Saddle

A type of arrangement of the trajectories of an autonomous system of planar ordinary differential equations

$$ \tag{* } \dot{x} = f ( x),\ \ x \in \mathbf R ^ {2} ,\ \ f: G \subset \mathbf R ^ {2} \rightarrow \mathbf R ^ {2} , $$

$ f \in C ( G) $, where $ G $ is the domain of uniqueness, in a neighbourhood of a singular point (equilibrium position) $ x _ {0} $. This type is characterized as follows. There is a neighbourhood $ U $ of $ x _ {0} $ such that for all trajectories of the system that start in $ U \setminus \{ x _ {0} \} $, both the positive and negative semi-trajectories are diverging (as time passes they leave any compact set $ V \subset U $). The exceptions are four trajectories, called the separatrices of the saddle. For two of these, the negative semi-trajectories are diverging and the positive semi-trajectories converge to $ x _ {0} $, and for the other two the situation is just the opposite. The first two separatrices are called stable, and the second two, unstable. The stable separatrices, completed by $ x _ {0} $, form a smooth curve through $ x _ {0} $— the stable manifold of the saddle. The unstable separatrices, together with $ x _ {0} $, form the smooth unstable manifold of the saddle. The point $ x _ {0} $ is also called the saddle.

The saddle $ x _ {0} $ is unstable in the sense of Lyapunov (cf. Lyapunov stability). Its Poincaré index is $ - 1 $( cf. Singular point). For a system (*) of class $ C ^ {1} $( $ f \in C ^ {1} ( G) $) with a non-zero matrix $ A = f ^ { \prime } ( x _ {0} ) $, a stationary point $ x _ {0} $ is a saddle when the eigenvalues $ \lambda _ {1} , \lambda _ {2} $ of $ A $ satisfy the condition $ \lambda _ {1} \lambda _ {2} < 0 $( a simple saddle; see Fig. a, where $ x _ {0} = 0 $) but it can also be a saddle when $ \lambda _ {1} = 0 \neq \lambda _ {2} $, or $ \lambda _ {1} = \lambda _ {2} = 0 $.

Figure: s083020a

In any of these cases the separatrices of the saddle are tangential at $ x _ {0} $ to the directions defined by the eigenvectors of $ A $.

If the system (*) is linear ( $ f ( x) = A ( x - x _ {0} ) $, where $ A $ is a constant matrix with eigenvalues $ \lambda _ {1} $ and $ \lambda _ {2} $), then a point $ x _ {0} $ of it is a saddle only when $ \lambda _ {1} \lambda _ {2} < 0 $. In this case the separatrices of the saddle $ x _ {0} $ are rectilinear, and all the other trajectories (distinct from $ x _ {0} $) are affine images of the hyperbolas of the form $ x _ {2} = c | x _ {1} | ^ {\lambda _ {2} / \lambda _ {1} } $, $ c \in \mathbf R \setminus \{ 0 \} $( Fig. b).

Figure: s083020b

The term "saddle" is also used for any of the arrangements of the trajectories of the system (*) in a neighbourhood $ U $ of an isolated stationary point $ x _ {0} $, where only a finite number $ m $( $ \geq 2 $) of trajectories in $ U \setminus \{ 0 \} $ approaches $ x _ {0} $ and where each of them, completed by $ x _ {0} $, touches at this point in a definite direction (an $ m $- separatrix saddle). Certain types of stationary points of autonomous systems of ordinary differential equations of order $ n \geq 3 $ are also called saddles. For references, see Singular point of a differential equation.

Comments

Cf. also Hyperbolic point and Hyperbolic set and the references given there.