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Parameterized stable/unstable manifolds for

periodic solutions of implicitly deﬁned dynamical systems

Archana Neupane Timsina ∗1and J.D. Mireles James †2

1,2Florida Atlantic University, Department of Mathematical Sciences

June 8, 2022

Abstract

We develop a multiple shooting parameterization method for studying stable/unstable man-

ifolds attached to periodic orbits of systems whose dynamics is determined by an implicit rule.

We represent the local invariant manifold using high order polynomials and show that the

method leads to eﬃcient numerical calculations. We implement the method for several exam-

ple systems in dimension two and three. The resulting manifolds provide useful information

about the orbit structure of the implicit system even in the case that the implicit relation is

neither invertible nor single-valued.

Key words. Implicitly deﬁned dynamical systems, computational methods, invariant manifolds,

periodic orbits, parameterization method

1 Introduction

A smooth diﬀeomorphism F:Rd→Rdgenerates a dynamical system by the rule

xn+1 =F(xn), n ∈N,

with initial condition x0∈Rd. The inﬁnite sequence {xn}∞

n=0 is called the (forward) orbit of x0

generated by F, and the fundamental objective of dynamical systems theory is to understand the

qualitative features of the set of all orbits generated by F. This analysis often begins by considering

simple invariant sets like ﬁxed points, periodic orbits, and their attached invariant manifolds. If

this program goes well one may move on to bifurcations of these objects, or to the study of more

exotic invariant sets like connecting orbits, horseshoes, strange attractors, and invariant tori.

A interesting generalization, whose motivation and literature are discussed brieﬂy in Section

1.2, is to consider a smooth map T:Rd×Rd→Rd, and to study the properties of the mapping F

deﬁned by the implicit rule

y=F(x)if T(y, x)=0.

∗Email: aneupanetims2016@fau.edu

†J.D.M.J partially supported by NSF grant DMS 1813501. Email: jmirelesjames@fau.edu

1

This immediately raises delicate questions about the domain of F, or whether Fis even single valued.

Of course the implicit function theorem provides valuable information. That is, if (¯y, ¯x)∈Rd×Rd

has T(¯y, ¯x) = 0, and D1T(¯y, ¯x)is an isomorphism, then there exists an open set U⊂Rd,¯x∈U,

and a smooth diﬀeomorphism F:U→Rdwith F(¯x) = ¯yand having that

T(F(x), x)=0,for all x∈U.

We say that Timplicitly deﬁnes the diﬀeomorphism Fnear ¯x.

We remark that for ﬁxed ¯x∈Rd, the solution ¯yof T(y, x)=0may not be unique. Hence F

depends on the choice of ¯y. Nevertheless, for a given choice of ¯ythe associated “branch” of Fis a

perfectly well deﬁned diﬀeomorphism. This and other fundamental notions for implicitly deﬁned

maps are discussed in Appendix B.

To iterate the procedure suppose ˜x∈F(U)and ˜y∈Rdhave T(˜y, ˜x)=0, with D1T(˜y, ˜x)

an isomorphism. Then there is an open set an open set U⊂Rd,˜x∈U, and a diﬀeomorphism

˜

F:˜

U→Rdwith ˜

F(x)a branch of solutions of T(y, x)=0having ˜

F(˜x) = ˜y. Now, for any

x∈U∩F−1(˜

U), the composition of xunder the two implicitly deﬁned maps Fand ˜

Fis well

deﬁned.

Continuing in this way, suppose that y16=. . . 6=yN∈Rdhave

T(y2, y1)=0

T(y3, y2)=0

.

.

.

T(yN, yN+1)=0

T(y1, yN)=0,

(1)

with the linear maps D1T(y2, y1),. . .,D1T(y1, yN)all isomorphisms. Then there exist neighbor-

hoods Uj⊂Rd, and diﬀeomorphisms Fj:Uj→Rdso that each Fjis a branch of solutions of

T(y, x)=0having Fj(yj) = yj+1 for j= 1, N (with the understanding that yN+1 =y1). Indeed,

by taking the Ujdisjoint, there is no reason to think of the Fjdiﬀerent maps. Rather, we think of

y1, . . . , yNas a period-Norbit of a single mapping F:U1∪. . . ∪UN→Rd, and have that Fis a

diﬀeomorphism in a neighborhood of each of the points y1, . . . , yN.

It makes sense in this context to consider the linear stability of the periodic orbit of F, and to

study in turn the attached invariant manifolds. The main goal of the present work is to develop

eﬃcient numerical procedures for computing high order polynomial approximations of the local

stable/unstable manifold attached to a periodic orbit of an implicitly deﬁned dynamical system.

We also implement and proﬁle the results in some example applications.

We stress that our approach does not require information about the implicitly deﬁned diﬀeo-

morphism F, much less any compositions of F. Rather, we develop a multiple shooting scheme

for the invariant manifolds which depends only on the mapping Tand the choice of periodic solu-

tion x1, . . . , xNfor the system given by Equation (1). A power matching argument leads to linear

systems of equations for the jets of the manifold, reducing the calculation of the stable/unstable

manifold parameterizations to a problem in linear algebra. As we will see, computing the local

invariant manifolds to high order leads to polynomial approximations valid in fairly large neigh-

borhoods of the periodic orbit. This in turn provides insights into the existence of more global

dynamical objects like heteroclinic/homoclinic connecting orbits for the implicitly deﬁned system.

2

The remainder of the paper is organized as follows. In the next section, Section 1.1, we introduce

the main examples considered in the remainder of the present work. In Section 1.2 we brieﬂy

discuss the literature concerning generalized notions of a dynamical system. In Section 2 we review

some basic notions from the parameterization method for invariant manifolds, and in Section 3 we

review how these notions work for ﬁxed points of implicitly deﬁned dynamical systems. We then

introduce the multiple shooting parameterization method for stable/unstable manifolds attached

to periodic orbits of implicitly deﬁned maps. In Section 4 we develop power series solutions for the

parameterized manifolds, and in Section 5 we implement numerical methods based on the power

series approach. Some conclusions are given in Section 6. The Appendices A, B, and C ﬁll in some

background details and develop some more technical extensions of the methods developed in the

main text.

All the MATLAB codes discussed in the present work are on github at

https://github.com/aneupanetims2016/Implicit-map-archana

1.1 A class of examples: perturbations of explicitly deﬁned maps

We now describe a class of examples suﬃcient for the needs of the present work. Other potential

applications are mentioned in Section 6

Let f:Rd→Rdbe a Ckdiﬀeomorphism (or real analytic with real analytic inverse if k=ω).

Then for any x0∈Rd,fdeﬁnes a dynamical system by the rule

f(xn) = xn+1,

for n= 0,1,2, . . . Deﬁne the function T:Rd×Rd→Rdby

T(y, x) = y−f(x),

and note that for a given ¯x∈Rd,¯ysolves the equation T(y, ¯x) = 0 if and only if

¯y=f(¯x).

In this case, the problem T(y, x)=0implicitly deﬁnes the dynamical system generated by the

diﬀeomorphism f(x). Now let U, V be open subsets of Rdand H:U×V→Rdbe a Ckfunction.

