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AMS Short Course on Rigorous Numerics in Dynamics:
The Parameterization Method for Stable/Unstable
Manifolds of Vector Fields
J.D. Mireles James, ∗
February 12, 2017
Abstract
This lecture builds on the validated numerical methods for periodic orbits presentedin the lecture of J. B. van den Berg. We discuss a functional analytic perspective onvalidated stability analysis for equilibria and periodic orbits as well as validated com-putation of their local stable/unstable manifolds. Building on this analysis we studyheteroclinic and homoclinic connecting orbits between equilibria and periodic orbitsof differential equations. We formulate the connecting orbits as solutions of certainprojected boundary value problems whcih are amenable to an a posteriori analysisvery similar to that already discussed for periodic orbits. The discussion will be drivenby several application problems including connecting orbits in the Lorenz system andexistence of standing and traveling waves.
Contents
1 Introduction 2
2 The Parameterization Method 42.1 Formal series solutions for stable/unstable manifolds . . . . . . . . . . . . . . 4
2.1.1 Manipulating power series: automatic differentiation and all that . . . 42.1.2 A first example: linearization of an analytic vector field in one complex
variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.3 Stable/unstable manifolds for equilibrium solutions of ordinary differ-
ential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.4 A more realistic example: stable/unstable manifolds in a restricted
four body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.5 Stable/unstable manifolds for periodic solutions of ordinary differen-
tial equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.1.6 Unstable manifolds for equilibrium solutions of partial differential
equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.1.7 Unstable manifolds for equilibrium and periodic solutions of delay
differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Equilibrium solutions for vector fields on Rn (slight return): beyond formalism 27
2.2.1 Justification of the invariance equation . . . . . . . . . . . . . . . . . . 27
∗Florida Atlantic University. Email: [email protected].
1
2.2.2 Non-resonant eigenvalues: existence . . . . . . . . . . . . . . . . . . . 272.2.3 Rescaling the eigenvectors: uniqueness . . . . . . . . . . . . . . . . . . 272.2.4 Homological equations: the polynomial case . . . . . . . . . . . . . . . 272.2.5 Homological equations: Faa Di Bruno formula and the general case . . 27
2.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Taylor methods and computer assisted proof 273.1 Analytic functions of several complex variables . . . . . . . . . . . . . . . . . 273.2 Banach algebras of infinite sequences . . . . . . . . . . . . . . . . . . . . . . . 303.3 Validated numerics for local Taylor methods . . . . . . . . . . . . . . . . . . . 36
3.3.1 A first example: one scalar equation in a single complex variable . . . 373.3.2 A second example: system of scalar equations in two complex variables
and stable/unstable manifolds for Lorenz . . . . . . . . . . . . . . . . 393.3.3 The more realistic example: stable/unstable manifolds of equilibrium
configurations in the four body problem . . . . . . . . . . . . . . . . . 423.3.4 A brief closing example: validated Taylor integrators . . . . . . . . . . 45
4 Connecting Orbits and the Method of Projected Boundaries 464.1 Heteroclinic connections between equilibria of vector fields . . . . . . . . . . . 464.2 Example: Heteroclinic Connections, Patterns, and Traveling Waves . . . . . . 474.3 Heteroclinic connections between periodic orbits of vector fields . . . . . . . . 51
4.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
1 Introduction
The qualitative theory of dynamical systems deals with the global orbit structure of nonlin-ear models. Owing to Poincare, this goal is reframed in terms of invariant sets. Questionsconcerning the existence, location, intrinsic geometric and topological properties, and in-ternal dynamics of invariant sets are at the core of dynamical systems theory. In fact, ashas already been discussed in this lecture series, Conley’s fundamental theorem makes pre-cise the claim that invariant sets and the connections between them completely classify thedynamics.
Another theme of this course is that problems in nonlinear analysis are often recastas solutions of functional equations, that these functional equations can be approximatelysolved by numerical methods, and that approximate solutions lead to mathematically rigor-ous results via a-posteriori analysis. The present lecture explores this theme in the contextof connecting orbits for differential equations. We begin with a brief discussion discretetime dynamical systems, which motivates the more detailed material in the remainder ofthe notes.
Studying linear and nonlinear stability of invariant sets, i.e. their local stable/unstablemanifolds, is the first step in understanding connecting orbits. Our approach to sta-ble/unstable manifolds is based on the Parameterization Method, a functional analyticframework for simultaneous study of both the embedding and the internal dynamics of theinvariant manifolds. The core of the parameterization method is the derivation of certaininvariance, or infinitesimal conjugacy equations, which are solved numerically to obtainchart or covering maps for the desired invariant manifold. Then, since the Parameteriza-tion Method is based on the study of operator equations it is also amiable to the kind ofa-posteriori analysis discussed in the notes for the introductory lecture by J.B. van den Berg[1].
2
Once local invariant manifolds are understood the next step is to connect them. Fordiscrete time dynamical systems this step is quite natural, as the evolution of the systemis governed by a known map. Continuous time dynamical systems require a little morework as the evolution of the system is only implicitly defined the differential equation. Webriefly review the boundary value formulation for connecting orbits between equilibriumand periodic orbits of differential equations. These boundary value problems are analyzedusing computer assisted techniques of proof as discussed in [1], though we only skim thedetails here and refer the interested reader to the literature on validated numerics for initialvalue problems. The main focus of these notes is on computation and validation for thestable/unstable manifolds themselves.
Remark 1.1 (Brief remarks on the literature). The reader interested in numerical meth-ods for dynamical systems will find the notes of Carles Simo [14] illuminating. We mentionalso that the seminal work of Lanford, Eckman, Koch, and Wittwer on the Feigenbaumconjectures (work which arguably launched the field of computer assisted analysis in dy-namical systems theory) is based on the study of a certain conjugacy equation (in this casethe Cvitanovic renormalization operator) via computer assisted means [15, 16]. The readerinterested in rigorous numerics for the computer assisted study of connecting orbits (andthe related study of topological horse shoes) will be interested in the work of [17] for planardiffeomorphisms, and should of course see the work of [18, 19, 20, 21]. We also mentionthat the seminal work of Warwick Tucker on the computer assisted solution of Smale’s14-th problem (existence of the Lorenz attractor) makes critical use of a normal form forthe dynamics at the origin of the system. This norm form was computed numerically, andvalidated numerical bounds obtained by studying a conjugacy equation as referred to above[22, 23].
The original references for the parameterization method, namely [24, 25, 26] for invariantmanifolds associated with fixed points, and [27, 28, 29] for invariant manifolds associatedwith invariant circles, have since launched a small industry. We mention briefly the appear-ance of KAM theories without action angle variables for area preserving and conformallysymplectic systems [30, 31], manifolds associated with invariant tori in Hamiltonian systems[32], phase resetting curves and isochrons [33], quasi-periodic solutions of PDEs [34], andmanifolds of mixed-stability [35]. Moreover even more applications are discussed in thereferences of these papers. We also mention the recent book by Alex Haro, Marta Candell,Jordi-Luis Figueras, and J.M. Mondelo [36].
The references discussed in the preceding paragraphs are by no means a thorough bibli-ography of the field of computer assisted proof in dynamical systems, and fail to even scratchthe surface of the literature on numerical methods for dynamical systems theory. Indeed,the field has exploded in the last decades and any short list of references cannot hope to hiteven the high points. We have referred only to the works most closely related to the presentdiscussion. The interested reader will find more complete coverage of the literature in theworks cited throughout this lecture.
3
2 The Parameterization Method
The Parameterization Method is a functional analytic framework for studying invariantmanifolds. The method has its roots in the work of Poincare, and congealed into a maturemathematical theory in a series of papers by de la Llave, Fontich, Cabre, and Haro [24, 25,26, 27, 28, 29]. An excellent historical overview is found in (CITATION).
The basic idea underpinning the Parameterization Method is simple and is, loosely speak-ing, based on two observations. First, the dynamical systems notion of equivalence is con-jugacy. Conjugacy is a notion expressing the fact that one dynamical system embeds inanother. For example in differential equations a conjugacy is a mapping which maps orbitsin a toy or model system to orbits in the full system of interest. This embedding is nowthought of as an unknown, in which case we think of conjugacy as an equation describingthe unknown. The second observation is this: once a conjugacy is framed as an equation,it is now amenable to analysis using all the tools of both classical nonlinear and numericalanalysis.
We will illustrate the flexibility and utility if these ideas in a number if examples, focusingon computational issues. But first we recall a few ideas concerning formal power series.
2.1 Formal series solutions for stable/unstable manifolds
In this section we illustrage computational aspects and applications of of the parameteriza-toin method. More precisely we give a number of examples of solving conjugacy equaitons.The examples are based on power series representations of the unknown functions/invariantobjects, so that it is worth reviewing the basic calculus of multivariate power series.
2.1.1 Manipulating power series: automatic differentiation and all that
Consider two power series P,Q : Cm → C
P (σ) =
∞∑
|α|=0
pασα, and Q(σ) =
∞∑
|α|=0
qασα.
Here α = (α1, . . . , αm) ∈ Nm is an m-dimensional multi-index, |α| = α1 +αm is the order ofthe multi-index, pα, qα ∈ C for all α ∈ Nm, σ = (σ1, . . . , σm) ∈ Cm, and σα := σα1
1 . . . σαmm .In this chapter we are not concerned with issues of convergence, i.e. we treat P and Q
as formal power series. We denote by V the space of all formal power series and note thatV is a vector space. Indeed let P,Q ∈ V and c ∈ C. The sum and scalar product on V aredefined term-by-term as
(P +Q)(σ) =
∞∑
|α|=0
(pα + qα)σα,
and
(cP )(σ) =
∞∑
|α|=0
cpασα.
