Validated Computation of Heteroclinic Sets

# Validated Computation of Heteroclinic Sets

Maciej J. Capiński AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków. Email: mcapinsk@agh.edu.pl. The work was partially supported by the Polish National Science Center grant 2012/05/B/ST1/00355.    J.D. Mireles James Florida Atlantic University, Department of Mathematical Sciences, 777 Glades Rd, Boca Raton, FL 33431, United States. Email: jmirelesjames@fau.edu. The work was partially supported by NSF grant DMS - 1318172.
###### Abstract

In this work we develop a method for computing mathematically rigorous enclosures of some one dimensional manifolds of heteroclinic orbits for nonlinear maps. Our method exploits a rigorous curve following argument build on high order Taylor approximation of the local stable/unstable manifolds. The curve following argument is a uniform interval Newton method applied on short line segments. The definition of the heteroclinic sets involve compositions of the map and we use a Lohner-type representation to overcome the accumulation of roundoff errors. Our argument requires precise control over the local unstable and stable manifolds so that we must first obtain validated a-posteriori error bounds on the truncation errors associated with the manifold approximations. We illustrate the utility of our method by proving some computer assisted theorems about heteroclinic invariant sets for a volume preserving map of .

Key words. invariant manifolds, heteroclinic orbits, rigorous enclosure of level sets, computer assisted proof, validated numerics, parameterization method, Newton’s method

AMS subject classifications. 34C30, 34C45, 37C05, 37D10, 37M99, 65G20, 70K44.

## 1 Introduction

The study of a nonlinear dynamical system begins by considering the fixed/periodic points, their linear stability, and their local stable and unstable manifolds. In order to patch this local information into a global picture of the dynamics one then wants to understand the connecting orbits. Questions about connecting orbits are naturally reformulated as questions about where and how the stable and unstable manifolds intersect.

A typical situation is that we consider a hyperbolic fixed point, so that the dimension of the stable manifold plus the dimension of the unstable manifold add up to the dimension of the whole space. A transverse intersection of the stable/unstable manifolds of a hyperbolic fixed point is again a single point. Such intersections give rise to homoclinic connecting orbits, and they are of special interest. For example Smale’s tangle theorem says that the existence of a transverse homoclinic intersection point implies the existence of hyperbolic chaotic motions [1].

More generally, consider a pair of distinct fixed points and assume that the dimensions and of their stable and unstable manifolds have , with being the dimension of the ambient space. In this case a transverse intersection of the manifolds results in a set which is (locally) a dimensional manifold of connecting orbits. For example we are sometimes interested in transport barriers or separatrices formed by codimension one stable or unstable manifolds [2].

In the present work we develop a computer assisted argument for proving the existence of one dimensional intersections between stable/unstable manifolds of distinct fixed points. Our arguments utilize the underlying dynamics of the map in order to obtain the alpha/omega limit sets of the intersection manifolds. In addition to abstract existence results our argument yields error bounds on the location of the intersection in phase space. Moreover we see that transversality of the intersections follows as a natural corollary, so that the heteroclinic intersections we obtain are in fact normally hyperbolic invariant manifolds.

The two main ingredients in our argument are high order numerical computation of the local stable/unstable manifolds with mathematically rigorous bounds on the truncation error, and a validated branch following algorithm used to rigorously enclose the intersections of these manifolds. Our treatment of the stable/unstable manifolds is based on the Parameterization method for invariant manifolds [3, 4, 5, 6, 7]. The Parameterization Method is a general functional analytic framework for studying invariant manifolds which reduces questions about the manifold to questions about the solutions of a certain nonlinear operator equation. Reformulating the problem in terms of an operator equation facilitates the design of efficient numerical algorithms for computing the manifold, and introduces a notion of a-posteriori error for the numerical approximations. A-posteriori analysis of the operator leads to computer assisted bounds on the truncation errors.

Next we use these parametric representations of the local stable/unstable manifolds to recast the desired heteroclinic intersection as the one-dimensional zero set of a certain finite dimensional map. An approximate zero set is computed numerically via a Newton/continuation scheme, and a uniform Newton-Krawczyk argument is applied along the dimensional approximate zero set yields a mathematically rigorous enclosure of the true solution. This validated branch following is complicated by the fact that the finite dimensional map contains terms given by multiple compositions of the underlying nonlinear dynamical system. A carefully chosen coordinate system is defined along the branch which allows us to mitigate the so called “wrapping effect”. We mention that a number of other authors have developed methods for validated computation of zero sets, and refer the interested reader to the works of [8, 9, 10, 11, 12, 13], and the references discussed therein for a more complete overview of the literature.