Consider the one parameter family of problems T:U×V→Rdby

T(y, x) = y−f(x) + H(y, x),(2)

and note that for any (¯y, ¯x)∈V×Uwe have that

D1T(¯y, ¯x) = Id +D1H(¯y, ¯x).

Note that D1T0(¯y, ¯x) = Id, so that –by the implicit function theorem – there is a δ > 0and a

smooth curve y: (−δ, δ)→V⊂Rdso that y(0) = ¯yand

T(y(),¯x) = 0,

for all ∈(−δ, δ).

3

Moreover, for a possibly smaller δ > 0we have that

D1T(y(),¯x) = Id +D1H(y(),¯x),

is invertible for each ∈(−δ, δ), by the Neumann theorem [44, 40]. Then there exists an r > 0and

a family of functions F:Br(¯x)×(−δ, δ)→Rdso that

F0(x) = f(x),

and

T(F(x), x)=0,

for all x∈Br(¯x)and ∈(−δ, δ). The family Fdepends smoothly on and is Ckfor each ﬁxed

. Moreover, for small 6= 0 and ¯x∈Uwe take ¯y=f(¯x)as an approximate zero for T(y, ¯x)and

apply Newton’s method to ﬁnd y()so that T(y(),¯x)=0. That is, for small we can compute

images of the implicitly deﬁned mapping F(x)using Newton’s method. For larger we perform

numerical continuation from the = 0 case.

Note that we make no attempt to guarantee that for a given the ¯ywe ﬁnd is globally unique.

Indeed, it may not be. What is required for our purposes is the local uniqueness given by the

implicit function theorem. This is expected to apply whenever the Newton method is successful

(as Newton will struggle or fail when a solution is degenerate) and is enough to give a unique local

branch of Fhaving F(¯x) = ¯y. Finally, suppose that ¯x∈Uis a hyperbolic ﬁxed point of f, and

recall that F0(x) = f(x). Since Fdepends smoothly on , it follows that for small 6= 0 the map

F(x)has a hyperbolic ﬁxed point near ¯xby the usual perturbation argument for maps.

The discussion just given shows that problems of the form given in Equation (2) provide a

natural class of examples - perturbations of diﬀeomorphisms - for which our method applies. Two

speciﬁc examples are when fis either the classic Hénon map or its three dimensional generalization

to the Lomelí map. These are discussed brieﬂy now.

1.1.1 Example 1: the Hénon map

Let θ1= (x1, y1), θ2= (x2, y2)denote points in the plane. The Hénon map is a two parameter

family of quadratic mappings deﬁned by

f(θ1) = f(x1, y1) =

1 + y1−αx2

1

βx1

(3)

The mapping is a classic example of a system with complicated dynamics, and was originally

introduced in [35]. See also the books of [22, 61]. We deﬁne an implicit Hénon system T:

R2×R2→R2given by

T(θ2, θ1) = T(x2, y2, x1, y1) =

x2−(1 −αx2

1+y1+x5

2)

y2−βx1+y5

2

.(4)

Here we have choosen, somewhat arbitrarily, the perturbation term

H(y, x) =

x5

y5

,

4

to be of high enough polynomial order that it is diﬃcult (if not impossible) to work out useful

formulas for the implicitly deﬁned system.

The equation for a ﬁxed point is T(x,x)=0, or

x1−(1 −αx2

1+y1+x5

1)

y1−βx1+y5

1

=

0

0

.

Similarly, the multiple shooting equations for a period two orbit are

T(θ2, θ1)=0

T(θ1, θ2)=0,(5)

or x2−(1 −αx2

1+y1+x5

2)=0

y2−βx1+y5

2= 0

x1−(1 −αx2

2+y2+x5

1)=0

y1−βx2+y5

1= 0

(6)

The implicit equations for ﬁxed points or periodic orbits are solved using Newton’s method. Eigen-

values and eigenvectors we compute using the approach outlined in Section B.1.

For classical parameters a, b ∈Rthe Hénon map has a pair of hyperbolic ﬁxed points, each with

one stable and one unstable eigenvalue. Then for small epsilon the same is true for the perturbation.

The numerical value of the unperturbed ﬁxed points serve as initial guesses for the perturbed ﬁxed

points in the Newton method. Similar comments hold for periodic orbits.

1.1.2 Example 2: the Lomelí map

We consider also the ﬁve parameter family of maps f:R3→R3given by

f(x, y, z) =

z+Q(x, y)

x

y

(7)

where Q(x, y) = ρ+γx+ax2+bxy +cy2and one usually takes a+b+c= 1.The system is known as

the Lomelí map, and it is a normal form quadratic volume preserving maps with quadratic inverse.

In that sense it can be thought of as a three dimensional generalization of the area preserving Hénon

map. The map was ﬁrst introduced in [49], and was subsequently studied by a number of authors

including [49, 23, 55, 57, 11, 28].

Let θ1= (x1, y1, z1), θ2= (x2, y2, z2)∈R3. We consider the dynamics implicitly deﬁned by the

map T:R3×R3→R3given by

T(θ2, θ1) = T(x2, y2, z2, x1, y1, z1)

=

x2−ρ−τx1−z1−ax2

1−bx1y1−cy2

2+αy5

2+βz5

2

y2−x1+γz5

2

z2−y1

.(8)

5

Note that Tis analytic in all variables. We remark that the perturbation is chosen so that the

system still preserves volume.

Fixed of points of the implicit Lomelí system (8) are obtained as solutions of

x−ρ−τx −z−ax2−bxy −cy2+(αy5+βz5)

y−x+γz5

z−y

=

0

0

0

(9)

Similarly, a period four orbit for the Lomelí system solves the equations

T(θ2, θ1)=0

T(θ3, θ2)=0

T(θ4, θ3)=0

T(θ1, θ4)=0.

(10)

More explicitly, this is

x2−ρ−τx1−z1−ax2

1−bx1y1−cy2

1+(αy5

2+βz5

2) = 0

y2−x1+γz5

2= 0

z2−y1= 0

x3−ρ−τx2−z2−ax2

2−bx2y2−cy2

2+(αy5

3+βz5

3) = 0

y3−x2+γz5

3= 0

z3−y2= 0

x4−ρ−τx3−z3−ax2

3−bx3y3−cy2

3+(αy5

4+βz5

4) = 0

y4−x3+γz5

4= 0

z4−y3= 0

x1−ρ−τx4−z4−ax2

4−bx4y4−cy2

4+(αy5

1+βz5

1) = 0

y1−x4+γz5

1= 0

z1−y4= 0

(11)

The equations for ﬁxed and periodic orbits are again amenable to Newton’s method, and the

multipliers λ1, λ2, λ3∈C, and associated eigenvectors ξj1, ξj2, ξj3∈C3,1≤i≤4are are computed

as discussed in Section A.3,

1.2 Generalized notions of dynamical systems

Generalizations of nonlinear dynamics to the setting of relations instead of functions, where neither

uniqueness of forward or backward iterates is required, appeared in the early 1990’s in the work of

Akin [2] and McGehee [53]. The Ph.D. dissertation of Sander generalized stable/unstable manifold

theory to the setting of relations [25], and work by Lerman [45] and Wather [65] studied transverse

homoclinic/heteroclinic phenomena in the setting of non-invertible dynamical systems, with a view

toward applications to semi-ﬂows in inﬁnite dimensions. Further work by Sander [63, 62, 64] studied

homoclinic bifurcations for noninvertible maps and relations.