The product of two power series is defined via the Cauchy product, i.e.
(P ·Q)(σ) =
∞∑
|α|=0
(p ∗ q)ασα,
4
where
(p ∗ q)α =
α1∑
β1=0
. . .
αm∑
βm=0
pα1−β1,...,αm−βmqβ1,...,βm .
We define formal derivatives on V via term by term differentiation. For example
∂
∂σjP (σ) :=
∞∑
α1=0
. . .
∞∑
αj=0
. . .
∞∑
αm=0
(αj + 1)pα1,...,αj+1,...,αmσα.
Two power series P,Q ∈ V are equal if an only their coefficients are equal at all orders.This fact can be used to define other operations on formal series. For example, Q is themultiplicative inverse of P if Q(σ)P (σ) = 1, where 1 denotes the series with first ordercoefficient one and all other coefficients zero. Then in order to work out the coefficients ofQ in terms of the coefficients of P , we assume for example that p00 6= 0, and write
(Q · P )(σ) =
∞∑
|α|=0
qασα
∞∑
|α|=0
pασα
=
∞∑
|α|=0
α1∑
β1=0
. . .
αm∑
βm=0
qα1−β1,...,αm−βmpβ1,...,βm
σα
= p00qα +
∞∑
|α|=1
α1∑
β1=0
. . .
αm∑
βm=0
qα1−β1,...,αm−βmpβ1,...,βm
σα
= 1.
Since the coefficients on both sides of the equation must be equal, we “match like powers”of σ and have
p0,...,0q0,...,0 = 1,
and
p0,...,0qα = −α1∑
β1=0
. . .
αm∑
βm=0
qα1−β1,...,αm−βmpβ1,...,βm .
Isolating qα gives
qα =
1
p0,...,0if α = (0, . . . , 0)
−1p0,...,0
(p ∗ q)α otherwise.
Other operations on power series, like composition with the elementary functions of math-ematical physics, are computed using automatic differentiation.
Examples of automatic differentiation in one dimension: Consider a power series ofone variable
P (σ) =
∞∑
n=0
pnσn,
and assume (for example) that we want to exponentiate P as a power series. More preciselywe want to find a power series
Q(σ) =
∞∑
n=0
qnσn,
5
so thatQ(σ) = eP (σ).
In other words we want to work out the coefficients qn in terms of the known coefficientspn.
The idea is to exploit:
• That differentiation terms composition of scalar functions into multiplication of scalarfunctions via the chain rule.
• That the exponential function itself solves a simple, linear, differential equation.
So, differentiating Q(σ) with respect to σ gives
Q′(σ) = eP (σ)P ′(σ) = Q(σ)P ′(σ),
or, on the level of power series
∞∑
n=0
(n+1)qn+1σn =
( ∞∑
n=0
qnσn
)( ∞∑
n=0
(n+ 1)pn+1σn
)=
∞∑
n=0
(n∑
k=0
(n− k + 1)pn−k+1qk
)σn.
Then, upon matching like powers of σ we have that
(n+ 1)qn+1 =
n∑
k=0
(n− k + 1)pn−k+1qk.
Finally, by reindexing and isolating qn we have that
qn =1
n
n−1∑
k=0
(n− k)pn−kqk.
Many other examples can be worked out in this way. For example let
Q(σ) = P (σ)α,
with α ∈ C. Then by the chain rule
Q′(σ) = αP (σ)α−1P ′(σ),
orP (σ)Q′(σ) = αQ(σ)P ′(σ).
Computing Cauchy products and matching like powers leads to
qn =1
np0
n−1∑
k=0
(nα− k(α+ 1))pn−kqk.
Many other examples can be computed similarily.
Remark 2.1 (Computational efficiency of automatic differentiation). The trick describedabove is often referred to as automatic differentiation for power series, though the ideasgo back centuries. The miricale of automatic differentiation is that it reduces the cost offunction composition to the cost of a Cauchy product, as long as one of the functions is itselfthe solution of a simple differential equation. Note that this includes all the usual functionsof elementary calculus, as well as others like Bessel functions and elliptic integrals. Thedown side is that automatic differentiation requires us to keep track of both the variables pnand qn. In many applications however memory is not the primary computational constraint.
6
The radial gradient and automatic differentiation for multivariate series: In twoor more dimensions we require a modification of the tricks above. The idea is to explot theso called radial gradient. More precisely Suppose that
P (σ1, σ2) =
∞∑
m=0
∞∑
n=0
pmnσm1 σ
n2 ,
and that we want to compute
Q(σ1, σ2) = eP (σ1,σ2) =
∞∑
m=0
∞∑
n=0
qmnσm1 σ
n2 .
Then we consider the radial gradient given by the expression
∇Q(σ1, σ2)
(σ1
σ2
)=
(σ1
∂
∂σ1+ σ2
∂
∂σ2
)Q(σ1, σ2) =
∞∑
m=0
∞∑
n=0
(m+ n)qmnσm1 σ
n2 .
At the same time, after using the chain rule, we have that the radial gradient is also givenby
∇Q(σ1, σ2)
(σ1
σ2
)= ∇
(eP (σ1,σ2)
)(σ1
σ2
)
= eP (σ1,σ2)∇P (σ1, σ2)
(σ1
σ2
)
= Q(σ1, σ2)
(σ1
∂
∂σ1+ σ2
∂
∂σ2
)P (σ1, σ2)
=
( ∞∑
m=0
∞∑
n=0
qmnσm1 σ
n2
)( ∞∑
m=0
∞∑
n=0
(m+ n)pmnσm1 σ
n2
)
=
∞∑
m=0
∞∑
n=0
m∑
j=0
n∑
k=0
(j + k)qm−j,n−kpjk
σm1 σ
n2 ,
Matching like powers gives
(m+ n)qmn =
m∑
j=0
n∑
k=0
(j + k)qm−j,n−kpjk,
or
qmn = q00pmn +1
(m+ n)
m∑
j=0
n∑
k=0
δmnjk (j + k)qm−jn−kpjk,
with
δmnjk :=
0 if j = m and k = n
0 if j = 0 and k = 0
1 otherwise
So given the coefficients of P we can compute the coefficients of Q at the cost of a Cauchyproduct, just as in the one dimensional case.
7
To do P (σ1, σ2)α, consider
P (σ1, σ2) =
∞∑
m=0
∞∑
n=0
pmnσm1 σ
n2 ,
and that we want to compute
Q(σ1, σ2) = P (σ1, σ2)α =
∞∑
m=0
∞∑
n=0
qmnσm1 σ
n2 .
Again, the radial gradient of Q is
∇Q(σ1, σ2)
(σ1
σ2
)=
∞∑
m=0
∞∑
n=0
(m+ n)qmnσm1 σ
n2 .
and is also given by
∇Q(σ1, σ2)
(σ1
σ2
)= ∇P (σ1, σ2)α
(σ1
σ2
)
= αP (σ1, σ2)α−1∇P (σ1, σ2)
(σ1
σ2
).
Now, multiply both sides by P get
P (σ1, σ2)∇Q(σ1, σ2)
(σ1
σ2
)= αQ(σ1, σ2)∇P (σ1, σ2)
(σ1
σ2
)
or( ∞∑m=0
∞∑n=0
pmnσm1 σ
n2
)( ∞∑m=0
∞∑n=0
(m+ n)qmnσm1 σ
n2
)=
( ∞∑m=0
∞∑n=0
αqmnσm1 σ
n2
)( ∞∑m=0
∞∑n=0
(m+ n)pmnσm1 σ
n2
)
and taking Cauchy products gives
∞∑
m=0
∞∑
n=0
m∑
j=0
n∑
k=0
(j + k)pm−j,n−kqjkσm1 σ
n2 =
∞∑
m=0
∞∑
n=0
m∑
j=0
n∑
k=0
α(j + k)qm−j,n−kpjkσm1 σ
n2 .
Matching like powers we have
m∑
j=0
n∑
k=0
(j + k)pm−j,n−kqjk =
m∑
j=0
n∑
k=0
α(j + k)qm−j,n−kpjk,
or
(m+n)p00qmn+
m∑
j=0
n∑
k=0
δmnjk (j+k)pm−j,n−kqjk = α(m+n)q00pmn+
m∑
j=0
n∑
k=0
δmnjk α(j+k)qm−j,n−kpjk.
Isolating qmn gives
qmn = αpα−100 pmn +
1
(m+ n)p00
m∑
j=0
n∑
k=0
δmnjk (j + k) (αqm−j,n−kpjk − pm−j,n−kqjk) .
8
2.1.2 A first example: linearization of an analytic vector field in one complexvariable
Perhaps the simplest example which illustrate the idea of the parameterization method isto consider a single first order ODE in one complex variable. So, suppose that U ⊂ C is anopen set and that F : U → C is complex analytic. We study the differential equation
d
dzu(z) = F (u(z)). (1)
Assume that that z0 ∈ U is an equilibrium solution, i.e. that
F (z0) = 0, andd
dzF (z0) = λ.
A basic dynamical question is: can we, at z0, analytically conjugate nonlinear differentialequation to the linear differential equation σ′ = λσ? An equivalent question is to look foran analytic function P having that P (0) = z0 and satisfying the invariance equation
λσP ′(σ) = F (P (σ)), (2)
for all σ in some neighborhood of z0?It turns out that the answer is yes, assuming that z0 is hyperbolic, i.e. that
real(λ) 6= 0.
When λ purely imaginary the answer is subtle, involving number theoretic properties of λ,and KAM type arguments (REFERENCES). For the present discussion we put aside issuesof convergence and focus on computing a polynomial approximation of P .
