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''structurally-stable (dynamical) system''

A smooth [[Dynamical system|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|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|Local structural stability]].

Structurally-stable systems were introduced by A.A. Andronov and L.S. Pontryagin [[#References|[1]]]. 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|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 [[#References|[1]]], . 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 [[#References|[3]]]. 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|Non-wandering point]]) should form a [[Hyperbolic set|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 [[#References|[5]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Andronov,   L.S. Pontryagin,   "Systèmes grossiers"  ''Dokl. Akad. Nauk SSSR'' , '''14''' :  5  (1937)  pp. 247–250</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  M.M. Peixoto,   "Structural stability on two-dimensional manifolds"  ''Topology'' , '''1''' :  2  (1962)  pp. 101–120</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  M.M. Peixoto,   "Structural stability on two-dimensional manifolds—a further remark"  ''Topology'' , '''2''' :  2  (1963)  pp. 179–180</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Smale,   "Differentiable dynamical systems"  ''Bull. Amer. Math. Soc.'' , '''73'''  (1967)  pp. 747–817</TD></TR><TR><TD valign="top">[4a]</TD> <TD valign="top">  A.G. Kushnirenko,   "Problems in the general theory of dynamical systems on a manifold"  ''Transl. Amer. Math. Soc.'' , '''116'''  (1981)  pp. 1–42  ''Ninth Math. Summer School''  (1976)  pp. 52–124</TD></TR><TR><TD valign="top">[4b]</TD> <TD valign="top">  A.B. Katok,   "Dynamical systems with hyperbolic structure"  ''Transl. Amer. Math. Soc.'' , '''116'''  (1981)  pp. 43–96  ''Ninth Math. Summer School''  (1976)  pp. 125–211</TD></TR><TR><TD valign="top">[4c]</TD> <TD valign="top">  V.M. Alekseev,   "Quasirandom oscillations and qualitative questions in celestial mechanics"  ''Transl. Amer. Math. Soc.'' , '''116'''  (1981)  pp. 97–169  ''Ninth Math. Summer School''  (1976)  pp. 212–341</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  Z. Nitecki,   "Differentiable dynamics. An introduction to the orbit structure of diffeomorphisms" , M.I.T.  (1971)</TD></TR></table>



====Comments====
The problem whether structural stability is equivalent with Axiom A and the strong transversality condition is known as the Palis–Smale conjecture. That these conditions are sufficient was known for a quite long time (cf. [[#References|[a4]]] for diffeomorphisms and [[#References|[a5]]] for flows defined by vector fields). As to their necessity, it was known that it suffices to prove that structural stability implies Axiom A. For $C^1$-diffeomorphisms on closed manifolds this was shown to be the case in [[#References|[a2]]]. (That $\Omega$-stability implies Axiom A was proven simultaneously in [[#References|[a3]]]; a diffeomorphism is said to be $\Omega$-stable whenever it is structurally stable on its $\Omega$-limit set, cf. [[Limit set of a trajectory|Limit set of a trajectory]].)

That the structurally-stable systems do not form a dense set (in the space of all systems with the $C^1$-metric) for dimensions $\geq3$ implies that they cannot provide that universal tool for applications as was originally hoped for, namely, that all physical processes can essentially be described by structurally-stable systems (cf. [[#References|[a6]]]). Important structurally-stable systems are the Anosov systems (cf. [[Y-system|$Y$-system]]; [[#References|[a1]]]) and hyperbolic strange attractors (cf. [[Hyperbolic set|Hyperbolic set]]; [[Strange attractor|Strange attractor]]). But most strange attractors and chaotic systems that arise in applications (e.g., the [[Lorenz attractor|Lorenz attractor]]) are not structurally stable. In [[#References|[a7]]] another definition of stability has been proposed, with respect to which most attractors are stable and such that the stable systems are dense. This definition is based on a concept of  "neighbourhood of a system"  that is closely related to applications and that can be used in numerical and physical experiments to model the data with $\epsilon$-error for given $\epsilon>0$.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  D.V. Anosov,   "Geodesic flows on compact Riemannian manifolds of negative curvature"  ''Proc. Steklov Inst.'' , '''90'''  (1969)  ''Trudy Mat. Inst. Steklov.'' , '''90'''  (1969)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Mañé,   "A proof of the $C^1$-stability conjecture"  ''Publ. Math. IHES'' , '''66'''  (1988)  pp. 161–210</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Palis,   "On the $C^1$ $\Omega$-stability conjecture"  ''Publ. Math. IHES'' , '''66'''  (1988)  pp. 211–215</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Robbin,   "A structural stability theorem"  ''Ann. of Math.'' , '''94'''  (1971)  pp. 447–493</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  R.C. Robinson,   "Structural stability of vector fields"  ''Ann. of Math.'' , '''99'''  (1974)  pp. 154–175</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  R. Thom,   "Structural stability and morphogenesis" , Benjamin  (1976)  (Translated from French)</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  E.C. Zeeman,   "Stability of dynamical systems"  ''Nonlinearity'' , '''1'''  (1988)  pp. 115–155 {{ZBL|0643.58005}}</TD></TR>
</table>