6

The ideas of the authors mentioned above have been applied to generalized dynamical systems

coming from applications to population dynamics [3], iterated diﬀerence methods/numerical algo-

rithms [50, 25], delay diﬀerential equations [65], adaptive control [1], discrete variational problems

[27, 67], and economic theory [41, 42, 43, 54]. Indeed, this list is far from comprehensive and the

interested reader will ﬁnd a wealth of additional references in the works just cited. We mention also

the recent book on dynamical systems deﬁned by implicit rules [51], where many further examples

and references are found.

A complementary approach to the study of generalized dynamics, based on functional analytic

rather than topological tools, is given by the parameterization method. The idea of the param-

eterization method is to consider the equation deﬁning a (semi-)conjugacy between a subset of

the given system, and some simpler model problem. Example include stable/unstable manifolds

attached to ﬁxed or periodic orbits, or a quasiperiodic family of orbits - that is an invariant torus.

The equations describing special solutions often have nicer properties than the Cauchy problem

describing a generic orbit. While this observation is important for a classical dynamical system

deﬁned by an invertible map, it can be even more useful when studying dynamical systems which

are not invertible, are ill posed, or are not even single valued.

The parameterization method was originally developed for studying non-resonant invariant man-

ifolds attached to ﬁxed points of inﬁnite dimensional maps between Banach spaces in a series of

papers by Cabré, Fontich, and de la Llave [6, 7, 8], though the approach has roots going back to the

Nineteenth Century (see appendix B of [8]). The method has since been extended to the study of

parabolic ﬁxed points [4], invariant tori and their stable/unstable ﬁbers [31, 30, 32, 10, 38], for sta-

ble/unstable manifolds attached to periodic solutions of ordinary diﬀerential equations [37, 13, 59],

and to develop KAM arguments without action angle variables [18, 9]. See also the recent book of

Haro, Canadell, Figueras, Luque and Mondelo [29] for much more complete overview.

The parameterization method can also be extended to generalized dynamical systems like those

mentioned in the ﬁrst paragraph of this section. We refer for example to the work of [14, 15] on

stable and center manifolds for ill-posed problem, the work of [20, 68] on invariant tori for ill-posed

PDEs and state dependent delay diﬀerential equations [34, 33], the work of [17, 16] on periodic orbits

and their isochrons in state dependent perturbations of ODEs, and the related work of [12, 26] on

computer assisted existence proofs for periodic orbits in the Boussinesq equation and in some state

dependent delay diﬀerential equations.

Remark 1.1. The work of [19], which develops numerical methods for computing stable/unstable

manifolds attached to ﬁxed points of implicitly deﬁned discrete time dynamical systems, is the

jumping oﬀ point for the present study – which extends their method to periodic orbits. Another

paper closely related to the present work is [28], where the authors develop a multiple shooting

parameterization method for computing stable/unstable manifolds attached to periodic orbits of

diﬀeomorphisms. The main contribution of the present work is to extend the methods of [28] to

the more general setting of implicitly deﬁned systems, and to illustrate their implementation in

examples. This also extends the applicability of the parameterization method for implicit systems

beyond the foundations laid in [19].

2 A brief overview of the parameterization methods for maps

We review some basic results about the parameterization method for maps.

7

2.1 Parameterization of stable/unstable manifolds attached to ﬁxed points

In this section we recall some basic results from the work of [6, 7, 8]. In fact, we paraphrase these

results, simplifying them to the ﬁnite dimensional setting of the present work. The reader interested

in inﬁnite dimensional dynamics can consult the references just cited for theorems formulated in

full generality. Moreover, we recall that spec(x)refers to the eigenvalues of DF (x), and refer to

Section A.2 for a review of notions and notation related to stability of ﬁxed points.

Lemma 2.1 (Parameterization method for ﬁxed points in Rd).Suppose that U⊂Rdis an open

set, that F:U→Rdis a Ck(U)mapping with k= 1,2,3,...,∞, ω, that x∗∈Uis a ﬁxed point of

F, and that DF (x∗)is invertible. Take ds=dim(Es)to be the dimension of the stable (generalized)

eigenspace/the number of stable eigenvalues (counted with multiplicity).

Let α, β > 0have that

|λ| ≤ α < 1,

for all λ∈specs(x∗)and

1< β ≤ |λ|,

for all λ∈specu(x∗). Let L∈Nbe the smallest natural number with

αL<1

β,

and assume that

L+ 1 < k.

Then there exists an open set Ds⊂Rdswith 0∈Ds, a polynomial K:Ds→Rdsof degree not

more than L, and a Ckmapping P:Ds→Rdso that

1.

P(0) = x∗,

2. The columns of DP (0) span Es, and

3.

F(P(θ)) = P(K(θ)),(12)

for all θ∈Ds.

Moreover, Pis unique up to the choice of the scalings of the columns of DP (0).

Several additional comments are in order. First, we remark that the columns of DP (0) can be

taken as stable (generalized) eigenvectors of DF (x∗), so that DP (0) is unique up to the choice of the

scalings of these vectors. The theorem says that once these scalings are ﬁxed, the parameterization

Pis uniquely determined.

Note also that if k=∞or k=ωthen L+ 1 < k is automatically satisﬁed. Consider the case

when k=ω, that is F(real) analytic at x∗, and suppose that the scalings of D P (0) are ﬁxed. Then

P, and hence its power series expansion at 0, is uniquely determined. In this case Kand Pare

worked out by power matching arguments, and these arguments lead in turn to practical numerical

schemes. In fact, the scalings of the eigenvectors can be chosen so that the power series coeﬃcients

of Pdecay at a desired exponential rate. Numerical schemes for determining the optimal scalings

of eigenvectors are developed in [5].

The following lemma allows us to determine the polynomial mapping Ka-priori, in the case

that some (generic) non-resonance conditions hold between the stable eigenvalues.

8

Lemma 2.2 (Non-resonant eigenvalues implies Klinear ).Let λ1, . . . , λds∈Cdenote the stable

eigenvalues of DF (x∗), and assume that each has multiplicity exactly one. Moreover, assume that

for all (n1, . . . , nds)∈Ndswith

2≤n1+. . . +nds≤L,

we have that

λn1

1. . . λnds

ds/∈specs(x∗).(13)

Then we can choose Kto be the linear mapping

K(θ)=Λθ,

where θ= (θ1, . . . , θds)∈Rdsand Λis the ds×dsmatrix

Λ =

λ10. . . 0 0

0λ2. . . 0 0

.

.

..

.

.....

.

..

.

.

0 0 . . . λds−10

0 0 . . . 0λds

.

