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Math Help - Stable by but not Asymptotically Stable

  1. #1
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    Stable by but not Asymptotically Stable

    Show that the equilibrium point x_0=0 for the differential equation x^{\prime}=0 is stable but not asymptotically stable.
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  2. #2
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    Quote Originally Posted by joestevens View Post
    Show that the equilibrium point x_0=0 for the differential equation x^{\prime}=0 is stable but not asymptotically stable.
    I don't know exactly how to do this but you probably need to use the ε, δ definition of stable and asymptotically stable.
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  3. #3
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    Quote Originally Posted by JJMC89 View Post
    I don't know exactly how to do this but you probably need to use the ε, δ definition of stable and asymptotically stable.
    I got as far as knowing that but I don't know where to do. Anyone have any suggestions as to how to do this?
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  4. #4
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    In this problem you can actually integrate and find the solution

    x(t) = x_0

    Now, with x_0 = 0 gives your solution. Now perturb it so

    x^{*}(t) = x_1.

    So,

    | x(t) - x^{*}(t)| = x_1 for all t (this a stable)

    However, we don't have for some \delta > 0

    | x(t) - x^{*}(t)| \le \delta \; \text{for}\; t > t* (this a asymptotic stability).
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