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# hw4 - 2(2 where α β γ are unknowns Find numerical values...

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Math 454 Problem Set #4 Fall 2008 Sect. 1 Due Thursday, 9/25, at the beginning of class. Reminder. You are encouraged to work together on homework problems. However, the ﬁnal write- up must be your own. You will be quizzed on some of these problems, so be sure you can do all the problems without help. Also, unless otherwise stated, all reading assignments are from the text. Read: Sects. 3.4, 3.6 Problems - Sect. 3.4 #1, 2, 5, 6, 7, 8, 11, 12 P1. Consider the differential equation ˙ x = r + ( x + r ) 2 (1) (a) Check that Eq. (1) undergoes a saddle-node bifurcation at r = 0 by drawing a (carefully- labelled!) bifurcation diagram. (b) Following the lecture on 9/15, we want to ﬁnd a change of variables to transform Eq. (1) into the saddle-node normal form. As in that lecture, we assume that | x | ≤ ε , | r | ≤ ε , where ε > 0 is a constant. By making ε small, we can make the error terms appearing below as small as we like. First, let X be our new variable, and suppose x = X + αX 2 + βXr + γr
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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 2-F ( 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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