That is, Λis the matrix with the stable eigenvalues on the diagonal entries and zeros in all other

entries.

We say that the stable eigenvalues are non-resonant when the condition given by Equation (13)

is satisﬁed. We say there is a resonance at (n1, . . . , nds)∈Ndsif

λn1

1. . . λnds

ds∈specs(x∗).

In this case, the polynomial Kis required to have a monomial term of the form c θn1

1. . . θnds

dswith

non-zero c∈Rds. That is, even in the resonant case the form of the polynomial Kcan be determined

by examining the resonances between the stable eigenvalues. Numerical procedures for determining

Pand Kin the resonant case are discussed in [66].

It is worth remarking that when the stable eigenvalues are non-resonant, Equation (12) reduces

to

F(P(θ)) = P(Λθ), θ ∈Ds⊂Rds,(14)

so that Pis now the only unknown in the equation. Indeed, the equation is viewed as requiring

a conjugacy between the dynamics on the image of Pand the diagonal linear map given by the

stable eigenvalues.

We also note that the domain Dscan be chosen so that ΛsDs⊂Ds. In this case, since Equation

(14) holds, it is easy to see that Pparameterizes a local stable manifold. To see this, let θ∈Ds.

Since Pis continuous (in fact Ck) we have that

lim

n→∞ Fn(P(θ)) = lim

n→∞ F(P(Λnθ))

=FPlim

n→∞ Λnθ

=F(P(0))

=F(x∗)

=x∗,

9

so that image(P)⊂Ws(x∗). Noting that image(P)is a dsdimensional manifold tangent to Esat

x∗gives equality rather than inclusion.

Remark 2.3 (Generality).Lemma 2.1 follows trivially from Theorem 1.1 of [6, 7, 8]. In the much

more general work just cited Uis taken to be an open subset of a Banach space, and the inﬁnite

dimensional complications result in more delicate spectral assumptions. The ﬁnite dimensional

setting of the present work, and the fact that we parameterize the full stable manifold simplify

somewhat the statement of Lemma.

Remark 2.4 (Unstable manifold parameterization).Note that in Lemma 2.1, the assumption that

DF (x∗)is invertible implies that Fis a local diﬀeomorphism. Then, in a small enough neighborhood

of x∗there is a well deﬁned Ckinverse mapping F−1. Let Σdenote the diagonal matrix of unstable

eigenvalues of DF (x∗), so that Σ−1is the matrix of stable eigenvalue of DF −1(x∗). Assume that

these stable eigenvalues (entries of Σ−1) are non-resonant. Then there exists an open set Duand a

Ckmapping Q:Du→Rdso that

F−1(Q(σ)) = Q(Σ−1σ), σ ∈Du.

Applying Fto both sides of the equation and composing with Σleads to the equation

Q(Σσ) = F(Q(σ)), σ ∈Du.

In other words, the unstable parameterization Qsatisﬁes exactly the same invariance equation as

the stable parameterization P. Only the conjugating matrix changes, in the sense that the matrix

of stable eigenvalues Λis replaced by the matrix of unstable eigenvalues Σ.

2.2 Stable/unstable manifolds attached to periodic orbits

The material in this section provides a brief review of the techniques developed in [28] for parameteri-

zation of stable/unstable manifolds attached to periodic orbits of an explicitly given diﬀeomorphism.

The main idea is to exploit multiple shooting schemes which avoid function compositions.

Let x1, , ..., xN∈Rdbe the points along a hyperbolic period Norbit. Let λ1, . . . , λdsdenote

the stable multipliers of the periodic orbit, and let

Λ =

λ10. . . 0 0

0λ2. . . 0 0

.

.

..

.

.....

.

..

.

.

0 0 . . . λds−10

0 0 . . . 0λds

,

denote the ds×dsdiagonal matrix of stable multipliers (similarly Σdenote the du×dudiagonal

matrix of unstable multipliers). For 1≤j≤ds, let ξj,1, . . . , ξj,ds⊂Cddenote the eigenvectors of

DF (xj)associated with the eigenvalue λj.

10

Assume that the stable multipliers are non-resonant, in the sense of Lemma 2.2. Then, by

Lemma 2.2, there is an open set Ds⊂Rdsand are unique P1,...PN:Ds→Rdso that

P1(0) = x1

.

.

.

PN(0) = xN

and

DP1(0) = [ξ1,1, . . . , ξ1,ds]

.

.

.

DPN(0) = [ξN,1, . . . , ξN ,ds],

having that

FN(P1(θ)) = P1(Λθ)

.

.

.

FN(PN(θ)) = PN(Λθ)

(15)

Note that we are treating the periodic point as a ﬁxed point of the composition map, so that

Lemmas 2.2 and 2.1 apply directly.

On the other hand, the presence of composition mapping FNis precisely what makes these

equations diﬃcult, as FNis in general a much more complicated map than F. The main result of

[28] (see Section 3) is that the parameterizations admit a composition free formulation.

Lemma 2.5 (Composition free invariance equations).Under the hypotheses above (non-degenerate

periodic orbit and non-resonant multipliers), the functions P1, . . . , PN:Ds→Rdsatisfy the system

of composition free equations

F(P1(θ)) = P2˜

Λθ

F(P2(θ)) = P3˜

Λθ

.

.

.

F(PN−1(θ)) = PN˜

Λθ

F(PN(θ)) = P1˜

Λθ

where

˜

Λ =

N

√λ10. . . 0 0

0N

√λ2. . . 0 0

.

.

..

.

.....

.

..

.

.

0 0 . . . N

pλds−10

0 0 . . . 0N

pλds

,

11

is the diagonal matrix of N-th roots of the multipliers. (Here it is suﬃcient to choose any branch

of the N-th root).

On easily checks that if P1, . . . , PNsatisfy the invariance equations in Lemma 2.5, then they

solve Equations (15). From the perspective of numerical calculations it is much easier to solve

simultaneously the system of equations given in Lemma 2.5 than it is to apply the parameterization

method directly to the composition mapping FN. This is illustrated by examples in [28]. Note also

that the N-th roots of the multipliers are the eigenvalues of the derivative of the multiple shooting

map, see Equation (31).

3 Parameterization methods for implicitly deﬁned maps

Recall that implicitly deﬁned dynamical system were discussed in the introduction and are reviewed

in more detail in Section B. We now discuss the parametrization method for ﬁxed points of implicit

maps as introduced in [19], and then extend these ideas via a multiple shooting scheme to periodic

orbits of implicit systems. For the sake of clarity let us recall that T:Rd×Rd→Rdis a smooth

mapping, and that we are interested in the implicitly deﬁned dynamical system Fis given by the

rule

F(x) = yif and only if T(y, x)=0.

Then x∗is a ﬁxed point if Fif and only if T(x∗, x∗)=0. See Equation (1) in the introduction for

the implicit equations satisﬁed by a periodic orbit.

3.1 Stable/unstable manifolds attached to implicit ﬁxed points

Before introducing new results for periodic orbits of implicitly deﬁned maps, we ﬁrst review the

main result of[19] for ﬁxed points.

