View source for Hyperbolic point

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A hyperbolic point of a surface is a point at which the [[Osculating paraboloid|osculating paraboloid]] is a [[Hyperbolic paraboloid|hyperbolic paraboloid]]. At a hyperbolic point the [[Dupin indicatrix|Dupin indicatrix]] is given by a pair of conjugate hyperbolas.



====Comments====
At a hyperbolic point the surface has negative Gaussian curvature, and conversely: If a surface has negative Gaussian curvature at a point, that point is hyperbolic.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Blaschke,   K. Leichtweiss,   "Elementare Differentialgeometrie" , Springer  (1973)</TD></TR></table>

A hyperbolic point of a dynamical system is a point $x=x^*$ in the domain of definition of a system

\begin{equation}\dot x=f(x),\quad x=(x_1,\dots,x_n),\label{*}\end{equation}

such that $f(x^*)=0$, while the matrix $A$, which is equal to the value of $\partial f/\partial x$ at $x=x^*$, has $k$ eigen values with positive real part and $n-k$ eigen values with negative real part, $0<k<n$. In a neighbourhood of a hyperbolic point there exists an $(n-k)$-dimensional invariant surface $S_+$, constituted by solutions of \eqref{*} which, as $t\to\infty$, asymptotically approach $x=x^*$, as well as a $k$-dimensional invariant surface $S_-$, formed by the solutions of \eqref{*} which asymptotically approach $x=x^*$ as $t\to-\infty$. The behaviour of the trajectories of \eqref{*} in a sufficiently small neighbourhood of a hyperbolic point may be described by means of the following theorem [[#References|[4]]]: There exists a homeomorphism of some neighbourhood of a hyperbolic point onto some neighbourhood of the point $u=0$, $u=(u_1,\dots,u_n)$, which converts the trajectories of \eqref{*} into trajectories of the linear system $\dot u=Au$.

For a diffeomorphism with a fixed point, a hyperbolic point is defined by the absence of eigen values of modulus one in the linear part of the diffeomorphism at the fixed point under consideration. Thus, a hyperbolic point of the system \eqref{*} remains a hyperbolic point of the diffeomorphism generated by a shift along a trajectory of the system \eqref{*}.

====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  H. Poincaré,   "Mémoire sur les courbes definiés par une équation différentielle"  ''J. de Math.'' , '''7'''  (1881)  pp. 375–422</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  H. Poincaré,   "Mémoire sur les courbes definiés par une équation différentielle"  ''J. de Math.'' , '''8'''  (1882)  pp. 251–296</TD></TR><TR><TD valign="top">[1c]</TD> <TD valign="top">  H. Poincaré,   "Mémoire sur les courbes definiés par une équation différentielle"  ''J. de Math.'' , '''1'''  (1885)  pp. 167–244</TD></TR><TR><TD valign="top">[1d]</TD> <TD valign="top">  H. Poincaré,   "Mémoire sur les courbes difiniés par une équation différentielle"  ''J. de Math.'' , '''2'''  (1886)  pp. 151–217</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.M. [A.M. Lyapunov] Liapunoff,   "Problème général de la stabilité du mouvement" , Princeton Univ. Press  (1947)  (Translated from Russian)  (Reprint: Kraus, 1950)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.A. Coddington,   N. Levinson,   "Theory of ordinary differential equations" , McGraw-Hill  (1955)  pp. Chapts. 13–17</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P. Hartman,   "Ordinary differential equations" , Birkhäuser  (1982)</TD></TR></table>

''V.K. Mel'nikov''

====Comments====
Often, an invariant point in the system \eqref{*} is said to be hyperbolic whenever the matrix $A$ has no eigen values with real part zero (i.e., in the above also $k=0$ and $k=n$ are admitted). See, e.g., [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.C. Irwin,   "Smooth dynamical systems" , Acad. Press  (1980)</TD></TR></table>