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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)