Theorem 3.1. Suppose that U, V ⊂Rdare open sets and that T:U×V→Rdis a Ckmapping

with ﬁxed point x∗∈U∩V, that is

T(x∗, x∗) = 0.

Assume that

•D1T(x∗, x∗)is invertible.

•Let λ1, . . . , λds∈Cdenote the stable eigenvalues and ξ1, . . . , ξds∈Cdassociated eigenvectors

of −D1T(x∗, x∗)−1D2T(x∗, x∗). Assume that the stable eigenvalues are distinct (otherwise

choose the appropriate ξjas generalized eigenvectors).

•Let

α= max

1≤j≤ds|λj|,

β= max

λ∈specu(x∗)λ−1,

and 2≤Lbe the smallest integer so that

αLβ < 1.

Assume that L+ 1 ≤k.

12

•Assume that for all (n1, . . . , nds)∈Ndswith 2≤n1+. . . +nds≤Lwe have that

λn1

1. . . λnds

ds6=λj

for 1≤j≤λds.

Then there exists an open set Ds⊂Rdswith 0∈Ds, and a Ckmapping P:Ds→Rdso that

P(0) = x∗,

DP (0) = [ξ1, . . . , ξds],

and

T(P(Λθ), P (θ))) = 0, θ ∈Ds(16)

where Λis the ds×dsmatrix with the stable eigenvalues on the diagonal entries and zero entries

elsewhere. Pparameterizes a local stable manifold attached to the ﬁxed point x∗of the implicitly

deﬁned mapping F.Pis unique up to the choices of the scalings of the eigenvectors.

The proof is a simple matter of translating the assumptions about T, its derivative, and its

eigenvalues/eigenvectors into equivalent statements about F, and then applying Lemma 2.1 to the

implicitly deﬁned mapping F. Recalling for example that F(x) = yif and only if T(y, x)=0, then

by letting y=P(Λθ)and x=P(θ), Equation (16), is equivalent to

F(P(θ)) = P(Λθ), θ ∈Ds,

and this is precisely Equation (14).

3.2 Stable/unstable manifolds attached to implicit periodic orbits

We now introduce a multiple shooting version of the parameterization method for periodic orbits

of implicitly deﬁned systems. We remark that the multipliers and eigenvectors for such an orbit

are computed as discussed in Section B.1.

Theorem 3.2. Suppose that U, V ⊂Rdare open sets and that T:U×V→Rdis a Ckmapping,

and that x1, . . . , xN∈U∩Vhave

T(x2, x1)=0

.

.

.

T(xN−1, xN)=0

T(x1, xN)=0

Assume that:

•the matrices D1T(x2, x1), . . . , D1T(xN, xN−1), D1T(x1, xN)are invertible.

•Let λ1, . . . , λds∈Cdenote the stable multipliers and for 1≤j≤Nlet ξj,1, . . . , ξj,ds∈Cd

denote associated eigenvectors. Assume that the stable multipliers are distinct (otherwise

choose the appropriate generalized eigenvectors).

13

•Let

α= max

1≤j≤ds|λj|,

β= max

λ∈specu(x∗)λ−1,

and 2≤Lbe the smallest integer so that

αLβ < 1.

Assume that L+ 1 ≤k.

•Assume that for all (n1, . . . , nds)∈Ndswith 2≤n1+. . . +nds≤Lwe have that

λn1

1. . . λnds

ds6=λj

for 1≤j≤λds.

Then there exists an open set Ds⊂Rdswith 0∈Ds, and Ckmappings P1, . . . , PN:Ds→Rdso

that

P1(0) = x1, . . . , PN(0) = xN

DP1(0) = [ξ1,1, . . . , ξ1,ds], . . . , DPN(0) = [ξN,1, . . . , ξN,ds],

and

T(P2(˜

Λθ), P1(θ))) = 0

T(P3(˜

Λθ), P2(θ))) = 0

.

.

.

T(PN(˜

Λθ), PN−1(θ))) = 0

T(P1(˜

Λθ), PN(θ))) = 0

(17)

for all θ∈Ds. Here ˜

Λis the ds×dsmatrix with N-th roots of the stable eigenvalues on the diagonal

entries and zero entries elsewhere. Pjparameterizes a local stable manifold attached to the periodic

point xjof the implicitly deﬁned mapping F. The Pjare unique up to the choices of the scalings

of the eigenvectors.

The theorem follows by applying Lemma 2.5 to the implicit map Fdeﬁned by T(y, x)=0. We

remark that the knowledge the Pjexist tells us that it is reasonable to develop numerical methods

to ﬁnd them. Moreover, the fact that they solve a functional equation leads to eﬃcient numerical

methods and a-posteriori error bounds. Indeed, if Tis analytic then the Pjare analytic as well, and

it makes sense to look for power series solutions of the functional equations. This topic is pursued

in the next section.

4 Formal series solution of equation (17)

In this section we illustrate the formal series calculations which allow us to compute stable/unstable

manifolds using the parameterization method. In particular, we derive the linear recurrence equa-

tions for the power series coeﬃcients of the functions solving Equation (17). We illustrate the

14

method for several examples of one dimensional stable/unstable manifolds attached to implicitly

deﬁned ﬁxed and periodic points. These calculations involve only power series of one variable. Sim-

ilar calculations for two dimensional manifolds, involving power series of two variables, are given in

the Appendices.

4.1 Operations on formal power series

We recall some basic facts about manipulating power series. Consider two inﬁnite sequences of

complex numbers {an}∞

n=0,{bn}∞

n=0 ⊂Cand the corresponding power series

f(z) =

∞

X

n=0

anznand g(z) =

∞

X

n=0

bnzn.

Suppose that λ∈C. Then

f(λz) =

∞

X

n=0

λnanzn.

Also, for any α, β ∈Cthe linear combination αf +βg has power series

αf(z) + βg(z) =

∞

X

n=0

(αan+βbn)zn.

Moreover, the product of two power series is given by the Cauchy product

f(z)g(z) =

∞

X

n=0

(a∗b)nzn,

where

(a∗b)n=X

k1+k2=n

ak1bk2

=

n

X

k=0

an−kbk.

Higher order products are deﬁned analogously. For example suppose that f1, . . . , fNare power

series given by

fi(z) =

∞

X

n=0

ai

nzn,1≤i≤N.

Then

f1(z). . . fN(z) =

∞

X

n=0 a1∗. . . ∗aNnzn,

where the N-th Cauchy product is given by

(a1∗. . . ∗aN)n=X

k1+...+kN=n

a1

k1. . . aN

kN

=

n

X

k1=0

k1

X

k2=0

. . .

kN−3

X

kN−2=0

kN−2

X

kN−1=0

a1

n−k1a2

k1−k2. . . aN−1

kN−2−kN−1aN

kN−1.

15

Note that the ﬁrst form of the sum is easier to read, but that the second form is easily implemented

in computer programs as a loop.

Another important operation is the extraction of the coeﬃcients of n-th order from the n-th

term of a Cauchy product. For example, we have that

(a∗b)n=b0an+a0bn+

n−1

X

k=1

an−kbk.

