In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, structural stability is a fundamental property of a
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
which means that the qualitative behavior of the trajectories is unaffected by small perturbations (to be exact
''C''1-small perturbations).
Examples of such qualitative properties are numbers of
fixed points and
periodic orbit In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.
Iterated functions
Given a ...
s (but not their periods). Unlike
Lyapunov stability
Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. ...
, 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
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
s,
vector fields on
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s and
flows generated by them, and
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two m ...
s.
Structurally stable systems were introduced by
Aleksandr Andronov
Aleksandr Aleksandrovich Andronov (russian: Алекса́ндр Алекса́ндрович Андро́нов; , Moscow – October 31, 1952, Gorky) was a Soviet physicist and member of the Soviet Academy of Sciences (1946). He worked exten ...
and
Lev Pontryagin
Lev Semenovich Pontryagin (russian: Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet mathematician. He was born in Moscow and lost his eyesight completely due ...
in 1937 under the name "systèmes grossiers", or rough systems. They announced a characterization of rough systems in the plane, the
Andronov–Pontryagin criterion The Andronov–Pontryagin criterion is a necessary and sufficient condition for the stability of dynamical systems in the plane. It was derived by Aleksandr Andronov and Lev Pontryagin in 1937.
Statement
A dynamical system
: \dot = v(x),
wh ...
. In this case, structurally stable systems are ''typical'', 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
strange attractor
In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...
). An important class of structurally stable systems in arbitrary dimensions is given by
Anosov diffeomorphism
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "cont ...
s and flows.
Definition
Let ''G'' be an
open domain in R
''n'' with
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
closure and smooth (''n''−1)-dimensional
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
*Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film
*Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
. Consider the space ''X''
1(''G'') consisting of restrictions to ''G'' of ''C''
1 vector fields on R
''n'' that are transversal to the boundary of ''G'' and are inward oriented. This space is endowed with the ''C''
1 metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathema ...
in the usual fashion. A vector field ''F'' ∈ ''X''
1(''G'') is weakly structurally stable if for any sufficiently small perturbation ''F''
1, the corresponding flows are
topologically equivalent In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct of flows, are important in the study of iterated fun ...
on ''G'': there exists a
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
''h'': ''G'' → ''G'' which transforms the oriented trajectories of ''F'' into the oriented trajectories of ''F''
1. If, moreover, for any ''ε'' > 0 the
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
''h'' may be chosen to be ''C''
0 ''ε''-close to the identity map when ''F''
1 belongs to a suitable neighborhood of ''F'' depending on ''ε'', then ''F'' is called (strongly) structurally stable. These definitions extend in a straightforward way to the case of ''n''-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 ''h'' must be a
topological conjugacy In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct of flows, are important in the study of iterated func ...
.
It is important to note that topological equivalence is realized with a loss of smoothness: the map ''h'' 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 naïve ''C''
1 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 ''X''
1(''G''), but it is unknown whether the same property holds in the strong case.
Examples
Necessary and sufficient conditions for the structural stability of ''C''
1 vector fields on the unit disk ''D'' that are transversal to the boundary and on the
two-sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
''S''
2 have been determined in the foundational paper of Andronov and Pontryagin. According to the
Andronov–Pontryagin criterion The Andronov–Pontryagin criterion is a necessary and sufficient condition for the stability of dynamical systems in the plane. It was derived by Aleksandr Andronov and Lev Pontryagin in 1937.
Statement
A dynamical system
: \dot = v(x),
wh ...
, such fields are structurally stable if and only if they have only finitely many singular points (
equilibrium states) and periodic trajectories (
limit cycle
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity ...
s), which are all non-degenerate (hyperbolic), and do not have saddle-to-saddle connections. Furthermore, the
non-wandering set
In dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and mixing. When a dynamical system has a wandering set of non-zero measure, then the system is a dissipative system. This is the opposit ...
of the system is precisely the union of singular points and periodic orbits. In particular, structurally stable vector fields in two dimensions cannot have
homoclinic
In mathematics, a homoclinic orbit is a trajectory of a flow of a dynamical system which joins a saddle equilibrium point to itself. More precisely, a homoclinic orbit lies in the intersection of the stable manifold and the unstable manifold of ...
trajectories, which enormously complicate the dynamics, as discovered by
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
.
Structural stability of non-singular smooth vector fields on the
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
can be investigated using the theory developed by Poincaré and
Arnaud Denjoy
Arnaud Denjoy (; 5 January 1884 – 21 January 1974) was a French mathematician.
Biography
Denjoy was born in Auch, Gers. His contributions include work in harmonic analysis and differential equations. His integral was the first to be able t ...