We illustrate our method in a concrete example, and study some global heteroclinic sets in , which are defined as the transverse intersection of the unstable and stable manifolds for a pair of distinct fixed points of the Lomelí map. This map is defined by Equation (LABEL:eq:LomeliMap) in Section LABEL:sec:vortexBubbles, where we discuss the map and its dynamics in more detail. Figures LABEL:fig:LomeliMap1 and LABEL:fig:LomeliMap2 illustrate the unstable and stable manifolds, as well as their intersections, for two different choices of parameters for the Lomelí map.

Figure LABEL:fig:LomeliMap1 illustrates the map with parameter values , , , , and . The map with these parameters was also studied in [14]. The upper left frame suggests that the intersection of and is a system of arcs beginning and ending at the fixed points. The numerically computed intersection of the manifolds is shown as a collection of green points in the lower right frame. In Section LABEL:sec:intersections (see Section LABEL:sec:arc-CAP, in particular) we prove the existence of two distinct intersection arcs whose iterates generate all six curves shown in the lower right frame.

Figure LABEL:fig:LomeliMap2 illustrates the map with parameter values , , , , and . Again, the map with these parameters was also studied in Section of [15], see especially the bottom left frame of figure in that reference. For these parameter values the heteroclinic intersections of and is a system of loops, as we see islands of red surrounded by blue in the upper left frame. The lower right frame illustrates the numerically computed intersection of the manifolds as a collection of green points. In Section LABEL:sec:intersections (see Section LABEL:sec:loop-CAP, in particular) we prove the existence of a single intersection loop whose iterates generate the system of loops shown in the lower right frame.

The methods of the present work provide computer assisted proof that the heteroclinic invariant sets suggested by these pictures actually exist. While we implement our methods only for intersection arcs for the Lomelí family in , it is clear that the theoretical framework developed here applies much more generally. Indeed, employing the rigorous numerical methods for multiparameter continuation recently developed in [11] it should be possible to adapt our methods to the study of higher dimensional manifold intersections.

The remainder of the paper is organized as follows. In Section LABEL:sec:prelim we establish some notation and review some preliminary material including the definitions and basic properties of the stable and unstable manifolds of fixed points, the definitions and basic properties of heteroclinic invariant sets. We also discuss the dynamics of the Lomelí map. In Section LABEL:sec:parmMeth we review the basic notions of the Parameterization Methods for stable/unstable manifolds of fixed points, and illustrate the formal computation of the Taylor expansion of the parameterization. We recall an a-posteriori validation theorem which allows us to obtain rigorous computer assisted bounds on the truncation errors. Finally in Section LABEL:sec:intersections we develop and implment the main tools of the paper, namely the curve following/continuation argument used to enclose the heteroclinic arcs. All of the codes used to obtain the results in this paper can be found at the web page [16].

## 2 Preliminaries

### 2.1 Notations

Throughout this paper we shall use to denote a ball of radius about . To simplify notations we will also write for a ball centered at zero, and for a ball centered at zero with radius . Similarly, for let denote the usual absolute value for complex numbers and for define the norm

 ∥z∥Ck=max1≤i≤k|zi|.

For and let

 Dk(z,R)={w∈Ck:∥w−z∥Ck

denote the poly-disk of radius about . We write for the ball centered at zero and for the unit ball centered at zero.

If we consider , then we will use , for the projections onto the coordinates, respectively.

We now write out the interval arithmetic notations conventions that will be used in the paper. Let be a subset of . We shall denote by an interval enclosure of the set , that is, a set

 [U]=Πki=1[ai,bi]⊂Rk,

such that . Similarly, for a family of matrixes we denote its interval enclosure as , that is, a set

 [A]=([aij,bij])i=1,...,kj=1,...,m⊂Rk×m,

such that . For , by we shall denote an interval enclosure

 [DF(U)]=[{A∈Rk×m|Aij∈[infx∈U∂Fi∂xj(x),supx∈U∂Fi∂xj(x)]}].

For a set and a family of matrixes we shall use the notation to denote an interval enclosure

 [A][U]=[{Au:A∈[A],u∈[U]}].

