Structural stability
If \(\mathcal{D}\) is the set of self-diffeomorphisms of a compact smooth manifold \(M\ ,\) and \(\mathcal{D}\) is equipped with the \(C^{1}\) topology then \(f \in \mathcal{D}\) is structurally stable if and only if for each \(g\) in some neighborhood of \(f\) in \(\mathcal{D}\) there is a homeomorphism \(h : M \rightarrow M\) such that
commutes. That is, \(f\) is topologically conjugate to each nearby \(g\ .\)
For flows, \(\mathcal{D}\) is replaced by \(\mathcal{X}\ ,\) the set of smooth vector fields on \(M\) equipped with the \(C^{1}\) topology, and the flow generated by \(X \in \mathcal{X}\) is structurally stable if and only if for each \(Y\) in some neighborhood of \(X\) in \(\mathcal{X}\) there is a homeomorphism \(h : M \rightarrow M\) that sends the orbits of \(X\) to the orbits of \(Y\ ,\) preserving the orientation of the orbits.
For every manifold, structurally stable diffeomorphisms and flows form non-empty open subsets of \(\mathcal{D}\) and \(\mathcal{X}\) (Palis and Smale 1970)
The Pre-History of Structural Stability
In 1881, Poincaré, motivated by problems in celestial mechanics, initiated the publication of a series of papers which had far reaching consequences (Poincaré 1882). For the first time, an autonomous ordinary differential equation on a differentiable manifold \(M\ ,\) also called a vector field, flow, or dynamical system on \(M\ ,\) was considered from the point of view of the geometry of the set of its trajectories. Typical of this point of view is the Poincaré-Bendixson Theorem. Thus was born the geometric or qualitative theory of flows on \(M\ .\)
If \(X\) is a vector field on \(M\ ,\) the partition of \(M\) into its set of trajectories of \(X\) is the phase portrait of \(X\ .\) So the qualitative or geometric theory of Poincarée deals with the geometry of the phase portrait of differential equations.
In 1892 there appeared the thesis of Liapunov creating the remarkable theory of stability of individual orbits of a flow (Liapunov 1947). Although heavily analytical in its methods, the theory also has a distinctive qualitative flavor.
During the first half of the last century we have the important work of G. D. Birkhoff on flows, and done very much in the spirit of Poincaré (Birkhoff 1927).
Since about 1890, following Cantor's ideas, there has been a slow inexorable trend to put all branches of mathematics on a set theoretic basis. By 1950 these ideas had been generally accepted by mathematicians and had become so to speak the law of the land, with emphasis on such things as sets, structures, equivalence relations, and so on. In this connection the name of Bourbaki comes to the fore because of his brave attempt to somehow control or direct this vast multifarious movement.
Nevertheless the qualitative theory of Poincaré and Birkhoff, marginal to these developments and lacking well defined goals, lost its luster (Peixoto 1989). We clarify this by making two points.
- No formal topological structure, no space, was built from the set of differential equations.
- Neither Poincaré nor Birkhoff ever made precise what they meant by saying that two differential equations were qualitatively equivalent.
Structural stability remedies both of these shortcomings. As a result, dynamical systems became a thriving subject of research and entered the mainstream of mathematics. Steve Smale is widely recognized as the main architect of this; see especially (Smale 1967).
The Foundational Paper of Andronov and Pontrjagin
Structural stability of dynamical systems was introduced in 1937, albeit with a different name, by Andronov and Pontrjagin in a short note (Andronov and Pontrjagin 1937). Consider the dynamical system \[ \frac{dx}{dt} = P(x,y) \quad \frac{dy}{dt} = Q(x,y) \] defined on the disc \(D^{2}\) in the xy-plane, the vector field \((P, Q)\) entering transversally across the boundary \(\partial D^{2}\ .\)
Definition. The system above is said to be rough (grossier since the note was written in French) if, given \(\epsilon > 0\ ,\) we can find a \(\delta > 0\) such that whenever \(p(x,y), q(x,y)\ ,\) together with their first derivatives, are \(< \delta \) in absolute value, then the perturbed system \[ \frac{dx}{dt} = P(x,y) + p(x, y) \quad \frac{dy}{dt} = Q(x,y) + q(x,y) \] is such that there exists an \(\epsilon\)-homeomorphism \(h : D^{2} \rightarrow D^{2}\) (i.e., \(h\) moves each point of \(D^{2}\) by less than \(\epsilon\)) which transforms the trajectories of the original system to the trajectories of the perturbed system. As discussed below, the word rough was changed to structurally stable.
Suppose now that the original system is such that:
- Singularities and closed orbits are hyperbolic. (See below.)
- There is no trajectory connecting saddle points.
