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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
April 1, 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:RdRdgenerates a dynamical system by the rule
xn+1 =F(xn), n N,
with initial condition x0Rd. 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, horse shoes, 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×RdRd, 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 imeadieatly raises delicate questions about the domain of F, or wether Fis even single valued.
Of course the implicit function theorem provides valuable information. That is, if (¯y, ¯x)Rd×Rd
has Ty, ¯x) = 0, and D1T(¯y, ¯x)is an isomorphism, then there exists an open set ¯xURdand a
smooth diffeomorphism F:URdwith F(¯x) = ¯yand having that
T(F(x), x)=0,for all xU.
We say that Timplicitly defines the diffeomorphism Fnear ¯x.
We remark that for fixed ¯xRd, 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 ˜xF(U)and ˜yRdhave Ty, ˜x) = 0, with D1T(˜y, ˜x)an
isomorphism. Then there is an open set ˜x˜
URdand a diffeomorphism ˜
F:˜
URdwith ˜
F(x)a
branch of solutions of T(y, x) = 0 having ˜
Fx) = ˜y. Now, for any xUF1(˜
U), the composition
of xunder the two implicitly defined maps Fand ˜
Fis well defined.
Continuing in this way, suppose that y16=. . . 6=yNRdhave
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 UjRd, and diffeomorphisms Fj:UjRdso 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. . . UNRd, 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 systems.
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 solution
x1, . . . , xNfor the system 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 neighborhoods 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.
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 discuss
2
briefly 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 found at
http://cosweb1.fau.edu/~jmirelesjames/parmImplicitMaps.html
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:RdRdbe a Ckdiffeomorphism (or real analytic with real analytic inverse if k=ω).
Then for any x0Rd,fdefines a dynamical system by the rule
f(xn) = xn+1,
for n= 0,1,2, . . . Define the function T:Rd×RdRdby
T(y, x) = yf(x),
and note that for a given ¯xRd,¯ysolves the equation T(y, ¯x) = 0 if and only if
¯y=fx).
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×VRdbe a Ckfunction.
Consider the one parameter family of problems T:U×VRdby
T(y, x) = yf(x) + H(y, x),(2)
and note that for any y, ¯x)V×Uwe have that
D1Ty, ¯x) = Id +D1H(¯y, ¯x).
Note that D1T0y, ¯x) = Id, so that –by the implicit function theorem – there is a δ > 0and a
smooth curve y: (δ, δ)VRdso that y(0) = ¯yand
T(y(),¯x) = 0,
for all (δ, δ).
Moreover, for a possibly smaller δ > 0we have that
D1T(y(),¯x) = Id +D1H(y(),¯x),
3
is invertible for each (δ, δ), by the Neumann theorem. Then there exists an r > 0and a family
of functions F:Brx)×(δ, δ)Rdso that
F0(x) = f(x),
and
T(F(x), x)=0,
for all xBrx)and (δ, δ). The family Fdepends smoothly on and is Ckfor each fixed
. Moreover, for small 6= 0 and ¯xUwe 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 Fx) = ¯y. Finally, suppose that ¯xUis 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 + y1αx2
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, 59]. We define an implicit Hénon system T:
R2×R2R2given by
T(θ2, θ1) = T(x2, y2, x1, y1) =
x2(1 αx2
1+y1+x5
2)
y2βx1+y5
2
.(4)
Here we have choosen, somewhat arbitrarily, the perturbation term
H(y, x) =
x5
y5
,
to be of high enough polynomial order that it is difficult (if not impossible) to work out useful
formulas for the implicitly defined system.
4
The equation for a fixed point is T(x,x)=0, or
x1(1 αx2
1+y1+x5
1)
y1βx1+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
y2βx1+y5
2= 0
x1(1 αx2
2+y2+x5
1)=0
y1βx2+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:R3R3given 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 [47], and was subsequently studied by a number of authors
including [47, 23, 53, 55, 11, 28].
