About 5 months ago I attended a conference at UNC, Chapel Hill organized by Idris Assani. As I mentioned in this post here, after a kind invitation of Idris, I got the task of writing a survey article (before August 1, 2012) on homoclinic/heteroclinic bifurcations to appear in the proceedings of the conference.
As it turns out, I finished a preliminary version of this survey within the deadline (and this was fortunate because I could take one week of vacations without bad conscious ) and, here I am accomplishing my promise (also made in this post here) of posting the (currently submitted) survey text here.
My plan was then to close this post at this point. However, after re-reading the text, I thought that a better idea is to start a series of posts including some deleted scenes from the initial text (and hence strictly containing the material of the survey). In order words, the idea is to have a series of posts giving access to the “original version” of the survey text above.
The posts of this series will be entitled “Homoclinic/heteroclinic bifurcations: X” where introduction, Newhouse phenomena, thin horseshoes, fat horseshoes, and slightly fat horseshoes. In particular, as the title of the today’s post indicates, this series begins right now (just below the fold).
In his seminal work (in 1890) on Celestial Mechanics, Henri Poincaré (29 April 1854 — 17 July 1912) emphasized the relevance of the concept of homoclinic orbits (bi-asymtoptic solutions in his notation) in Dynamical Systems by stating:
“Rien n’est plus propre à nous donner une idée de la complication du problème des trois corps et en général de tous les problèmes de dynamique …”
(in a free translation to English: “Nothing is more adequate to give us an idea of the complexity of the -body problem and in general of all problems in dynamics …”)
In fact, the history behind the introduction of this notion is fascinating: in a few words, H. Poincaré submitted a first version of his work to a concours in honor of G. Mittag-Leffler and financially support by the king Oscar of Sweden, but, after some comments of L. Phragmén, it was discovered a mistake in part of his text related to the presence of homoclinic orbits. For nice accounts in English and French (resp.) on this beautiful chapter of the history of Dynamical Systems, see the book of June Barrow-Green and the article of J.-C. Yoccoz (resp.) on this subject.
In modern language, we define an homoclinic orbit as follows. Given a diffeomorphism of a compact (boundaryless) manifold , denote by the -th iterate of , . Let be a periodic of with minimal period , i.e., and is minimal with this property. We say that the orbit of a point is homoclinic to whenever as , that is, the orbit of is accumulates the orbit of the periodic point both in the past and the future.
Similarly, given two periodic points with (minimal) periods (resp.), and in distinct orbits, i.e., for all , we say that the orbit of a point is heteroclinic to and whenever as and as .
G. Birkhoff was one of the first to “confirm” the predictions of H. Poincaré on homoclinic orbits by proving in 1935 that, “in general”, one can find periodic orbits of very high period near homoclinic orbits.
Later on, by taking as a source of inspiration the works of G. Birkhoff on homoclinic orbits, and Cartwright and Littlewood (here, here and here), and Levinson on differential equations similar to the Van der Pol equation (a differential equation steaming from Engineering problems related to nonlinear oscillators and radio waves), Steve Smale proposed in 1967 a geometrical model (discovered in the beaches of Rio) nowadays called Smale’s horseshoe explaining in a very satisfactory way the mechanism responsible for the dynamical complexity near a “general” homoclinic orbit.
In the subsection below, we will quickly revisit some features of Smale’s horseshoe as a paradigm of hyperbolic set of a dynamical system. The basic references for historical and mathematical details on the material in the remainder of this post is the classical book of J. Palis and F. Takens.
2. Transverse homoclinic orbits and Smale’s horseshoes
Let be a diffeomorphism, and let be a periodic point of . For sake of simplicity, let’s assume that is a fixed point, i.e., : this can be achieved by replacing by some iterate , and, as far as the discussion of this section is concerned, this replacement has no serious effect. The stable and unstable sets of are
In this notation, is homoclinic to if and only if , and is heteroclinic to and if and only if .