Note that from a geometric perspective, Equation (2) asks that the push forward of thevector field λσ under P is tangent to (in fact equal to) the vector field f restricted to theimage of P . This condition guarantees that P takes solution curves of the equation σ′ = λσto solution curves of u′ = F (u). We revisit this claim in greater generality in Section(REFERENCE). For the moment we are more concerned with computational aspects of thequestion.
Let
F (z) =
∞∑
n=1
an(z − z0)n,
be the power series expansion of f at the equilibrium z0. Note that a0 = 0 as F (z0) = 0.Since we are interested in P analytic we make the power series ansatz
P (σ) =
∞∑
n=0
pnσn.
and try to work out the unknown coefficients. Note that, since we want P (0) = z0 we havethat p0 = z0.
One sees that
λσP ′(σ) =
∞∑
n=0
λnpnσn,
and lets
Q(σ) =
∞∑
n=1
qnσn = F (P (σ)).
9
Then, matching like powers of σ we obtain the deceptively simple looking recursion relations
nλpn = qn. (3)
Note that for n = 1 we have that
λp1 = q1 = (F (P (0)))′ = F ′(P (0))P ′(0) = F ′(z0)pn = λp1.
In other words, p1 is completely arbitrary. However, for n ≥ 2 the qn depend on thecoefficients of P in a rather involved way.
Indeed it turns out that the coefficients pn are formally well defined to all orders andthat at each fixed order n ≥ 2 the coefficient pn depends only on the lower order termsp0, . . . , pn−1. The argument establishing this fact is a bit of a technical diversion, but isworth including as it motivates the formalism employed throughout the remainder of theSection.
First, note that by Taylor’s theorem
P (σ) =
∞∑
n=0
pnσn =
∞∑
n=0
1
n!
dn
dσnP (0)σn,
and
f(P (σ)) =
∞∑
n=1
qnσn =
∞∑
n=0
(1
n!
dn
dσn(F (P (0)))
)σn,
so thatdn
dσnP (0) = n!pn.
To treat the composition term we recall the Faa di Bruno formula, which says that fordifferentiable functions f and g, the n-th derivative of the composition is
dn
dxnf(g(x)) =
n∑
k=1
fk(g(x))Bn,k(g′(x), g′′(x), gn−k+1(x)), (4)
where fm, gm denote m-th derivatives. Here Bn,k are the bell polynomials defined by
Bn,k(u1, . . . , un−k+1) =∑ n!
α!
(u1
1!
)α1
. . .
(un−k+1
(n− k + 1)!
)αn−k+1
, (5)
where the sum is over all multi-indices α = (α1, . . . , αn−k+1) ∈ Nn−k+1 which simultaneouslysatisfy that
α1 + . . .+ αn−k+1 = k,
and thatα1 + 2α2 + . . .+ (n− k + 1)αn−k+1 = n.
We are interested in the dependance of (f(g(x)))n on gn(x). Inspection of Equation (4)shows that gn−k+1 equals gn if and only if k = 1. Then the sums with k = 2, . . . , n containonly derivatives g of order lower than n.
Further inspection of the k = 1 term shows that
f ′(g(x))Bn,1(g′(x), . . . , gn−k+1(x)) = f ′(g(x))gn(x).
10
To see this, consider again Equation (5) with k = 1, and note that
α1 + . . .+ αn−k+1 = α1 + . . .+ αn = 1,
if and only if exactly one of the indices is 1 and the remaining indices are zero. Similarly,when k = 1 we have that
α1 + 2α2 + . . .+ (n− k + 1)αn−k+1 = α1 + 2α2 + . . .+ nαn = n,
if and only if α1 = α2 = . . . = αn−1 = 0 and αn = 1. Taking these constraints intoconsideration we see that α = (0, . . . , 1) ∈ Nn is the only multi-index involved in the sum,so that
Bn,1(u1, . . . , un) =∑ n!
α!
(u1
1!
)α1
. . .(unn!
)αn=
n!
0! . . . 0!1!
(u1
1!
)0
. . .(unn!
)1
= un,
and indeedBn,1(g′(x), . . . , gn(x)) = gn(x),
as claimed.Now, taking x = 0, g(0) = P (0) = z0, and f(g(0)) = F (P (0)) = F (z0) gives
dn
dσn(F (P (0))) = F (P (0))
dn
dznP (0) +
n∑
k=2
Rnk (P (0))
= F ′(P (0))dn
dznP (0) +
n∑
k=2
Rnk (P (0))
= n!F ′(z0)pn +
n∑
k=2
Rnk (P (0))
= n!λpn +
n∑
k=2
Rnk (P (0)),
The terms Rnk (P (0)) involve derivatives of F of order two or more, and derivatives of P oforder strictly less than n. Of course the derivatives of F in these formulae are evaluated atP (0) = z0, and derivatives of P are evaluated at 0: but derivatives evaluated at these pointsare just the Taylor coefficients of f and P . Then, in fact the Rk(P (0)) depend only on theTaylor coefficients a1, . . . , an−1 and p0, . . . , pn−1 as desired.
Then we have that
qn =1
n!
dn
dσn(f(P (0))) = λpn +
1
n!
Nn∑
k=1
Rnk (P (0)),
so that the recursion relations of Equation (3) are
nλpn = λpn +1
n!
Nn∑
k=1
Rnk (P (0)),
or
pn =1
λ(n− 1)sn, (6)
11
when n ≥ 2. We refer to Equation (6) as the homological equations for P . Here sn dependsrecursively on p0, . . . , pn−1. In specific examples it is usually better to work out the form ofsn from scratch, exploiting the specific form of the problem. Note that p0 = z0 and that p1
is arbitrary.Even though the argument above is rarely of value in actual computations, it shows
that the coefficients of P (σ) are formally well defined to all orders and that each coefficientdepends only on lower order terms. The argument above insures that in practice we will beable to computer these coefficients recursively.
Consider for example the complex logistic equation equation
u(z)′ = cu(z)− u(z)2, (7)
i.e. Equation (1) withF (z) = cz − z2 = z(c− z),
andF ′(z) = c− 2z.
The equilibria occur at z = 0 and z = c. Lets focus on the case of z0 = c. Then λ = F ′(c) =c− 2c = −c. Assume that λ does not lie on the imaginary axis
The linearized dynamics are given by σ′ = λσ, so that σ(t) = σ0eλt is the linearized
flow. We would like to analytically conjugate Equation (7) to the linear flow at z0 = c. Inthis setting Equation (2) becomes
σλP ′(σ) = cP (σ)− P (σ)2.
Since we are interested in P analytic we make the power series ansatz
P (σ) =
∞∑
n=0
pnσn.
and try to work out the unknown coefficients. We have that
∞∑
n=0
nλpnσn =
∞∑
n=0
(cpn −
n∑
k=0
pn−kpk
)σn.
Matching like powers of σ gives the recursion relations
nλpn = cpn −n∑
k=0
pn−kpk
= cpn − 2p0pn −n−1∑
k=1
pn−kpk
= cpn − 2cpn −n−1∑
k=1
pn−kpk
= −cpn −n−1∑
k=1
pn−kpk
= λpn −n−1∑
k=1
pn−kpk
12
or
pn =−1
λ(n− 1)
n−1∑
k=1
pn−kpk. (8)
when n ≥ 2. Recall that p0 = z0 = c and that p1 is arbitrary.Of course Equation (8) describes pn in exactly the fashion anticipated by the homological
equations of Equation (6). The advantage being that Equation (8) reveals the explicit formof the dependence of pn on lower order terms, while in Equation (8) this dependence is stilllocked up in the notoriously complicated Faa di Bruno formula.
It is worth remarking that this deceptively simple example tells more or less the entirestory, thanks to automatic differentiation. For example take
u′(z) = cu(z)− eu(z),
i.e. Equation (1) withF (z) = cz − ez.
Assume that z0 ∈ C has F (z0) = 0 and that λ = F ′(z0) has non-zero real part. ThenEquation (2) becomes
σλP ′(σ) = cP (σ)− eP (σ),
and letting
P (σ) =
∞∑
n=0
pnσn,
and
Q(σ) =
∞∑
n=0
qnσn,
one recalls from Section (REFERENCE) that q0 = ep0 = ez0 , and that for n ≥ 1
qn =1
n
n−1∑
k=0
(n− k)pn−kqk
= q0pn +1
n
n−1∑
k=1
(n− k)pn−kqk.
Then, matching like powers in Equation (2.1.2) leads to
nλpn = cpn − q0pn −1
n
n−1∑
k=1
(n− k)pn−kqk.
The interested reader can now check that the homological equations are
(pnqn
)=
(−1
λ(n−1)
∑n−1k=1
(n−k)n pn−kqk
q0pn + 1n
∑n−1k=1(n− k)pn−kqk
).
Other problems involving elementary functions are worked out similarly.
We return to Equation (7), with an eye to numerical considerations. The main pointto be made, and one which appears again and again when the parameterization method is
13
Figure 1: Dependence of the coefficient decay on the scaling of p1: ...
used to compute stable/unstable manifolds, is that the freedom of choice in the first orderTaylor coefficient of P is useful for stabilizing numerics.
To illustrate this point take c = 1/2, so that p0 = c = 0.5, and use Equation (8) we thencompute the coefficient of P for all 2 ≤ n ≤ N , and obtain the polynomial approximation
PN (σ) =
N∑
n=0
pnσn.
For example choosing p1 = 1 and N = 25 we obtain the Taylor coefficients with magnitudeswhose base ten logarithms are illustrated in the left Frame of Figure 1. Note then that thecoefficients grow exponentially fast, and that |p25| ≈ 107. From a theoretical point of viewthere is no problem with this at all. It only suggests that the radius of convergence of theP is something less than one. However from a numerical point of view it is undesirable towork with polynomials whose coefficients are very large.