Let θ1= (x1, y1, z1), θ2= (x2, y2, z2)R3. We consider the dynamics implicitly defined by the
map T:R3×R3R3given by
T(θ2, θ1) = T(x2, y2, z2, x1, y1, z1)
=
x2ρτx1z1ax2
1bx1y1cy2
2+αy5
2+βz5
2
y2x1+γz5
2
z2y1
.(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
xρτx zax2bxy cy2+(αy5+βz5)
yx+γz5
zy
=
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
x2ρτx1z1ax2
1bx1y1cy2
1+(αy5
2+βz5
2) = 0
y2x1+γz5
2= 0
z2y1= 0
x3ρτx2z2ax2
2bx2y2cy2
2+(αy5
3+βz5
3) = 0
y3x2+γz5
3= 0
z3y2= 0
x4ρτx3z3ax2
3bx3y3cy2
3+(αy5
4+βz5
4) = 0
y4x3+γz5
4= 0
z4y3= 0
x1ρτx4z4ax2
4bx4y4cy2
4+(αy5
1+βz5
1) = 0
y1x4+γz5
1= 0
z1y4= 0
(11)
The equations for fixed and periodic orbits are again amenable to Newton’s method, and the
multipliers λ1, λ2, λ3C, and associated eigenvectors ξj1, ξj2, ξj3C3,1i4are 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 [51]. The Ph.D. dissertation of Sander generalized stable/unstable manifold
theory to the setting of relations [25], and work by Lerman [43] and Wather [63] 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 [61, 60, 62] 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 [48, 25], delay differential equations [63], adaptive control [1], discrete variational problems
[27, 65], and economic theory [40, 41, 42, 52]. 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 [49], 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, 57],
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, 66] 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 URdis an open
set, that F:URdis a Ck(U)mapping with k= 1,2,3,...,, ω, that xUis 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 λspecs(x)and
1< β ≤ |λ|,
for all λspecu(x). Let LNbe the smallest natural number with
αL<1
β,
and assume that
L+ 1 < k.
Then there exists an open set DsRdswith 0Ds, a polynomial K:DsRdsof degree not
more than L, and a Ckmapping P:DsRdso that
1.
P(0) = x,
2. The columns of DP (0) span Es, and
3.
F(P(θ)) = P(K(θ)),(12)
for all θDs.
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, . . . , λdsCdenote the stable
eigenvalues of DF (x), and assume that each has multiplicity exactly one. Moreover, assume that
for all (n1, . . . , nds)Ndswith
2n1+. . . +ndsL,
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 . . . λds10
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
dsspecs(x).
In this case, the polynomial Kis required to have a monomial term of the form c θn1
1. . . θnds
dswith
non-zero cRds. 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 [64].
It is worth remarking that when the stable eigenvalues are non-resonant, Equation (12) reduces
to
F(P(θ)) = Pθ), θ DsRds,(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 ΛsDsDs. In this case, since Equation
(14) holds, it is easy to see that Pparameterizes a local stable manifold. To see this, let θDs.
Since Pis continuous (in fact Ck) we have that
lim
n→∞ Fn(P(θ)) = lim
n→∞ F(Pnθ))
=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
xgives 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 xthere is a well defined Ckinverse mapping F1. Let Σdenote the diagonal matrix of unstable
eigenvalues of DF (x), so that Σ1is the matrix of stable eigenvalue of DF 1(x). Assume that
these stable eigenvalues (entries of Σ1) are non-resonant. Then there exists an open set Duand a
Ckmapping Q:DuRdso that
F1(Q(σ)) = Q1σ), σ Du.
Applying Fto both sides of the equation and composing with Σleads to the equation
Qσ) = 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, , ..., xNRdbe 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 . . . λds10
0 0 . . . 0λds
,
denote the ds×dsdiagonal matrix of stable multipliers (similarly Σdenote the du×dudiagonal
matrix of unstable multipliers). For 1jds, let ξj,1, . . . , ξj,dsCddenote 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 DsRdsand are unique P1,...PN:DsRdso 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)) = P1θ)
.
.
.
FN(PN(θ)) = PNθ)
(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:DsRdsatisfy the system
of composition free equations
F(P1(θ)) = P2˜
Λθ
F(P2(θ)) = P3˜
Λθ
.
.
.
F(PN1(θ)) = PN˜
Λθ
F(PN(θ)) = P1˜
Λθ
where
˜
Λ =
N
λ10. . . 0 0
0N
λ2. . . 0 0
.
.
..
.
.....
.
..
.
.
0 0 . . . N
pλds10
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×RdRdis 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 xis 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×VRdis a Ckmapping
with fixed point xUV, that is
T(x, x) = 0.
Assume that
D1T(x, x)is invertible.
Let λ1, . . . , λdsCdenote the stable eigenvalues and ξ1, . . . , ξdsCdassociated 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
1jds|λj|,
β= max
λspecu(x)λ1,
and 2Lbe the smallest integer so that
αLβ < 1.
Assume that L+ 1 k.
12
Assume that for all (n1, . . . , nds)Ndswith 2n1+. . . +ndsLwe have that
λn1
1. . . λnds
ds6=λj
for 1jλds.
Then there exists an open set DsRdswith 0Ds, and a Ckmapping P:DsRdso that
P(0) = x,
DP (0) = [ξ1, . . . , ξds],
and
T(Pθ), 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 xof 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=Pθ)and x=P(θ), Equation (16), is equivalent to
F(P(θ)) = Pθ), 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×VRdis a Ckmapping,
and that x1, . . . , xNUVhave
T(x2, x1)=0
.