For a “generic” , the fixed point is hyperbolic, i.e., the differential is a linear map without eigenvalues of norm . In this case, by the stable manifold theorem (see Appendix 1 of Palis-Takens book), they are injectively immersed submanifolds of of dimensions and , where , , and , resp. , are the stable and unstable subspaces of , i.e., the generalized eigenspaces of associated to the eigenvalues of norm strictly smaller, resp. larger, than .
We say that is a transverse homoclinic orbit to a hyperbolic fixed point when the stable and unstable manifolds of intersect transversally at , that is, and . By transversality theory (or more precisely, Kupka-Smale’s theorem), for a “generic” , all homoclinic orbits to fixed/periodic points are transverse.
The basic picture near a transverse homoclinic orbit is the following one:
Henri Poincaré said (also in the work) the following about the dynamics of this basic picture:
“On sera frappé de la complexité de cette figure, que je ne cherche même pas à tracer.”
(in a free translation to English: “One would be extremely surprised by the complexity of this picture, that I’ll not even try to draw.”)
The “complicated picture” that H. Poincaré is talking about can be visualized by iterating the compact pieces of the stable and unstable manifolds of in the basic picture. By doing so, one gets the following figure:The formal justification for this picture comes from the so-called -lemma of J. Palis (see page 155 of Palis-Takens book) saying that given any -dimensional disk transverse to , then the future iterates , , of converge to . Pictorially, the -lemma can be illustrated as follows:
From the -lemma one can justify Poincaré’s picture by starting with the red segment drawn below (see the topmost picture) and iterating it. We know (by the -lemma) that the iterates of the red segment tend to grow and approach . In particular, by iterating sufficiently many times, a new transverse homoclinic orbit is created (as indicated in the center picture below), so that one can apply the -lemma again and eventually one ends up with the complicated figure mentioned by Poincaré (see the bottommost picture below):
Remark 1 A more complete quotation to the paragraph (in the work) where H. Poincaré talks about homoclinic orbits (or bi-asymptotic solutions) is:
“Que l’on cherche à se représenter la figure formée par ces deux courbes et leurs intersections en nombre infini dont chacune correspond à une solution doublement asymptotique. Ces intersections forment une sorte de treillis, de tissu, de réseau à maille infiniment serrées ; chacune de ces deux courbes ne doit jamais se recouper elle-même, mais elle doit se replier sur elle-même de manière infiniment complexe pour venir recouper une infinité de fois toutes les mailles du réseau. On sera frappé de la complexité de cette figure, que je ne cherche même pas à tracer. Rien n’est plus propre à nous donner une idée de la complication du problème des trois corps et en général de tous les problèmes de dynamique où il n’y a pas d’intégrale uniforme et où les séries de Bohlin sont divergentes.”
Remark 2 As Jacob explained to me once, the name -lemma came from the fact that the original proof involved a quantity measuring the inclination (slope) of the disk with respect to . Then, Jacob noticed that it was a bad idea to call -lemma and he proposed to rename it “inclination lemma”, but it was too late to change. By the way, as one of Jacob’s students, Ricardo Mañé knew this history and he, Paulo Sad and Dennis Sullivan decided to tease Jacob by calling -lemma one of the main results of this paper here.
In theory, this complicated picture is extremely discouraging when trying to understand the dynamics near transverse homoclinic orbits. However, by carefully inspecting this picture, S. Smale discovered the fundamental picture (near a transverse homoclinic orbits to hyperbolic fixed points):
In a nutshell, this picture means that, near a transverse homoclinic point to a hyperbolic fixed point , one can find a rectangle containing and such that some iterate of maps in the “horseshoe”-shaped region shown above. Moreover, the picture was drawn to “convince” the reader that the action of the differential of on uniformly contracts any “almost” horizontal direction and uniformly expand any “almost” vertical direction.
Using these facts, S. Smale proved that the maximal invariant set consisting of all points in whose orbit under never escapes is a hyperbolic set, that is, there are constants , and a splitting for each such that:
- the splitting is -invariant: and ;
- is uniformly contracted and is uniformly expanded: for all , , , where is a norm associated to some choice of Riemannian metric on .
