I'm wondering if someone can provide a clarification between 2 seemingly opposing definitions from reputable sources on dynamical systems!

My Russian textbook, "Dynamical Systems I: Ordinary Differential Equations and Smooth Dynamical Systems" by Anosov, Arnol'd, Aronson, et al., says the following to determine whether a singular point of a dynamical system is asymptotically stable:

Theorem 4.2: If all eigenvalues of the linear part of a vector field $v$ at a singular point have negative real part, then the singular point is asymptotically stable.

To me, this means for any arbitrary dynamical system, say, $\dot{x} = f(x)$, where $x \in \mathbb{R}^{n}$, one can find where $f(x) = 0$, and solve the corresponding Jacobian for the eigenvalues of to determine stability. Further, if one finds that $\lambda_{i} < 0$, for $i = 1,2,...n$ then, this point is locally stable, by this theorem. But, is this theorem now suggesting that this is point is now asymptotically stable as well?

Almost every single textbook on ODEs that I have checked says to determine whether an equilibrium point is asymptotically stable, some more general method is required like constructing Lyapunov functions, determining limit sets, etc...

Why is there such a difference? Is there a difference?

  • 1
    $\begingroup$ You might not have enough regularity to linearize. Your limit set might not be a point. The eigenvalues might be too hard to calculate. $\endgroup$
    – Ian
    Commented Jul 17, 2016 at 13:35
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    $\begingroup$ Theorem 4.2 gives a sufficient condition for asymptotic stability not necessary. Consider for example the simple first order system $\dot{x}=-x^3$ which is asymptotically stable but fails to meet conditions of the Theorem. $\endgroup$
    – RTJ
    Commented Jul 17, 2016 at 13:37
  • $\begingroup$ Hi. Yes, I thought of that example too. So, this seems to be a "shortcut" theorem, that avoids the use of Lyapunov functions, etc... when they can be avoided? $\endgroup$ Commented Jul 17, 2016 at 13:39
  • $\begingroup$ Yes, it is called the first Lyapunov theorem. $\endgroup$
    – RTJ
    Commented Jul 17, 2016 at 13:40
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    $\begingroup$ Asymptotic stability=(Lyapunov stability)+(convergence to the singular point). Local asymptotic stability means that there exists a region around the equilibrium s.t. the above two properties hold true for perturbations within this region. Global means that they hold for arbitrarily large perturbations. $\endgroup$
    – RTJ
    Commented Jul 17, 2016 at 13:55


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