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Math Help - Why bounded ode solution tends to a constant

  1. #1
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    Why bounded ode solution tends to a constant

    Dear all, consider now a solution
    x(t):\ [0,\infty)\to R
    to an ode
    x'=f(x)
    If x(t) is bounded. Show \lim_{t\to\infty}x(t)=c
    for some constant c
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  2. #2
    MHF Contributor chisigma's Avatar
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    Re: Why bounded ode solution tends to a constant

    Quote Originally Posted by xinglongdada View Post
    Dear all, consider now a solution
    x(t):\ [0,\infty)\to R
    to an ode
    x'=f(x)
    If x(t) is bounded. Show \lim_{t\to\infty}x(t)=c
    for some constant c
    A counterxample: x(t)= \sin t is bounded and obeys to ODE x^{'}= \sqrt{1-x^{2}} , but the \lim_{t \rightarrow \infty} x(t) doesn't exsist...

    Kind regards

    \chi \sigma
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Re: Why bounded ode solution tends to a constant

    Quote Originally Posted by xinglongdada View Post
    Dear all, consider now a solution x(t):\ [0,\infty)\to R to an ode x'=f(x) . If x(t) is bounded. Show \lim_{t\to\infty}x(t)=c for some constant c
    Chisigma provided you an excellent answer. It would be interesting to transcribe the exact formulation of the problem.
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  4. #4
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    Re: Why bounded ode solution tends to a constant

    Indeed, this is a very nice counterexample...Good luck
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