Remark 3 In the case of our picture above, there is no mystery behind the choice of the splitting: is an almost horizontal direction and is an almost vertical direction.
Furthermore, by using the hyperbolicity of the set , S. Smale showed that the dynamics of restricted to is topologically conjugated to a Bernoulli shift in two symbols, that is, there exists a homeomorphism such that where is given by . Indeed, the construction of is done by analyzing the itinerary of the orbits of points . More precisely, denoting by and the connected components of containing and (resp.), see Figure 1, one associates to each , the itinerary of with respect to , i.e., one defines by the condition . Then, by definition, , and, by using the hyperbolicity properties of , one can show that is an homeomorphism (see pages 27–30 of Palis-Takes book). In resume, the dynamics of can be modeled by a Markov process.
Among the several striking consequences of S. Smale’s results, we observe that the set of periodic orbits of is dense in and the dynamical system is sensitive to initial conditions (that is, two nearby distinct points tend to get far apart after an appropriate number of iterations of the dynamics) simply because the same is true for the Bernoulli shift ! In particular, S. Smale’s results allows to recover the result of G. Birkhoff (mentioned above) that the transverse homoclinic point of the hyperbolic periodic point is accumulated by periodic orbits of of arbitrarily high period.
By obvious reason, the maximal invariant set was baptized horseshoe by S. Smale. Partly motivated by this, we introduce the following concepts:
- is -invariant, that is, ;
- there are constants , and a splitting for each with:
- and ;
- for all , , , where is a norm associated to some choice of Riemannian metric on .
In other words, is a hyperbolic set of a diffeomorphism whenever is -invariant and the infinitesimal dynamics of , i.e., the dynamics of the differential , over (or, more precisely, on ) completely decomposes into two -equivariant subbundles and such that is a stable subbundle (that is, it is forwardly contracted by ) and is an unstable subbundle (that is, it is backwardly contracted by ).
Example 1 The orbit of a hyperbolic period point of period is a trivial (i.e., finite) hyperbolic set, while Smale’s horseshoes are non-trivial (i.e., infinite) hyperbolic sets.
One of the key features of hyperbolic sets is the fact that the infinitesimal information on the structure of over a hyperbolic set imposes a certain number of global geometrical constraints on the dynamics of on . For example, given , denote by and the stable and unstable sets of . In general, the stable and unstable sets of an arbitrary point of an arbitrary diffeomorphism may have a wild geometry. On the other hand, as we already mentioned, it is known that the stable and unstable sets of hyperbolic periodic points are injectively immersed submanifolds thanks to the stable manifold theorem. In other words, the geometry of stable and unstable sets improves under appropriate hyperbolicity conditions, and, as it turns out, it is possible to generalize the stable manifold theorem to show that the stable and unstable sets of any point in a hyperbolic set has well-behaved stable and unstable sets:
Theorem 2 (Generalized stable manifold theorem) Let be a hyperbolic set of a -diffeomorphism , . Then, the stable set of any is an injectively immersed -submanifold of dimension and, for all sufficiently small , the local stable set
is a embedded disk in of dimension . map associating to stable sets depend continuously on and . Furthermore, the map is continuous.
Another way of phrasing the previous theorem is: given a hyperbolic set , the family of stable sets of points form a continuous lamination with leaves.
Actually, this is not the full statement of the generalized stable manifold theorem. For more complete statements see Appendix 1 of Palis-Takens book and references therein (especially the classical book of M. Hirsch, C. Pugh and M. Shub).
Coming back to the discussion of Smale’s horseshoes, it turns out that they are not arbitrary hyperbolic sets in the sense that they fit the following definitions:
Definition 3 A set is a basic set of a diffeomorphism if is an infinitehyperbolic set such that
- is transitive, i.e., there exists whose orbit is dense in ;
- is locally maximal, i.e., there exists a neighborhood of such that the maximal invariant set of coincides with , that is, .