The problem is overcome by adjusting the scaling of P . Consider the problem againwith c = 1/2 but this time choosing p1 = 0.4 and N = 130. The base ten logarithms ofthe magnitudes of the resulting Taylor coefficients are shown in the right frame of Figure1. This time we see that the coefficients decay exponentially fast and there are no largecoefficients. The last coefficient has magnitude on the order of machine epsilon. Then it isreasonable to hope that the radius of convergence of the series is at least one.
In order to see that PN gives results are dynamically meaningful, i.e. that we obtain(at least approximately) the desired conjugacy, choose σ0 = −1 (a point on the boundaryof the interval [−1, 1]) and compute
z1 = PN (σ0) =
150∑
n=0
pnσn0 = 0.277777777777834
as well asz2 = PN (eλTσ0) = 0.336649436264490
with T = 1. Then we integrate the initial condition z1 numerically for T = 1 time unitsin order to simulate the flow. We obtain that the difference between z2 and the numericalflow is smaller than 4.2 × 10−14, a discrepancy of about 200 multiples of machine epsilon.
14
We remark that checking the conjugacy error with T = 2, . . . , 10 produces errors no worsethan this, i.e. T = 1 is not “cherry picked”. It is also worth remarking that repeating theexperiment with N = 120 yields conjugacy errors on the order of 10−13, i.e. we actually seea difference using this many terms.
2.1.3 Stable/unstable manifolds for equilibrium solutions of ordinary differen-tial equations
A more important, but only somewhat more complicated class of problems is to considernonlinear stability in the vicinity of an equilibrium point of a vector field. So, supposethat f : Rn → Rn is a (real analytic) vector field and that p0 ∈ Rn is a hyperbolic fixedpoint of f with, let us say, m ≤ n-stable eigenvalues. Let λ1, . . . , λm ∈ C denote the stableeigenvalues of Df(p0) and ξ1, . . . , ξm ∈ Rn be associated eigenvectors. We are interested inthe m-dimensional local stable manifold attached to p0.
The parameterization method studies the infinitesimal invariance equation
∂
∂σ1P (σ1, . . . , σm) + . . .+
∂
∂σmP (σ1, . . . , σm) = f(P (σ1, . . . , σm)). (9)
As we will see in Section (CITATION), if P : (−1, 1)m → Rn is a solution of Equation (9)subject to the first order constraints
P (0, . . . , 0) = p0, and∂
∂σjP (0, . . . , 0) = ξj ,
for 1 ≤ j ≤ m, then P parameterizes a local stable manifold patch at p0. Indeed, P is asolution of Equation (9) if and only of
Φ(P (σ1, . . . , σm), t) = P (eλ1tσ1, . . . , eλmtσm), (10)
i.e. P recovers the dynamics on the manifold via this conjugacy. The geometric meaning ofEquation (10) is illustrated in Figure
Consider for example the Lorenz system defined by the vector field f : R3 → R3 givenby the formula
f(x, y, z) =
σ(y − x)x(ρ− z)− yxy − βz
.
For ρ > 1 there are three equilibrium points
p0 =
000
, and p1,2 =
±√β(ρ− 1)
±√β(ρ− 1)ρ− 1
.
Choose one of the three fixed points above and denote it by p ∈ R3. Let λ denote aneigenvalue of Df(p), and let ξ denote an associated choice of eigenvector. If λ is a stableeigenvalue, assume it is the only stable eigenvalue, so that the remaining two eigenvaluesare unstable (or vice versa if λ is stable). The invariance equation reduces to
λσ∂
∂θP (σ) = f [P (σ)],
15
pP
Rn
Pp
Rn
Rm Rm
et
et
P ()
(P (), t) = Pet
Figure 2: ...
and we look for a power series solution
P (σ) =
∞∑
n=0
p1n
p2n
p3n
σn.
Imposing the first order constraints leads to
p0 = p, and p1 = ξ.
Then
λσ∂
∂σP (σ) =
∞∑
n=0
nλ
p1n
p2n
p3n
σn,
and
f(P (σ)) =
∞∑
n=0
σ(p2n − p1
n)ρ p1
n − p2n −
∑nk=0 p
1n−k p
3k
−β p3n +
∑nk=0 p
1n−k p
2k
σn.
Matching like powers gives
nλ
p1n
p2n
p3n
=
σ(p2n − p1
n)ρ p1
n − p2n −
∑nk=0 p
1n−k p
3k
−β p3n +
∑nk=0 p
1n−k p
2k
,
16
for all n ≥ 2. Isolate the terms of order n on the left and we have
σ(p2n − p1
n)− nλp1n
ρ p1n − p2
n − p10p
3n − p3
0p1n − nλp2
n
−β p3n + p1
0p2n + p2
0p1n − nλp3
n
=
0∑n−1k=1 p
1n−k p
3k
−∑n−1k=1 p
1n−k p
2k
.
When expressed in matrix form the above system of equations is
[Df(p0)− nλId] pn = sn, (11)
where pn = (p1n, p
2n, p
3n)T and
sn :=
0∑n−1k=1 p
1n−k p
3k
−∑n−1k=1 p
1n−k p
2k
.
The main observation is that Equation (11) is a linear equation in pn, and the right handside is invertible as long as nλ is not an eigenvalue of Df(p0). But we assumed that λ isthe only stable eigenvalue, so that nλ < λ < 0 is never an eigenvalue when n ≥ 2. SolvingEquation (11) recursively leads to Taylor coefficients for P to any finite order. Let
PN (σ) =
N∑
n=0
pnσn,
denote the N -th order approximation obtained by solving Equation (11) for 2 ≤ n ≤ N beour approximation of the stable/unstable manifold.
Now let λ1 and λ2 have the same stability (both stable or both unstable) and assumethat the remaining eigenvalue has opposite stability (unstable or stable). The invarianceequation reduces to
λ1σ1∂
∂σ1P (σ1, σ2) + λ2σ2
∂
∂σ2P (σ1, σ2) = f [P (σ1, σ2)],
and we look for
P (σ1, σ2) =
∞∑
m=0
∞∑
n=0
p1mn
p2mn
p3mn
σm1 σ
n2 .
Power matching as above gives
[Df(p0)− (mλ1 + nλ2)Id] pmn = smn, (12)
where
smn =
0∑mk=0
∑nl=0 δ
mnkl p
1(m−k)(n−l)p
3kl
−∑mk=0
∑nl=0 δ
mnkl p
1(m−k)(n−l)p
2kl
.
Here δmnkl = 0 when either k = 0 and l = 0 or when k = m and l = n. Otherwise it is one(so δmnkl simply extracts terms of order mn from the sum).
Again, if mλ1 + nλ2 6= λ1,2 then the matrix is invertible and the formal series solutionP is defined to all orders. Then solving the homological equations for all 2 ≤ |α| ≤ N leadsto our numerical approximation
PN (σ1, σ2) =
N∑
n=0
n∑
m=0
pn−m,mσn−m1 σm2 .
17
2.1.4 A more realistic example: stable/unstable manifolds in a restricted fourbody problem
For a less elementary example we consider the following model from celestial mechanics.Consider three particles with masses m1, m2, m3 > 0, normalized so that
m1 +m2 +m3 = 1.
We assume that the particles are located at the vertices of a planar equilateral triangle.These masses are called the primaries. We choose rotating coordinates which fix the barycen-ter of the triangle is at the origin and cause the x-axis to bisect the triangle. This placesthe primaries at positions
p1 = (x1, y1), p2 = (x2, y2), and p3 = (x3, y3),
with
x1 =−|K|
√m2
2 +m2m3 +m23
Ky1 = 0
x2 =|K| [(m2 −m3)m3 +m1(2m2 +m3)]
2K√m2
2 +m2m3 +m23
y2 =−√
3m3
2m3/22
√m3
2
m22 +m2m3 +m2
3
and
x3 =|K|
2√m2
2 +m2m3 +m23
y3 =
√3
2√m2
√m3
2
m22 +m2m3 +m2
3
whereK = m2(m3 −m2) +m1(m2 + 2m3).
We are interested in the equations of motion for a fourth, massless particle p withcoordinates (x, y) whose trajectory is governed by the gravitational field of the primaries.Define
Ω(x, y) :=1
2(x2 + y2) +
m1
r1(x, y)+
m2
r2(x, y)+
m3
r3(x, y),
wherer1(x, y) :=
√(x− x1)2 + (y − y1)2,
r2(x, y) :=√
(x− x2)2 + (y − y2)2,
andr3(x, y) :=
√(x− x3)2 + (y − y3)2.
Then∂
∂xΩ = Ωx(x, y) = x− m1(x− x1)
r1(x, y)3− m2(x− x2)
r2(x, y)3− m3(x− x3)
r3(x, y)3,
18
and∂
∂yΩ = Ωy(x, y) = y − m1(y − y1)
r1(x, y)3− m2(y − y2)
r2(x, y)3− m3(y − y3)
r3(x, y)3,
andWe introduce the variables
u1 = x, u2 = x′, u3 = y, u4 = y′,
The vector field in the co-rotating frame is given by
u′1u′2u′3u′4
=
u2
2u4 + Ωx(u1, u3)u4
−2u2 + Ωy(u1, u3)
=: f(u1, u2, u3, u4). (13)
This time we consider p0 = (p10, p
20, p
30, p
40) ∈ R4 an equilibrium point with two unsta-
ble eigenvalues λ1, λ2 ∈ C. Choose associated eigenvectors ξ1, ξ2 ∈ R4. Now we seek aparameterization P : (−1, 1)× (−1, 1)→ R4 satisfying
λ1σ1∂
∂σ1P (σ1, σ2) + λ2σ2
∂
∂σ2P (σ1, σ2) = f(P (σ1, σ2)),
with
P (0, 0) = p0,∂
∂σ1P (0, 0) = ξ1, and
∂
∂σ2P (0, 0) = ξ2.
