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Math Help - Number of isolated equilibria

  1. #1
    M.A
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    Number of isolated equilibria

    Hi
    Suppose for a dynamical system \dot x = f(x), x \in \mathbb R^n there exists a finite number of isolated equilibria, all of them are locally stable (i.e eigenvalues of the associated Jacobian for each equilibrium have negative real parts).
    My question is: Can the number of the of equilibria in the statement above exceed one? (sorry if it is a trivial question)


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  2. #2
    MHF Contributor

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    Re: Number of isolated equilibria

    No. If p and q are two locally stable equilibria, there must exist a locally unstable equilibrium some where "between" them (lying on some curve from one to the other.)
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  3. #3
    M.A
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    Re: Number of isolated equilibria

    thanks for participation but can you provide a proof or a reference to a proof ?
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