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By Patton, C.C. and Campbell, J.M

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Un )T said to be generalized normalized variables or generalized normalized coordinates (cf. [90]). 4 The generalized normalized coordinates depend on the special choice of right eigenvectors ri (u) (i = 1, . . , n). Obviously, for any strictly hyperbolic system, generalized normalized coordinates must be normalized coordinates. 3 are only two extreme cases. 3). In such a case, it is necessary to introduce a new concept—the weak linear degeneracy (cf. [50], [75]). 2) namely, λi (u(i) (s)) ≡ λi (u0 ), ∀|s| small.

2 Estimates Independent of Weak Linear Degeneracy 57 Now we consider the case that there exists j = i such that (τ, ξi (τ )) ∈ DjT . When j < i, we have |ξi (τ ) − λj (0)τ | ≥ |ξi (τ ) − λi (0)τ | ≥ δ0 xi0 . 23) When j > i, since (τ, ξi (τ )) ∈ DjT , we have ξi (τ ) ≥ (λj (0) + δ0 )τ or ξi (τ ) ≤ (λj (0) − δ0 )τ. 25) whereas, if ξi (τ ) ≤ (λj (0) − δ0 )τ , it is easy to get |ξi (τ ) − λj (0)τ | ≥ δ0 xi0 . 16) follows. 16) in a similar way. 2) holds. 1). 29) and where κ1 , κ2 , and κ3 are positive constants independent of θ and T .

P; j = p + 1, . . , n), 1 ∂rik (uk ek + σ(u − uk ek )) dσ 0 ∂uj (k = p + 1, . . , n, j = k). 49) with Fijk (u) ≡ 0 Fijj (u) ≡ 0 (j, k = 1, . . 62) (j = p + 1, . . 52). If λi (u) is WLD, when i = 1, . . , p, we have p Fijk uh eh h=1 ≡0 (j, k = 1, . . 64) whereas, when i = p + 1, . . , n, we have Fiii (ui ei ) ≡ 0. 65) Furthermore, if λi (u) is LD, when i = 1, . . , p, we have Fijk (u) ≡ 0 (j, k = 1, . . 66) whereas, when i = p + 1, . . , n, we have Fiii (u) ≡ 0. 7 Two Lemmas on Ordinary Differential Equations 49 where T > 0 is a given number, ai (t) ∈ C[0, T ] (i = 0, 1, 2), and a0 (t) ≥ 0 for 0 ≤ t ≤ T .

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