Then Andronov and Pontrjagin announced the following:
Theorem. A necessary and sufficient condition for the system \[ \frac{dx}{dt} = P(x,y) \quad \frac{dy}{dt} = Q(x,y) \] to be structurally stable is that the conditions (i) and (ii) are satisfied.
Proofs were given by DeBaggis (1952) but mistakes there made it necessary to write another proof (Peixoto and Peixoto 1959).
Hyperbolicity is an eigenvalue condition. The eigenvalues of the linear part of a vector field at a hyperbolic singularity have nonzero real part. In the case of a closed orbit \(\gamma\ ,\) we choose a point \(p \in \gamma\) and a transversal \(\tau\) to \(\gamma\) at \( p \ .\) The vector field's flow defines a local diffeomorphism (the first-return map) of \(\tau\) to itself having \(p\) as a fixed point. Hyperbolicity of \(\gamma\) means that the eigenvalues of the linear part of the first return map at \(p\) have absolute value different from 1.
The \(\epsilon\)-homeomorphism \(h\) considered in the definition above goes in the direction of giving remedy to item (b) in the preceding section. The no saddle connection condition (ii) above prompted Smale to introduce Thom transversality theory in dynamical systems.
In 1937 A. Andronov and S. Khaikin published in Russian a book, Theory of Oscillations, in which "rough" equations are mentioned many times in the context of physical applications (Andronov, Vitt, and Khaikin 1949). This work reflects the activities of a number of people in the former Soviet Union referred to as the Gorki School.
In 1949, Solomon Lefschetz directed the translation of this book into English and had the word "rough" translated as "structurally stable." By doing so, he made an important contribution to the subject, namely conveying the interplay between the concepts of stability and a certain structure, the phase portrait, subject to topological equivalence. Besides, the very concepts of "structure" and "stability" are fundamental for so many human endeavors, inside and outside mathematics, and the expression "structural stability" is such a happy choice of words that its use has spread far and wide. A Google search reports more than a million webpages with the expression, in fields that include civil engineering, soil science, thermodynamics, human physiology, electrical engineering, cosmology, and psychology, to name a few.
Structural Stability in Dimension Two
In 1959 Peixoto introduced the space of \(C^{r}\) flows, \(r \geq 1\ ,\) on the 2-disc \(D^{2}\ ,\) an infinite dimensional Banach space \(\mathcal{X} = \mathcal{X}^{r}(D^{2})\ ,\) thus addressing point (1) of #The Pre-History of Structural Stability (Peixoto 1959). He then made the definition of structural stability that appears at the outset of this article\[X \in \mathcal{X}\] is structurally stable if it has a neighborhood \(\mathcal{N} \subset \mathcal{X}\) such that for each \(Y \in \mathcal{N}\) there is a homeomorphism of the disc to itself sending orbits of \(X\) to orbits of \(Y\ .\) It was then shown that on the 2-disc the new definition is equivalent to the Andronov-Pontrjagin \(\epsilon\)-definition and that the set SS of structurally stable systems is open in \(\mathcal{X}^{r}(D^{2})\ .\) In (Peixoto 1959a), it is further shown that SS is dense in \(\mathcal{X}\ ,\) and investigation of its topological properties was begun, even for flows on the \(n\)-disc (Peixoto 1959b).
On a compact two dimensional manifold \(M^{2}\ ,\) Peixoto considered the space \(\mathcal{X} = \mathcal{X}^{r}(M^{2})\) of \(C^{r}\) flows equipped with the \(C^{r}\) topology, \(r \geq 1\) (Peixoto 1962). When \(M^{2}\) is orientable he showed that SS is open-dense in \(\mathcal{X} \ ,\) and he characterized the structurally stable flows as those having hyperbolic periodic orbits, no other recurrence, and no saddle connections. (An orbit is recurrent if it is contained in its own \(\omega \)-limit set or its own \(\alpha \)-limit set.)
When \(M^{2}\) is not orientable and \(r \geq 2\ ,\) the density of SS in \(\mathcal{X}^{r}\) remains an open question, except for three specific surfaces — the projective plane, the Klein bottle, and the pretzel (Gutierrez 1978). When \(r = 1\ ,\) local use of Pugh's Closing Lemma makes orientability irrelevant in the proof of Peixoto's theorems (Pugh 1967). In the same vein, it should be noted that perturbations in the \(C^{r}\) topology, \(r \geq 2\ ,\) may need to be global, not local (Gutierrez 1987).
