11.2 Motivating Example
393
Φ(t)
.
= f MC (σ MC (t)) − ϕ M (t) = Φ 0
for any t ≥ t 0 , where
Φ 0 = Φ 0 (σ MC 0 )
.
= f MC (σ MC 0 ) − ϕ M 0 .
Using these notations, (11.1) can be conveniently rewritten as
˙
σ MC (t) = −f M
f MC (σ MC (t)) − Φ 0
+ Q 0 + q a (t; t 0 )
(11.2)
for t ≥ t 0 , with IC σ MC (t 0 ) = σ MC 0 .
11.2.2 Invariant Manifolds, Coexisting Dynamics, and
Bifurcations Without Parameters
Consider first the autonomous MC − M circuit where a(t) = 0, hence q a (t; t 0 ) =
0, t ≥ t 0 . Then, the previous expressions of Q(t) and Φ(t) simplify as follows (for
t ≥ t 0 ):
Q(t) = q MC (t) + f M (ϕ M (t)) = Q 0
Φ(t) = f MC (σ MC (t)) − ϕ M (t) = Φ 0
i.e., Q(t) and Φ(t), which depend only on the state variables in the (v, i)-domain,
are invariants of motion for the dynamics. Then, the three-dimensional state space
(σ MC (t), q MC (t), ϕ M (t)) ∈ R 3 can be foliated in ∞ 2 1D manifolds (each manifold
is a curve in R 3 ), which are (positively) invariant for the dynamics in the (v, i)domain, given by
M(Q 0 , Φ 0 ) = {(σ MC (t), q MC (t), ϕ M (t))
τ
∈ R
3
: q MC (t) + f M (ϕ M (t)) = Q 0 ,
f MC (σ MC (t)) − ϕ M (t) = Φ 0 }
(11.3)
where Q 0 , Φ 0 ∈ R.
On each manifold the MC − M circuit has a first-order dynamics described by
the SE
˙
σ MC (t) = −f M
f MC (σ MC (t)) − Φ 0
+ Q 0 .
(11.4)
From (11.4) it is clear that the nonlinear dynamics (number of equilibrium points
(EPs) and their stability properties) is strongly dependent upon the terms Φ 0 and
Q 0 , which in turn depend upon the ICs in the (v, i)-domain. Hence, in the (v, i)domain there coexist ∞ 2 different first-order dynamics, one for each manifold M
specified by Q 0 and Φ 0 .