We write

(d

a∗b)n=

n−1

X

k=1

an−kbk,

to denote the terms in the Cauchy product depending only on lower order terms. Note that this is

(d

a∗b)n= (a∗b)n−a0bn−b0an=X

k1+k2=n

k1,k26=n

ak1bk2

Similarly, deﬁne

(\

a1∗. . . ∗aN)n= (a1∗. . . ∗aN)n−a1

0. . . aN−1

0aN

n−. . . −a2

0. . . aN

0a1

n,

which is equivalent to

(\

a1∗. . . ∗aN)n=X

k1+...+kN=n

k1,...,kN6=n

a1

k1. . . aN

kN.

4.2 An overview of the power matching strategy

In pursuit of a formal series solution of Equation (17), suppose that x1, . . . , xN∈Rdis a period

N-orbit for the implicitly deﬁned dynamics. That is, we assume that T:Rd×Rd→Rdis a

smooth function and that x1, . . . , xNsolve Equation (1). In the discussion to follow, let λdenote

a stable/unstable multiplier and ξ1, . . . , ξN∈Rdbe an associated collection of stable/unstable

eigenvectors, computed as described in Section A.3.

Since we want to solve a functional equation (Equation (17)) with prescribed ﬁrst order data,

we look for a power series solution of form

Pj(θ) =

∞

X

n=0

pj

nθn.1≤j≤N.

Here, for each n∈Nand 1≤j≤N, the power series coeﬃcient pj

n∈Rd. Note that if the Pjare

the solutions of Equation (17), then for 1≤j≤Nwe have that

pj

0=xj,and pj

1=ξj.

Supposing that Tis analytic in both variables (otherwise we proceed formally) write

Qj(θ) = T(Pj+1(˜

Λθ), Pj(θ)) =

∞

X

n=0

qj

nθn= 0,(18)

16

where it is understood that jN+1 =j1, and where the qj

ndepend on the coeﬃcients of the Pjin a

possibly complicated way. Nevertheless, since Qj(θ)=0, we have that

qj

n= 0,(19)

for all n≥0. Since our unknowns are the coeﬃcients pj

n, and since the qj

ndepend on them, we use

Equation (19) to derive recurrence relations for the coeﬃcients of the Pj. The following example is

meant to provide some insight into this procedure. Detailed calculations for non-trivial examples

are given in the following sections, and the appendices.

Example 4.1. As a simple example, consider the nonlinear mapping T:R2→Rgiven by

T(y, x) = x+y+xy +1

2x2.

Let F:U⊂R→Rdenote the mapping deﬁned by the implicitly by the requirement that F(x) = y

if and only if

T(y, x) = 0.

Suppose now that xj∈R,1≤j≤Nare periodic for F. That is, assume that

T(x2, x1)=0

.

.

.

T(x1, xN)=0,

with

∂1T(xj+1, xj)6= 0,

for 1≤j≤N, again with the understanding that xN+1 =x1. Suppose in addition that the periodic

orbit has multiplier −1<λ<1. In this case the stable manifold is the union of a one dimensional

neighborhoods of the points xj, and we seek

P1(θ) =

∞

X

n=0

p1

nθn

.

.

.

PN(θ) =

∞

X

n=0

pN

nθn,

satisfying Equation (17).

Our aim is to work out the coeﬃcients of the Qj(θ)deﬁned in Equation (18). To this end,

consider the component equation

Qj(θ) =

∞

X

n=0

qj

nθn=T(Pj+1(λθ), Pj(θ)) = 0,

17

which becomes

Qj(θ) = Pj(θ) + Pj+1(λθ) + Pj(θ)Pj+1(λθ) + 1

2Pj(θ)2

=

∞

X

n=0

pj

nθn+

∞

X

n=0

pj+1

nλnθn+ ∞

X

n=0

pj

nθn! ∞

X

n=0

pj+1

nλnθn!+1

2 ∞

X

n=0

pj

nθn!2

=

∞

X

n=0 pn

n+λnpj+1

n+

n

X

k=0

λkpj

n−kpj+1

k+

n

X

k=0

1

2pj

n−kpj

k!θn.

Matching like powers results in

qj

n=pj

n+λnpj+1

n+

n

X

k=0

λkpj

n−kpj+1

k+

n

X

k=0

1

2pj

n−kpj

k,

for n≥2(the ﬁrst order coeﬃcients are already constrained). Recalling that qj

n= 0 and isolating

the pj

nand pj+1

nterms on the left hand side of the equality leads to

pj

n+λnpj+1

n+λnpj

0pj+1

n+pj+1

0pj

n+pj

0pj

n=−

n−1

X

k=1

λkpj

n−kpj+1

k−

n−1

X

k=1

1

2pj

n−kpj

k

=−(\

pj∗pj+1)n−1

2(\

pj∗pj)n

or

1 + pj+1

0+pj

0λn(1 + pj

0)

pj

n

pj+1

n

=sj

n

where sndepends only on lower order coeﬃcients.

Since pj

0=xjfor 1≤j≤N, one easily checks that the entries of the row vector on the left hand

side of the equation depend on derivatives of Tevaluated along the periodic orbit. More precisely,

we have that

∂

∂x T(xj+1 , xj)λn∂

∂y T(xj+1 , xj)

pj

n

pj+1

n

=sj

n

By combining the results for each of the components, we see that the n-th order coeﬃcients solve

a linear equation of the form

An

p1

n

.

.

.

pN

n

=

s1

n

.

.

.

sN

n

,

for n≥2. Since the ﬁrst order terms are known, we can solve for n= 2. Once these have been

obtained, we solve for n= 3. And so on.

18

The problem of determining the power series coeﬃcients can be solved quite generally for multi-

variable power series by exploiting the Faa di Bruno formula (multivariable generalization of the

Leibniz rule). See for example the arguments for maps in [6], or the arguments for unstable manifolds

of delay diﬀerential equations in [36]. This approach however leads to formulas which may be

cumbersome in practice, and we ﬁnd it illuminating to consider the procedure in the context of

speciﬁc examples. We illustrate the formal series computation of the power series coeﬃcients for

parameterizations of some one and two dimensional stable/unstable manifolds attached to ﬁxed

and periodic orbits in polynomial examples in two and three dimensions. It is fairly straightforward

to generalize these computations to any polynomial system. Computations for non-polynomial

systems are handled using automatic diﬀerentiation for power series. Non-polynomial nonlinearities

are discussed in detail in [29]. See also [39, 19, 28].

4.3 A worked example: ﬁxed points of an implicit Hénon system

We now derive a formal series solution of the invariance equation given in Equation (16) for the sta-

ble/unstable manifold attached to a ﬁxed point of the implicit Hénon system given in Equation (4).

Since the Hénon mapping is on R2and the ﬁxed points will have one dimensional stable/unstable

eigenspace, this provides a simple example where the attached invariant manifolds have dimension

less than that of the phase space.