Structural Stability in Dimension Greater Than Two
A Morse-Smale dynamical system has only trivial recurrence, only finitely many periodic orbits, all of which are hyperbolic, and the invariant manifolds of the periodic orbits meet transversally. For flows on surfaces, Peixoto's Theorem can be summarized as \[\textrm{MS = SS}\] and \(\textrm{SS}\) is generic. Smale initially speculated that the same holds in higher dimensions, but he soon realized dynamics gets more interesting there. Although he and Jacob Palis proved that \(\textrm{MS } \Rightarrow \textrm{ SS}\) (Palis and Smale 1970), his horseshoe map (as well as the Arnold Cat Map, also known as the Thom diffeomorphism of the 2-torus) shows that a structurally stable system can have infinitely many periodic orbits (Smale 1965). That is, \(\textrm{MS } \not \!\!\Leftarrow \textrm{ SS}\ .\) Furthermore, structural stability is not generic: there are some dynamical systems that cannot be approximated by structurally stable ones (Smale 1966). The set of structurally stable systems is open but not dense.
Trying to unify Morse-Smale systems, the horseshoe, and Anosov's globally hyperbolic systems (Anosov 1967), Smale devised Axiom A (Smale 1967). For a diffeomorphism \(f : M \rightarrow M\) it requires
- The nonwandering set of \(f\ ,\) \(\Omega (f)\ ,\) is hyperbolic.
- The periodic orbits are dense in \(\Omega (f) \ .\)
If, in addition, the stable and unstable manifolds of all points of \(\Omega\) meet one another transversally then \(f\) is said to satisfy Axiom A and Strong Transversality. A similar definition holds for flows. These systems are called AS systems, an abbreviation that can also be read as "Anosov-Smale." Clearly \(\textrm{MS } \subset \textrm{ AS}\ .\) Palis and Smale formally make the conjecture that \(\textrm{AS = SS}\) in (Palis and Smale 1970).
In 1971 Joel Robbin proved that if \(f\) is \(C^{2}\) and AS then it is structurally stable, and in 1976 Clark Robinson reduced the \(C^{2}\) hypothesis to \(C^{1}\ .\) They also proved the result for flows. Thus they showed \(\textrm{AS } \Rightarrow \textrm{ SS}\) (Robbin 1971, Robinson 1976, Robinson 1975). Then, in 1988 Ricardo Mañé proved the converse for diffeomorphisms, and later S. Hu and S. Hayashi completed the picture for flows, yielding the satisfying characterization of structural stability in all dimensions: SS = AS (Mañé 1988, Hayashi 1997, Hu 1994).
A byproduct of these proofs is that the \(\epsilon\)-definition and the non-\(\epsilon\)-definition of structural stability are equivalent. In fact, John Guckenheimer, John Franks, Ricardo Mañé, and others had already proved that AS is equivalent to stronger forms of the \(\epsilon\)-definition in which the conjugacy is required to depend nicely on the perturbation (Franks 1973, Franks 1974, Guckenheimer 1972, Mañé 1975).
By the way, in (Anosov 1967) Dimitrii Anosov gives a thorough discussion of structural stability, emphasizing the \(\epsilon\) approach. It is reasonable to say that every proof of structural stability succeeded or failed according to the \(\epsilon\) definition, but that the non-\(\epsilon\) definition is more natural when considering the class of all structurally stable systems.
Finally we mention that for non-invertible maps, the definition of structural stability makes sense. Notable theorems in this regard are that Anosov endomorphisms of compact manifolds and generic rational maps of the Riemann sphere are structurally stable (Mañé and Pugh 1975, Mañé et. al. 1983, Lyubich 1983).
Theorems of the Structural Stability Type
Structural stability is a natural concept and demands broader influence. Here is a Bourbaki-style definition into which the dynamics structural stability notion fits nicely.
Defn. If a set is equipped with a topology and an equivalence relation then its structurally stable elements are those interior to the equivalence classes. The "structure" is whatever is preserved by the equivalence relation; its structure remains the same when a structurally stable element is perturbed.
For discrete dynamical systems the set is \(\mathcal{D} = \operatorname{Diffeo}(M)\ ,\) equipped with the \(C^{1}\) topology, and the equivalence relation is topological conjugacy. For flows the space is \(\mathcal{X}\) and the equivalence relation is orbit equivalence. Discrete dynamical systems and flows are actions by the groups \(\mathbb{Z}\) and \(\mathbb{R}\ .\) For actions of more general groups the equivalence relation is similar: orbits are sent to orbits by a homeomorphism.
Structural stability behaves non-trivially on restrictions or extensions of the space and equivalence relation. If \(X\) is the rotation vector field on the 2-sphere (its orbits are the latitudes and the poles) then \(X\) is structurally stable when considered in the subspace of \(\mathcal{X}\) consisting of divergence free vector fields, but it is not structurally stable in \(\mathcal{X}\ .\) Likewise, if the equivalence relation of topological conjugacy in \(\mathcal{D}\) is changed to smooth conjugacy then the set of structurally stable diffeomorphisms is empty.