Let
P (σ1, σ2) =
P1(σ1, σ2)P2(σ1, σ2)P3(σ1, σ2)P4(σ1, σ2)
with
P1(σ1, σ2) =
∞∑
m=0
∞∑
n=0
amnσm1 σ
n2 ,
P2(σ1, σ2) =
∞∑
m=0
∞∑
n=0
bmnσm1 σ
n2 ,
P3(σ1, σ2) =
∞∑
m=0
∞∑
n=0
cmnσm1 σ
n2 ,
P4(σ1, σ2) =
∞∑
m=0
∞∑
n=0
dmnσm1 σ
n2 ,
and have thata00 = p1
0, b00 = 0, c00 = p30, and d00,
and thata10 = ξ1
1 , b10 = ξ21 , c10 = ξ3
1 , d10 = ξ41 ,
a01 = ξ12 , b01 = ξ2
2 , c01 = ξ32 , d01 = ξ4
2 ,
19
Then
λ1σ1∂
∂σ1P (σ1, σ2) + λ2σ2
∂
∂σ2P (σ1, σ2) =
∞∑
m=0
∞∑
n=0
(mλ1 + nλ2)
amnbmncmndmn
σm1 σ
n2 .
Let
(P1(σ1, σ2)− x1)2
+ (P3(σ1, σ2)− y1)2
= Q(σ1, σ2) =
∞∑
m=0
∞∑
n=0
Qmnσm1 σ
n2 ,
(P1(σ1, σ2)− x2)2
+ (P3(σ1, σ2)− y2)2
= R(σ1, σ2) =
∞∑
m=0
∞∑
n=0
Rmnσm1 σ
n2 ,
(P1(σ1, σ2)− x3)2
+ (P3(σ1, σ2)− y3)2
= S(σ1, σ2) =
∞∑
m=0
∞∑
n=0
Smnσm1 σ
n2 ,
Define
δmnjk =
0 if j = m and k = n
0 if j = 0 and k = 0
1 otherwise
.
ThenQ00 = (a00 − x1)2 + (c00 − y1)2,
Qmn = 2(a00 − x1)amn + 2(c00 − y1)cmn +
m∑
j=0
n∑
k=0
(am−j,n−kajk + cm−j,n−kcjk)
R00 = (a00 − x2)2 + (c00 − y2)2,
Rmn = 2(a00 − x2)amn + 2(c00 − y2)cmn +
m∑
j=0
n∑
k=0
(am−j,n−kajk + cm−j,n−kcjk)
S00 = (a00 − x3)2 + (c00 − y3)2,
Smn = 2(a00 − x3)amn + 2(c00 − y3)cmn +
m∑
j=0
n∑
k=0
(am−j,n−kajk + cm−j,n−kcjk) .
We also define
T (σ1, σ2) = Q(σ1, σ2)−3/2 =
∞∑
m=0
∞∑
n=0
Tmnσm1 σ
n2 ,
U(σ1, σ2) = R(σ1, σ2)−3/2 =
∞∑
m=0
∞∑
n=0
Umnσm1 σ
n2 ,
V (σ1, σ2) = S(σ1, σ2)−3/2 =
∞∑
m=0
∞∑
n=0
Vmnσm1 σ
n2 ,
and have thatT00 = Q
−3/20 = ((a0 − x1)2 + (c0 − y1)2)−3/2
20
and
Tmn =−3
2Q−5/200 Qmn +
1
(m+ n)Q00
m∑
j=0
n∑
k=0
δmnjk (j+k)
(−3
2Tm−j,n−kQjk −Qm−j,n−kTjk
)
U00 = R−3/20 = ((a0 − x2)2 + (c0 − y2)2)−3/2
and
Umn =−3
2R−5/200 Rmn+
1
(m+ n)R00
m∑
j=0
n∑
k=0
δmnjk (j+k)
(−3
2Um−j,n−kRjk −Rm−j,n−kUjk
)
and finally
V00 = S−3/20 = ((a0 − x3)2 + (c0 − y3)2)−3/2
and
Vmn =−3
2S−5/200 Smn +
1
(m+ n)S00
m∑
j=0
n∑
k=0
δmnjk (j + k)
(−3
2Vm−j,n−kSjk − Sm−j,n−kVjk
).
Define
s2mn := −3
2Q−5/200 (a00 − x1)
m∑
j=0
n∑
k=0
δmnjk (am−j,n−kajk + cm−j,n−kcjk)
+a00 − x1
(m+ n)Q00
m∑
j=0
n∑
k=0
δmnjk (j+k)
(−3
2Tm−j,n−kQjk −Qm−j,n−kTjk
)+
m∑
j=0
n∑
k=0
δmnjk am−j,n−kTjk,
s2mn := −3
2R−5/200 (a00 − x2)
m∑
j=0
n∑
k=0
δmnjk (am−j,n−kajk + cm−j,n−kcjk)
+a00 − x2
(m+ n)R00
m∑
j=0
n∑
k=0
δmnjk (j+k)
(−3
2Um−j,n−kRjk −Rm−j,n−kUjk
)+
m∑
j=0
n∑
k=0
δmnjk am−j,n−kUjk
s2mn := −3
2S−5/200 (a00 − x3)
m∑
j=0
n∑
k=0
δmnjk (am−j,n−kajk + cm−j,n−kcjk)
+a00 − x3
(n+m)S00
m∑
j=0
n∑
k=0
δmnjk (j+k)
(−3
2Vm−j,n−kSjk − Sm−j,n−kVjk
)+
m∑
j=0
n∑
k=0
δmnjk am−j,n−kVjk,
s4mn := −3
2Q−5/200 (c00 − y1)
m∑
j=0
n∑
k=0
δmnjk (am−j,n−kajk + cm−j,n−kcjk)
+c00 − y1
(m+ n)Q00
m∑
j=0
n∑
k=0
δmnjk (j+k)
(−3
2Tm−j,n−kQjk −Qm−j,n−kTjk
)+
m∑
j=0
n∑
k=0
δmnjk cm−j,n−kTjk
s4mn := −3
2R−5/200 (c00 − y2)
m∑
j=0
n∑
k=0
δmnjk (am−j,n−kajk + cm−j,n−kcjk)
21
+c00 − y2
(m+ n)R00
m∑
j=0
n∑
k=0
δmnjk (j+k)
(−3
2Um−j,n−kRjk −Rm−j,n−kUjk
)+
m∑
j=0
n∑
k=0
δmnjk cm−j,n−kUjk
s4n := −3
2S−5/200 (c00 − y3)
m∑
j=0
n∑
k=0
δmnjk (am−j,n−kajk + cm−j,n−kcjk)
+c00 − y3
(m+ n)S00
m∑
j=0
n∑
k=0
δmnjk (j+k)
(−3
2Vm−j,n−kSjk − Sm−j,n−kVjk
)+
m∑
j=0
n∑
k=0
δmnjk cm−j,n−kVjk.
The homological equations for pmn = (amn, bmn, cmn, dmn) are
[Df(p0)− (mλ1 + nλ2)Id] pmn = Hmn,
where
Hmn =
0h2mn
0h4mn
,
withh2mn = m1s
2mn +m2s
2mn +m3s
2mn,
andh4mn = m1s
4mn +m2s
4mn +m3s
4mn.
Remark 2.2. Several technical details deserve remark.
• (Complex conjugate eigenvalues) When there are complex conjugate eigenvalues infact none of the preceding discussion changes. The only modification is that, if wechoose complex conjugate eigenvectors, then the coefficients will appear in complexconjugate pairs, i.e.
pmn = pmn.
Then taking the complex conjugate variables gives the parameterization of the realinvariant manifold,
P (σ1, σ2) := P (σ1 + iσ2, σ1 − iσ2),
where P is the formal series defined in the preceding discussion. For more details seealso [42, 4, 5].
• (Resonance and non-resonance) When the eigenvalues are resonant, i.e. when there isa fixed pair (m, n) ∈ N2 so that
mλ1 + nλ2 = λj ,
for either j = 1 or j = 2, then all is not lost. In this case we cannot conjugate analyt-ically to the diagonalized linear vector field. However by modifying the model vectorfiled to include a polynomial term which “kills” the resonance the formal computationgoes through. The theoretical details are in [24], and numerical implementation withcomputer assisted error bounds are discussed and implemented in [40].
22
2.1.5 Stable/unstable manifolds for periodic solutions of ordinary differentialequations
Utility of the parameterization method is not limited to the study of stable/unstable mani-folds of equilibria. The method can also be used to compute invariant manifolds associatedwith periodic orbits. Suppose for example that γ : [0, T ]Rn is a smooth periodic orbit forthe (real analytic) vector field f : Rn → Rn, i.e. suppose that
γ(0) = γ(T ),
and thatd
dtγ(t) = f(γ(t)).
The stability of the periodic orbits is determined by its Floquet exponents. A thoroughreview of Floquet theory would is beyond the scope of the present discussion. Suffice to say,λ ∈ C is a stable Floquet exponent for γ, and ξ : [0, 2T ]→ Rn is the parameterization of anassociated stable normal bundle for γ, if ξ is smooth and 2T periodic, and the pair solvethe eigenvalue problem
d
dtξ(t) + λξ(t)−Df(γ(t))ξ(t) = 0, (14)
for all t ∈ [0, 2T ].Suppose now that γ(t) has m < n stable Floquet exponents λ1, . . . , λm ∈ C and let
ξ1, . . . , ξm : [0, 2T ] → Rn be associated stable normal bundles. (Stability for Floquet expo-nents means that the real part of the exponent is strictly less than zero). We are interestedin the local stable manifold attached to γ and tangent to the span of ξ1(t), . . . , ξm(t) foreach t ∈ [0, 2T ].
