258
6 Memristor Circuits: Invariant Manifolds, Coexisting Attractors, Extreme. . .
The EPs of (6.48) are in the form P = ( ¯
x, 0, − ¯
x) where ¯
x is the solution of the
algebraic equation
¯
x + n( ¯
x) = m 1 ¯
x
3
− (m 0 − 1) ¯
x = X 0 .
(6.50)
The graphical intersection between constant X 0 and curve m 1 ¯
x 3 −(m 0 −1) ¯
x permits
to derive that:
• there exist three EPs iff
|X 0 | <
2
3
(m 0 − 1)
m 0 − 1
3m 1
(6.51)
• there exist two EPs iff
|X 0 | =
2
3
(m 0 − 1)
m 0 − 1
3m 1
(6.52)
• there exists only one EP iff
|X 0 | >
2
3
(m 0 − 1)
m 0 − 1
3m 1
.
(6.53)
If X 0 = 0, the dynamics evolve on manifold M(0) and is the same as that
of Chua’s oscillator. In particular, if β = 15, m 0 = 8/7, and m 1 = 4/63, then
the EPs P + = (3/2, 0, −3/2), P − = P + are asymptotically stable if α < 7 (a
Hopf bifurcation occurs at α = 7), whereas P 0 = (0, 0, 0) is always an unstable
saddle point (see for instance the analysis in [15]). On the other hand, changing
X 0 implies that the EPs change their location in the phase-space, as well as their
stability properties, with respect to Chua’s oscillator.
Numerical simulations confirm the analysis described above and provide insights
into the complex behavior in the MCC of Fig. 6.26. Hereinafter the following values
are assumed to carry out the numerical study: β = 15, m 0 = 8/7, and m 1 = 4/63.
The circuit parameter α and the constant X 0 depending on the initials conditions are
varied to make clear bifurcation phenomena. Two main cases are identified:
• bifurcations on a fixed manifold: the initial conditions (v C 1 0 , v C 2 0 , i L 0 , ϕ M 0 ) are
such that X 0 in (6.49) is fixed; qualitative changes in the global phase portrait and
bifurcations of the SEs (6.48) occur due to changes of the circuit parameter α;
• bifurcations without parameters: the circuit parameter α is kept constant,
whereas the initial conditions (v C 1 0 , v C 2 0 , i L 0 , ϕ M 0 ) are changed in such a way
that X 0 varies and the variation of X 0 in turn gives rise to qualitative changes in
the global phase portrait and bifurcations of the SEs (6.48).
6 Memristor Circuits: Invariant Manifolds, Coexisting Attractors, Extreme. . .
The EPs of (6.48) are in the form P = ( ¯
x, 0, − ¯
x) where ¯
x is the solution of the
algebraic equation
¯
x + n( ¯
x) = m 1 ¯
x
3
− (m 0 − 1) ¯
x = X 0 .
(6.50)
The graphical intersection between constant X 0 and curve m 1 ¯
x 3 −(m 0 −1) ¯
x permits
to derive that:
• there exist three EPs iff
|X 0 | <
2
3
(m 0 − 1)
m 0 − 1
3m 1
(6.51)
• there exist two EPs iff
|X 0 | =
2
3
(m 0 − 1)
m 0 − 1
3m 1
(6.52)
• there exists only one EP iff
|X 0 | >
2
3
(m 0 − 1)
m 0 − 1
3m 1
.
(6.53)
If X 0 = 0, the dynamics evolve on manifold M(0) and is the same as that
of Chua’s oscillator. In particular, if β = 15, m 0 = 8/7, and m 1 = 4/63, then
the EPs P + = (3/2, 0, −3/2), P − = P + are asymptotically stable if α < 7 (a
Hopf bifurcation occurs at α = 7), whereas P 0 = (0, 0, 0) is always an unstable
saddle point (see for instance the analysis in [15]). On the other hand, changing
X 0 implies that the EPs change their location in the phase-space, as well as their
stability properties, with respect to Chua’s oscillator.
Numerical simulations confirm the analysis described above and provide insights
into the complex behavior in the MCC of Fig. 6.26. Hereinafter the following values
are assumed to carry out the numerical study: β = 15, m 0 = 8/7, and m 1 = 4/63.
The circuit parameter α and the constant X 0 depending on the initials conditions are
varied to make clear bifurcation phenomena. Two main cases are identified:
• bifurcations on a fixed manifold: the initial conditions (v C 1 0 , v C 2 0 , i L 0 , ϕ M 0 ) are
such that X 0 in (6.49) is fixed; qualitative changes in the global phase portrait and
bifurcations of the SEs (6.48) occur due to changes of the circuit parameter α;
• bifurcations without parameters: the circuit parameter α is kept constant,
whereas the initial conditions (v C 1 0 , v C 2 0 , i L 0 , ϕ M 0 ) are changed in such a way
that X 0 varies and the variation of X 0 in turn gives rise to qualitative changes in
the global phase portrait and bifurcations of the SEs (6.48).