Saddle at infinity

singular saddle point

A type of arrangement of the trajectories of a dynamical system. A dynamical system $ f ^ { t } $( or, $ f ( \cdot , t) $, see [1]) defined on $ \mathbf R ^ {n} $ is said to have a saddle at infinity if there are points $ x _ {k} $ and numbers $ \tau _ {k} < 0 $ and $ \theta _ {k} > 0 $, $ k \in \mathbf N $, such that the sequences

$$ \{ f ^ { \tau _ {k} } x _ {k} \} _ {k \in \mathbf N } ,\ \ \{ f ^ { \theta _ {k} } x _ {k} \} _ {k \in \mathbf N } $$

are convergent and $ | x _ {k} | \rightarrow \infty $ as $ k \rightarrow \infty $. This definition, which is due to V.V. Nemytskii, was generalized by M.V. Bebutov to dynamical systems defined on an arbitrary metric space; here the condition "xk∞ as k∞" is replaced by "the sequence xkkN does not contain any convergent subsequence" .

The absence of a saddle at infinity is a necessary condition for the possibility of global rectification of the dynamical system (see Complete instability). For a completely-unstable dynamical system defined on a metric space not to have a saddle at infinity it is necessary and sufficient that the quotient space of the dynamical system be Hausdorff.

References

[1]	V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)

Comments

The "quotient space of a dynamical systemquotient space of a dynamical system" is usually called its orbit space.