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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 .
By substitution, these yield the third-order SE in the (v, i)-domain
˙
ϕ L = f
MC (σ MC )q MC − e
˙
σ MC = q MC
˙
q MC = −f L (ϕ L ) + a
in the state variables ϕ L , σ MC and q MC .
In the (ϕ, q)-domain we can write:
KϕL:
ϕ MC (t; t 0 ) = ϕ L (t; t 0 ) + ϕ e (t; t 0 )
KqL:
q a (t; t 0 ) = q MC (t; t 0 ) + q L (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 )
˙
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 ).
By substitution, we obtain the second-order SE in the (ϕ, q)-domain