We shall say that a family of matrixes is invertible, if each matrix is invertible. We shall also use the notation

 [A]−1[U]=[{A−1u:A∈[A],u∈[U]}].

### 2.2 Stable/unstable manifolds of fixed points

The material in this section is standard and can be found in any textbook on the qualitative theory of dynamical systems. We refer for example to [17, 18]. Let be a smooth (in our case analytic) map and assume that is a fixed point of . Let be a neighborhood of , and define the set

 Wsloc(p,U)={w∈U:fn(w)∈Ufor all n≥0}.

This set is referred to as the local stable set of relative to .

If is a hyperbolic fixed point, i.e. if none of the eigenvalues of are on the unit circle, then the stable manifold theorem gives that there exists a neighbourhood of so that is an -dimensional embedded disk tangent to the stable eigenspace at . If is analytic then the embedding is analytic.

The invariant set

 Ws(p) ={w∈Rn:limn→∞fn(w)=p},

is the stable manifold of , and consists of all orbits which accumulate under forward iteration of the map to the fixed point . If is invertible then we have that

 Ws(p)=∞⋃n=0f−n[Wsloc(p,U)],

i.e. the stable manifold is obtained as the union of all backwards iterates of a local stable manifold.

We say that the sequence is a backward orbit of if

 f(xj)=xj+1,

for all . If and for all we say that has a backward orbit in . If

 limj→−∞xj=p,

we say that has a backward orbit accumulating at . Let be an open neighborhood of and define the set

 Wuloc(p,U)={w∈U:w has a backward orbit in U}.

We refer to this as a local unstable manifold of relative to . If is hyperbolic, then the unstable manifold theorem gives that there exists an open neighborhood of so that is a dimensional embedded disk, tangent at to the unstable eigenspace of . If is analytic the embedding is analytic. If is invertible then the unstable manifold of under can be seen as the stable manifold of under . In the case that is invertible, the unstable manifold of is

 Wu(p) ={w∈Rn:limn→∞f−n(w)=p} =∞⋃n=0fn[Wuloc(p,U)].

### 2.3 Heteroclinic Arcs

Suppose that are hyperbolic fixed points of an invertible map , and that the invariant manifolds and are of dimension and respectively. Assume that . If is a point in the transverse intersection of and it then follows that there is an arc having that and that

 γ(s)⊂Wu(p1)∩Ws(p2),for alls∈[−a,a].

Moreover this intersection is transverse, hence is as smooth as . We refer to as a heteroclinic arc.

Since the point , is heteroclinic from to , it follows that the set

 S(γ)=⋃n∈Zfn(γ)⊂Wu(p1)∩Ws(p2),

is invariant. Moreover the entire set is heteroclinic from to so that

 ¯¯¯¯¯¯¯¯¯¯¯¯S(γ)=S(γ)∪{p1}∪{p2},

i.e. accumulates only at the fixed points. Then is a compact invariant set. We refer to as the heteroclinic invariant set generated by .

Suppose that can be continued to a longer curve , i.e. suppose that there is with and . Then

 ¯¯¯¯¯¯¯¯¯¯¯¯S(γ)⊂¯¯¯¯¯¯¯¯¯¯¯¯S(~γ).

We are interested in the largest compact invariant set so obtained, and single out two cases of particular interest.

###### Definition 2.1 (fundamental heteroclinic loop)

Suppose that , i.e. is a closed loop. Then the heteroclinic invariant set is the union of countably many closed loops accumulating at and . If each point is a point of transverse intersection of and then the closed loop has no self intersections. Moreover the lack of self intersections propagates under forward and backward iteration as is a diffeomorphism. In this case we refer to as a fundamental heteroclinic loop.

An example of a heteroclinic loop connection for the Lomelí map is given in Figure LABEL:fig:LomeliMap2. This connection is generated by a single loop, propagated under forward and backward iterations of .

###### Definition 2.2 (m-fold fundamental heteroclinic arc)

If for some and for , then the -th iterate of the arc is a continuation of . (If then reparameterize.) We refer to as an -fold fundamental heteroclinic arc. Now

 Sm(γ):=⋃i∈Zfim(γ)

is itself an arc connecting and , which we refer to as the heteroclinic path generated by .

