Saddle at infinity

singular saddle point

A type of arrangement of the trajectories of a dynamical system. A dynamical system (or, , see [1]) defined on is said to have a saddle at infinity if there are points and numbers and , , such that the sequences

are convergent and as . 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.