6.1 First-Order Memristor Circuits
233
and stays in S thereafter. This easily implies that any solution of (6.13) is bounded
and hence defined for any t ≥ t 0 .
The state space of (6.13) is one dimensional and coincides with the ϕ M -axis.
Equation (6.13) is a first-order autonomous ODE with bounded solutions. Then due
to known properties, any solution necessarily converges to an EP as t → +∞
(Chap. 4).
Remark 6.8 We have given a mathematical proof that any solution of (6.13) is
bounded and convergent toward an EP. It is instructive to notice that the same
properties can be immediately verified from a geometric viewpoint by using the
dynamic route (cf. Fig. 6.10).
Summing up, the global dynamics in the (ϕ, q)-domain can be described as
follows.
For any |Q 0 | < Q sn
0 , the SE (6.13) has two asymptotically stable EPs ¯
ϕ α
M and
¯
ϕ
γ
M . If the initial condition ϕ M 0 ∈ (−∞, ¯
ϕ
β
M ), then the corresponding solution
converges to ¯
ϕ α
M . If ϕ M 0 ∈ ( ¯
ϕ
β
M , +∞), then the solution converges to ¯
ϕ
γ
M . The
attraction basin A(·) of an asymptotically stable EP is defined as the set of initial
conditions such that the corresponding solution converges to the considered EP.
Then, we have
A( ¯
ϕ
α
M ) = (−∞, ¯
ϕ
β
M )
while
A( ¯
ϕ
γ
M ) = ( ¯
ϕ
β
M , +∞).
For these values of Q 0 the M − C circuit is in bistable mode.
When Q 0 < −Q sn
0 (resp., Q 0 > Q sn
0 ), the SE (6.13) has only one asymptotically
stable EP ¯
ϕ α
M (resp., ¯
ϕ
γ
M ) attracting all solutions. We have
A( ¯
ϕ
α
M ) = R
when Q 0 < −Q sn
0 , while
A( ¯
ϕ
γ
M ) = R
when Q 0 > Q sn
0 . For all these values of Q 0 the M–C circuit is in monostable mode.
The special case Q 0 = ±Q sn
0 can be dealt with in a similar way.
Note that in any case the location of the EPs of (6.13) changes when Q 0 changes.
Clearly, by varying Q 0 in R we obtain infinitely many different first-order dynamics
for (6.13), some of which are bistable, some monostable.
The concept of foliation in invariant manifolds permits to investigate not only
EPs and stability properties of (6.13), but bifurcation phenomena as well. To this
end, it is convenient to identify two logically different causes leading to bifurcations.
233
and stays in S thereafter. This easily implies that any solution of (6.13) is bounded
and hence defined for any t ≥ t 0 .
The state space of (6.13) is one dimensional and coincides with the ϕ M -axis.
Equation (6.13) is a first-order autonomous ODE with bounded solutions. Then due
to known properties, any solution necessarily converges to an EP as t → +∞
(Chap. 4).
Remark 6.8 We have given a mathematical proof that any solution of (6.13) is
bounded and convergent toward an EP. It is instructive to notice that the same
properties can be immediately verified from a geometric viewpoint by using the
dynamic route (cf. Fig. 6.10).
Summing up, the global dynamics in the (ϕ, q)-domain can be described as
follows.
For any |Q 0 | < Q sn
0 , the SE (6.13) has two asymptotically stable EPs ¯
ϕ α
M and
¯
ϕ
γ
M . If the initial condition ϕ M 0 ∈ (−∞, ¯
ϕ
β
M ), then the corresponding solution
converges to ¯
ϕ α
M . If ϕ M 0 ∈ ( ¯
ϕ
β
M , +∞), then the solution converges to ¯
ϕ
γ
M . The
attraction basin A(·) of an asymptotically stable EP is defined as the set of initial
conditions such that the corresponding solution converges to the considered EP.
Then, we have
A( ¯
ϕ
α
M ) = (−∞, ¯
ϕ
β
M )
while
A( ¯
ϕ
γ
M ) = ( ¯
ϕ
β
M , +∞).
For these values of Q 0 the M − C circuit is in bistable mode.
When Q 0 < −Q sn
0 (resp., Q 0 > Q sn
0 ), the SE (6.13) has only one asymptotically
stable EP ¯
ϕ α
M (resp., ¯
ϕ
γ
M ) attracting all solutions. We have
A( ¯
ϕ
α
M ) = R
when Q 0 < −Q sn
0 , while
A( ¯
ϕ
γ
M ) = R
when Q 0 > Q sn
0 . For all these values of Q 0 the M–C circuit is in monostable mode.
The special case Q 0 = ±Q sn
0 can be dealt with in a similar way.
Note that in any case the location of the EPs of (6.13) changes when Q 0 changes.
Clearly, by varying Q 0 in R we obtain infinitely many different first-order dynamics
for (6.13), some of which are bistable, some monostable.
The concept of foliation in invariant manifolds permits to investigate not only
EPs and stability properties of (6.13), but bifurcation phenomena as well. To this
end, it is convenient to identify two logically different causes leading to bifurcations.