6.2 Second-Order Memristor Oscillators
239
study and permits to obtain qualitative and analytical results that cannot be easily
derived from a direct analysis of the second-order SEs in the (v, i)-domain.
6.2 Second-Order Memristor Oscillators
In Sect. 6.1.2, we have considered a simple autonomous M − C circuit where the
memristor is flux-controlled. We have been able to write an SE describing the
dynamics and, on this basis, a number of salient features have been studied. In
particular, the dynamics in the (ϕ, q)-domain has been proved to be first-order, so
that any solution is convergent toward an EP and the circuit is either mono-stable or
bistable for the considered cubic nonlinearity.
In this section, we consider a variant of the M − C circuit studied in Sect. 6.1.2,
where we assume that the memristor is charge-controlled but not flux-controlled.
This apparently simple modification leads to substantial structural changes in the
dynamics. Indeed, we first observe that in the charge-controlled case we are unable
to write an SE description and the circuit displays impasse points (cf. Chap. 4),
i.e., points where the solution cannot be continued either forward or backward
in time. We then show that we can break impasse points by considering a more
realistic circuit including a parasitic element. This also shows that the circuit is
intrinsically second order and is able to display relaxation oscillation. Also for
this circuit we study invariant manifolds in the (v, i)-domain, coexisting dynamics
and bifurcations without parameters. In particular, we show that in this case there
coexist oscillatory and convergent dynamics; moreover, the circuit can display Hopf
bifurcations without parameters.
6.2.1 M − C Circuit with Impasse Points
Consider for t ≥ t 0 , where −∞ < t 0 < ∞, a simple memristor-based circuit
in the class LM (see the M–C circuit in Fig. 6.14), composed of one memristor
M connected to a capacitor C. Suppose the memristor is charge-controlled and is
defined by a smooth non-monotone function
ϕ M (t) = h(q M (t)) = −aq M (t) + bq
3
M (t)
(6.22)
with a, b > 0. Note that the memristor is not flux-controlled and is (locally) active
since the memristance R(q M (t)) = h (q M (t)) < 0 for |q M | <
√
a/3b.
Let v C (t 0 ) = v C 0 , q M (t 0 ) = q M 0 be the initial conditions at t 0 for the
state variables in the (v, i)-domain. It follows that q C (t 0 ) = q C 0 = Cv C 0 . The
corresponding circuit in the (ϕ, q)-domain obtained via FCAM is represented in
Fig. 6.15.
Analysis by inspection permits to write the following KϕL, KqL, and CRs
239
study and permits to obtain qualitative and analytical results that cannot be easily
derived from a direct analysis of the second-order SEs in the (v, i)-domain.
6.2 Second-Order Memristor Oscillators
In Sect. 6.1.2, we have considered a simple autonomous M − C circuit where the
memristor is flux-controlled. We have been able to write an SE describing the
dynamics and, on this basis, a number of salient features have been studied. In
particular, the dynamics in the (ϕ, q)-domain has been proved to be first-order, so
that any solution is convergent toward an EP and the circuit is either mono-stable or
bistable for the considered cubic nonlinearity.
In this section, we consider a variant of the M − C circuit studied in Sect. 6.1.2,
where we assume that the memristor is charge-controlled but not flux-controlled.
This apparently simple modification leads to substantial structural changes in the
dynamics. Indeed, we first observe that in the charge-controlled case we are unable
to write an SE description and the circuit displays impasse points (cf. Chap. 4),
i.e., points where the solution cannot be continued either forward or backward
in time. We then show that we can break impasse points by considering a more
realistic circuit including a parasitic element. This also shows that the circuit is
intrinsically second order and is able to display relaxation oscillation. Also for
this circuit we study invariant manifolds in the (v, i)-domain, coexisting dynamics
and bifurcations without parameters. In particular, we show that in this case there
coexist oscillatory and convergent dynamics; moreover, the circuit can display Hopf
bifurcations without parameters.
6.2.1 M − C Circuit with Impasse Points
Consider for t ≥ t 0 , where −∞ < t 0 < ∞, a simple memristor-based circuit
in the class LM (see the M–C circuit in Fig. 6.14), composed of one memristor
M connected to a capacitor C. Suppose the memristor is charge-controlled and is
defined by a smooth non-monotone function
ϕ M (t) = h(q M (t)) = −aq M (t) + bq
3
M (t)
(6.22)
with a, b > 0. Note that the memristor is not flux-controlled and is (locally) active
since the memristance R(q M (t)) = h (q M (t)) < 0 for |q M | <
√
a/3b.
Let v C (t 0 ) = v C 0 , q M (t 0 ) = q M 0 be the initial conditions at t 0 for the
state variables in the (v, i)-domain. It follows that q C (t 0 ) = q C 0 = Cv C 0 . The
corresponding circuit in the (ϕ, q)-domain obtained via FCAM is represented in
Fig. 6.15.
Analysis by inspection permits to write the following KϕL, KqL, and CRs