Unformatted text preview: 2 , (2) where α, β, γ are unknowns. Find numerical values for α, β, γ so that in terms of the new variable X , Eq. (1) becomes ˙ X = r + X 2 + O ( ε 3 ) . (3) (c) Solve Eq. (2) to ﬁnd an explicit expression for X in terms of x . (d) There’s a simpler way to deﬁne X in terms of x and r which gives Eq. (1) with no error terms . What is it? (e) Extra credit: If ε = 1 / 10 , roughly how big is the size of the O ( ε 3 ) remainder in Eq. (3)? Hint: First, ﬁnd the exact formula for the transformed equation ˙ X = F ( X, r ) . Since X = x + O ( ε 2 ) , it’s reasonable to guess that  X  is not much bigger than  x  itself, say  X  ≤ cε for some reasonably small value of c . (You will have to guess a value for c .) How big can  r + X 2F ( X, r )  get for  X  ≤ cε and  r  ≤ ε ? A numerical estimate (by plugging in a few values of X and r ) is ﬁne....
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 Spring '10
 GLASNER
 Math, Bifurcation theory, Saddlenode bifurcation, error terms, saddlenode normal form

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