For any the set is another heteroclinic path from to . If each point is a point of transverse intersection of and then for each , the sets are disjoint. We refer to

 ¯¯¯¯¯¯¯¯¯¯¯¯S(γ)=m−1⋃j=0fj(¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Sm(γ))=⋃i∈Zfi(γ)∪{p1}∪{p2},

as an -fold heteroclinic branched manifold, as is composed of paths (or branches).

An example of a heteroclinic branched manifold for the Lomelí map is given in Figure LABEL:fig:LomeliMap1. It consists of six paths, which are generated by two distinct -fold fundamental heteroclinic arcs.

In both the case of the heteroclinic arcs, and the case of heteroclinic loops, the compact invariant sets are maximal with respect to , in the sense that no larger invariant set can be obtained by continuation of the arc . In either case we refer to as a fundamental heteroclinic arc. Note that under the assumption that arises as the one dimensional transverse intersection of smooth manifolds, the classification theorem for one dimensional manifolds gives that only these two cases occur.

### 2.4 Vortex bubbles and the Lomelí map

In the sequel we restrict our attention to a particular dynamical configuration known as a vortex bubble. Heteroclinic arcs play an important role in the study of vortex bubbles, and vortex bubbles in provide a non-trivial application, which can still be completely visualized. Vortex bubbles appear in the fluid dynamics and plasma physics literature as a model of turbulent circulation. See for example [19, 20, 21] and the references discussed therein. At present we provide a brief qualitative sketch sufficient to our needs.

The main features of a vortex bubble are as follows. Consider a volume preserving diffeomorphism (which could arise as a time one map of a volume preserving flow) having a pair of distinct hyperbolic fixed points . We suppose that has two dimensional unstable manifold, and that has two dimensional stable manifold. Moreover we suppose that the unstable eigenvalues at and the stable eigenvalues at occur in complex conjugate pair, hence the linear dynamics at each fixed point is rotational.

We assume that the rotation is in the same direction at the fixed points, and also that the curvature at and of the local unstable/stable manifolds is such that the manifolds bend or “cup” toward one another. Should the two dimensional global stable/unstable manifolds enclose a region we say that a bubble (or resonance bubble) is formed. Under these conditions it is not unusual that the circulation at the fixed points drives a circulation throughout the bubble, in which case we say that there is a vortex bubble. Inside the vortex bubble one may find invariant circles and tori, as well as complicated chaotic motions.

The situation just described is sketched in Figure LABEL:fig:bubbleSketch. An important global consideration is the intersection of the two dimensional unstable and stable manifolds which, if transverse, gives rise to heteroclinic arcs as discussed in Section LABEL:sec:heteroArcs.

One elementary mathematical model exhibiting vortex bubble dynamics is the five parameter family of quadratic volume preserving maps

 f(x,y,z)=⎛⎜ ⎜⎝z+α+τx+ax2+bxy+cy2xy⎞⎟ ⎟⎠, \hb@xt@.01(2.1)

with . We refer to this as the Lomelí family, or simply as the Lomelí map. The map is a natural generalization of the Hénon map from two to three dimensions, and is the subject of a number of studies [22, 23, 24, 15]. In particular the Lomelí map provides a normal form for volume preserving quadratic diffeomorphisms with quadratic inverse, and is a toy model for turbulent fluid/plasma flow near a vortex [23, 15, 14, 25].

For typical parameters the map has two hyperbolic fixed points and with stability as discussed above. The global embedding of the stable and unstable manifolds for the system is illustrated in two specific instances in Figures LABEL:fig:LomeliMap1 and LABEL:fig:LomeliMap2 of the Introduction. These computations suggest that both case 1 of heteroclinic arcs (in this case) 3-fold, and case 2 of heteroclinic loops occur for the system as defined in Section LABEL:sec:heteroArcs occur for the Lomelí map. In the sequel we prove, by a computer assisted argument, that this is indeed the case.

## 3 Review of the parameterization method for stable/unstable manifolds of fixed points

Let be a smooth map and suppose that is a hyperbolic fixed point as in Section LABEL:sec:stableMan. Then the eigenvalues for have

 |λj|≠1,for all1≤j≤k,

i.e. none of the eigenvalues are on the unit circle. Let denote the stable eigenvalues of , where . We order the stable eigenvalues so that

 |λs|≤…≤|λ1|<1,

and so that is unstable when .