Let x∗∈R2have T(x∗,x∗) = 0, and suppose that λ∈Cis the stable eigenvalue and that

ξ∈C2is an associated eigenvector. Indeed, note that λ∈R(as the only other eigenvalue is

unstable), so that we can choose ξ∈R2. The eigendata is computed numerically following the

discussion in Section 1.1.1.

Motivated by Theorem 3.1 we seek P: (−τ, τ )→R2so that

P(0) = x∗, P 0(0) = ξ,

and

T(P(λθ), P (θ)) = 0,

for θ∈(−τ, τ ). Observe that since λis the only stable eigenvalue, the resonance conditions of

Theorem 3.1 are automatically satisﬁed.

Since Tis analytic in both variables we look for analytic Pof the form

P(θ) =

P∞

n=0 anθn

P∞

n=0 bnθn

,

and note that

T(P(λθ), P (θ)) = T ∞

X

n=0

λnanθn,

∞

X

n=0

λnbnθn,

∞

X

n=0

anθn,

∞

X

n=0

bnθn!= 0

has component equations

∞

X

n=0

λnanθn−1 + α"∞

X

n=0

anθn#2

−

∞

X

n=0

bnθn−"∞

X

n=0

λnanθn#5

= 0

∞

X

n=0

λnbnθn−β

∞

X

n=0

anθn+"∞

X

n=0

λnbnθn#5

= 0.

(20)

19

Deﬁne the inﬁnite sequence {δn}∞

n=0 by

δn=(1n= 0

0n≥1,

to represent the power series coeﬃcients of the constant function taking the value 1. We rewrite

Equation (20) in terms of Cauchy products as

∞

X

n=0

[λnan−δn+α(a∗a)n−bn−λn(a∗a∗a∗a∗a)n]θn= 0

∞

X

n=0

[λnbn−βan+λn(b∗b∗b∗b∗b)n]θn= 0.

Recalling the Cauchy “hat products” deﬁned in Section 4.1, we observe that

(a∗a)n= 2a0an+ ([

a∗a)n,

and that

(a∗a∗a∗a∗a)n= 5a4

0an+ ( \

a∗a∗a∗a∗a)n,

and similarly for the coeﬃcients involving the 5-th power of b. Matching like powers of θin both

sides of (20), and recalling that the ﬁrst order coeﬃcients n= 0 and n= 1 are already known, we

obtain for n≥2

anλn−bn+ 2αa0an+α([

a∗a)n−5a4

0λnan−λn(\

a∗a∗a∗a∗a)n= 0

λnbn−βan+ 5b4

0λnbn+λn(\

b∗b∗b∗b∗b)n= 0 (21)

and note that the “hat” products depend only on terms of order lower that n.

Isolating terms of order non the left and lower order terms on the right leads to the Homological

equations

λn+ 2αa0−5a4

0λn−1

−β5b4

0λn+λn

an

bn

=

S1

n

S2

n

(22)

for n≥2, where,

S1

n=−α([

a∗a)n+λn(\

a∗a∗a∗a∗a)n

S2

n=−λn(\

b∗b∗b∗b∗b)n.(23)

This is a linear equation for (an, bn), where the right hand side depends only on terms of lower

order. We can solve the homological equations to any desired order, provided that the matrices are

invertible.

Remark 4.2 (Non-resonances and uniqueness).Again, if the ﬁxed point is a saddle, then λnis

never resonant, and Equation (22) has a unique solution for all n≥2. It follows that the formal

power series solution is unique up to the choice of the scaling of the eigenvector. This comment in

fact holds generally. See [6].

20

4.4 A second worked example: period two orbit of implicit Hénon

Suppose now that x1= (x1, y1)and x2= (x2, y2)is a period two point for the implicit Hénon

system, which is computed numerically – along with its ﬁrst order data – as discussed in Section

1.1.1. Motivated by Theorem 3.2, we seek parameterizations P, Q : (−τ, τ )→R2so that

T(Q(λθ), P (θ)) = 0

T(P(λθ), Q(θ)) = 0.(24)

Letting

P(θ) =

∞

X

n=0

an

bn

θn, Q(θ) =

∞

X

n=0

cn

dn

θn,

Equation (24) becomes

∞

X

n=0

anλnθn−

1−α"∞

X

n=0

cnθn#2

+

∞

X

n=0

dnθn+"∞

X

n=0

anλnθn#5

= 0

∞

X

n=0

bnλnθn−β

∞

X

n=0

cnθn+"∞

X

n=0

bnλnθn#5

= 0

∞

X

n=0

cnλnθn−

1−α"∞

X

n=0

anθn#2

+

∞

X

n=0

bnθn+"∞

X

n=0

cnλnθn#5

= 0

∞

X

n=0

dnλnθn−β

∞

X

n=0

anθn+"∞

X

n=0

dnλnθn#5

= 0

.(25)

Expanding the powers as Cauchy products and extracting the terms of order n, we have

∞

X

n=0

λnan−δn+α(c∗c)n−dn−λn(a∗a∗a∗a∗a)n

λnbn−βcn+λn(b∗b∗b∗b∗b)n

λncn−δn+α(a∗a)n−bn−λn(c∗c∗c∗c∗c)n

λndn−βan+(d∗d∗d∗d∗d)n

θn=

0

0

0

0

.(26)

Extracting from the Cauchy products terms of order nand matching like powers of θleads to the

equations

λnan+ 2αc0cn+α(d

c∗c)n−dn−λn5a4

0an−λn(\

a∗a∗a∗a∗a)n= 0

λnbn−βcn+λn5b4

0bn+λn(\

b∗b∗b∗b∗b)n= 0

λncn+ 2αa0an+α([

a∗a)n−bn−λn5c4

0cn−λn(\

c∗c∗c∗c∗c)n= 0

λndn−βan+λn5d4

0dn+λn(\

d∗d∗d∗d∗d)n= 0

21

for n≥2. Observing that these equations are linear in (an, bn, cn, dn)we isolate the terms of order

non the left and have the homological equations

λn−5a4

0λn0 2αc0−1

0λn+ 5b4

0λn−β0

2αa0−1λn−5c4

0λn0

−β0 0 λn+ 5d4

0λn

an

bn

cn

dn

=

S1

S2

S3

S4

(27)

Where S1=−α(d

c∗c)n+λn(\

a∗a∗a∗a∗a)n

S2=−λn(\

b∗b∗b∗b∗b)n

S3=−α([

a∗a)n+λn(\

c∗c∗c∗c∗c)n

S4=−λn(\

d∗d∗d∗d∗d)n.

(28)

Once the period two point and its eigenvectors are known, so that we have the ﬁrst and second

order coeﬃcients, we solve the homological equations for 2≤n≤Nto ﬁnd the coeﬃcients of the

parameterization to order N. Indeed, the scheme just described generalizes to manifolds attached

to periodic orbits of any period in an obvious way.

5 Numerical Results

We illustrate the utility of the explicit homological equations derived in the previous section with

some example calculations.

