6.2 Second-Order Memristor Oscillators
241
with ϕ M (t 0 ; t 0 ) = 0. This can be put in the simplified form
d x(t)
dt
= −
1
C
y(t) +
1
C
Y 0
(6.25a)
x(t) = h(y(t)) = −ay(t) + by
3 (t)
(6.25b)
with x(t 0 ) = X 0 , by letting x(t) = ϕ M (t; t 0 )+h(q M 0 ) = ϕ M (t), y(t) = q M (t; t 0 )+
q M 0 = q M (t), X 0 = h(q M 0 ) and
Y 0 = q M 0 + q C 0 = q M 0 + Cv C 0 .
(6.26)
Note that Y 0 is a term depending on the initial conditions v C 0 and q M 0 for the state
variables in the (v, i)-domain.
It turns out that according to (6.25) the dynamics of the M–C circuit in Fig. 6.14
evolve, starting from any initial condition X 0 , onto the constraint x = h(y). For any
Y 0 there is only one EP
P = ( ¯
x, ¯
y) = (h(Y 0 ), Y 0 ).
(6.27)
The global dynamics, and the stability of the EP, can be easily assessed from the
following rule (see (6.25a)):
d x(t)
dt
> 0
y < Y 0
d x(t)
dt
< 0
y > Y 0 .
That is, for all t > t 0 , the solution of the M–C circuit in Fig. 6.14 must follow the
curve along the direction indicated by the arrowheads in the dynamic route reported
in Figs. 6.16 and 6.17 for the cases |Y 0 | <
√
a/3b and |Y 0 | >
√
a/3b, respectively
(Fig. 6.17 reports only the case Y 0 >
√
a/3b, but dual considerations hold for Y 0 <
−
√
a/3b). It is apparent in Figs. 6.16 and 6.17 that x(t) is increasing in the solid
part of the dynamic route (i.e., when y < Y 0 ) but decreasing in the dashed part.
The analysis of the dynamic route allows us to draw the following results:
1. if the initial conditions v C 0 , q M 0 for the state variables in the (v, i)-domain are
such that
|Y 0 | = |q M 0 + q C 0 | <
a/3b
then the unique EP P is unstable (Fig. 6.16). Moreover, there are two forward
impasse points at P 1 =
√
a/3b(−(2/3)a, 1) and P 2 = −P 1 . Indeed, if we
consider a solution starting at ( ˆ
X, ˆ
Y ) with ˆ
Y ∈ (Y 0 ,
√
a/3b) ∪ (
√
a/3b, ∞)
and ˆ
X = h( ˆ
Y ), it can be checked that the solution reaches P 1 in finite time ˆ
t.
However, P 1 is not an EP and there is no way to prolong the solution for t > ˆ
t
241
with ϕ M (t 0 ; t 0 ) = 0. This can be put in the simplified form
d x(t)
dt
= −
1
C
y(t) +
1
C
Y 0
(6.25a)
x(t) = h(y(t)) = −ay(t) + by
3 (t)
(6.25b)
with x(t 0 ) = X 0 , by letting x(t) = ϕ M (t; t 0 )+h(q M 0 ) = ϕ M (t), y(t) = q M (t; t 0 )+
q M 0 = q M (t), X 0 = h(q M 0 ) and
Y 0 = q M 0 + q C 0 = q M 0 + Cv C 0 .
(6.26)
Note that Y 0 is a term depending on the initial conditions v C 0 and q M 0 for the state
variables in the (v, i)-domain.
It turns out that according to (6.25) the dynamics of the M–C circuit in Fig. 6.14
evolve, starting from any initial condition X 0 , onto the constraint x = h(y). For any
Y 0 there is only one EP
P = ( ¯
x, ¯
y) = (h(Y 0 ), Y 0 ).
(6.27)
The global dynamics, and the stability of the EP, can be easily assessed from the
following rule (see (6.25a)):
d x(t)
dt
> 0
y < Y 0
d x(t)
dt
< 0
y > Y 0 .
That is, for all t > t 0 , the solution of the M–C circuit in Fig. 6.14 must follow the
curve along the direction indicated by the arrowheads in the dynamic route reported
in Figs. 6.16 and 6.17 for the cases |Y 0 | <
√
a/3b and |Y 0 | >
√
a/3b, respectively
(Fig. 6.17 reports only the case Y 0 >
√
a/3b, but dual considerations hold for Y 0 <
−
√
a/3b). It is apparent in Figs. 6.16 and 6.17 that x(t) is increasing in the solid
part of the dynamic route (i.e., when y < Y 0 ) but decreasing in the dashed part.
The analysis of the dynamic route allows us to draw the following results:
1. if the initial conditions v C 0 , q M 0 for the state variables in the (v, i)-domain are
such that
|Y 0 | = |q M 0 + q C 0 | <
a/3b
then the unique EP P is unstable (Fig. 6.16). Moreover, there are two forward
impasse points at P 1 =
√
a/3b(−(2/3)a, 1) and P 2 = −P 1 . Indeed, if we
consider a solution starting at ( ˆ
X, ˆ
Y ) with ˆ
Y ∈ (Y 0 ,
√
a/3b) ∪ (
√
a/3b, ∞)
and ˆ
X = h( ˆ
Y ), it can be checked that the solution reaches P 1 in finite time ˆ
t.
However, P 1 is not an EP and there is no way to prolong the solution for t > ˆ
t