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My understanding of hyperbolic points (correct me if I'm wrong) is that there must be an unstable and stable manifold in the neighborhood of the hyperbolic point. So essentially the hyperbolic point is the saddle point of the manifolds intersecting. Does this mean that all hyperbolic points are unstable? Furthermore, are all hyperbolic orbits also unstable orbits?

Thanks!

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  • $\begingroup$ Since "hyperbolic-geometry" is about the geometry of hyperbolic space, while your question is about the behavior of trajectories near critical points of ODEs, I have retagged your question as differential-equations and dynamical-systems. $\endgroup$
    – Neal
    Aug 17, 2013 at 19:44

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The standard definition of a hyperbolic equilibrium point $p \in M$ of a sufficiently smooth vector field $X$ on a sufficiently smooth manifold $M$ is that i.) $X(p) = 0$; and ii.) the eigenvalues $\lambda_k = \sigma_k + i\omega_k$ of the Jacobian matrix of $X$ at $p$, i.e. the matrix $\begin{bmatrix}\frac{\partial X^i}{\partial x^j} \end{bmatrix}(p)$, where the $x^j$ are local coordinates near $p$, all have non-zero real parts. Thus the equilibria at which all eigenvalues have positive or negative real parts are admitted as hyperbolic under this definition. Purely attracting equilibria (all $\sigma_k < 0$) as well as purely repelling equilibria (all $\sigma_k > 0$) are included in the general class of "hyperbolic equilibria", even though the flow $\phi_X$ of $X$ near these points lacks the saddle-like structure one is used to associating with the term "hyperbolic". Of course, in the event that some $\sigma_k < 0$ and some $\sigma_k > 0$ at $p$ then the usual saddle-like structure appears in the dynamics of $X$ and $\phi_X$ near $p$, and most of the integral curves will look in some ways like the hyperbolas we see in classic analytic geometry. Stable and unstable manifolds of $p$ are defined in all cases with $\sigma_k \ne 0$, though they may indeed be vacuous or, at the other extreme, open subsets of the manifold $M$.

The choice of the term "hyperbolic" to cover the case all $\sigma_k \ne 0$ is apparently of historical origin, though it is perhaps not always geometrically lucid. Check out this widipedia page for more details. Any any good book which covers the qualitative theory of ordinary differential equations should also treat these matters; one of may favorites is Jack K. Hale's Ordinary Differential Equations which is available in paperback from Dover Press. See section III.6, pp. 106-117, for a very detailed discussion.

Similar terminology applies to periodic orbits and invariant submanifolds, if I am not mistaken, though the analogous definitions may be considerably more complicated. The references I cited should provide many leads into these subjects.

Hope this helps!

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  • $\begingroup$ Ohhh ok, thanks for the good answer. So does this mean that a hyperbolic point cannot have a "center manifold"....or I think it's called "asymptotically stable" $\endgroup$ Aug 17, 2013 at 20:58
  • $\begingroup$ @John Dodson: And I thank you for your kind words and, of course, for "acceptance"! But to your point: right, a hyperbolic zero of $X$ cannot have a "center", since this is precluded by the fact that the $\sigma_k \ne 0$. Check out the wiki page I cited and the links therein, or just google around a bit--the net is overflowing with this kind of stuff! Cheers! $\endgroup$ Aug 17, 2013 at 21:11

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