.
.
T(xN1, xN)=0
T(x1, xN)=0
Assume that:
the matrices D1T(x2, x1), . . . , D1T(xN, xN1), D1T(x1, xN)are invertible.
Let λ1, . . . , λdsCdenote the stable multipliers and for 1jNlet ξj,1, . . . , ξj,dsCd
denote associated eigenvectors. Assume that the stable multipliers are distinct (otherwise
choose the appropriate generalized eigenvectors).
13
Let
α= max
1jds|λj|,
β= max
λspecu(x)λ1,
and 2Lbe the smallest integer so that
αLβ < 1.
Assume that L+ 1 k.
Assume that for all (n1, . . . , nds)Ndswith 2n1+. . . +ndsLwe have that
λn1
1. . . λnds
ds6=λj
for 1jλds.
Then there exists an open set DsRdswith 0Ds, and Ckmappings P1, . . . , PN:DsRdso
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), PN1(θ))) = 0
T(P1(˜
Λθ), PN(θ))) = 0
(17)
for all θDs. 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 λC. 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
(ab)nzn,
where
(ab)n=X
k1+k2=n
ak1bk2
=
n
X
k=0
ankbk.
Higher order products are defined analogously. For example suppose that f1, . . . , fNare power
series given by
fi(z) =
X
n=0
ai
nzn,1iN.
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
. . .
kN3
X
kN2=0
kN2
X
kN1=0
a1
nk1a2
k1k2. . . aN1
kN2kN1aN
kN1.
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
(ab)n=b0an+a0bn+
n1
X
k=1
ankbk.
We write
(d
ab)n=
n1
X
k=1
ankbk,
to denote the terms in the Cauchy product depending only on lower order terms. Note that this is
(d
ab)n= (ab)na0bnb0an=X
k1+k2=n
k1,k26=n
ak1bk2
Similarly, define
(\
a1. . . aN)n= (a1. . . aN)na1
0. . . aN1
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, . . . , xNRdis a period
N-orbit for the implicitly defined dynamics. That is, we assume that T:Rd×RdRdis a
smooth function and that x1, . . . , xNsolve Equation (1). In the discussion to follow, let λdenote
a stable/unstable multiplier and ξ1, . . . , ξNRdbe 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.1jN.
Here, for each nNand 1jN, the power series coefficient pj
nRd. Note that if the Pjare
the solutions of Equation (17), then for 1jNwe 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 n0. 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:R2Rgiven by
T(y, x) = x+y+xy +1
2x2.
Let F:URRdenote the mapping defined by the implicitly by the requirement that F(x) = y
if and only if
T(y, x) = 0.
Suppose now that the points xjR,1jNare a periodic for F. That is, we assume that
T(x2, x1) = 0
.
.
.
T(x1, xN) = 0,
with
1T(xj+1, xj)6= 0,
for 1jN, 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
nkpj+1
k+
n
X
k=0
1
2pj
nkpj
k!θn.
Matching like powers results in
qj
n=pj
n+λnpj+1
n+
n
X
k=0
λkpj
nkpj+1
k+
n
X
k=0
1
2pj
nkpj
k,
for n2(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=
n1
X
k=1
λkpj
nkpj+1
k
n1
X
k=1
1
2pj
nkpj
k
=(\
pjpj+1)n1
2(\
pjpj)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 1jN, 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 n2. 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. 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 explicit 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 xR2have T(x,x) = 0, and suppose that λCis the stable eigenvalue and that
ξC2is an associated eigenvector. Indeed, note that λR(as the only other eigenvalue is
unstable), so that we can choose ξR2. 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
λnanθn1 + α"
X
n=0
anθn#2
X
n=0
bnθn"
X
n=0
λnanθn#5
= 0
X
n=0
λnbnθnβ
X
n=0
anθn+"
X
n=0
λnbnθn#5
= 0.
(20)
19
Define the infinite sequence {δn}
n=0 by
δn=(1n= 0
0n1,
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
[λnanδn+α(aa)nbnλn(aaaaa)n]θn= 0
X
n=0
[λnbnβan+λn(bbbbb)n]θn= 0.
