Difference between revisions of "Hyperbolic point"
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− | A hyperbolic point of a surface is a point at which the [[ | + | {{TEX|done}} |
− | + | 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 [[#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">[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 | + | 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) |
Hyperbolic point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hyperbolic_point&oldid=18331