6.2 Second-Order Memristor Oscillators
247
6.2.4 Nonlinear Dynamics and Hopf Bifurcations Without
Parameters
Let us study the dynamics of the M −C −L circuit in the (ϕ, q)-domain. The unique
EP of (6.30) is given by ¯
x = h(Q 0 ) and ¯
y = Q 0 . The Jacobian of the vector field
defining (6.30) at the EP is
J ( ¯
x, ¯
y) =
0 −
1
C
1
L −
h (Q 0 )
L
=
0 −
1
C
1
L
a−3bQ 2
0
L
.
(6.34)
By denoting with
α = trJ ( ¯
x, ¯
y) =
a − 3bQ 2
0
L
Δ = detJ ( ¯
x, ¯
y) =
1
LC
> 0
the eigenvalues of J ( ¯
x, ¯
y) are
λ 1,2 =
−α ±
√
α 2 − 4Δ
2
.
(6.35)
For α = 0, i.e., Q 0 = ±
√
a/3b, the vector field defining (6.30) is in Normal
Form and the linearization at the EP exhibits a center (i.e., a pair of purely imaginary
eigenvalues λ 1,2 = ±j
√
Δ). By evaluating the real and imaginary part of the
eigenvalues λ 1,2 , we can easily conclude the following.
1. If α < 0, i.e.,
|Q 0 | <
a/3b
then the unique EP is unstable (J ( ¯
x, ¯
y) has two eigenvalues with positive real part)
and it is seen that (6.30) presents persistent oscillations. For verification, Fig. 6.20
reports the real and imaginary parts of the eigenvalues λ 1,2 in (6.35) as a function
of Q 0 when the circuit parameters are C = 9/2, L = 1, a = 1, and b = 1/3.
2. If α > 0, i.e.,
|Q 0 | >
a/3b
then the unique EP is asymptotically stable (J ( ¯
x, ¯
y) has two eigenvalues with
negative real part—see Fig. 6.20). It is seen that (6.30) exhibits no oscillatory
behavior and the only EP is also globally attracting for the trajectories.
247
6.2.4 Nonlinear Dynamics and Hopf Bifurcations Without
Parameters
Let us study the dynamics of the M −C −L circuit in the (ϕ, q)-domain. The unique
EP of (6.30) is given by ¯
x = h(Q 0 ) and ¯
y = Q 0 . The Jacobian of the vector field
defining (6.30) at the EP is
J ( ¯
x, ¯
y) =
0 −
1
C
1
L −
h (Q 0 )
L
=
0 −
1
C
1
L
a−3bQ 2
0
L
.
(6.34)
By denoting with
α = trJ ( ¯
x, ¯
y) =
a − 3bQ 2
0
L
Δ = detJ ( ¯
x, ¯
y) =
1
LC
> 0
the eigenvalues of J ( ¯
x, ¯
y) are
λ 1,2 =
−α ±
√
α 2 − 4Δ
2
.
(6.35)
For α = 0, i.e., Q 0 = ±
√
a/3b, the vector field defining (6.30) is in Normal
Form and the linearization at the EP exhibits a center (i.e., a pair of purely imaginary
eigenvalues λ 1,2 = ±j
√
Δ). By evaluating the real and imaginary part of the
eigenvalues λ 1,2 , we can easily conclude the following.
1. If α < 0, i.e.,
|Q 0 | <
a/3b
then the unique EP is unstable (J ( ¯
x, ¯
y) has two eigenvalues with positive real part)
and it is seen that (6.30) presents persistent oscillations. For verification, Fig. 6.20
reports the real and imaginary parts of the eigenvalues λ 1,2 in (6.35) as a function
of Q 0 when the circuit parameters are C = 9/2, L = 1, a = 1, and b = 1/3.
2. If α > 0, i.e.,
|Q 0 | >
a/3b
then the unique EP is asymptotically stable (J ( ¯
x, ¯
y) has two eigenvalues with
negative real part—see Fig. 6.20). It is seen that (6.30) exhibits no oscillatory
behavior and the only EP is also globally attracting for the trajectories.