420
11 Nonlinear Dynamics of Circuits with Mem-Elements
= h 21 [f MC (σ MC (t; t 0 ) + σ MC 0 ) − f MC (σ MC 0 )]
+h 22 q L (t; t 0 ) + ϕ e (t; t 0 ).
Then, the mem-element circuit is described in the (ϕ, q)-domain by the secondorder SE
˙
σ MC (t; t 0 ) = −h 11 [f MC (σ MC (t; t 0 ) + σ MC 0 ) − f MC (σ MC 0 )]
−h 12 q L (t; t 0 ) − q a (t; t 0 ) − f M
f MC (σ MC (t; t 0 ) + σ MC 0 )
−f MC (σ MC 0 ) + ϕ M 0
+ f M (ϕ M 0 ) + q MC 0
L ˙
q L (t; t 0 ) = h 21 [f MC (σ MC (t; t 0 ) + σ MC 0 ) − f MC (σ MC 0 )]
+h 22 q L (t; t 0 ) + ϕ e (t; t 0 ).
By time differentiation we can find the fourth-order SE in the state variables
σ MC , q MC , ϕ M , and i L that describes the circuit dynamics in the (v, i)-domain. On
this basis it is possible to study invariant manifolds, reduced-order dynamics on
manifolds, and bifurcations without parameters. The mathematical details are left to
the interested reader.
11.7 Examples Concerning Class LME
In this section we present a number of examples to illustrate how we can write an
SE description in the (ϕ, q)-domain and in the (v, i)-domain for circuits with memelements belonging to the class LME . The examples enable one to write the SEs by
inspection but a systematic procedure for writing the SEs may be developed along
similar lines as for the class LME. The reader is referred to [26] for more details.
The main new aspect of LME , with respect to LME, is the presence of nonlinear charge-controlled capacitors or nonlinear flux-controlled inductors. Before
illustrating the examples we then describe their CRs in the (ϕ, q)-domain and also
in the (v, i)-domain.
11.7.1 Charge-Controlled Nonlinear Capacitor C q
A charge-controlled nonlinear capacitor C q is defined by the algebraic CR v C (t) =
f C (q C (t)) in the (v, q)-domain, i.e., α = 0 and β = −1. The corresponding CR in
the (ϕ, q)-domain is
˙
ϕ C (t; t 0 ) = f C (q C (t; t 0 ) + q C 0 )
11 Nonlinear Dynamics of Circuits with Mem-Elements
= h 21 [f MC (σ MC (t; t 0 ) + σ MC 0 ) − f MC (σ MC 0 )]
+h 22 q L (t; t 0 ) + ϕ e (t; t 0 ).
Then, the mem-element circuit is described in the (ϕ, q)-domain by the secondorder SE
˙
σ MC (t; t 0 ) = −h 11 [f MC (σ MC (t; t 0 ) + σ MC 0 ) − f MC (σ MC 0 )]
−h 12 q L (t; t 0 ) − q a (t; t 0 ) − f M
f MC (σ MC (t; t 0 ) + σ MC 0 )
−f MC (σ MC 0 ) + ϕ M 0
+ f M (ϕ M 0 ) + q MC 0
L ˙
q L (t; t 0 ) = h 21 [f MC (σ MC (t; t 0 ) + σ MC 0 ) − f MC (σ MC 0 )]
+h 22 q L (t; t 0 ) + ϕ e (t; t 0 ).
By time differentiation we can find the fourth-order SE in the state variables
σ MC , q MC , ϕ M , and i L that describes the circuit dynamics in the (v, i)-domain. On
this basis it is possible to study invariant manifolds, reduced-order dynamics on
manifolds, and bifurcations without parameters. The mathematical details are left to
the interested reader.
11.7 Examples Concerning Class LME
In this section we present a number of examples to illustrate how we can write an
SE description in the (ϕ, q)-domain and in the (v, i)-domain for circuits with memelements belonging to the class LME . The examples enable one to write the SEs by
inspection but a systematic procedure for writing the SEs may be developed along
similar lines as for the class LME. The reader is referred to [26] for more details.
The main new aspect of LME , with respect to LME, is the presence of nonlinear charge-controlled capacitors or nonlinear flux-controlled inductors. Before
illustrating the examples we then describe their CRs in the (ϕ, q)-domain and also
in the (v, i)-domain.
11.7.1 Charge-Controlled Nonlinear Capacitor C q
A charge-controlled nonlinear capacitor C q is defined by the algebraic CR v C (t) =
f C (q C (t)) in the (v, q)-domain, i.e., α = 0 and β = −1. The corresponding CR in
the (ϕ, q)-domain is
˙
ϕ C (t; t 0 ) = f C (q C (t; t 0 ) + q C 0 )