Recalling the Cauchy “hat products” defined in Section 4.1, we observe that
(aa)n= 2a0an+ ([
aa)n,
and that
(aaaaa)n= 5a4
0an+ ( \
aaaaa)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 n2
anλnbn+ 2αa0an+α([
aa)n5a4
0λnanλn(\
aaaaa)n= 0
λnbnβan+ 5b4
0λnbn+λn(\
bbbbb)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+ 2αa05a4
0λn1
β5b4
0λn+λn
an
bn
=
S1
n
S2
n
(22)
for n2, where,
S1
n=α([
aa)n+λn(\
aaaaa)n
S2
n=λn(\
bbbbb)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 n2. 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
1α"
X
n=0
cnθn#2
+
X
n=0
dnθn+"
X
n=0
anλnθn#5
= 0
X
n=0
bnλnθnβ
X
n=0
cnθn+"
X
n=0
bnλnθn#5
= 0
X
n=0
cnλnθn
1α"
X
n=0
anθn#2
+
X
n=0
bnθn+"
X
n=0
cnλnθn#5
= 0
X
n=0
dnλnθnβ
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
λnanδn+α(cc)ndnλn(aaaaa)n
λnbnβcn+λn(bbbbb)n
λncnδn+α(aa)nbnλn(ccccc)n
λndnβan+(ddddd)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
cc)ndnλn5a4
0anλn(\
aaaaa)n= 0
λnbnβcn+λn5b4
0bn+λn(\
bbbbb)n= 0
λncn+ 2αa0an+α([
aa)nbnλn5c4
0cnλn(\
ccccc)n= 0
λndnβan+λn5d4
0dn+λn(\
ddddd)n= 0
21
for n2. 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
λn5a4
0λn0 2αc01
0λn+ 5b4
0λnβ0
2αa01λn5c4
0λn0
β0 0 λn+ 5d4
0λn
an
bn
cn
dn
=
S1
S2
S3
S4
(27)
Where S1=α(d
cc)n+λn(\
aaaaa)n
S2=λn(\
bbbbb)n
S3=α([
aa)n+λn(\
ccccc)n
S4=λn(\
ddddd)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 2nNto 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
λs0.1559
ξu
0.9882
0.1529
ξs
0.4612
0.8873
= 0.03 p0
0.6326
0.1898
λu≈ −1.971
λs0.1559
ξu
0.9886
0.1505
ξs
0.4613
0.8873
= 0.0315 p0
0.6326
0.1898
λu≈ −1.973
λs0.1559
ξu
0.9886
0.1503
ξs
0.4613
0.8872
= 0.04 p0
0.6330
0.1900
λu≈ −1.987
λs0.1560
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 +
5x40
0 5y4
,
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) 1.59,
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
˜
λu1.951i, and ˜
λs0.1670i.
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
˜
λu2.020i, and ˜
λs0.165i.
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 2nNare 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 “tear” 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=
0.344444444,τ= 1.333333333,a= 0.5,b= 0.5,c= 1,α= 1,β= 1,γ= 1, and = 0.01. We also
compute the two dimensional local stable/unstable manifolds associated with a period four orbit.
The results illustrated in Figures 5 and 6 are obtained by solving order by order the homological
equations given in Equations (37) and (39) respectively.
The local manifolds in Figures 5 have been iterated (forward for the unstable manifolds and
backwards for the stable) and seem to intersect transversally. This suggests that the heteroclinic
arcs of the = 0 system studied in [55] persist into the implicit system at least for small . Numerical
values of the fixed points, period orbits, and their first order data can be found by running the
computer programs.
The program generating the results discussed here is
TwoD_Manifold_period4.m
26
Figure 3: Implicit Hénon – connecting orbits: Stable and unstable manifolds when = 0.04.
The green curves represent the unstable manifolds of the two fixed points. The blue curves represent
unstable manifolds attached to the period two orbit. Similarly, the cyan curves represent the
stable manifolds of the two fixed points, and the red curves the stable manifolds of the period two
orbit. All curves are plots of polynomial approximations of the local manifolds computed using the
parameterization method; no iteration has been applied to “grow” the manifolds. Note that the
blue and cyan curves, as well as the green and the red curves already intersect. These intersections
provide numerical evidence for the existence of transverse connecting orbits from the period two
orbit to the fixed point and from the fixed point to the period two. These connections also appear
to be isolated away from the singular set, so that their existence would imply the existence of a
geometric horseshoe (heteroclinic cycle).
6 Conclusions
In this work we have developed a multiple shooting method for studying invariant manifolds at-
tached to periodic orbits of implicitly defined dynamical systems, effectively extending the param-
eterization method to this setting. After some preliminary formal series calculations are performed
“by hand”, our approach reduces the computation of the parameterizations the basic linear algebra
and facilitates polynomial approximation to any desired order. By judiciously adjusting the scalings
of the eigenvectors, the method can be used to compute fairly large portions of the attached local
stable/unstable manifolds of the fixed/periodic orbits. In some examples these large local manifolds
parameterizations already indicate the existence of heteroclinic and homoclinic connecting orbits-
for the implicitly defined dynamics. In other examples, some globalization methods can be applied
after the initial parameterization.
