# Rough system

structurally-stable (dynamical) system

A smooth dynamical system with the following property: For any $\epsilon>0$ there is a $\delta>0$ such that for any perturbation of the system by not more than $\delta$ in the $C^1$-metric, there exists a homeomorphism of the phase space which displaces the points by not more than $\epsilon$ and converts the trajectories of the unperturbed system into trajectories of the perturbed system. Formally, this definition assumes that a certain Riemannian metric is given on the phase manifold. In fact, one speaks of a structurally-stable system when the phase manifold is closed, or else if the trajectories form part of some compact domain $G$ with a smooth boundary not tangent to the trajectories; here the perturbation and the homeomorphism are considered on $G$ only. In view of the compactness, the selection of the metric is immaterial.

Thus, a small (in the sense of $C^1$) perturbation of a structurally-stable system yields a system equivalent to the initial one as regards all its topological properties (however, this definition comprises one additional requirement, viz. this equivalence must be realized by a homeomorphism close to the identity). The terms "roughness" and "(structural) stability" are used in a broader sense, e.g. to mean merely the preservation of some property of the system under a small perturbation (in such a case it is preferable to speak of the structural stability of the property in question). See also Local structural stability.

Structurally-stable systems were introduced by A.A. Andronov and L.S. Pontryagin . If the dimension of the phase manifold is small (one for discrete time and one or two for continuous time), structurally-stable systems can be simply characterized in terms of the qualitative properties of behaviour of trajectories (then they are the so-called Morse–Smale systems, cf. Morse–Smale system); in that case they form an open everywhere-dense set in the space of all dynamical systems, provided with the $C^1$-topology , . Thus, systems whose trajectories display a behaviour which is more complex and more sensitive to small perturbations are considered here as exceptional. If the dimensions are larger, none of these facts hold, as was established by S. Smale . He advanced the hypothesis according to which, irrespective of all these complications, it is possible in the general case to formulate the following necessary and sufficient conditions for structural stability in terms of a qualitative picture of the behaviour of the trajectories: 1) the non-wandering points (cf. Non-wandering point) should form a hyperbolic set $\Omega$, in which the periodic trajectories are everywhere dense (the so-called Smale's Axiom A); and 2) the stable and unstable manifolds of any two trajectories from $\Omega$ should intersect transversally (the strong transversality condition). That these conditions are sufficient has now been proved in almost all cases; as regards their necessity, proof is now (the 1970s) only available if the definition of structural stability is somewhat changed (see, e.g.,

or ).

How to Cite This Entry:
Rough system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rough_system&oldid=32549
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article