multiple time scale dynamics with two fast variables and one slow ...

where
R1= (∇Λ·dz)∧dy1
R7= (∇Γ·dz)∧dy1
R2= da∧(∇g1· dz) R8= db∧(∇g1· dz)
R3= (∇Λ·dz)∧dy2
R9= (∇Γ·dz)∧dy2
R4= da∧(∇g2· dz) R10= db∧(∇g2· dz)
R5= (∇Λ·dz)∧db R11= dy1∧ (∇g1· dz)
R6= da∧(∇Γ·dz) R12= (∇g2· dz)∧dy2
Note that we only considered the six 2-forms Pay1, Pay2, Pab, Pby1, Pby2, Py1y2 as they
span the space of two-forms 2 R 4 . To simplify notation we let
Z := (Pay1, Pay2) T = (Z1, Z2)
X := (Pab, Pby1, Pby2, Py1y2) T = (X1, X2, X3, X4) T .
Step 2: The next lemma provides fundamental estimates which we shall use
throughout this section.
Lemma 2.5.1. The equations for (Z, X) can be written in the form:
Z ′ = ΛZ+η1(Z, X, t)
X ′ = BX+η2(Z, X, t) (2.24)
where B= B(z,ǫ) is a 4×4 matrix-valued function. The following conditions hold:
(i) For a=0, b=0,ǫ= 0 we haveη1= 0,η2= 0.
(ii) The matrix B satisfies:
exp
t
for some ¯M≥ 1,µ>0 and t> s.
s
(B−Λ(Id))dξ ≤ ¯M exp(−µ(t− s)) (2.25)
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