5.1 Numerical example: stable/unstable manifolds attached to ﬁxed

points of the implicit Hénon system

As a ﬁrst example we consider stable/unstable manifolds attached to ﬁxed points of the implicit

Hénon system deﬁned in Equation (4). We compute a ﬁxed point, and its stable/unstable eigen-

values and eigenvectors as discussed in Section B.1. The results are summarized in Figure 1. This

ﬁrst order data allows us to compute the Taylor coeﬃcients of parameterizations of the manifolds

order by order, by recursively solving the homological equations. Some results are reported for the

unstable manifold in Figure 1.

The results in the Figure illustrate the fact that, while small changes in result in small changes

in the ﬁrst order data, the global dynamics are greatly aﬀected. Note also that the scaling of the

eigenvector has to be decreased as increases. This reﬂects the fact that the domain of analyticity

of the parameterization shrinks as increases. See also the remark below. We note that while

the parameterized manifold is not terribly large (roughly order one) many terms are needed to

conjugate the nonlinear to the linear dynamics.

The program which generates the results discussed here is

henonPaperEx_fixedPoint.m

22

First order data: implicit Hénon

parameter ﬁxed point eigenvalues eigenvectors

= 0.01 p0≈

0.6317

0.1895

λu≈ −1.939

λs≈0.1559

ξu≈

−0.9882

0.1529

ξs≈

−0.4612

−0.8873

= 0.03 p0≈

0.6326

0.1898

λu≈ −1.971

λs≈0.1559

ξu≈

−0.9886

0.1505

ξs≈

−0.4613

−0.8873

= 0.0315 p0≈

0.6326

0.1898

λu≈ −1.973

λs≈0.1559

ξu≈

−0.9886

0.1503

ξs≈

−0.4613

−0.8872

= 0.04 p0≈

0.6330

0.1900

λu≈ −1.987

λs≈0.1560

ξu≈

−0.9888

0.1492

ξs≈

−0.4613

−0.8872

Table 1: Fixed point/stability data: the table reports the location and stability of one of

the ﬁxed points of the implicit Hénon system as the parameter varies. Data is given to four

decimal places. More accurate values (approximately machine precision) are obtained by running

the programs.

Remark 5.1 (Loss of the hypotheses of the implicit function theorem).Following the discussion

in Section 1.1, we see that the implicit Hénon equations deﬁne a local diﬀeomorphism whenever

D1T(x2, y2) = Id +

−5x40

0 5y4

,

is invertible. For > 0the matrix is singular on the vertical line through

x∗() = 1

51/4

.

Note that when = 0.01 we have that

x∗(0.01) ≈2.115,

and the singular line is far from the attractor. However as increases the singular line moves closer

to the attractor, disrupting the assymptotic dynamics dramatically. In particular note that

x∗(0.0315) ≈1.587,

and

x∗(0.04) ≈1.495,

so that the singular line eventually moves into the attractor, creating the jumps, or breaks see in

the bottom left and right frames of Figure 1.

23

Figure 1: Implicit Hénon –stable/unstable manifolds attached to ﬁxed points: four calcu-

lations of the local unstable manifold of the ﬁxed point with data as in Table 1. The local unstable

manifold is colored dark blue, and eight of its forward iterates are lighter. In each case we computed

N= 75 Taylor coeﬃcients, with the eigenvector scalings as reported below. Top left: = 0.01.

The eigenvector is scaled by α= 1.0. Top right: = 0.031. The eigenvector is scaled by α= 0.85.

Bottom left: = 0.0315. The eigenvector is scaled by α= 0.8. Bottom right: = 0.04. The

eigenvector is scaled by α= 0.6. These scalings insure that the highest order coeﬃcient computed

has magnitude on the order of machine epsilon.

5.2 Numerical example: stable/unstable manifolds attached to periodic

orbits of the implicit Hénon system

We now illustrate the computation of the stable/unstable manifolds attached a period two point

for the implicit Hénon systems. For the period two problem we consider only the two larger values

of . When = 0.0315 there is a period two orbit located at

p1≈

−0.4945

0.2940

p2≈

0.9802

−0.1483

24

Figure 2: Implicit Hénon –stable/unstable manifolds attached to period 2 orbits: two

calculations of the local unstable manifolds colored with light blue attached to a period two orbit of

the implicit Hénon system. In each case we computed N= 50 Taylor coeﬃcients, with eigenvector

scalings as reported below. Left: = 0.0315. The eigenvector is scaled by α= 0.75. Right:

= 0.04. The eigenvector is scaled by α= 0.5. These scalings ensure that the highest order

coeﬃcient computed has magnitude on the order of machine epsilon.

with multipliers

λu≈ −3.807,and λs≈ −0.0279.

We choose the square roots

˜

λu≈1.951i, and ˜

λs≈0.1670i.

and eigenvectors

ξu

1≈

0.7868

−0.0919

ξu

2≈

−0.5982

−0.1210

ξs

1≈

0.3958

−0.5076

and ξs

2≈

0.2829

0.7110

.

Similarly, when = 0.04 the data is

p1≈

−0.4995

0.2943

p2≈

0.9814

−0.1499

with multipliers

λu≈ −4.080,and λs≈ −0.0274.

We choose the square roots

˜

λu≈2.020i, and ˜

λs≈0.165i.

25

and eigenvectors

ξu

1≈

0.7800

−0.0902

ξu

2≈

−0.6083

−0.1158

ξs

1≈

0.3923

−0.5107

and ξs

2≈

0.2821

0.711

.

The results are reported with only four signiﬁcant ﬁgures. More accurate data is obtained by

running the computer programs.

In both cases these are taken as initial data for computation of the stable/unstable parame-

terizations, whose Taylor coeﬃcients for orders 2≤n≤Nare found by recursive solution or the

homological equations deﬁned explicitloy in Equations (27) and (28). The resulting local manifolds

and a number of forward iterations are illustrated in Figure 2. See Remark 5.1 for the explication

of the “jump/break” in the attractor.

The programs which generate the results discussed here are

more_iteration.m

and

henonForPaper_per2.m

Remark 5.2 (Heteroclinic/homoclinic connections: inﬁnite forward and backward time orbits).

Figures 3 and 4 illustrate local parameterizations of the the stable and unstable manifolds attached

to the ﬁxed points and the period two orbit of the implicit Hénon system with = 0.04, without

and with that application of two iterates of the implicit dynamics. At this parameter value the

singular value has moved into the basin of attraction and strongly disrupts the system. Nevertheless,

the intersection of unstable and stable manifolds illustrated in the ﬁgure suggest the existence of

heteroclinic and homoclinic orbits: that is, dynamics which exist for all forward and backward time.

The ﬁgures illustrates that, even though simulating the system for long times is very diﬃcult (the

intersection of the singular set with the attractor disrupts iteration schemes based on Newton’s

method) we nevertheless obtain a great deal of useful information about the global dynamics by

studying the parameterized manifolds.

5.3 Numerical example: stable/unstable manifolds attached to ﬁxed

points of the implicit Lomelí system

In this section we compute and extend the two dimensional local stable/unstable manifolds attached

to ﬁxed points of the implicit Lomelí system deﬁned by Equation (8) with parameter values ρ= 0.34,

τ= 1.3,a= 0.5,b= 0.5,