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11 Nonlinear Dynamics of Circuits with Mem-Elements
CR of MC σ :
v MC = f
MC (σ MC )q MC
˙
σ MC = q MC
˙
q MC = i MC
CR of L ϕ :
i L = f L (ϕ L )
where
˙
ϕ L = v L
CR of M ϕ :
v M = f
M (q M )i M
and
˙
q M = i M .
By substitution we obtain the fourth-order SE
˙
ϕ L = −f
M (q M )f L (ϕ L ) − f
MC (σ MC )q MC
˙
q MC = f L (ϕ L )
˙
σ MC = q MC
˙
q M = −f L (ϕ L )
in the state variables ϕ L , q MC , σ MC , q M .
In the (ϕ, q)-domain we have
KϕL:
ϕ M (t; t 0 ) = ϕ L (t; t 0 ) + ϕ MC (t; t 0 )
KqL:
q M (t; t 0 ) + q L (t; t 0 ) = 0; q L (t; t 0 ) = q MC (t; t 0 )
CR of MC σ :
ϕ MC (t; t 0 ) = f MC (σ MC (t; t 0 ) + σ MC 0 ) − f MC (σ MC 0 )