An interesting direction for further research would be to use the methods developed here to
27
Figure 4: Implicit Hénon – more connecting orbits: In this figure the local unstable manifolds
of the period two points have been iterated twice, again for the = 0.04 system. Iterates of different
manifold segments are shown in matching colors, so that yellow is the image of yellow, orange of
orange, brown of brown, and black of black. After “growing” the local manifolds under iteration we
now see that the unstable manifolds of the period two points intersect the stable manifolds of the
period two, providing evidence for another geometric horseshoe (homoclinic tangle).
study problems in crystalline lattices, like the Frenkel Kontorova model [24, 19]. While constant
solutions of such models can be studied by finding fixed points, non-trivial equilibrium solutions
appear as periodic solutions of some implicitly defined maps. Connecting orbits between periodic
solutions describe traveling waves in the lattice. Moreover, the methods developed in the present
work are amenable to mathematically rigorous computer assisted validation methods similar to
those discussed in [56, 54, 5, 45]. Combining the methods of the present work with the techniques
of the references just cited would lead computer assisted methods of proof for theorems about
Frenkel Kontorova and other such problems.
Another interesting direction of research is to extend the methods of the present work to infinite
dimensional implicitly defined dynamical systems, like delay differential equations. For example,
with τ > 0and f:Rd×RdRda smooth function, a delay differential equation of the form
y0(t) = f(y(t), y(tτ)),
28
Figure 5: Implicit Lomelí systems– stable/unstable manifolds attached to fixed points:
the local invariant manifold parameterizations and a number of forward/backward iterations. The
image on the right illustrates both manifolds superimposed together, and suggests that the manifolds
intersect transversally.
can be rewritten as a step map
T(y(t), x(t)) = y(t)x(0) Zt
τ
f(y(s), x(s)) ds, (29)
where x(s)is the history function defined on [τ, 0]. That is, given x, if yhas T(y , x)=0then
y(tτ)is a solution of the delay differential equation on the interval [0, τ]with history x(t)given
on [τ, 0].
The interested reader can consult the papers [21, 46, 44] where the authors study the dynamics
generated by some delay differential equations by considering discretization of the implicitly defined
dynamical system defined by the zeros of Equation (29). In particular, computer assisted proofs
of periodic orbits for delay equations are given in the last reference just cited, using a multiple
shooting setup much like the one considered in the present work. The authors of [36] are currently
adapting the methods of the present work to the infinite dimensional setting of delay equations to
study homoclinic chaos in systems like Mackey-Galss [50].
29
Figure 6: Implicit Lomelí systems– stable/unstable manifolds attached to a period 4
orbit: the local invariant manifold parameterizations.
7 Acknowledgments
The authors would like to acknowledge the contribution of two anonymous referees who carefully
read the submitted version of this manuscript. Their comments and suggestions greatly improved
the final version. Conversations with Hector Lomrlí, Rafael de la Llave, Emmanuel Fleurantin and
Jorge Gonzalez are also gratefully acknowledged. The second author was partially supported by
NSF grant DMS 1813501 during work on this project.
A Definitions and Background
In this section we review some basic definitions from the qualitative theory of nonlinear dynamical systems. We also
review the main results from [6, 7, 8] about the parameterization method for fixed points of local diffeomorphisms,
and results from [28] extending these results to periodic orbits. The reader familiar with this material may want to
skim or skip this section upon first reading, referring back to it only as needed.
30
A.1 Discrete time semi-dynamical systems: Maps
The material in this section is standard, and an excellent reference is [59]. Suppose that URdis an open set and
F:UUis a Ck(U)mapping, with k= 0,1,2,...,, ω. For x0U, define the sequence x1=F(x0),x2=F(x1),
and in general xn+1 =F(xn)for n0. We refer to the set {xn}
n=0 as the forward orbit of x0under F, and write
orbit(x0, F )to denote this set. Let F0(x) = x,F1(x) = F(x),F2(x) = F(F(x)) and in general Fn(x)denote the
composition of Fwith itself ntimes applied to x. When Fis understood we simply write orbit(x0)and talk about
the orbit of x0. Then
orbit(x0) =
[
n=0
Fn(x0).
A sequence {xn}0
n=−∞ Uwith F(x1) = x0and F(xn) = xn+1 for all n < 0is a backward orbit of x0
under F. The pair (U, F )is referred to as a semi-dynamical system, as, while forward orbits are uniquely defined,
backwards orbits need not exist and when they do exist they need not be unique.
A.2 Local stable/unstable manifolds for fixed points/periodic orbits
Let FCk(U)with k1and suppose that xUis a fixed point, so that
F(x) = x.
We write spec(x) = {λ1,...,λd} ⊂ Cto denote the set of eigenvalues of DF (x). Let ξ1,...,ξdCdbe an
associated choice of (possibly generalized) eigenvectors. Let D1Cdenote the open unit disk in the complex plane,
and S1denote the unit circle. Define
specs(x) = spec(x)D1
specc(x) = spec(x)S1
specu(x) = spec(x)\(specs(x)specc(x)) ,
and note that specs(x)is the set of eigenvalues with complex absolute value less than one, specc(x)is the set of
eigenvalues with complex absolute value equal to on, and specu(x)is the set of eigenvalues with complex absolute
value greater than one. There are referred to as the stable, center, and unstable eigenvalues respectively, and we
note that any of two of these sets could be empty. If specc(x) = then we say that xis a hyperbolic fixed point.
Define the vector spaces Es,Ec, and Euto be the span of the stable, the center, and the unstable eigenvectors
respectively. These are referred to as the stable, center, and unstable eigenspaces of DF (x), and they are invariant
linear subspaces for the dynamics induced by DF (x). It is a classical fact that they are tangent to corresponding
locally invariant nonlinear manifolds of Fin a neighborhood of x. Let ds=dim(Es),dc=dim(Ec), and du=
dim(Eu)denote the dimension of the stable/center/unstable eigenspaces, or equivalently the number (counted with
multiplicity) of stable/center/unstable eigenvalues.
Define the sets
Ws(x) = nxU: lim
n→∞
Fn(x) = xo
Wu(x) = xU:there exists a backward orbit {xn}of xwith lim
n→−∞
xn=x.
These are referred to as the stable and unstable sets for xrespectively. In a similar fashion, for any open set VU
with xV, define
Ws
loc(x, V ) = {xV:Fn(x)Vfor all n0,and Fn(x)xas n→ ∞}
Wu
loc(x, V ) = xV:there is a backward orbit for xin Vwith lim
n→−∞
xnx,
and note that for any VUwe have that Ws
loc(x, V )Ws(x), and Wu
loc(x, V )Wu(x).
The following stable manifold theorem says that if xis hyperbolic then there exist local stable/unstable sets
with especially nice properties.
Theorem A.1 (Local stable manifold theorem).Suppose that xis a hyperbolic fixed point for F. Then there
exists an open set VUwith xVso that Ws
loc(x, V )and Wu
loc(x, V )are respectively dsand dudimensional
embedded disks – as smooth as F– and tangent at xto Esand Eurespectively.
31
The theorem gives that the stable/unstable sets are locally smooth manifolds. If Fis a diffeomorphism then the
full stable/unstable sets are obtained by iterating Fand F1, hence the stable/unstable sets are smooth manifolds
(which can nevertheless be embedded in Uin very complicated ways). However, if Fis not invertible the global
stable/unstable sets might misbehave in a number of ways.
Connectedness: While the unstable set must be connected ( image of a disk is connected under iteration of
a continuous map) the stable set can in general be disconnected. The unstable set can have self intersections.
Dimension: both the stable/unstable sets can increase in dimension outside a neighborhood of x.
Smoothness: the stable/unstable sets need not be smooth manifolds away from x. At points where DF (x)
has an isolated non-singularity the set can develop corners or cusps.
Examples of each of these phenomena are discussed in [62], and many explicit examples are given. See also [25].
A.3 Multiple shooting for periodic orbits
With URdan open set, and F:URda smooth mapp, suppose that x1,...,xNUhave
F(x1) = x2
.
.
.
F(xN1) = xN
F(xN) = x1
Then {x1,...,xN}is a periodic orbit for F. If the xj,1jNare distinct, then Nis the least period. We refer
to xj,1jNas a period Npoint. If D F (xj)is invertible for each 1jNwe say that the periodic orbit is
non-degenerate.
Note that ¯xUis a period Npoint for Fif and only if ¯xis a fixed point of the composition FN. If the orbit of
¯xis non-degenerate and least period Nthen DF N( ¯x)is invertible. We note that if {x1,...,xN}is a non-degenerate
periodic orbit then the matrices DF N(xj),1jNhave the same eigenvalues. These are also referred to as the
multipliers of the periodic orbit.
If DF Nx)has no eigenvalues on the unit circle we say that the periodic orbit is hyperbolic and Theorem A.1
applies to the composition mapping FN. In particular, there are local stable and unstable manifolds attached to the
points of the periodic orbit.
Let UN=U×...×URN d denote the product of Ncopies of U. Define G:UNRNd by
G(x1, x2,...,xN1, xN) =
F(xN)
F(x1)
F(x2)
.
.
.
F(xN1)
.(30)
and observe that if (x1,...,xN)RNd is a fixed point of Gthen {x1,...,xN}is a period Norbit for F. We refer
to Gas a multiple shooting map for a period Norbit of F. In practice numerically computing fixed points of G
using Newton’s method is more stable than computing fixed points of FN[28] also with Newton. This is because the
condition number of DF Ngrows exponentially with N. While D G is a larger matrix, it has a much better condition
number, and modern linear algebra routines easily solve the Newton step. Taking advantage of the sparse structure
of DG (which we have not done) leads to even great improvements.
Note also that if
DG(x1,...,xN) =
0 0 ... 0DF (xN)
DF (x1) 0 ... 0 0
0DF (x2)... 0 0
0 0 . . . DF (xN1) 0
(31)
is invertible then the periodic orbit is non-degenerate. In fact, λCis an eigenvalue of DG(x1,...,xN)if and
only if λNis an eigenvalue of DFN(xj). Moreover, one can check that if ξ= (ξ1,...,ξN)CdN is an eigenvector
32
associated with the eigenvalue λof the matrix DG(x1,...,xN), then for 1jNwe have that (λN, ξj)is an
eigenvalue/eigenvector pair for the matrix DF N(xj). That is, the multipliers of the periodic orbit and the eigenspaces
of DF N(xj)are easily recovered from the eigenvalues/eigenvectors of DG(x1,...,xN). The interested reader will
find a more thorough discussion of the relationship between multiple shooting maps and periodic orbits in [28].
B Implicitly defined dynamical systems
Let U, V Rdbe open sets and suppose that T:V×URdis a smooth function. We are interested in the existence
of open sets DU,RVand a mapping F:DRRddefined by the rule
F(x) = y, (32)
if and only if for a fixed given input xD,ysolves the equation
T(y, x)=0,(33)
with yR. We say that the mapping Fis implicitly defined by the rule given in Equation (33). Note that Fneed
not be one-to-one or even single valued globally. However, if for a fixed ¯xUwe have that there are ¯y, ˜yUso that
Ty, ¯x) = Ty, ¯x)=0, then we require that there are neighborhoods ¯
R, ˜
RVwith ¯y¯
R,˜y˜
Rand ¯
R˜
R=.
Existence and regularity of implicitly defined maps is subtle, yet– as already mentioned in the introduction – the
implicit function theorem comes to our aid. Let D1T(y , x)and D2T(y, x)denote the partial derivatives of Twith
respect to the first and second variables respectively, and suppose that T(x1, x0)=0(note that D1T(y , x), D2T(y, x)
are d×dmatrices). If D1T(x1, x0)is invertible then, by the implicit function theorem [58], there exists an r > 0
and a function F:Br(x0)URdso that F(x0) = x1and
T(F(x), x)=0,(34)
for all xBr(x0). Moreover, the mapping Fis as smooth as T. By differentiating (34) we have that DF (x0)solves
the equation
D1T(x1, x0)DF (x0) = D2T(x1, x2),(35)
with D1T(x1, x0)invertible. Of course the map Fdepends on the choice of x1. However, once we choose a solution
x1, the deffeomorphism Fis well defined and unique locally. The discussion above motivates the following definition.
Definition 1. We say that ¯xUis a regular point for Tif there exists ¯yVsuch that
Ty, ¯x)=0,
and D1Ty, ¯x)is invertible. Note that, by the implicit function theorem as above, ¯xis in the interior of D=dom(F).
Moreover, Fis a local diffeomorphism of a neighborhood of ¯xinto a neighborhood of ¯y.
Remark B.1 (Numerical evaluation of F).Evaluation of F(x)requires solving the nonlinear equation T(y , x) = 0
with xgiven. In practice we use Newton’s method as follows. Let ¯xbe fixed and y0be an approximate solution in
the sense that
kT(y0,¯x)k ≈ 0.
For n0, define
yn+1 =yn+ ∆n,
where nsolves the linear equation
D1T(yn,¯x)∆n=T(yn,¯x).
Convergence and error analysis for the algorithm is a classic topic (see any book on numerical analysis). For the
moment it is enough to remark that the algorithm is expected to perform well close enough to a regular point, as
invertibility of the derivative is an open property. In practice, if after Nsteps of the algorithm we have a numerical
approximation yNwith kT(yN,¯x)k< τtol (i.e. defect smaller than some prescribed tolerance) then we consider the
algorithm to have converged. We take yNas our numerical solution and have Fx)yN.
B.1 Fixed and periodic points
Assume that xUVRdis a regular point for Thaving
T(x, x)=0.
33
Then there exists an open neighborhood DUof xand a diffromorphism F:DRdwith
F(x) = x.
In this case, there may be points near xwith well defined forward orbits under F. To further study this question
we consider the stability of x.
Exploiting the formula for the derivative in Equation (35), we have that
DF (x) = D1T(x, x)1D2T(x, x),
where D1T(x, x)is invertible thanks to the assumption that xis a regular point for F. If the stable and/or center
eigenspaces of DF (x)are non-empty, then the attached local stable and/or center manifolds are natural places to
look for orbits which remain in a neighborhood of xunder forward iteration of F. (Likewise the unstable manifold
is a natural place to look for points with backward orbits). We focus for a moment on the stable manifold.
Let λ1,...,λdsCbe the stable eigenvalues of DF (x)and ξ1,...,ξdsCddenote associated stable eigenvec-
tors. Note (λ, ξ)is an eigenpair for DF (x)if and only if they solve the generalized eigenvalue problem
D2T(x, x)ξ=λD1T(x, x)ξ.
From a numerical point of view, this equation has the advantage of not requiring the inversion of any matrix. Once
the the fixed point and eigendata are known we can apply the algorithms based on the parameterization method
discussed in the main body of the paper to compute the stable (or unstale) manifolds.
In a similar fashion, suppose that x1,...,xNUVhave
T(x2, x1)=0
T(x3, x2)=0
.
.
.
T(xN, xN1)=0
T(x1, xN)=0,
with the x1,...,xNdistinct. If each of x1,...,xNis a regular point for T, then the collection is a periodic orbit
(of least period N) for an implicitly defined map F, whose domain can be taken as a union of neighborhoods of the
periodic orbit. Again, we are interested in the existence of well defined orbits near x1,...,xN, so we consider the
stability of the periodic orbit.
To find the multipliers and eigenvectors, proceed as follows. Recall from Section A.2 that the multipliers are
found by computing the eigenvalues and eigenvectors of the derivative of the multiple shooting map. The formula for
the derivative is in Equation (31), and exploiting again the formula for the derivative of Fgiven in Equation (35),
and the fact that the periodic orbit is non-degenerate, the non-zero entries of DG(x1,...,xN)are
DF (x1) = D1T(x2, x1)1D2T(x2, x1)
DF (x2) = D1T(x3, x2)1D2T(x3, x2)
.
.
.
DF (xN1) = D1T(xN, xN1)1D2T(xN, xN1)
DF (xN) = D1T(x1, xN)1D2T(x1, xN).
It is an exercise to write the associated generalized eigenvalue problem and avoid the matrix inversion. Once the stable
(or unstable) eigendata is determined, the stable (or unstable) manifold can be computed using the parameterization
method developed in the body of the present work.
C Parameterization of two dimensional invariant manifolds
In this appendix we provide the details for higher dimensional stable/unstable manifolds, focusing on the case of two
dimensions. These calculations involve power series of two variables.
34
C.1 Formal power series of two variables
Let fand gbe two variable power series if the form
f(z1, z2) =
X
m=0
X
n=0
amnzm
1zn
2,and g(z1, z2) =
X
m=0
X
n=0
bmnzm
1zn
2.
We have that
αf(z1, z2) + βg(z1, z2) =
X
m=0
X
n=0
(αamn +βbmn )zm
1zn
2,
f(λ1z1, λ2z2) =
X
m=0
X
n=0
λm
1λn
2amnzm
1zn
2,
and
f(z1, z2)g(z1, z2) =
X
m=0
X
n=0
(ab)mnzm
1zn
2,
where the coefficients of the two variable Cauchy product are given by
(ab)mn =X
j1+j2=m
k1+k2=n
aj1k1bj2k2
=
m
X
j=0
n
X
k=0
amj,nkbjk .
If f1,...,fNare power series given by
fi(z1, z2) =
X
m=0
X
n=0
ai
mnzm
1zn
21iN,
then
f1(z1, z2)...fN(z1, z2) =
X
m=0
X
n=0
(a1...aN)mn zm
1zn
2,
where
(a1...aN)mn =X
j1+...+jN=m
k1+...+kN=n
a1
j1k1...aN
jNkN
=
m
X
j1=0
j1
X
j2=0
...
jN2
X
jN1=0
n
X
k1=0
k1
X
k2=0
...
kN2
X
kN1=0
a1
mj1,nk1...aN
jN1kN1
For coefficient extraction define
(d
ab)mn = (ab)mn b00amn a00 bmn,
and similarly
(\
a1...aN)mn = (a1... aN)mn a1
00 ...aN1
00 aN
mn ...a2
00 ...aN
00a1
mn,
to be the Cauchy product of order m, n with the m, n-th order coefficients removed.
C.2 Parameterized stable/unstable manifolds attached to fixed points of
the implicit Lomelí system
Consider the implicit Lomelí system defined in Equation (8). At the parameter values studied in the present work
the Lomelí map has a pair of hyperbolic fixed points. One of the fixed points has 2d unstable and 1d stable manifold,
while for the other it is vice versa. For small 6= 0 these features persist into the implicit system, and we will compute
the formal series expansion for the parameterization of a two dimensional stable manifold of the implicit system. We
focus on the case of complex conjugate eigenvalues, but the real distinct case is similar.
35