Here are five other examples of the structural stability type in dynamics.
- Ω-stability. A dynamical system is \(\Omega\)-stable if, under perturbations, the dynamics on the perturbed nonwandering set is conjugate to the dynamics on the original nonwandering set. Clearly \(\textrm{SS} \Rightarrow \Omega S\ ,\) and it was hoped that although SS is not generic, perhaps \(\Omega\)S is. Ralph Abraham and Steve Smale demonstrate that this is not true (Abraham and Smale 1970). Nevertheless, Palis adapted Mañé's techniques to show that \(\Omega\)-stability is equivalent to Axiom A and no cycles among Smale's basic sets (Palis 1987).
- Finite stability. Jorge Sotomajor proposed that a diffeomorphism \(f\) be called finitely structurally stable if there exist a \(C^{1}\) neighborhood \(\mathcal{N}\) of \(f\) and a \(C^{1}\) embedded \(n\)-disc \(D \subset \mathcal{N}\) such that \(f \in D\) and each \(g \in \mathcal{N}\) is conjugate to some member of \(D\ .\) Robinson and Williams show that finite structural stability is not generic (Robinson and Williams 1973).
- Future stability. If the decomposition of the phase space into sets of points with mutually asymptotic orbits is stable under perturbation, the system is said to be future stable. Mike Shub and Bob Williams show that future stability is not generic (Shub and Williams 1969).
- Prevalence stability. Jim Yorke and others proposed an idea to speak of "almost every" in function spaces (Hunt et al. 1992, Kaloshin 1997). Then a system is prevalence stable if almost every element of its neighborhood is conjugate to it.
- Tolerance stability. Christopher Zeeman proposed an equivalence idea that weakens the conjugacy requirement inspired by (Zeeman 1962). For each \(\epsilon > 0\) the diffeomorphism \(f\) is required to have a neighborhood \(\mathcal{N}\) such that for all pairs \(g, g^{\prime} \in \mathcal{N}\ ,\) each \(g\)-orbit lies in the \(\epsilon\)-neighborhood of a \(g^{\prime}\)-orbit. See also (Ikegami 1983, Takens 1971).
Here are a few examples outside dynamics in which the structural stability viewpoint makes sense.
- The space consists of smooth maps of the plane to itself and the equivalence relation is that there is a diffeomorphism of the plane to itself carrying the singular set of one map to the singular set of the other. Whitney showed that the \(C^{3}\) structurally stable maps are those with cusp and fold singularities (Whitney 1955).
- The space consists of all smooth maps of the circle into the plane and two maps are considered equivalent if there is a diffeomorphism of the plane to itself carrying the image of one map to the image of the other. The structurally stable elements in this context are the embeddings and self-transverse immersions.
- The space consists of all Riemann metrics on a smooth compact surface and the equivalence relation is that the web of curvature lines in one metric are homeomorphic the web in the other. The set of structurally stable metrics in this sense are open and dense in the set of all Riemann metrics (Gutierrez and Sotomajor 1982).
- The space consists of all Riemann metrics on a smooth compact surface and the equivalence relation is that the focal decompositions of the tangent bundle are homeomorphic. The set of structurally stable metrics in this sense is open and dense in the set of all Riemann metrics of negative curvature (Peixoto and Pugh 2007).
- The space consists of parameterized curves into the space of vector fields, and the equivalence relation is that for \(\beta , \gamma : [0, 1] \rightarrow \mathcal{X}\) there is a homeomorphism \(h : [0, 1] \rightarrow [0, 1]\) such that for each \(s \in [0, 1]\ ,\) \(\beta (s)\) is conjugate to \(\gamma (s)\ .\) This is the realm of bifurcation theory.
- René Thom proposed a theory of morphogenesis — the time-evolution of form — viewing it as structural stability punctuated by catastrophes (Thom 1976).
The inventive reader can extend structural stability further throughout mathematics.
References
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- Zeeman C. (1962) The topology of the brain and visual perception. Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) pp. 240-256 Prentice-Hall, Englewood Cliffs, N.J.
Internal references
- Yuri A. Kuznetsov (2007) Conjugate maps. Scholarpedia, 2(12):5420.
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- Jonathan Bard (2008) Morphogenesis. Scholarpedia, 3(6):2422.
- Michael Shub (2007) Morse-Smale systems. Scholarpedia, 2(3):1785.
- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
- David H. Terman and Eugene M. Izhikevich (2008) State space. Scholarpedia, 3(3):1924.
External Links
See Also
Anosov Diffeomorphism, Bifurcations, Morse-Smale Systems, Stability