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.

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

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

A hyperbolic point of a dynamical system is a point in the domain of definition of a system

(*)

such that , while the matrix , which is equal to the value of at , has eigen values with positive real part and eigen values with negative real part, . In a neighbourhood of a hyperbolic point there exists an -dimensional invariant surface , constituted by solutions of (*) which, as , asymptotically approach , as well as a -dimensional invariant surface , formed by the solutions of (*) which asymptotically approach as . The behaviour of the trajectories of (*) 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 , , which converts the trajectories of (*) into trajectories of the linear system .

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 (*).

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 (*) is said to be hyperbolic whenever the matrix has no eigen values with real part zero (i.e., in the above also and are admitted). See, e.g., [a1].

References

[a1]	M.C. Irwin, "Smooth dynamical systems" , Acad. Press (1980)