8.3 One-Dimensional Arrays of MCCs
329
Fig. 8.6 One-dimensional
array of N diffusively
coupled identical MCCs
MOC i−1
ϕ C 1,i−1
G i,i−1
MOC i
ϕ C 1,i
MOC i+1
ϕ C 1,i+i
G i,i+1
with ICs v C 1,i (t 0 ) = v C 1,i (t 0 ), v C 2,i (t 0 ) = v C 2,i (t 0 ), i L,i (t 0 ) = i L,i (t 0 ), ϕ M,i (t 0 ) =
ϕ M,i (t 0 ) and i = 1, . . . , N. The SEs in the (v, i)-domain are an IVP for a system of
4N coupled first-order ODEs in the state variables
w c (t) = (v C 1,1 , v C 2,1 , i L,1 , ϕ M,1 , . . . , v C 1,N , v C 2,N , i L,N , ϕ M,N )
T
∈ R
4N .
Invariant manifolds in the state space R 4N in the (v, i)-domain of (8.11) can be
identified as follows. Define functions Q i : R 4N → R, i = 1, 2, . . . , N as
Q i (w c ) = f (ϕ M,i )+
1
R
ϕ M,i +C 1 v C 1,i −
L
R
i L,i −
k∈N i
1
R ik
(ϕ M,k −ϕ M,i ) (8.12)
and, for any Q 0 = (Q 0,1 , . . . , Q 0,N ) T ∈ R N , let
M c (Q 0 ) = {w c ∈ R
4N
: Q i (w c ) = Q 0,i , i = 1, 2, . . . , N}.