. Using the
Poincaré recurrence map
Poincaré is a French surname. Notable people with the surname include:
* Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science
* Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré
* Luc ...
, the question is reduced to determining structural stability of diffeomorphisms of the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
. As a consequence of the
Denjoy theorem, an orientation preserving ''C''
2 diffeomorphism ''ƒ'' of the circle is structurally stable if and only if its
rotation number
In mathematics, the rotation number is an invariant of homeomorphisms of the circle.
History
It was first defined by Henri Poincaré in 1885, in relation to the precession of the perihelion of a planetary orbit. Poincaré later proved a theore ...
is rational, ''ρ''(''ƒ'') = ''p''/''q'', and the periodic trajectories, which all have period ''q'', are non-degenerate: the
Jacobian of ''ƒ''
''q'' at the periodic points is different from 1, see
circle map
In mathematics, particularly in dynamical systems, Arnold tongues (named after Vladimir Arnold) Section 12 in page 78 has a figure showing Arnold tongues. are a pictorial phenomenon that occur when visualizing how the rotation number of a dynamica ...
.
Dmitri Anosov
Dmitri Victorovich Anosov (russian: Дми́трий Ви́кторович Ано́сов; November 30, 1936 – August 7, 2014) was a Russian mathematician active during the Soviet Union, he is best known for his contributions to dynamical syste ...
discovered that hyperbolic automorphisms of the torus, such as the
Arnold's cat map, are structurally stable. He then generalized this statement to a wider class of systems, which have since been called
Anosov diffeomorphism
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "cont ...
s and Anosov flows. One celebrated example of Anosov flow is given by the geodesic flow on a surface of constant negative curvature, cf
Hadamard billiards
In physics and mathematics, the Hadamard dynamical system (also called Hadamard's billiard or the Hadamard–Gutzwiller model) is a chaotic dynamical system, a type of dynamical billiards. Introduced by Jacques Hadamard in 1898, and studied by Ma ...
.
History and significance
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
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
on the
three-body problem
In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...
in
celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
. Around the same time,
Aleksandr Lyapunov
Aleksandr Mikhailovich Lyapunov (russian: Алекса́ндр Миха́йлович Ляпуно́в, ; – 3 November 1918) was a Russian mathematician, mechanician and physicist. His surname is variously romanized as Ljapunov, Liapunov, Liapo ...
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 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
George Birkhoff
George David Birkhoff (March 21, 1884 – November 12, 1944) was an American mathematician best known for what is now called the ergodic theorem. Birkhoff was one of the most important leaders in American mathematics in his generation, and during ...
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
oscillation
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
s by Andronov, Witt, and Khaikin. The term "structural stability" is due to
Solomon Lefschetz
Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear o ...
, who oversaw translation of their monograph into English. Ideas of structural stability were taken up by
Stephen Smale
Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics facult ...
and his school in the 1960s in the context of hyperbolic dynamics. Earlier,
Marston Morse
Harold Calvin Marston Morse (March 24, 1892 – June 22, 1977) was an American mathematician best known for his work on the ''calculus of variations in the large'', a subject where he introduced the technique of differential topology now known a ...
and
Hassler Whitney
Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, characteristic classes, and geometric integration t ...
initiated and
René Thom
René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958.
He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became w ...
developed a parallel theory of stability for differentiable maps, which forms a key part of
singularity theory
In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...
. Thom envisaged applications of this theory to biological systems. Both Smale and Thom worked in direct contact with
Maurício Peixoto
Maurício Matos Peixoto, (April 15, 1921, in Fortaleza, Ceará – April 28, 2019, in Rio de Janeiro), was a Brazilian engineer and mathematician. He pioneered the studies on structural stability, and was the author of Peixoto's theorem.
Biogra ...
, who developed
Peixoto's theorem In the theory of dynamical systems, Peixoto's theorem, proved by Maurício Peixoto, states that among all smooth flows on surfaces, i.e. compact two-dimensional manifolds, structurally stable systems may be characterized by the following properti ...
in the late 1950s.
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 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
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
. 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
Morse–Smale systems.
See also
*
Homeostasis
In biology, homeostasis (British English, British also homoeostasis) Help:IPA/English, (/hɒmɪə(ʊ)ˈsteɪsɪs/) is the state of steady internal, physics, physical, and chemistry, chemical conditions maintained by organism, living systems. Thi ...
*
Self-stabilization
Self-stabilization is a concept of fault-tolerance in distributed systems. Given any initial state, a self-stabilizing distributed system will end up in a correct state in a finite number of execution steps.
At first glance, the guarantee of self ...
,
superstabilization
*
Stability theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial diffe ...
References
*
*
*
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Differential equations
Dynamical systems
Stability theory