dx(t)
dt
= α(−x(t) + y(t) − n(x(t))) + X 0
(8.7a)
dy(t)
dt
= x(t) − y(t) + z(t)
(8.7b)
dz(t)
dt
= −βy(t)
(8.7c)
where we let n(x(t)) = Rf (x(t)) and
X 0 = αRQ(w(t 0 )) = α(n(ϕ M (t 0 )) + ϕ M (t 0 ) + RC 1 v C 1 (t 0 ) − Li L (t 0 ))
(8.8)
which is a term depending on the ICs for the state variables in the (v, i)-domain.
The ICs are x(t 0 ) = ϕ M (t 0 ), y(t 0 ) = ϕ L (t 0 ) = Li L (t 0 ), z(t 0 ) = ϕ L (t 0 ) − ϕ M (t 0 ) +
Rq C 2 (t 0 ) = Li L (t 0 ) − ϕ M (t 0 ) + RC 2 v C 2 (t 0 ).
We stress that, when the ICs v C 1 (t 0 ), i L (t 0 ) and ϕ M (t 0 ) are such that X 0 = 0,
i.e., Q(w(t 0 )) = 0, and the nonlinear dynamics of the MCC is on the invariant
zero-manifold M(0), the SEs (8.7) formally coincide with those of the classical
Chua’s oscillator (Chap. 4). Therefore, the MCC can undergo standard bifurcations
on the fixed manifold M(0) by varying the circuit parameters α and β. On the other
hand, due to the term X 0 at the right-hand side of (8.7), MCC exhibits a rich variety
of dynamic behaviors and different coexisting attractors. The dynamic behavior of
a nonlinear system (number and stability of EPs and limit cycles, etc.) is indeed
known to be heavily dependent also on constant terms in the vector field [33, 34]. We
can then envisage an alternative mechanism to induce bifurcations, i.e., bifurcations
due to the change of ICs and X 0 for fixed circuit parameters (bifurcations without
parameters).
Bifurcations in the MCC have been investigated in Sect. 6.3 of Chap. 6, where,
in particular, the memristor characteristic is chosen as
f (ϕ M ) = −
8
7
ϕ M +
4
63
ϕ
3
M
(8.9)
which results to be a good smooth approximation of the nonlinear Chua’s diode
characteristic [35]. For the sake of completeness, and to make the chapter more selfconsistent, next we briefly summarize the Hopf and period-doubling bifurcations
without parameters leading to a scenario where complex dynamics depending on
the choice of ICs and invariant manifold are observed. Some selected numerical
simulations of (8.7) by varying Q 0 (and, hence, X 0 ), for suitable fixed sets of circuit
parameters (L, C 1 , C 2 , R), are reported.
325
y(t) = ϕ C 2 (t; t 0 ) + ϕ L (t 0 )
(8.6b)
z(t) = −Rq L (t; t 0 ) + ϕ L (t 0 ) − ϕ M (t 0 ) + Rq C 2 (t 0 ).
(8.6c)
The following SEs in adimensional form for t ≥ t 0 are obtained
dx(t)
dt
= α(−x(t) + y(t) − n(x(t))) + X 0
(8.7a)
dy(t)
dt
= x(t) − y(t) + z(t)
(8.7b)
dz(t)
dt
= −βy(t)
(8.7c)
where we let n(x(t)) = Rf (x(t)) and
X 0 = αRQ(w(t 0 )) = α(n(ϕ M (t 0 )) + ϕ M (t 0 ) + RC 1 v C 1 (t 0 ) − Li L (t 0 ))
(8.8)
which is a term depending on the ICs for the state variables in the (v, i)-domain.
The ICs are x(t 0 ) = ϕ M (t 0 ), y(t 0 ) = ϕ L (t 0 ) = Li L (t 0 ), z(t 0 ) = ϕ L (t 0 ) − ϕ M (t 0 ) +
Rq C 2 (t 0 ) = Li L (t 0 ) − ϕ M (t 0 ) + RC 2 v C 2 (t 0 ).
We stress that, when the ICs v C 1 (t 0 ), i L (t 0 ) and ϕ M (t 0 ) are such that X 0 = 0,
i.e., Q(w(t 0 )) = 0, and the nonlinear dynamics of the MCC is on the invariant
zero-manifold M(0), the SEs (8.7) formally coincide with those of the classical
Chua’s oscillator (Chap. 4). Therefore, the MCC can undergo standard bifurcations
on the fixed manifold M(0) by varying the circuit parameters α and β. On the other
hand, due to the term X 0 at the right-hand side of (8.7), MCC exhibits a rich variety
of dynamic behaviors and different coexisting attractors. The dynamic behavior of
a nonlinear system (number and stability of EPs and limit cycles, etc.) is indeed
known to be heavily dependent also on constant terms in the vector field [33, 34]. We
can then envisage an alternative mechanism to induce bifurcations, i.e., bifurcations
due to the change of ICs and X 0 for fixed circuit parameters (bifurcations without
parameters).
Bifurcations in the MCC have been investigated in Sect. 6.3 of Chap. 6, where,
in particular, the memristor characteristic is chosen as
f (ϕ M ) = −
8
7
ϕ M +
4
63
ϕ
3
M
(8.9)
which results to be a good smooth approximation of the nonlinear Chua’s diode
characteristic [35]. For the sake of completeness, and to make the chapter more selfconsistent, next we briefly summarize the Hopf and period-doubling bifurcations
without parameters leading to a scenario where complex dynamics depending on
the choice of ICs and invariant manifold are observed. Some selected numerical
simulations of (8.7) by varying Q 0 (and, hence, X 0 ), for suitable fixed sets of circuit
parameters (L, C 1 , C 2 , R), are reported.