In this case the parameterization method seeks a covering map P : [0, 2T ]× (−1, 1)m →Rn satisfying the invariance equation
∂
∂tP (t, σ1, . . . , σm)+
∂
∂σ1P (t, σ1, . . . , σm)+. . .+
∂
∂σmP (t, σ1, . . . , σm) = f(P (t, σ1, . . . , σm)),
(15)subject to the first order constraints
P (t, 0, . . . , 0) = γ(t), and∂
∂σjP (t, 0, . . . , 0) = ξj(t),
for all t ∈ [0, 2T ].It can be shown that P , satisfying the first order constraints, is a solution of Equation
(15) if an only if P satisfies the flow conjugacy
Φ(P (s, σ1, . . . , σm), t) = P (s+ t, eλ1tσ1, . . . , eλmt). (16)
for all s ∈ [0, 2T ] and all t ≥ 0. From this it follows that P parameterizes a local stablemanifold for γ. Then, as in the equilibrium case, we recover the dynamics on the manifoldin addition to the embedding. The geometric meaning of Equation (16) is illustrated inFigure 16.
Suppose now that we wish to numerically approximate P . Since P is analytic, periodicin t, and satisfies first order constraints in σ1, . . . , σm, the Fourier-Taylor ansatz
P (t, σ1, . . . , σm) =
∞∑
α1=0
. . .
∞∑
αm=0
∑
n∈Zpα1,...,αm,ne
2πiωktσα11 . . . σαmm
=
∞∑
|α|=0
∑
k∈Zpα,ne
2πiωktσα,
23
x
W sloc()
L(s,, t) =
s + tet
(s,)
P
x
W sloc()
P
Figure 3: ...
is appropriate. Here the last line exploits the multi-index notation for power series.Now one inserts the Fourier-Taylor ansatz into the invariance equation (15), expands the
nonlinearities using Cauchy products or automatic differentiation, and matches like powersof σ = (σ1, . . . , σm) to obtain the appropriate homological equations It is illuminating toexamine these ideas further in the context of a specific example.
So, returning to the Lorenz equation suppose that γ : [0, T ] → R3 is a periodic solutionwith a single real stable Floquet exponent γ < 0, and let ξ : [0, T ]→ R3 denote the associatedstable normal bundle. Let
γ(t) =∑
k∈Zγke
2πiωkt, and ξ(t) =∑
k∈Zξke
2πiωkt,
denote the Fourier series of this data.In this case we want to solve the equation
∂
∂tP (t, σ) + λσ
∂
∂σP (t, σ) = f(P (t, σ)),
where t ∈ [0, T ] and σ ∈ (−1, 1). For the moment we suppress the Fourier series notationand write
P (t, σ) =
∞∑
n=0
pn(t)σn,
and have∂
∂tP (t, σ) + λσ
∂
∂σP (t, σ) =
∞∑
n=0
(d
dtpn(t) + nλpn(t)
)σn,
24
Figure 4: ...
on the left and
f(P (t, σ)) =
∞∑
n=0
σp2n(t)− σp1
n(t)ρp1n(t)− p2
n(t)−∑nj=0 p
1n−j(t)p
3j (t)
−βp3n(t) +
∑nj=0 p
1n−j(t)p
2j (t)
σn,
on the right. Matching like powers and isolating the n-th order term (a computation whichgoes almost exactly as the one illustrated in section (REFERENCE)) leads to the homolog-ical equations
d
dtpn(t) + nλpn(t)−Df(γ(t))pn(t) = Sn(t), (17)
where
Sn(t) =
0
−∑n−1j=1 p
1n−j(t)p
3j (t)∑n−1
j=1 p1n−j(t)p
2j (t)
.
In other words, we obtain that each coefficient pn(t) for n ≥ 2 solves a linear ordinarydifferential equation with T -periodic data. These equations are solved recursively in Fourierspace to obtain an approximation of P to as many Fourier and Taylor modes as we wish.
Figure 4 illustrates a pair of local stable/unstable manifolds computed using the methodsketched above. The reader interested in the details may be interested in consulting (CITA-TION). In that reference efficient numerical solution of the homological equations in Fourierspace is accomplished using a change of coordinates which exploits a numerically computedFloquet normal form.
Remark 2.3 (First order Fourier data and homological equations). Note that it is alreadya nontrivial problem to obtain the Fourier expansions of the periodic orbit and its normalbundles. We note that this can be done using methods similar to those being discussedhere. Namely one expands the differential equation or the eigenvalue problem in terms
25
of the unknown Fourier series and tries to work out the unknown coefficients by powermatching.
In the case of Fourier series we cannot solver order by order as in the Taylor case,because the discrete convolution product (analog of the Cauchy product of power series)couples every term to every other. Instead one truncates to a finite Fourier series and appliesa numerical solver to the resulting system of scalar algebraic equations.
Such methods are discussed in much greater detail, even with computer assisted methodsfor obtaining mathematically rigorous error bounds, in the notes of (JP and JB). The meth-ods discussed in those lectures can easily be applied to the (linear!) homological equationsdiscussed above. Therefore in the present discussion we simply assume that we are able tosolve Fourier equations. The interested reader may also want to consult (CITATIONS).
2.1.6 Unstable manifolds for equilibrium solutions of partial differential equa-tions
The utility of the parameterization method is not limited to finite dimensional dynamicalsystem. For example the method is used to study unstable manifolds associated with steadystate solutions of parabolic partial differential equations.
Proceeding somewhat informally, let Ω ⊂ Rn be an open set and consider the parabolicreaction diffusion equation
∂
∂tu(x, t) = ∆u(x, t) + F (u(x, t)). (18)
Here u : Ω× [0,∞)→ R, x ∈ Ω, t ∈ [0,∞) and F : R→ R is a smooth function (in fact letsassume that it is analytic).
The problem is well-posed if we impose an initial and boundary conditions. Suppose forexample we require that
u(x, 0) = u0(x),
with u0 : Ω→ R in an appropriate Sobolev space, and the Neumann boundary conditions
∂
∂ηu(x, t) = 0,
for all t ≥ 0 and x ∈ ∂Ω. Here ∂/∂η is the normal derivative and ∂Ω denotes the boundaryof the domain. Then Equation (18) generates a (compact) semiflow on a Sobolev space. Wedenote the Sobolev space by X and the semiflow by Φ+ : X × R+ → X .
Suppose that u0 ∈ X is a hyperbolic fixed point for Equation (18), i.e. that u0 satisfiesthe boundary conditions, has
∆u0(x) + F (u0(x)) = 0,
at least in a weak sense. In this context, hyperbolicity means that the spectrum of the linearoperator L := ∆ + DF (u0) (subject to the boundary conditions) does not accumulate atthe imaginary axis. Here L is typically an unbounded, densely defined linear operator onX . Note in fact that L is sectorial, hence the spectrum is pure point and accumulates onlyat minus infinity.
Then L has at most finitely many unstable eigenvalues each with finite multiplicity. Weassume for the sake of simplicity that L has exactly m unstable eigenvalues λ1, . . . , λm ∈ C,and that ξ1, . . . , ξm : Ω → R are associated eigenfunctions. More precisely suppose that
26
the λj have positive real part, that each has multiplicity exactly one, that the pair (λj , ξj)(weakly) solves the eigenvalue problem
∆ξj(x) + F (u0(x))ξj(x)− λjξj(x) = 0,
for each1 ≤ j ≤ m, and that each ξj satisfies the boundary conditions.The classical stable/unstable manifolds theorems extend to the situation just described
(see for example REFERENCE), hence there exists a local stable/unstable manifolds at-tached to u0. We will describe briefly the use of the parameterization method to approxi-mation the local unstable manifold.
We will look for a function P : (−1, 1)m → X having that
m∑
j=1
λjσj∂
∂σjP (x, σ1, . . . , σm) = ∆P (x, σ1, . . . , σm) + F (P (x, σ1, . . . , σm)), (19)
where for each fixed (σ1, . . . , σm) ∈ (−1, 1)m, the function P (x, σ1, . . . , σm) satisfies theboundary conditions. We impose also the first order constraints
P (x, 0, . . . , 0) = u0(x), and∂
∂σjP (x, 0, . . . , 0) = ξj(x),
for all x ∈ Ω and 1 ≤ j ≤ m.We make the power series ansatz
P (x, σ) =
∞∑
|α|=0
pα(x)σα,
where each pα : Ω→ R is in X (hence each satisfies the boundary conditions).Consider for example the Fisher equation
(20)
2.1.7 Unstable manifolds for equilibrium and periodic solutions of delay dif-ferential equations
2.2 Equilibrium solutions for vector fields on Rn (slight return):beyond formalism
2.2.1 Justification of the invariance equation
2.2.2 Non-resonant eigenvalues: existence
2.2.3 Rescaling the eigenvectors: uniqueness
2.2.4 Homological equations: the polynomial case
2.2.5 Homological equations: Faa Di Bruno formula and the general case
2.3 Further reading
3 Taylor methods and computer assisted proof
3.1 Analytic functions of several complex variables
Let m ∈ N and z = (z1, . . . , zm) ∈ Cm. We endow Cm with the norm
‖z‖ = max1≤j≤m
|zj |,
27
where |zj | =√
real(zj)2 + imag(zj)2 is the usual complex absolute value. Let r > 0 andz0 = (z0
1 , . . . , z0m) ∈ Cm. The set
Dmr (z0) :=
z = (z1, . . . , zm) ∈ Cm : |zj − z0
j | < r for all 1 ≤ j ≤ m,
is called the polydisk of radius r centered at z0 ∈ Cm. When d, m, r, and z0 are understoodwe sometimes abbreviate to D := Dm
r (z0).Let U ⊂ Cm be an open set and z ∈ U . A continuous function f : U → C is said to be
holomorphic at z if for each 1 ≤ j ≤ m the complex partial derivative
∂
∂zjf(z) := lim
h→0
f(z1, . . . , zj + h, . . . , zm)− f(z1, . . . , zj , . . . , zm)
h,
exists and is finite. In other words, f is holomorphic at z if it is holomorphic in theone dimensional sense separately in each variable with the others fixed. We say that f isholomorphic on the open set Ω ⊂ Cm if f is holomorphic at z for each z ∈ Ω.
A fundamental result from the theory of functions of one complex variable is that afunction is holomorphic if and only if it is analytic. For functions of several complex variablesthis is still the case that, i.e. we have that f : Ω → Cm is holomorphic if and only if f hasa convergent power series representation about each point in Ω. To be more precise, letα = (α1, . . . , αm) ∈ Nm denote an m-dimensional multi-index, |α| := α1 + . . .+ αm denotethe “norm” or “order” of the multi-index, and (z − z0)α := (z1 − z0
1)α1 . . . (zm − z0m)αm to
denote (z − z0) ∈ Cm raised to the α-power. A continuous function f : Ω → C is analyticif and only if for each z0 ∈ Ω there is an r > 0 and a multi-indexed sequence of complexnumbers aαα∈Nm ⊂ C so that Dm
r (z0) ⊂ Ω and
f(z) =
∞∑
α1=0
. . .
∞∑
αm=0
aα1,...,αm
(z1 − z0
1
)α1. . .(zm − z0
m
)αm.
with the series converging absolutely and uniformly for all z ∈ Dmr (z0). We often employ
the convenient short hand
∞∑
|α|=0
aα(z − z0)α :=
∞∑
α1=0
. . .
∞∑
αm=0
aα1,...,αm
(z1 − z0
1
)α1. . .(zm − z0
m
)αm.
As in the one dimensional case the power series coefficients – or Taylor coefficients – aredetermined by certain Cauchy integrals. More precisely, suppose that f : Ω→ C is analyticand Dm
r (z0) ⊂ Ω. Then the α-th Taylor coefficient of the power series expansion of f at z0
is given by the iterated line integral
aα :=1
(2πi)m
∫
|z1−z01 |=r. . .
∫
|zm−z0m|=r
f(z1, . . . , zm)
(z1 − z01)α1+1 . . . (zm − z0
m)αm+1dz1 . . . dzm,
where the curves |zj − z0j | = r, 1 ≤ j ≤ m are parameterized with with positive orientation,
i.e. each curve has winding number +1 about z0j . Just as in the one dimensional case,
the function f is uniquely determined by its Taylor coefficients and in particular if Ω isconnected then these coefficients are all zero if and only if f is identically zero on Ω.
Define α! := α1! . . . αm! and let
∂|α|
∂zαf(z0) :=
∂α1+...+αm
∂zα11 . . . ∂zαmm
f(z0),
28
denote the α-th partial derivative of f at z0. In analogy with the one dimensional case wehave that if f is analytic on Ω then so is the function ∂|α|/∂zαf for every α ∈ Nm. Moreover,the power series of the partial derivatives are given by “term-by-term” differentiation of thepower series for f and we have the expression
aα =1
α!
∂|α|
∂zα(z0),
for the Taylor series coefficients. We also have the following Cauchy estimates.
Lemma 3.1 (Cauchy Estimates). Suppose that f is analytic on an open set Ω ⊂ C, andthat there are z0 ∈ Ω and R > 0 such that
DmR (z0) ⊂ Ω.
Then
|aα| ≤1
R|α|sup
z∈DmR (z0)
|f(z)|.
It follows, just as in the one dimensional case that
∞∑
|α|=0
|aα|ν|α| <M
(1− νR )m
<∞,
where M is any bound on |f(z)| in Dm
R (z0) and 0 < ν < R.There is also a Cauchy product in the setting of several complex variables. More precisely,
suppose that Ω ⊂ Cm is an open set, that z0 ∈ Ω, and that f, g : Ω→ C are analytic. Let
f(z) =
∞∑
|α|=0
aα(z − z0)α, and g(z) =
∞∑
|α|=0
bα(z − z0)α,
denote the power series expansions of f and g at z0, and assume that these series convergeabsolutely and uniformly on some poly-disk Dm
r (z0) ⊂ C. Then the pointwise product f · gis analytic on Ω and has power series expansion at z0 given by
(f · g)(z) =
∞∑
|α|=0
cα(z − z0)α, (21)
where
cα =
α1∑
β1=0
. . .
αm∑
βm=0
aα1−β1,...,αm−βmbβ1,...,βm .
The power series defined by the right hand side of Equation (21) is called the multivariableCauchy product of two power series (or just the Cauchy product). The Cauchy productconverges absolutely and uniformly at least on the poly-disk Dm
r (z0).Finally we define the function spaces used in this Chapter. Let Ω ⊂ Cm be an open set
and f : Ω→ C be a function. The norm
‖f‖C0(Ω,C) := supz∈Ω|f(z1, . . . , zm)|,
29
is referred to as the supremum norm, or “norm of uniform convergence” on Ω. We oftenabbreviate this simply to ‖f‖∞ when there is no cause for confusion. Let
Cω(Ω,C) := f : Ω→ C : f is analytic on Ω, and ‖f‖∞ <∞ ,
denote the set of bounded analytic functions on Ω. As in the one dimensional case we havethat if f, fn∞n=0 : Ω→ C are functions and the fn∞n=0 are analytic on Ω, then
limn→∞
‖f − fn‖∞ = 0,
implies that f is analytic, i.e. Cω(Ω,C) is a Banach space when endowed with the ‖ · ‖∞norm. In fact Cω(Ω,C) is a Banach algebra when endowed with pointwise multiplication offunctions.
3.2 Banach algebras of infinite sequences
Define Sm to be the collection of all complexm-dimensional multi-indexed sequences aαα∈Nm .For a = aαα∈Nm ∈ Sm and ν > 0 define the norm
‖a‖1ν,m :=
∞∑
α1=0
. . .
∞∑
αm=0
|aα1,...,αm |να1+...+αm =
∞∑
|α|=0
|aα|ν|α|,
and let`1ν,m :=
a = aα ∈ Sm : ‖a‖1ν,m <∞
.
This defines a Banach space structure.Suppose that f is analytic at z0 ∈ Cm. We define the Taylor transform of f at z0 to
be the resulting sequence of Taylor coefficients. More formally define the map Tz0 whichassociates to the f its sequence of Taylor coefficient aαα∈Nm . Here the power seriesexpansion is taken at z0. We have that
‖f‖∞ ≤ ‖T (f)‖ν ,
even when one of the two is infinite. So, let a = aαα∈Nm ∈ `1ν,m. The function f(z)defined by
f(z) =
∞∑
|α|=0
aαzα,
has f ∈ Cω(Dmν ). Conversely, if f : Dm
r → C is analytic then
T (f) ∈ `1ν ,
for all 0 < ν < r. As in the one dimensional case we have that the Taylor transform is awell defined mapping from Cω(Dm
r (z0),C) into `1ν,m for any 0 < ν < r, that the mapping isone-to-one, and bounded with bounded inverse.
One easily checks that `1ν,m inherits a Banach algebra structure from pointwise multipli-cation of functions through the Taylor transform. This fact is critical in nonlinear analysis.More precisely, given a, b ∈ `1ν we define the binary operator ∗ : `1ν × `1ν → Sd by
(a ∗ b)α =
α1∑
β1=0
. . .
αm∑
βm=0
aα1−β1,...,αm−βmbβ1,...,βm .
We refer to ∗ as the Cauchy product, and have that
30
•‖a ∗ b‖ν ≤ ‖a‖ν‖b‖ν ,
for all a, b ∈ `1ν , i.e. ∗ actually takes values in `1ν , making it a Banach algebra.
• For all f, g ∈ Cω(Ddr ),
T (fg) = T (f) ∗ T (g),
i.e. pointwise multiplication in the function domain corresponds to Cauchy productsin the transform domain.
Finally we note that, just as in the one variable case, differentiation is not a boundedoperation on Cω(Dm
r (z0),C) and hence does not induce a bounded linear map on `1ν,m. But,just as before, we recover boundedness by giving up domain. More precisely we have thefollowing Lemma, whose proof we leave as an exercise.
Lemma 3.2 (Cauchy Bounds in Several Complex Variables). Let ν > 0, z0 ∈ Cm anda = aαα∈Nm ∈ `1ν,m. Define f : Dm
ν (z0)→ C
f(z) :=
∞∑
|α|=0
aα(z − z0)α.
Let 0 < σ ≤ 1 andν = e−σν.
Then
supz∈Dmν (z0)
∣∣∣∣∂
∂zjf(z)
∣∣∣∣ ≤1
νσ‖a‖1ν,m.
In the validated numerical arguments to follow it makes sense to split `1ν,m into a finitedimensional vector space where we can do numerical computations, and an infinite dimen-sional space of “tails” in which we must do some analysis. To this end, define the Banachspaces
XNν,m =
a = aα ∈ `1ν,m| aα = 0 when |α| ≥ N + 1
,
andX∞ν,m =
a = aα ∈ `1ν,m|amn = 0 when 0 ≤ |α| ≤ N
,
Note that`1ν,m = XN
ν,m ⊕X∞ν,m,
and each a ∈ `1ν,m has a unique representation
a = aN + h,
with aN ∈ XNν,m and h ∈ X∞ν,m. (It might be natural to write h∞ to denote elements of X∞
but the “infinity” superscript becomes cumbersome).The Cauchy product is the fundamental nonlinear operation appearing in the discussion
of validated Taylor methods to follow. The following considerations will simplify our dis-cussion of truncation error analysis. Note that in the case of the Lorenz system as well asthe case of the restricted four body problem (after automatic differentiation), we encounteronly quadratic nonlinearities. Hence we focus only on the quadratic case now.
