11.6 Chaotic Circuit with Memcapacitor and Memristor
411
The following remarks follow from Corollary 11.1 and extend the results in
Chap. 7 to mem-circuits in LME:
• if a mem-circuit has no input source, then any manifold (11.28) is positively
invariant for the nonlinear dynamics in the (v, i)-domain;
• being K( 0 ) dependent on the vector 0 of ICs in the (v, i)-domain (see (11.27)),
then a change in ICs results in general in a different invariant manifold on which
the nonlinear dynamics take place. Two different invariant manifolds M(K(
0 ))
and M(K(
0 )) are characterized by two different vectors K(
0 ) and K(
0 )
associated with the ICs
0 and
0 . Therefore, by (11.30), on each of the
∞ n different invariant manifolds the mem-circuit exhibits a different nonlinear
dynamics. This explains theoretically the coexistence of infinitely many different
reduced-order nonlinear dynamics. Moreover, in addition to standard bifurcations
due to circuit parameter changes, also bifurcations due to changes in 0 for a
fixed set of circuit parameters, i.e., bifurcations without parameters, can occur.
Nonlinear dynamics can be also tuned by acting on 0 via suitable sequences of
finite time duration voltage or current pulses, as illustrated by means of a specific
example in the next section.
11.6 Chaotic Circuit with Memcapacitor and Memristor
Sustained oscillations and complex dynamics are observed in several mem-circuits
belonging to LME (see [20–22, 24] and references therein). In order to illustrate
the application of the methodology in this chapter, let us consider a variant of the
mem-circuit with MC σ and a M ϕ studied in [24], where a current source a(t) and a
voltage source e(t) are also introduced for programming purposes (see Fig. 11.7).
The hybrid representation of N R is obtained via (11.6) with
C
MCσ
qa
−G
R
ϕe
L
Mρ
N R
+
−
ϕ C
q MC
q L
Fig. 11.7 Chaotic circuit with memcapacitor and memristor
411
The following remarks follow from Corollary 11.1 and extend the results in
Chap. 7 to mem-circuits in LME:
• if a mem-circuit has no input source, then any manifold (11.28) is positively
invariant for the nonlinear dynamics in the (v, i)-domain;
• being K( 0 ) dependent on the vector 0 of ICs in the (v, i)-domain (see (11.27)),
then a change in ICs results in general in a different invariant manifold on which
the nonlinear dynamics take place. Two different invariant manifolds M(K(
0 ))
and M(K(
0 )) are characterized by two different vectors K(
0 ) and K(
0 )
associated with the ICs
0 and
0 . Therefore, by (11.30), on each of the
∞ n different invariant manifolds the mem-circuit exhibits a different nonlinear
dynamics. This explains theoretically the coexistence of infinitely many different
reduced-order nonlinear dynamics. Moreover, in addition to standard bifurcations
due to circuit parameter changes, also bifurcations due to changes in 0 for a
fixed set of circuit parameters, i.e., bifurcations without parameters, can occur.
Nonlinear dynamics can be also tuned by acting on 0 via suitable sequences of
finite time duration voltage or current pulses, as illustrated by means of a specific
example in the next section.
11.6 Chaotic Circuit with Memcapacitor and Memristor
Sustained oscillations and complex dynamics are observed in several mem-circuits
belonging to LME (see [20–22, 24] and references therein). In order to illustrate
the application of the methodology in this chapter, let us consider a variant of the
mem-circuit with MC σ and a M ϕ studied in [24], where a current source a(t) and a
voltage source e(t) are also introduced for programming purposes (see Fig. 11.7).
The hybrid representation of N R is obtained via (11.6) with
C
MCσ
qa
−G
R
ϕe
L
Mρ
N R
+
−
ϕ C
q MC
q L
Fig. 11.7 Chaotic circuit with memcapacitor and memristor