For the sake of simplicity, suppose that is diagonalizable and let denote a choice of associated eigenvectors. Then

 Df(p)=QΛQ−1,

where

 Λ=⎛⎜ ⎜⎝λ1…0⋮⋱⋮0…λk⎞⎟ ⎟⎠,

is the diagonal matrix of eigenvalues and

 Q=[ξ1,…,ξk],

is the matrix whose -th column is the eigenvector associated with . We write

 Λs=⎛⎜ ⎜⎝λ1…0⋮⋱⋮0…λs⎞⎟ ⎟⎠,

to denote the diagonal matrix of stable eigenvalues.

In the present work we assume that is analytic in a neighborhood of . Let

 Ds={z=(z1,…,zs)∈Cs:|zj|<1for % each1≤j≤s},

denote the -dimensional unit poly-disk in . The goal of the Parameterization Method is to find an analytic map having that

 P(0)=p,DP(0)=[ξ1,…,ξs], \hb@xt@.01(3.1)

and

 f[P(z1,…,zs)]=P(λ1z1,…,λszs), \hb@xt@.01(3.2)

for all . Such a map parameterizes a local stable manifold for at , as the following lemma makes precise.

###### Lemma 3.1

Suppose that is an analytic map satisfying the first order constraints in (LABEL:eq:parmLinearConstraints) and solving Equation (LABEL:eq:parmInvEq) in . Then is a chart map for a local stable manifold at , i.e.

1. for all the orbit of accumulates at ,

2. is tangent to the stable eigenspace at ,

3. is one-to-one on , i.e. is a chart map.

The proof of Lemma LABEL:lem:Par-lem follows along the same lines as Section of [15]. Here we sketch the argument: Point is seen by iterating the invariance equation (LABEL:eq:parmInvEq), considering the continuity of and the requirment that . Point follows directly from the first order constraint on given in Equation (LABEL:eq:parmLinearConstraints), and point is seen by applying the implicit function theorem to show that is one-to-one in a small neighborhood of the origin in , and then using that and are invertible maps to show that is one-to-one anywhere that Equation (LABEL:eq:parmInvEq) holds.

Since we seek analytic on a disk and satisfying first order constraints its natural to look for a power series representation

 P(z1,…,zs)=∞∑α1=0…∞∑αs=0pα1,…,αszα11…zαss=∞∑|α|=0pαzα, \hb@xt@.01(3.3)

where is an -dimensional multi-index,

 |α|=α1+…+αs,

for each , and for each , . Imposing the first order constraints (LABEL:eq:parmLinearConstraints) gives

 p0,…,0=0,andpej=ξj,

where for each the multi-index is given by , i.e. has a one in the -th entry and zeros elsewhere. The Taylor coefficients for can be determined by a power matching scheme. This procedure is illustrated by example in the next section.

###### Remark 3.2 (Unstable manifold parameterization)

The considerations above apply to the unstable manifold of at by considering the stable manifold of the inverse map at . In fact the equation becomes

 f−1[P(z1,…,zu)]=P(σ1z1,…,σuzu),

where is the number of stable eigenvalues of and denote these stable eigenvalues. Of course the stable eigenvalues for are the reciprocals of the unstable eigenvalues for so that . Applying to both sides of the equation and pre-composing with we obtain

 P(σ−11z1,…,σ−1uzu)=f[P(z1,…,zu)],

with the unstable eigenvalues of . This shows that chart maps for the stable and unstable manifolds satisfy the same Equation (LABEL:eq:parmInvEq). The difference is that in one case we conjugate to the linear map given by the stable eigenvalues of and in the other case the linear map given by the unstable eigenvalues.

### 3.1 Example: 2D manifolds associated with complex conjugate eigenvalues for the Lomelí family

In this section we write out in full detail how the parameterization method works on the concrete example of the Lomelí map. We include these formal computations for the sake of completeness.

Suppose that is a fixed point of the Lomelí map and that has a pair of stable complex conjugate eigenvalues , i.e. . Let denote the complex conjugate eigenvectors. Note that since the Lomelí map is volume preserving, it is the case that the remaining eigenvalue is real and unstable.

Take in the unit disk in and write

 P(v,w)=⎛⎜⎝P1(v,w)P2(v,w)P3(v,w)⎞⎟⎠=∞∑k=0∞∑l=0⎛⎜ ⎜⎝p1klp2klp3kl⎞⎟ ⎟⎠vkwl.

