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Hirsch and Smale Stability Definition Confusion

  1. May 24, 2010 #1
    So, I'm studying Dynamical Systems from Hirsch and Smale's "Differential Equations, Dynamical Systems, and Linear Algebra." For those who are acquainted with the book, the book is filled with typos. However, otherwise, it's great. I obtained this book from my University library and it appeared the reader before was very troubled by the typos and so he fixed most of them with pen/pencil. However, it seems the typo fixer got lazy a third of the way in, so I've kinda sorta fixed the errors I found. However, I'm currently stuck in one, that I'm just a little confused on whether it's an error or just a misunderstanding of mine. Straight from Hirsch and Smale:

    Definition 1: Suppose [tex]\bar{x} \in W [/tex] is an equilibrium of the differential equation

    (1) [tex]x' = f(x)[/tex]

    where [tex]f : W \rightarrow E [/tex] is a [tex]C^1[/tex] map from an open set W of the vector space [tex]E[/tex] into [tex]E[/tex].

    Then [tex]\bar{x}[/tex] is a stable equilibrium if for every neighborhood [tex]U[/tex] of [tex]\bar{x}[/tex] in [tex]W[/tex] there is a neighborhood [tex]U_1[/tex] of [tex]\bar{x}[/tex] in [tex]U[/tex] such that every solution [tex]x(t)[/tex] with [tex]x(0)[/tex] in [tex]U_1[/tex] is defined and in [tex]U[/tex] for all [tex] t > 0[/tex]. (See Fig. A.)

    Fig A:
    http://i50.photobucket.com/albums/f348/XavvaX/Smale1.png?t=1274719313

    Should it be U instead of U_1 in the definition and vice versa? I say this because x(t) is not defined in U for all t > 0, but rather, defined in U_1. However, I don't know if I'm just misreading the definition and/or the picture. However, in the other picture describing asymptotic stability, they also use the same notation.

    Fig B:
    http://i50.photobucket.com/albums/f348/XavvaX/Smale2.png?t=1274720012
     
  2. jcsd
  3. May 25, 2010 #2

    HallsofIvy

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    No, what they give is correct. There is nothing said about x(t) being defined in U or in U1. It says that x(0) is defined in U1. The point is that solutions which start in some set U1 may go into a larger set, U, but don't get too far away.
     
  4. May 25, 2010 #3
    I understand the intuitive idea, but your very posts confuses me. If the solution starts in U_1 and go to a set bigger than that, then it couldn't be U, since U is in U_1 (as per Fig A). Unless you mean to say that it starts U instead of U_1, which seems to go back to my initial problem.
     
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