11.7 Examples Concerning Class LME
427
˙
σ MC (t; t 0 ) = q MC (t; t 0 ) + q MC 0
CR of L ϕ :
˙
q L (t; t 0 ) = f L (ϕ L (t; t 0 ) + ϕ L 0 )
CR of M ϕ :
ϕ M (t; t 0 ) = f M (q M (t; t 0 ) + q M 0 ) − f M (q M 0 ).
By substitution we obtain the second-order SE
˙
q L (t; t 0 ) = f L
f M (q L (t; t 0 ) + q M 0 ) − f M (q M 0 )
−f MC (σ MC (t; t 0 ) + σ MC 0 ) + f MC (σ MC 0 ) + ϕ L 0
˙
σ MC (t; t 0 ) = −q L (t; t 0 ) + q MC 0
in the state variables q L (t; t 0 ), σ MC (t; t 0 ).
To study the invariant manifolds for the dynamics in the (v, i)-domain, note that
from KCL i M (t) + i MC (t) = 0 we obtain
d
dt
(q M (t) + q MC (t)) = 0.
Moreover, KVL v M (t) = v L (t) + v MC (t) yields
d
dt
ϕ M (t) =
d
dt
ϕ L (t) + f
MC (σ MC (t))q MC (t)
and hence
d
dt
(f M (q M (t)) =
d
dt
ϕ L (t) +
d
dt
(f MC (σ MC (t))).
Then, we can identify the two invariants of motion
q M (t) + q MC (t)
and
f M (q M (t)) − ϕ L (t) − f MC (σ MC (t))
which in turn permit to define ∞ 2 two-dimensional invariant manifolds for the
dynamics in the (v, i)-domain
M(Q 0 , Φ 0 ) = {(ϕ L (t), q MC (t), σ MC (t), q M (t))
T
∈ R
4
: