Posted by: matheuscmss | August 25, 2012

Homoclinic/heteroclinic bifurcations: introduction

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 {5} 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 {X=} 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).

1. Introduction

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 {3}-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 {f:M\rightarrow M} of a compact (boundaryless) manifold {M}, denote by {f^n = \underbrace{f \circ\dots\circ f}_{n}} the {n}-th iterate of {f}, {n\in\mathbb{Z}}. Let {p\in M} be a periodic of {f} with minimal period {k}, i.e., {f^{k}(p)=p} and {k\in\mathbb{N}} is minimal with this property. We say that the orbit {\{f^n(q): n\in\mathbb{Z}\}} of a point {q\neq p} is homoclinic to {p} whenever {f^{nk}(q)\rightarrow p} as {n\rightarrow\pm\infty}, that is, the orbit of {q} is accumulates the orbit of the periodic point {p} both in the past and the future.

Similarly, given two periodic points {p_1, p_2\in M} with (minimal) periods {k_1, k_2} (resp.), and in distinct orbits, i.e., {p_2\neq f^n(p_1)} for all {n\in\mathbb{N}}, we say that the orbit of a point {q\neq p_1, p_2} is heteroclinic to {p_1} and {p_2} whenever {f^{n k_1}(q)\rightarrow p_1} as {n\rightarrow-\infty} and {f^{n k_2}(q)\rightarrow p_2} as {n\rightarrow+\infty}.

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 {f:M\rightarrow M} be a {C^k} diffeomorphism, {k\geq 1} and let {p\in M} be a periodic point of {f}. For sake of simplicity, let’s assume that {p} is a fixed point, i.e., {f(p)=p}: this can be achieved by replacing {f} by some iterate {f^n}, and, as far as the discussion of this section is concerned, this replacement has no serious effect. The stable and unstable sets of {p} are

\displaystyle W^s(p):=\{q\in M: f^n(q)\rightarrow p \textrm{ as } n\rightarrow+\infty\}

and

\displaystyle W^u(p):=\{q\in M: f^n(q)\rightarrow p \textrm{ as } n\rightarrow-\infty\}

In this notation, {q} is homoclinic to {p} if and only if {q\in (W^s(p)\cap W^u(p))-\{p\}}, and {q} is heteroclinic to {p_1} and {p_2} if and only if {q\in (W^u(p_1)-\{p_1\})\cap (W^s(p_2)-\{p_2\})}.

For a “generic” {f}, the fixed point is hyperbolic, i.e., the differential {df(p):T_pM\rightarrow T_pM} is a linear map without eigenvalues of norm {1}. In this case, by the stable manifold theorem (see Appendix 1 of Palis-Takens book), they are injectively immersed {C^k} submanifolds of {M} of dimensions {s} and {u}, where {s=\textrm{dim}(E^s)}, {u=\textrm{dim}(E^u)}, and {E^s}, resp. {E^u}, are the stable and unstable subspaces of {df(p)}, i.e., the generalized eigenspaces of {df(p)} associated to the eigenvalues of norm strictly smaller, resp. larger, than {1}.

We say that {q} is a transverse homoclinic orbit to a hyperbolic fixed point {p} when the stable and unstable manifolds of {p} intersect transversally at {q\neq p}, that is, {q\in (W^s(p)\cap W^u(p))-\{p\}} and {T_qM=T_q W^s(p)\oplus T_qW^u(p)}. By transversality theory (or more precisely, Kupka-Smale’s theorem), for a “generic” {f}, 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 {p} in the basic picture. By doing so, one gets the following figure:The formal justification for this picture comes from the so-called {\lambda}-lemma of J. Palis (see page 155 of Palis-Takens book) saying that given any {u}-dimensional disk {D} transverse to {W^s(p)}, then the future iterates {f^n(D)}, {n\in\mathbb{N}}, of {D} converge to {W^u(p)}. Pictorially, the {\lambda}-lemma can be illustrated as follows:

From the {\lambda}-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 {\lambda}-lemma) that the iterates of the red segment tend to grow and approach {W^u(p)}. 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 {\lambda}-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 {\lambda}-lemma came from the fact that the original proof involved a quantity {\lambda} measuring the inclination (slope) of the disk {D} with respect to {W^s(p)}. Then, Jacob noticed that it was a bad idea to call {\lambda}-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 {\lambda}-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):

Figure 1. Smale’s horseshoe.

In a nutshell, this picture means that, near a transverse homoclinic point {q} to a hyperbolic fixed point {p}, one can find a rectangle {R} containing {p} and {q} such that some iterate {F=f^N} of {f} maps {R} in the “horseshoe”-shaped region {f^N(R)} shown above. Moreover, the picture was drawn to “convince” the reader that the action of the differential {dF} of {F} on {R} uniformly contracts any “almost” horizontal direction and uniformly expand any “almost” vertical direction.

Using these facts, S. Smale proved that the maximal invariant set {\Lambda:=\bigcap\limits_{n\in\mathbb{Z}}f^{nN}(R) = \bigcap\limits_{n\in\mathbb{Z}}F^{n}(R)} consisting of all points in {R} whose orbit under {F} never escapes {R} is a hyperbolic set, that is, there are constants {C>0}, {0<\lambda<1} and a splitting {T_xM=E^s(x)\oplus E^u(x)} for each {x\in\Lambda} such that:

  • the splitting is {dF}-invariant: {dF(x)\cdot E^s(x)=E^s(F(x))} and {dF(E^u(x))=E^u(F(x))};
  • {E^s} is uniformly contracted and {E^u} is uniformly expanded: {\|dF^n(x)\cdot v^s\|, \|dF^{-n}(x)\cdot v^u\|\leq C\lambda^n\|v\|} for all {n\geq 0}, {v^s\in E^s(x)}, {v^u\in E^u(x)}, where {\|.\|} is a norm associated to some choice of Riemannian metric on {M}.

Remark 3 In the case of our picture above, there is no mystery behind the choice of the splitting: {E^s(x)} is an almost horizontal direction and {E^u(x)} is an almost vertical direction.

Furthermore, by using the hyperbolicity of the set {\Lambda}, S. Smale showed that the dynamics of {F} restricted to {\Lambda} is topologically conjugated to a Bernoulli shift in two symbols, that is, there exists a homeomorphism {h:\Lambda\rightarrow\Sigma:=\{0,1\}^{\mathbb{Z}}} such that {h(F(x))=\sigma(h(x))} where {\sigma:\Sigma\rightarrow\Sigma} is given by {\sigma((a_i)_{i\in\mathbb{Z}})=(a_{i+1})_{i\in\mathbb{Z}}}. Indeed, the construction of {h} is done by analyzing the itinerary of the orbits of points {x\in\Lambda}. More precisely, denoting by {R_0} and {R_1} the connected components of {R\cap f^N(R)} containing {p} and {q} (resp.), see Figure 1, one associates to each {x\in\Lambda}, the itinerary {h(x)=(a_i)_{i\in\mathbb{Z}}} of {\{F^n(x): n\in\mathbb{Z}\}} with respect to {R_0\cup R_1}, i.e., one defines {a_n\in\{0,1\}} by the condition {F^n(x)\in R_{a_n}}. Then, by definition, {h(F(x))=\sigma(h(x))}, and, by using the hyperbolicity properties of {\Lambda}, one can show that {h} is an homeomorphism (see pages 27–30 of Palis-Takes book). In resume, the dynamics of {F|_{\Lambda}} 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 {F} is dense in {\Lambda} and the dynamical system {F|_{\Lambda}} 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 {\sigma}! In particular, S. Smale’s results allows to recover the result of G. Birkhoff (mentioned above) that the transverse homoclinic point {q} of the hyperbolic periodic point {p} is accumulated by periodic orbits of {f} of arbitrarily high period.

By obvious reason, the maximal invariant set {\Lambda} was baptized horseshoe by S. Smale. Partly motivated by this, we introduce the following concepts:

Definition 1 We say that a compact subset {\Lambda\subset M} is a hyperbolic setof a diffeomorphism {f:M\rightarrow M} if

  • {\Lambda} is {f}-invariant, that is, {f(\Lambda)=\Lambda};
  • there are constants {C>0}, {0<\lambda<1} and a splitting {T_xM=E^s(x)\oplus E^u(x)} for each {x\in\Lambda} with:
    • {df(x)\cdot E^s(x)=E^s(f(x))} and {df(E^u(x))=E^u(f(x))};
    • {\|df^n(x)\cdot v^s\|, \|df^{-n}(x)\cdot v^u\|\leq C\lambda^n\|v\|} for all {n\geq 0}, {v^s\in E^s(x)}, {v^u\in E^u(x)}, where {\|.\|} is a norm associated to some choice of Riemannian metric on {M}.

In other words, {\Lambda\subset M} is a hyperbolic set of a diffeomorphism {f:M\rightarrow M} whenever {\Lambda} is {f}-invariant and the infinitesimal dynamics of {f}, i.e., the dynamics of the differential {df}, over {\Lambda} (or, more precisely, on {T_{\Lambda}M}) completely decomposes into two {df}-equivariant subbundles {E^s} and {E^u} such that {E^s} is a stable subbundle (that is, it is forwardly contracted by {df}) and {E^u} is an unstable subbundle (that is, it is backwardly contracted by {df}).

