25:
606:. In addition, a structurally stable system may have transversal homoclinic trajectories of hyperbolic saddle closed orbits and infinitely many periodic orbits, even though the phase space is compact. The closest higher-dimensional analogue of structurally stable systems considered by Andronov and Pontryagin is given by the
601:
When Smale started to develop the theory of hyperbolic dynamical systems, he hoped that structurally stable systems would be "typical". This would have been consistent with the situation in low dimensions: dimension two for flows and dimension one for diffeomorphisms. However, he soon found examples
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rigorously investigated stability of small perturbations of an individual system. In practice, the evolution law of the system (i.e. the differential equations) is never known exactly, due to the presence of various small interactions. It is, therefore, crucial to know that basic features of the
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of vector fields on higher-dimensional manifolds that cannot be made structurally stable by an arbitrarily small perturbation (such examples have been later constructed on manifolds of dimension three). This means that in higher dimensions, structurally stable systems are not
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cannot, in general, be a diffeomorphism. Moreover, although topological equivalence respects the oriented trajectories, unlike topological conjugacy, it is not time-compatible. Thus, the relevant notion of topological equivalence is a considerable weakening of the naĂŻve
375:-dimensional compact smooth manifolds with boundary. Andronov and Pontryagin originally considered the strong property. Analogous definitions can be given for diffeomorphisms in place of vector fields and flows: in this setting, the homeomorphism
544:
Structural stability of the system provides a justification for applying the qualitative theory of dynamical systems to analysis of concrete physical systems. The idea of such qualitative analysis goes back to the work of
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conjugacy of vector fields. Without these restrictions, no continuous time system with fixed points or periodic orbits could have been structurally stable. Weakly structurally stable systems form an open set in
187:, they form an open dense set in the space of all systems endowed with appropriate topology. In higher dimensions, this is no longer true, indicating that typical dynamics can be very complex (cf.
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in the 1920s, but was first formalized with introduction of the concept of rough system by
Andronov and Pontryagin in 1937. This was immediately applied to analysis of physical systems with
144:, which considers perturbations of initial conditions for a fixed system, structural stability deals with perturbations of the system itself. Variants of this notion apply to systems of
562:
dynamics are the same for any small perturbation of the "model" system, whose evolution is governed by a certain known physical law. Qualitative analysis was further developed by
443:
of the system is precisely the union of singular points and periodic orbits. In particular, structurally stable vector fields in two dimensions cannot have
594:. Thom envisaged applications of this theory to biological systems. Both Smale and Thom worked in direct contact with MaurĂcio Peixoto, who developed
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and Anosov flows. One celebrated example of Anosov flow is given by the geodesic flow on a surface of constant negative curvature, cf
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which means that the qualitative behavior of the trajectories is unaffected by small perturbations (to be exact
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439:), which are all non-degenerate (hyperbolic), and do not have saddle-to-saddle connections. Furthermore, the
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574:, who oversaw translation of their monograph into English. Ideas of structural stability were taken up by
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It is important to note that topological equivalence is realized with a loss of smoothness: the map
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developed a parallel theory of stability for differentiable maps, which forms a key part of
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have been determined in the foundational paper of
Andronov and Pontryagin. According to the
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191:). An important class of structurally stable systems in arbitrary dimensions is given by
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and his school in the 1960s in the context of hyperbolic dynamics. Earlier,
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by
Andronov, Witt, and Khaikin. The term "structural stability" is due to
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trajectories, which enormously complicate the dynamics, as discovered by
404:), but it is unknown whether the same property holds in the strong case.
220:
203:, motivated by the work of Andronov and Pontryagin, developed and proved
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179:. They announced a characterization of rough systems in the plane, the
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Necessary and sufficient conditions for the structural stability of
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discovered that hyperbolic automorphisms of the torus, such as the
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Structural stability of non-singular smooth vector fields on the
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can be investigated using the theory developed by
Poincaré and
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Geometric
Methods in the Theory of Differential Equations
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of the circle is structurally stable if and only if its
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and are inward oriented. This space is endowed with the
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Examples of such qualitative properties are numbers of
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and flows. During the late 1950s and the early 1960s,
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514:at the periodic points is different from 1, see
420:that are transversal to the boundary and on the
183:. In this case, structurally stable systems are
175:in 1937 under the name "systĂšmes grossiers", or
167:Structurally stable systems were introduced by
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48:introducing citations to additional sources
289:if for any sufficiently small perturbation
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262:that are transversal to the boundary of
38:Relevant discussion may be found on the
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654:Rahman, Aminur; Blackmore, D. (2023).
355:belongs to a suitable neighborhood of
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709:(1988) . V. I. Arnold (ed.).
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429:AndronovâPontryagin criterion
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741:Encyclopedia of Mathematics
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715:[Coarse systems].
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330:. If, moreover, for any
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129:-small perturbations).
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758:"Structural stability"
754:MaurĂcio Matos Peixoto
734:D. V. Anosov (2001) ,
703:Andronov, Aleksandr A.
530:Anosov diffeomorphisms
235:−1)-dimensional
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59:"Structural stability"
16:Concept in mathematics
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367:is called (strongly)
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555:celestial mechanics
369:structurally stable
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682:10.1137/21M1426572
630:Superstabilization
625:Self-stabilization
592:singularity theory
559:Aleksandr Lyapunov
551:three-body problem
534:Hadamard billiards
433:equilibrium states
169:Aleksandr Andronov
142:Lyapunov stability
786:Dynamical systems
752:Charles Pugh and
707:Lev S. Pontryagin
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304:: there exists a
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211:Definition
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408:Examples
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