Hyperbolic point

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)