Example 1 The orbit {\Lambda=\{p,\dots, f^{k-1}(p)\}} of a hyperbolic period point {p} of period {k} 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 {df} over a hyperbolic set {\Lambda} imposes a certain number of global geometrical constraints on the dynamics of {f} on {\Lambda}. For example, given {x\in M}, denote by {W^s(x)=\{y\in M: \textrm{dist}(f^n(y), f^n(x))\rightarrow 0 \textrm{ as }n\rightarrow+\infty\}} and {W^u(x)=\{y\in M: \textrm{dist}(f^n(y), f^n(x))\rightarrow 0 \textrm{ as }n\rightarrow-\infty\}} the stable and unstable sets of {x}. 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 {\Lambda\subset M} be a hyperbolic set of a {C^k}-diffeomorphism {f:M\rightarrow M}, {k\geq 1}. Then, the stable set {W^s(x)} of any {x\in\Lambda} is an injectively immersed {C^k}-submanifold of dimension {\textrm{dim}(E^s(x))} and, for all sufficiently small {\varepsilon>0}, the local stable set

\displaystyle W^s_{loc}(x)=\{y\in W^s(x): \textrm{dist}(f^n(y),f^n(x))\leq\varepsilon\}

is a {C^k} embedded disk in {W^s(x)} of dimension {\textrm{dim}(E^s(x))}. map associating to {x\in\Lambda} stable sets {W^s(x)} depend continuously on {x\in\Lambda} and {f}. Furthermore, the map {\Lambda\ni x\mapsto W^s_{loc}(x)\subset M} is continuous.

Another way of phrasing the previous theorem is: given a hyperbolic set {\Lambda}, the family of stable sets {W^s(x)} of points {x\in\Lambda} form a continuous lamination with {C^k} 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 {\Lambda\subset M} is a basic set of a diffeomorphism {f:M\rightarrow M} if {\Lambda} is an infinitehyperbolic set such that

  • {\Lambda} is transitive, i.e., there exists {x\in\Lambda} whose orbit {\{f^n(x)\}_{n\in\mathbb{Z}}} is dense in {\Lambda};
  • {\Lambda} is locally maximal, i.e., there exists a neighborhood {U} of {\Lambda} such that the maximal invariant set {\bigcap\limits_{n\in\mathbb{Z}}f^n(U)} of {U} coincides with {\Lambda}, that is, {\bigcap\limits_{n\in\mathbb{Z}}f^n(U)=\Lambda}.

Definition 4 A set {\Lambda\subset M} is a (uniformly hyperbolic) horseshoe of a diffeomorphism {f:M\rightarrow M} if {\Lambda} is a basic set of {f} of saddle-type, i.e., both subbundles {E^s} and {E^u} appearing in Definition 1 are non-trivial.

From the qualitative point of view, a (uniformly hyperbolic) horseshoe {\Lambda} of a diffeomorphism {f} behaves exactly as a Smale’s horseshoe near a transverse homoclinic orbit. For instance, it is possible to show that the restriction {f} to {\Lambda} is topologically conjugated to a Markov shift of finite type. In particular, {\Lambda} is topologically a Cantor set, and, despite the fact that the dynamics of {f|_{\Lambda}} is chaotic (e.g., in the sense that it is sensitive to initial conditions), one can reasonably understand {f|_{\Lambda}} because it topologically modeled by a Markov process. Actually, it is possible to prove that {f|_{\Lambda}} also has plenty of interesting properties from the statistical (ergodic) point of view: for example, {\Lambda} supports several ergodic {f}-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 {K} be a (uniformly hyperbolic) horseshoe of a {C^k}, {k\geq 2}, diffeomorphism {f:M\rightarrow M} of a compact surface (i.e., {2}-dimensional manifold) {M} possessing a periodic point {p\in K} associated to a quadratic homoclinic tangency, that is, the stable and unstable manifolds (i.e, curves in our current setting) {W^s(p)} and {W^u(p)} of {p} meet tangentially at a point {q\in (W^s(p)\cap W^u(p))-K} and the order of contact between {W^s(p)} and {W^u(p)} at {q} is {1}, that is, the curves {W^s(p)} and {W^u(p)} are tangent at {q} but their curvatures differ at {q}.

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 {U} of the horseshoe {K} and {V} of the homoclinic orbit {\mathcal{O}(q):=\{f^n(q):n\in\mathbb{Z}\}} of {q} such that

\displaystyle \bigcap\limits_{n\in\mathbb{Z}} f^n(U\cup V)=K\cup \mathcal{O}(q).

