Saddle node

A type of arrangement of the trajectories in a neighbourhood of a singular point $ x _ {0} $ of an autonomous system of second-order ordinary differential equations

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

$ f \in C ( G) $, where $ G $ is the domain of uniqueness. This type is characterized as follows. Suppose that a certain neighbourhood $ U $ of $ x _ {0} $ is partitioned into $ m $( $ 3 \leq m < + \infty $) curvilinear sectors (cf. Sector in the theory of ordinary differential equations) by semi-trajectories (the separatrices of the saddle node) approaching $ x _ {0} $. Suppose that $ h $ of these sectors, $ 2 \leq h < m $, are saddle sectors and that the others are open nodal sectors, and suppose also that each semi-trajectory approaching $ x _ {0} $, completed with $ x _ {0} $, touches it in a definite direction. Then $ x _ {0} $ is called a saddle node.

A saddle node is unstable in the sense of Lyapunov (cf. Lyapunov stability). Its Poincaré index is $ 1 - ( h/2) $( cf. Singular point). If $ f \in C ^ {1} ( G) $ and the matrix $ A = f ^ { \prime } ( x _ {0} ) \neq 0 $, then the singular point $ x _ {0} $ can be a saddle node for (*) only when the eigenvalues $ \lambda _ {1} , \lambda _ {2} $ of $ A $ satisfy one of the following conditions:

a) $ \lambda _ {1} = 0 \neq \lambda _ {2} $;

b) $ \lambda _ {1} = \lambda _ {2} = 0 $.

In any of these cases $ x _ {0} $ can also be a saddle or a node for (*), and in case b), also a point of another type. If it is a saddle node, then $ m = 3 $, $ h = 2 $, and all the semi-trajectories of the system that approach $ x _ {0} $ touch at this point the directions defined by the eigenvectors of $ A $( see Fig. aand Fig. b, where the heavy lines are the separatrices at the saddle node $ x _ {0} = 0 $, and the arrows indicate the direction of motion along the trajectories of the system as $ t $ increases; they can also be in the opposite direction).

Figure: s083040a

Figure: s083040b

Comments

The flow near a saddle node does not enjoy structural stability: If $ x _ {0} $ is a saddle node for (*), there is a neighbourhood $ N $ of $ x _ {0} $ in $ \mathbf R ^ {2} $ such that for any $ \epsilon > 0 $ there is a system $ \dot{x} = y( x) $ having no equilibrium in $ N $, such that $ | f- g | < \epsilon $ and $ | ( \partial f / \partial x _ {i} ) - ( \partial g/ \partial x _ {i} ) | < \epsilon $, $ i = 1, 2 $. However, the saddle node bifurcation is robust and cannot be perturbed away ([a1]) (cf. also Rough system).

References

[1]	N.N. Bautin, E.A. Leontovich, "Methods and means for a qualitative investigation of dynamical systems on the plane" , Moscow (1976) (In Russian)
[a1]	J. Guckenheimer, P. Holmes, "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer (1983)
[a2]	A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, "Qualitative theory of second-order dynamic systems" , Wiley (1973) (Translated from Russian)

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