6.1 First-Order Memristor Circuits
235
Remark 6.10 The bifurcations due to changing the initial conditions for the state
variables in the (v, i)-domain, for a fixed set of circuit parameters, are a new
dynamic phenomenon peculiar to memristor circuits and are strictly related to
the property of foliation of the state space in the (v, i)-domain. In the following,
we will refer to them as bifurcations without (changing circuit) parameters,
or, simply, bifurcations without parameters, to differentiate them from standard
bifurcations due to changing the circuit parameters. By analogy, we will use the term
bifurcations without parameters also when studying bifurcations due to changing
initial conditions in higher-order memristor circuits in the next sections.
Remark 6.11 It is worth to remark that the mathematical concept of bifurcations
without parameters has been originally introduced in [6]. We refer the reader to
Appendix 1 for a discussion on the link between the bifurcations studied for the
M − C circuit and bifurcations without parameters introduced in [6].
Let us now study the dynamics of the M − C circuit in the (v, i)-domain. We
have the following.
Property 6.3 Consider the M − C circuit with the cubic memristor nonlinearity (6.15). Any solution of the SEs (6.7) in the (v, i)-domain is bounded and hence
defined for t ≥ t 0 , moreover it converges to an EP as t → +∞.
Proof It is quite simple to verify this result in mathematical terms. In fact, we have
seen that any solution ϕ M (t) of the reduced system (6.13) is bounded (for t ≥
t 0 ), hence also v C (t) = dϕ M (t)/dt = −f (ϕ M (t)) + Q 0 is bounded, implying
that this is true also of any solution (v C (t), ϕ M (t)) of (6.7). Moreover, the reduced
system (6.13) is convergent, i.e., we have ϕ M (t) → ¯
ϕ M and dϕ M (t)/dt → 0 as
t → +∞, where ¯
ϕ M is an EP of (6.13). Since dϕ M (t)/dt = v C (t), it follows that
the corresponding solution of the SE in the (v, i)-domain tends to the EP (0, ¯
ϕ M ) as
t → +∞.
The EPs of the SEs (6.7) in the (v, i)-domain are given by ¯
v C = 0 and ¯
ϕ M ∈ R,
i.e., there is a manifold (a continuum of EPs) given by the ϕ M -axis. Note that at an
EP the memristor flux is an arbitrary constant while the capacitor voltage vanishes.
As remarked before, the state space (v C , ϕ M ) in the (v, i)-domain can be foliated
in ∞ 1 1D invariant manifolds where the reduced dynamics is first order. Moreover,
on any fixed manifold, the M − C circuit always has isolated EPs. If |Q 0 | < Q sn
0 ,
the EPs on a given manifold M(Q 0 ) of (6.7) are (0, ¯
ϕ α
M ), (0, ¯
ϕ
β
M ) and (0, ¯
ϕ
γ
M ).
Hence, due to Property 6.3, in the (v C , ϕ M ) state-space any trajectory leaving from
(v C 0 , ϕ M 0 ) = (0, ¯
ϕ
β
M ) lies on the manifold M(Q 0 ) and converges toward one of
the stable EPs, i.e., toward either (0, ¯
ϕ α
M ) or (0, ¯
ϕ
γ
M ). Such analytical results are
confirmed by numerical simulations (see Fig. 6.11) of the SE (6.7) in the (v, i)domain with circuit parameters a = 1, b =
1
3 in (6.15) and C = 1. The dynamics
for other values of Q 0 can be analyzed similarly.
Remark 6.12 We stress that although the M − C circuit is described by a
second-order SE in the (v, i)-domain, it never oscillates. Actually, any solution
235
Remark 6.10 The bifurcations due to changing the initial conditions for the state
variables in the (v, i)-domain, for a fixed set of circuit parameters, are a new
dynamic phenomenon peculiar to memristor circuits and are strictly related to
the property of foliation of the state space in the (v, i)-domain. In the following,
we will refer to them as bifurcations without (changing circuit) parameters,
or, simply, bifurcations without parameters, to differentiate them from standard
bifurcations due to changing the circuit parameters. By analogy, we will use the term
bifurcations without parameters also when studying bifurcations due to changing
initial conditions in higher-order memristor circuits in the next sections.
Remark 6.11 It is worth to remark that the mathematical concept of bifurcations
without parameters has been originally introduced in [6]. We refer the reader to
Appendix 1 for a discussion on the link between the bifurcations studied for the
M − C circuit and bifurcations without parameters introduced in [6].
Let us now study the dynamics of the M − C circuit in the (v, i)-domain. We
have the following.
Property 6.3 Consider the M − C circuit with the cubic memristor nonlinearity (6.15). Any solution of the SEs (6.7) in the (v, i)-domain is bounded and hence
defined for t ≥ t 0 , moreover it converges to an EP as t → +∞.
Proof It is quite simple to verify this result in mathematical terms. In fact, we have
seen that any solution ϕ M (t) of the reduced system (6.13) is bounded (for t ≥
t 0 ), hence also v C (t) = dϕ M (t)/dt = −f (ϕ M (t)) + Q 0 is bounded, implying
that this is true also of any solution (v C (t), ϕ M (t)) of (6.7). Moreover, the reduced
system (6.13) is convergent, i.e., we have ϕ M (t) → ¯
ϕ M and dϕ M (t)/dt → 0 as
t → +∞, where ¯
ϕ M is an EP of (6.13). Since dϕ M (t)/dt = v C (t), it follows that
the corresponding solution of the SE in the (v, i)-domain tends to the EP (0, ¯
ϕ M ) as
t → +∞.
The EPs of the SEs (6.7) in the (v, i)-domain are given by ¯
v C = 0 and ¯
ϕ M ∈ R,
i.e., there is a manifold (a continuum of EPs) given by the ϕ M -axis. Note that at an
EP the memristor flux is an arbitrary constant while the capacitor voltage vanishes.
As remarked before, the state space (v C , ϕ M ) in the (v, i)-domain can be foliated
in ∞ 1 1D invariant manifolds where the reduced dynamics is first order. Moreover,
on any fixed manifold, the M − C circuit always has isolated EPs. If |Q 0 | < Q sn
0 ,
the EPs on a given manifold M(Q 0 ) of (6.7) are (0, ¯
ϕ α
M ), (0, ¯
ϕ
β
M ) and (0, ¯
ϕ
γ
M ).
Hence, due to Property 6.3, in the (v C , ϕ M ) state-space any trajectory leaving from
(v C 0 , ϕ M 0 ) = (0, ¯
ϕ
β
M ) lies on the manifold M(Q 0 ) and converges toward one of
the stable EPs, i.e., toward either (0, ¯
ϕ α
M ) or (0, ¯
ϕ
γ
M ). Such analytical results are
confirmed by numerical simulations (see Fig. 6.11) of the SE (6.7) in the (v, i)domain with circuit parameters a = 1, b =
1
3 in (6.15) and C = 1. The dynamics
for other values of Q 0 can be analyzed similarly.
Remark 6.12 We stress that although the M − C circuit is described by a
second-order SE in the (v, i)-domain, it never oscillates. Actually, any solution