Complete instability

A property of a dynamical system. A dynamical system is called completely unstable if all its points are wandering points (cf. Wandering point).

For a dynamical system given in $\mathbf R^n$ to be globally straightenable (or globally rectifiable) (i.e. there exists a topological imbedding $\mathbf R^n\to\mathbf R^n\times\mathbf R^n$ that maps each trajectory of the system into some straight line $\{a\}\times\mathbf R$, where the point $a\in\mathbf R^n$ depends on the trajectory) it is necessary and sufficient that it is completely unstable and has no saddle at infinity (Nemytskii's theorem [1]).

References

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

Comments

For a slightly different formulation of Nemytskii's theorem (avoiding the notion of saddle at infinity), see [a2]. An easily accessible proof is given in [a1]. The property of being globally straightenable is closely related to that of being globally parallelizable: A dynamical system is said to be (globally) parallelizable whenever it is isomorphic to a system of the form $S\times\mathbf R$ where all points move with speed 1 along the lines $\{x\}\times\mathbf R$ ($x\in S$).

References

[a1]	J. Dugundji, H.A. Antosiewicz, "Parallelizable flows and Liapunov's second method" Ann. of Math. , 73 (1961) pp. 543–555
[a2]	V.V. Nemytskii, "Topological problems in the theory of dynamical systems" AMS Transl. Series 1 , 5 (1954) pp. 414–497 Uspekhi Mat. Nauk , 4 (1949) pp. 91–153