We see that

 P(0,0)=⎛⎜ ⎜⎝p100p200p300⎞⎟ ⎟⎠=p,∂∂vP(0,0)=⎛⎜ ⎜⎝p110p210p310⎞⎟ ⎟⎠=ξ,∂∂wP(0,0)=⎛⎜ ⎜⎝p101p201p301⎞⎟ ⎟⎠=¯ξ,

by imposing the first order constraints of Equation (LABEL:eq:parmLinearConstraints). Plugging the unknown power series expansion of into the invariance Equation (LABEL:eq:parmInvEq) gives

 f[P(v,w)]= =⎛⎜⎝P3(v,w)+α+τP1(v,w)+aP1(v,w)2+bP1(v,w)P2(v,w)+cP2(v,w)2P1(v,w)P2(v,w)⎞⎟⎠ =∞∑k=0∞∑l=0⎛⎜ ⎜ ⎜⎝p3kl+δklα+τp1kl+∑ki=0∑lj=0(ap1k−il−jp1ij+bp1k−il−jp2ij+cp2k−il−jp2ij)p1klp2kl⎞⎟ ⎟ ⎟⎠vkwl,

on the left (where if and and otherwise), and

 P(λv,¯λw)=∞∑k=0∞∑l=0λk¯λl⎛⎜ ⎜⎝p1klp2klp3kl⎞⎟ ⎟⎠vkwl,

on the right. Matching like powers leads to

 ⎛⎜ ⎜⎝p3kl+δklα+τp1kl+∑ki=0∑lj=0ap1k−il−jp1ij+bp1k−il−jp2ij+cp2k−il−jp2ijp1klp2kl⎞⎟ ⎟⎠=λk¯λl⎛⎜ ⎜⎝cp1klp2klp3kl⎞⎟ ⎟⎠

for all . Extracting terms of order and isolating them on the left hand side leads to the linear homological equations

 ⎡⎢ ⎢ ⎢⎣τ+2a+b−λk¯λlb+2c11−λk¯λl001−λk¯λl⎤⎥ ⎥ ⎥⎦⎛⎜ ⎜⎝p1klp2klp3kl⎞⎟ ⎟⎠=skl, \hb@xt@.01(3.4)

where

 skl=⎛⎜ ⎜⎝−∑ki=0∑lj=0^δij(ap1k−il−jp1ij+bp1k−il−jp2ij+cp2k−il−jp2ij)00⎞⎟ ⎟⎠

and the coefficient

 ^δij=⎧⎨⎩0i=0 and j=0,0i=k and j=l,1otherwise,

accounts for the fact that terms of order have been extracted from the sums. In other words: the depend only on terms where .

The question arises: for what with does the linear Equation (LABEL:eq:parmLinEqnsLomeli) have a unique solution? Direct inspection of the formula for when is the Lomelí map allows us to rewrite the homological equation as

 (Df(p)−λk¯λlId)pkl=skl. \hb@xt@.01(3.5)

Note that the matrix on the left hand side is characteristic for . In other words, this matrix is invertible as long as is not an eigenvalue of . Since both and are impossible for , and since the remaining eigenvalue is unstable, we see that this matrix is invertible for all cases of concern. Then the Taylor coefficients of are formally well defined to all orders. Moreover we obtain a numerical algorithm by recursively solving the homological equations to any desired order. We write

 PN(v,w)=N∑n=0∑k+l=npklvkwl,

to denote the -th order polynomial obtained by solving the homological equations to order . We remark that by solving these homological equations using interval arithmetic we obtain validated interval enclosures of the true coefficients. This discussion goes through unchanged if are unstable rather than stable eigenvalues.

###### Remark 3.3

Note that the Lomelí map is real analytic, and for real fixed points we are interested in the real image of . Using the fact that is a real matrix when is a real fixed point, we see that

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Df(p)−λk¯λlId=Df(p)−¯λkλlId,

and similarly it is easy to check that for all . Since we choose the first order coefficients so that

 ¯¯¯¯¯¯¯p10=¯¯¯ξ=p01,

it follows that the solutions of all homological equations inherit this property, i.e. that

 ¯¯¯¯¯¯pkl=plk

for all .