In other words, we’ll suppose that the local dynamics of {f} on {U\cup V} consists precisely of the horseshoe {K} and the homoclinic orbit of tangency {\mathcal{O}(q)}, that is, locally (on {U\cup V}) the interesting dynamical phenomena come exclusively from the horseshoe and {\mathcal{O}(q)}.

Figure 2. Localization of the dynamics via the neighborhoods U (of the horseshoe) and V (of the tangency).

Note that the maximal invariant set

\displaystyle \Lambda_f:=\bigcap\limits_{n\in\mathbb{Z}} f^n(U\cup V)

capturing the local dynamics of {f} on {U\cup V} is not a hyperbolic set. Indeed, it is not hard to convince oneself that the natural candidate for the stable {E^s(q)}, resp. unstable {E^u(q)}, direction at {q} in Definition 1 is the {1}-dimensional direction {T_qW^s(p)}, resp. {T_qW^u(p)}. However, since {W^s(p)} and {W^u(p)} meet tangentially at {q}, one would have {E^s(q)=T_qW^s(p)=T_qW^u(p)=E^u(q)}, so that the condition {T_qM = E^s(q)\oplus E^u(q)} in Definition 1 is never fulfilled.

On the other hand, since {\Lambda_f=K\cup\mathcal{O}(q)} and the single orbit {\mathcal{O}(q)} is the sole responsible for the non-hyperbolicity of {\Lambda_f}, we still completely understand the local dynamics of {f} on {U\cup V}.

Now, let’s try to understand the local dynamics on {U\cup V} of a {C^k}-diffeomorphism {g:M\rightarrow M} {C^k}-close to {f}. Consider {\mathcal{U}} a sufficiently small {C^k}-neighborhood of {f} such that the dynamically relevant objects in Figure 2 above admit a continuation for any {g\in\mathcal{U}}: more precisely, we select {\mathcal{U}} so that, for any {g\in\mathcal{U}}, the maximal invariant set

\displaystyle K_g=\bigcap\limits_{n\in\mathbb{Z}}g^{n}(U)

is a (uniformly hyperbolic) horseshoe, the periodic point {p} has a continuation into a nearby (hyperbolic) periodic point {p_g} of {g}, and the compact curve {c^s(f)}, resp. {c^u(f)}, inside the stable, resp. unstable, manifold {W^s(p)}, resp. {W^u(p)} containing {p} and {q} and crossing {V} has a continuation into a nearby compact curve {c^s(g)}, resp. {c^u(g)}, in the stable, resp. unstable, manifold of {p_g} crossing {V}.

Using these dynamical objects associated to {g\in\mathcal{U}}, we can organize the parameter space {\mathcal{U}} by writing {\mathcal{U}=\mathcal{U}_-\cup \mathcal{U}_0\cup \mathcal{U}_+} where

  • {g\in\mathcal{U}_-} whenever {c^s(g)} and {c^u(g)} don’t intersect;
  • {g\in\mathcal{U}_0} whenever {c^s(g)} and {c^u(g)} have a quadratic tangency at a point {q_g} in {V};
  • {g\in\mathcal{U}_+} whenever {c^s(g)} and {c^u(g)} have two transverse intersection points in {V}.

Since {q} corresponds to a quadratic tangency of {f}, we have that {\mathcal{U}_0} is a codimension {1} hypersurface dividing {\mathcal{U}} into the two connected open subsets {\mathcal{U_-}} and {\mathcal{U}_+}. The picture below illustrates the decomposition {\mathcal{U}=\mathcal{U}_-\cup \mathcal{U}_0\cup \mathcal{U}_+} of the parameter space and the features on phase space of the elements of {\mathcal{U}_-}, {\mathcal{U}_0} and {\mathcal{U}_+}.

Organization of the parameter space \mathcal{U}

From the (local) dynamical point of view, the regions {\mathcal{U}_-} and {\mathcal{U}_0} of the parameter space {\mathcal{U}} are not particularly interesting: in fact, by inspecting the definitions, it is not hard to show that

  • {\Lambda_g=K_g} for any {g\in\mathcal{U}_-}, and
  • {\Lambda_g=K_g\cup \mathcal{O}(q_g)} for any {g\in\mathcal{U}_0}.

In other words, all potentially new dynamical phenomena, i.e., homoclinic bifurcations, come from {\mathcal{U}_+}, that is, after non-trivially unfolding the quadratic tangency associated to diffeomorphisms in {\mathcal{U}_0}.

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 {\mathcal{U}_0}.


Responses

  1. A very concise and clear post! I would like to read the following posts on these topics.


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