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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.
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A hyperbolic point of a surface is a point at which the [[osculating paraboloid]] is a [[hyperbolic paraboloid]]. At a hyperbolic point the [[Dupin indicatrix]] is given by a pair of conjugate hyperbolas.
  
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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.
  
====Comments====
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== In dynamical systems ==
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====
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A hyperbolic point of a dynamical system is a point $x=x^*$ in the domain of definition of a system
<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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h0483201.png" /> in the domain of definition of a system
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\begin{equation}\dot x=f(x),\quad x=(x_1,\dots,x_n),\label{*}\end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h0483202.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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such that $f(x^*)=0$, while the matrix $A$, which is equal to the value of $\partial f/\partial x$ at $x=x^*$, has $k$ eigenvalues with positive real part and $n-k$ eigenvalues 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 $\dot u=Au$.
  
such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h0483203.png" />, while the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h0483204.png" />, which is equal to the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h0483205.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h0483206.png" />, has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h0483207.png" /> eigen values with positive real part and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h0483208.png" /> eigen values with negative real part, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h0483209.png" />. In a neighbourhood of a hyperbolic point there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h04832010.png" />-dimensional invariant surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h04832011.png" />, constituted by solutions of (*) which, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h04832012.png" />, asymptotically approach <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h04832013.png" />, as well as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h04832014.png" />-dimensional invariant surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h04832015.png" />, formed by the solutions of (*) which asymptotically approach <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h04832016.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h04832017.png" />. The behaviour of the trajectories of (*) 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h04832018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h04832019.png" />, which converts the trajectories of (*) into trajectories of the linear system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h04832020.png" />.
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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{*}.
  
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 (*) remains a hyperbolic point of the diffeomorphism generated by a shift along a trajectory of the system (*).
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====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Blaschke,   K. Leichtweiss,   "Elementare Differentialgeometrie" , Springer  (1973)</TD></TR></table>
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
Often, an invariant point in the system (*) is said to be hyperbolic whenever the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h04832021.png" /> has no eigen values with real part zero (i.e., in the above also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h04832022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048320/h04832023.png" /> are admitted). See, e.g., [[#References|[a1]]].
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Often, an invariant point in the system \eqref{*} is said to be hyperbolic whenever the matrix $A$ has no eigenvalues with real part zero (i.e., in the above also $k=0$ and $k=n$ are admitted). See, e.g., [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.C. Irwin,  "Smooth dynamical systems" , Acad. Press  (1980)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.C. Irwin,  "Smooth dynamical systems" , Acad. Press  (1980)</TD></TR></table>

Latest revision as of 11:50, 1 May 2023

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

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.

In dynamical systems

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 $k$ eigenvalues with positive real part and $n-k$ eigenvalues 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 [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

[a1] W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)

References

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

V.K. Mel'nikov

Comments

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

References

[a1] M.C. Irwin, "Smooth dynamical systems" , Acad. Press (1980)
How to Cite This Entry:
Hyperbolic point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hyperbolic_point&oldid=18331
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article