Choosing complex conjugate variables

 v=s+itandw=s−it,

with real we see that the map given by

 ^P(s,t)=P(s+it,s−it),

is real valued. Moreover the constraint that lie in the unit Poly-disk in imposes that

 B2={(s,t)∈R2:√s2+t2<1},

is the natural domain for . These remarks show also that the truncated polynomial approximation has complex conjugate coefficients, and hence a real image when we make use the complex conjugate variables, as long as we include each coefficients with in our approximation.

###### Remark 3.4 (Uniqueness and decay rate of the coefficients)

The discussion of the homological equation above shows that the Taylor coefficients are unique up to the choice of the length of . In fact it can be shown that if are the coefficients associated with the scaling , and are the coefficients associated with the scaling , then we have the relationship

 pkl=τk+l^pkl.

A simple proof of this fact is found for example in [26]. Then the length of adjusts the decay rate of the power series coefficients. This freedom is exploited in order to stabilize numerical computations.

###### Remark 3.5 (A-posteriori error)

The computations above are purely formal, and in practice we would like to measure the accuracy of the approximation on a fixed domain. In order to make such a measurement we exploit that the lengths of the eigenvectors tune the decay rates of the power series coefficient, so that we can always fix the domain of to be the unit disk. Equation (LABEL:eq:parmInvEq) then suggests we define the a-posteriori error functional

 ϵN=sup|v|,|w|<1∥f[PN(v,w)]−PN(λv,¯λw)∥C3.

This quantity provides a heuristic indicator of the quality of the approximation on the unit disk. Of course small defects do not necessarily imply small truncation errors. In Section LABEL:sec:aPosParm we discuss a method which makes this heuristic indicator precise.

Note that if is fixed then is a function of the length of only. Then by varying the length we can make as small as we wish (up to machine errors). To see this we simply note that is exactly equal to to zero and first order, so that the function is zero to second order and has higher order coeffieicnts decaying as fast as we wish. Once a desired error is achieved fixed we increase and repeat the procedure. In this way one can optimize the size of the image of relative to a fixed desired error tolerance. Again we refer the interested reader to [26], where such algorithms are discussed in more detail.

###### Remark 3.6 (Generalizations)

The computations sketched above succeed in much greater generality. For example when we study an dimensional manifold of an analytic mapping , then looking for a parameterization of the form

 P(z)=∞∑|α|=0pαzα,

leads to a homological equation of the form

 [Df(p)−λα11…λαssId]pα=sα,

for the coefficients with . Here again depends only on terms of order lower than , and the form of is determined by the nonlinearity of . Note that the general case leads to the non-resonance conditions

 λα11…λαss≠λjfor1≤j≤s.

Inspection of these conditions shows that the non-resonance conditions hold generically, i.e. they reduce to a finite collection of constraint equations. Moreover, should a resonance occur it is still possible for the Parameterization Method to succeed. However when there is a resonance, rather than conjugating to the linear map generated by the stable eigenvalues, it is necessary to conjugate the dynamics on the manifold to a certain polynomial which “kills” the resonant terms. The general development of the Parameterization Method for stable/unstable manifolds of non-resonant fixed points is in [3, 4, 5]. Validated numerical methods for the resonant case as well as the non-diagonalizable case are developed in [27].

### 3.2 A-posteriori validation and computer assisted error bounds

We say that an analytic function is an analytic -tail if the Taylor coefficients of are zero to -th order, i.e. if

 h(z)=∞∑|α|=0hαzα,

and

 hα=0,for 0≤|α|≤N.

Such functions are used to represent truncation errors in power series methods. For and let

 Dk(p,R)={z∈Ck:∥z−p∥

We make the following assumptions.

A1: Assume that is analytic and that has .

A2: Assume that is nonsingular, diagonalizable, and hyperbolic. Let denote the stable eigenvalues, denote a choice of corresponding eigenvectors, denote the diagonal matrix of stable eigenvalues, and denote the matrix whose columns are the stable eigenvectors.

A3: Assume that is an -th order polynomial, with . Assume that is an exact formal solution of the equation

 f∘PN=PN∘Λs,

to -th order, that is, we assume that the power series of the right hand side is equal to the power series of the left hand side exactly up to -th order.

The following definition collects some constants which are critical in the a-posteriori validation theorem to follow.