Definition 4 A set is a (uniformly hyperbolic) horseshoe of a diffeomorphism if is a basic set of of saddle-type, i.e., both subbundles and appearing in Definition 1 are non-trivial.
From the qualitative point of view, a (uniformly hyperbolic) horseshoe of a diffeomorphism behaves exactly as a Smale’s horseshoe near a transverse homoclinic orbit. For instance, it is possible to show that the restriction to is topologically conjugated to a Markov shift of finite type. In particular, is topologically a Cantor set, and, despite the fact that the dynamics of is chaotic (e.g., in the sense that it is sensitive to initial conditions), one can reasonably understand because it topologically modeled by a Markov process. Actually, it is possible to prove that also has plenty of interesting properties from the statistical (ergodic) point of view: for example, supports several ergodic -invariant probabilities coming from the so-called thermodynamical formalism of R. Bowen, D. Ruelle and Y. Sinai. See e.g. the book of R. Bowen for a nice account on this subject.
Therefore, we can “declare” that the local dynamics near transverse homoclinic orbits, or, more generally, uniformly hyperbolic horseshoes, is well-understood, and hence we can start the discussion of the local dynamics near homoclinic tangencies (i.e., non-transverse homoclinic orbits).
2.1. Homoclinic tangencies: an introduction
Let be a (uniformly hyperbolic) horseshoe of a , , diffeomorphism of a compact surface (i.e., -dimensional manifold) possessing a periodic point associated to a quadratic homoclinic tangency, that is, the stable and unstable manifolds (i.e, curves in our current setting) and of meet tangentially at a point and the order of contact between and at is , that is, the curves and are tangent at but their curvatures differ at .
The main geometrical features of a quadratic homoclinic tangency are captured by the following picture:
For sake of simplicity, we’ll assume that there are two neighborhoods of the horseshoe and of the homoclinic orbit of such that
In other words, we’ll suppose that the local dynamics of on consists precisely of the horseshoe and the homoclinic orbit of tangency , that is, locally (on ) the interesting dynamical phenomena come exclusively from the horseshoe and .
Note that the maximal invariant set
capturing the local dynamics of on is not a hyperbolic set. Indeed, it is not hard to convince oneself that the natural candidate for the stable , resp. unstable , direction at in Definition 1 is the -dimensional direction , resp. . However, since and meet tangentially at , one would have , so that the condition in Definition 1 is never fulfilled.
On the other hand, since and the single orbit is the sole responsible for the non-hyperbolicity of , we still completely understand the local dynamics of on .
Now, let’s try to understand the local dynamics on of a -diffeomorphism -close to . Consider a sufficiently small -neighborhood of such that the dynamically relevant objects in Figure 2 above admit a continuation for any : more precisely, we select so that, for any , the maximal invariant set
is a (uniformly hyperbolic) horseshoe, the periodic point has a continuation into a nearby (hyperbolic) periodic point of , and the compact curve , resp. , inside the stable, resp. unstable, manifold , resp. containing and and crossing has a continuation into a nearby compact curve , resp. , in the stable, resp. unstable, manifold of crossing .
Using these dynamical objects associated to , we can organize the parameter space by writing where
- whenever and don’t intersect;
- whenever and have a quadratic tangency at a point in ;
- whenever and have two transverse intersection points in .
Since corresponds to a quadratic tangency of , we have that is a codimension hypersurface dividing into the two connected open subsets and . The picture below illustrates the decomposition of the parameter space and the features on phase space of the elements of , and .
From the (local) dynamical point of view, the regions and of the parameter space are not particularly interesting: in fact, by inspecting the definitions, it is not hard to show that
- for any , and
- for any .
In other words, all potentially new dynamical phenomena, i.e., homoclinic bifurcations, come from , that is, after non-trivially unfolding the quadratic tangency associated to diffeomorphisms in .
Next time, we will start our discussion from this point and our goal will be to explain (via the so-called Newhouse phenomena) why one can’t hope to understand the dynamics of all diffeomorphisms in .