Saddle point

A point on a smooth surface such that the surface near the point lies on different sides of the tangent plane. If a point on a twice continuously-differentiable surface is a saddle point, then the Gaussian curvature of the surface at the point is non-positive. A saddle point is a generalization of a hyperbolic point.

Comments

A surface all of whose points are saddle points is a saddle surface.

A saddle point of a differentiable function $f : M \to \mathbf{R}$ is a point $x$ of the differentiable manifold $M$ which is critical, i.e. $\mathrm{d} f (x) = 0$, non-degenerate, i.e. the Hessian matrix $\left({ \partial^2 f / \partial x^i \partial x^j }\right)$ is non-singular, and such that $x$ is not a local maximum or a local minimum, i.e. the Hessian matrix is indefinite. Thus, a non-degenerate critical point of $f : M \to \mathbf{R}$ is a saddle point if its index (the number of negative eigenvalues of the Hessian matrix at that point) is $\ne 0,\,\dim M$. (The index does not depend on the local coordinates chosen.) The graph of a real-valued function of two variables $f : \mathbf{R}^2 \to \mathbf{R}$ near a saddle point looks like a saddle. See also Saddle point in game theory.

A saddle of a differential equation on $\mathbf{R}^2$ is also often called a saddle point of that differential equation. More generally, given a dynamical system $\dot x = f(x)$ on $\mathbf{R}^n$ (or on a differentiable manifold) one considers the eigenvalues of $D F(x_0)$ at an equilibrium point $x_0$. If both positive and negative real parts occur, $x_0$ is called a saddle, a saddle point or, sometimes, a Poincaré saddle point.

References

[a1]	M.W. Hirsch, S. Smale, "Differential equations, dynamical systems, and linear algebra" , Acad. Press (1974) pp. 190ff MR0486784 Zbl 0309.34001
[a2]	D.R.J. Chillingworth, "Differential topology with a view to applications" , Pitman (1976) pp. 150ff MR0646088 Zbl 0336.58001