###### Definition 3.7 (Validation values for the Stable Manifold)

Let and be as in assumptions - . A collection of positive constants , , , , and are called validations values for if

 supz∈Ds∥f[PN(z)]−PN(Λsz)∥Ck≤ϵtol, \hb@xt@.01(3.6)
 supz∈Ds∥PN(z)−p∥Ck≤R′
 0
 supz∈Ds∥[Df]−1(PN(z))∥≤K1, \hb@xt@.01(3.9)
 max\lx@stackrelβ∈Nm|β|=2max1≤j≤ksupq∈Dk(p,R)∥∂βfj(q)∥C≤K2. \hb@xt@.01(3.10)

Some explanation of the meaning of these constants is appropriate. In Equation (LABEL:eq:valVal1), we see that measures the defect associated with the approximation on . The requirement that in Equation (LABEL:eq:valVal2) guarantees that the image of the approximate parameterization is contained in the disk , i.e. strictly interior to the disk on which we have control over derivatives. The image of the true parameterization will live in the larger disk . Note that there is no assumption that either or are small. Equation (LABEL:eq:valVal3) postulates a uniform bound on the absolute values of the stable eigenvalues, while Equation (LABEL:eq:valVal4) requires a uniform bound on the inverse of the Jacobian derivative of holding over all of . Finally Equation (LABEL:eq:valVal5) requires a uniform bound on the second derivatives of which is valid over all of . Interval arithmetic computation of validation values is discussed in [25], and such computations are implemented for the Lomelí map in the same reference.

The following theorem is the main result of [25]. Its proof is found in the same reference.

###### Theorem 3.8 (A-posteriori error bounds)

Given assumptions , suppose that , , , , , and are validation values for . Let

 Nf=#{β∈Nk:|β|=2and∂βfj(q)≠0forq∈Dk(p,R),1≤j≤k},

count the number of not identically zero partial derivatives of , and suppose that and satisfy the three inequalities

 N+1>−ln(K1)ln(μ∗), \hb@xt@.01(3.11)
 δ
 2K1ϵtol1−K1(μ∗)N+1<δ. \hb@xt@.01(3.13)

Then there is a unique analytic -tail having that

 supz∈Dm∥h(z)∥Ck≤δ,

and that

 P(z)=PN(z)+h(z),

is the exact solution of Equation (LABEL:eq:parmInvEq).

The theorem provides explicit conditions, all checkable by finite computations using interval arithmetic, sufficient to insure that there is an analytic -tail so that solve the invariance equation for the Parameterization Method. In this case is the truncation error on associated with stopping our Taylor approximation at -th order. then provides an explicit bound on the truncation error. Note also that the size of is related to the a-posteriori error times the quantity , which in a sense measures how far from singular is on the image of . The theorem also gives an indication of how large the order of approximation must be taken.

In practice once validation values for are found one then checks that satisfies the condition given by Equation (LABEL:eq:aPosEq1), computes a bound on the quantity given in the right hand side of Equation (LABEL:eq:aPosEq2), computes a bound on the quantity given in the left hand side of Equation (LABEL:eq:aPosEq3), and then checks that . Each of these computations and checks is done using interval arithmetic. If this procedure succeeds then the theorem holds for any . Of course in practice we then take as small as possible, i.e. very close to . Again these matters are discussed in more detail in [25].

###### Remark 3.9 (Cauchy bounds on the derivative)

Since the truncation error is an analytic function bounded by on the fixed disk , we can bound derivatives of on any smaller disk using classical estimates of complex analysis. Indeed, let be an analytic function with

 supz∈Dm(ν)∥f(z)∥≤M.

Then for any the Cauchy bounds

 supz∈Dm(νe−σ)∥∥∥∂∂zjf(z)∥∥∥≤2πνσM,

translate bounds on the size of a function into bounds on the size of its derivative on a strictly smaller disk. An elementary proof of the Cauchy bounds can be found for example in [25]. Repeated application of the Cauchy bounds leads to estimates of -th order derivatives inverse proportional to .

Now suppose that and are as in Theorem LABEL:thm:validateManifolds, so that parameterizes a local stable manifold for , and for . Then for , , and we have that

 ∂∂zjP(z)=∂∂zjPN(z)+∂∂zjh(z).

In practice the first term on the right is computed explicitly as the partial derivative of the known polynomial , while the Cauchy bounds applied to the second term on the right yield that

This decomposition is used in order to control derivatives of the truncation errors of the Parameterizations in the heteroclinic arc computations to follow. We just have to make sure that our heteroclinic arcs are contained in the interior of the domain .