31
Let a, b ∈ `1ν,m and a = aN + u and b = bN + v with aN , bN ∈ XNν,m and u, v ∈ X∞ν,m.
Consider the convolution
a ∗ b = (aN + u) ∗ (bN + v) = (aN ∗ bN ) + (aN ∗ v) + (bN ∗ u) + (u ∗ v).
Note that if b = a this reduces to
(a ∗ a) = (aN ∗ aN ) + 2(aN ∗ v) + (v ∗ v).
For the applications to follow it is desirable to define the two additional binary operators∗, ∗ : `1ν,m ⊕ `1ν,m → `1ν,m given by
(a∗b)α =
α1∑
β1=0
. . .
αm∑
βm=0
δαβaα1−β1,...,αm−βmbβ1,...,βm , (22)
and
(a∗b)α =1
|α|
α1∑
β1=0
. . .
αm∑
βm=0
δαβ |β|aα1−β1,...,αm−βmbβ1,...,βm , (23)
for |α| > 0 and (a∗b)0 = 0. Here, for α, β ∈ Nm with 0 ≤ |β| ≤ |α| we define
δαβ :=
0 if βj = 0 for 1 ≤ j ≤ m0 if βj = αj for 1 ≤ j ≤ m1 otherwise
,
i.e. δαβ is zero if β is the zero multi-index, or if β = α, but is one otherwise. Note that if δαβis the traditional Kronecker delta then δαβ = 1− δαβ . Note that
(a∗b)α = (a ∗ b)α − a0bα − aαb0,
i.e. that (a∗b)α is the α term of the Cauchy product with all dependance on aα and bαremoved. Similar for (a∗b).
Note that for these operators we have the estimates
‖a∗b‖1ν,m ≤
∞∑
|α|=1
|aα|ν|α|∞∑
|α|=1
|bα|ν|α| , (24)
and
‖a∗b‖1ν,m ≤
∞∑
|α|=1
|aα|ν|α|∞∑
|α|=1
|bα|ν|α| . (25)
For the first of these we define the sequences
a =
0 if α = 0
aα otherwise, and b =
0 if α = 0
bα otherwise,
and note that
‖a∗b‖1ν,m = ‖a ∗ b‖1ν,m ≤ ‖a‖1ν,m‖b‖1ν,m =
∞∑
|α|=1
|aα|ν|α|∞∑
|α|=1
|bα|ν|α|
32
For the second bound, let |a|, |b| ∈ `1ν,m denote the sequences whose terms are given by theabsolute values of the terms of a and b. Then
‖a∗b‖1ν,m =
∞∑
|α|=0
∣∣∣∣∣∣1
|α|
α1∑
β1=0
. . .
αm∑
βm=0
δαβ |β|aα1−β1,...,αm−βmbβ1,...,βm
∣∣∣∣∣∣ν|α|
≤∞∑
|α|=0
α1∑
β1=0
. . .
αm∑
βm=0
δαβ|β||α||aα1−β1,...,αm−βm ||bβ1,...,βm |ν|α|
≤∞∑
|α|=0
α1∑
β1=0
. . .
αm∑
βm=0
δαβ |aα1−β1,...,αm−βm ||bβ1,...,βm |ν|α|
≤ ‖|a|∗|b|‖1ν,m
≤
∞∑
|α|=1
|aα|ν|α|∞∑
|α|=1
|bα|ν|α| ,
as desired.Now, with aN , bN ∈ XN
ν,m fixed, we define the operators C, C : XNν,m ⊕ XN
ν,m → X∞ν,m,
L, L : XNν,m ⊕X∞ν,m → X∞ν,m and N , N : X∞ν,m ⊕X∞ν,m → X∞ν,m by
C(aN , bN )α =
0 for 0 ≤ |α| ≤ N(aN ∗bN )α for N + 1 ≤ |α| ≤ 2N
0 otherwise
, (26)
C(aN , bN )α =
0 for 0 ≤ |α| ≤ N(aN ∗bN )α for N + 1 ≤ |α| ≤ 2N
0 otherwise
, (27)
L(aN , v)α =
0 for 0 ≤ |α| ≤ N(aN ∗v)α |α| ≥ N + 1
, (28)
L(aN , v)α =
0 for 0 ≤ |α| ≤ N(aN ∗v)α |α| ≥ N + 1
, (29)
N(u, v)α =
0 for 0 ≤ |α| ≤ N(u∗v)α |α| ≥ N + 1
, (30)
and
N(u, v)α =
0 for 0 ≤ |α| ≤ N(u∗v)α |α| ≥ N + 1
, (31)
Now for |α| ≥ N + 1 the tail terms of the binary operators have
(a∗b)α = C(aN , bN )α + L(aN , v)α + L(bN , u)α + N(u, v)α,
(a∗b)α = C(aN , bN )α + L(aN , v)α + L(bN , u)α + N(u, v)α,
(a∗a)α = C(aN , aN ) + 2L(aN , u) + N(u, u).
and(a∗a)α = C(aN , aN ) + 2L(aN , u) + N(u, u).
The following Lemma records some useful bounds.
33
Lemma 3.3. Fix aN , bN ∈ XNν,m. We have the bounds
‖C(aN , bN )‖X∞ν,m ≤2N∑
|α|=N+1
∣∣∣∣∣∣
α1∑
β1=0
. . .
αm∑
βm=0
δαβaα1−β1,...,αm−βmbβ1,...,βm
∣∣∣∣∣∣ν|α|,
‖C(aN , bN )‖X∞ν,m ≤2N∑
|α|=N+1
1
|α|
∣∣∣∣∣∣
α1∑
β1=0
. . .
αm∑
βm=0
δαβ |β|aα1−β1,...,αm−βmbβ1,...,βm
∣∣∣∣∣∣ν|α|,
‖L(aN , v)‖X∞ν,m ≤
N∑
|α|=1
|aα|ν|α|,
‖v‖1ν,m
‖L(aN , v)‖X∞ν,m ≤
N∑
|α|=1
|aα|ν|α|,
‖v‖1ν,m
‖N(u, v)‖X∞ν,m ≤ ‖u‖1ν,m‖v‖1ν,m.
and‖N(u, v)‖X∞ν,m ≤ ‖u‖
1ν,m‖v‖1ν,m.
Proof. For aN , bN ∈ XNν,m fixed simply note that
‖C(aN , bN )‖X∞ =
∞∑
|α|=N+1
|C(aN , bN )α|ν|α|
=
2N∑
|α|=N+1
|(aN ∗bN )α|ν|α|,
just by the definition of C(aN , bN ), and similarly for the C(aN , bN ) estimate.Now, take aN ∈ XN
ν,m and v ∈ X∞ν,m. Then
‖L(aN , v)‖X∞ =
∞∑
|α|=N+1
∣∣(L(aN , v)α
∣∣ ν|α|
=
∞∑
|α|=N+1
∣∣(aN ∗v)α
∣∣ ν|α|
≤∞∑
|α|=0
∣∣(aN ∗ v)α
∣∣ ν|α|
= ‖aN ∗v‖1ν,m
≤
N∑
|α|=1
|aNα |ν|α| ‖v‖X∞ ,
exploiting the product estimate for the operator ∗, the fact that aN is zero at orders aboveN , and the fact that v is zero at orders below N + 1. Again, the L estimate is similar.
Finally if u, v ∈ X∞ν,m then ‖N(u, v)‖X∞ν,m ≤ ‖u‖ν,m1‖v‖1ν,m, just by the Banach algebra
estimate for the Cauchy product. The N estimate is similar.
34
Holding aN ∈ XNν,m fixed, and considering the action of the derivatives of L, L, N , N on
h ∈ X∞ν,m, we see that
DL(aN , v)h = L(aN , h),
DL(aN , v)h = L(aN , h),
as L, L are linear in v. Similarily
DuN(u, v)h = N(h, v),
DuN(u, v)h = N(h, v),
DvN(u, v)h = N(u, h),
DvN(u, v)h = N(u, h),
and alsoDN(u, u)h = 2N(u, h),
andDN(u, u)h = 2N(u, h),
by the Banach algebra properties of `1ν,m. Bounds on the derivatives follow immediately.
Corollary 3.4. Fix aN , bN ∈ XNν,m. Then
‖DL(aN , v)‖B(X∞ν,m) ≤N∑
|α|=1
|aα|ν|α|,
‖DL(aN , v)‖B(X∞ν,m) ≤N∑
|α|=1
|aα|ν|α|,
‖DuN(u, v)‖B(X∞ν,m) ≤ ‖v‖1ν,m,
‖DuN(u, v)‖B(X∞ν,m) ≤ ‖v‖1ν,m,
‖DvN(u, v)‖B(X∞ν,m) ≤ ‖u‖1ν,m,
‖DvN(u, v)‖B(X∞ν,m) ≤ ‖u‖1ν,m,
and‖DN(u, u)‖B(X∞ν,m) ≤ 2‖u‖1ν,m,
‖DN(u, u)‖B(X∞ν,m) ≤ 2‖u‖1ν,m.
Remark 3.5. Note that in the estimates above, the quantity
‖aN‖1ν,m =
N∑
|α|=1
|aNα |ν|α|,
is made as small as we wish, by taking ν small enough. This is due to the removal of theconstant term from the norm.
35
3.3 Validated numerics for local Taylor methods
The observation driving our computer assisted error analysis is that, the recursion relationsgive rise to a fixed point problem for the unknown Taylor series, and the fixed point operatorbecomes a contraction if we take the projection dimension high enough. Rather than provingthis maxim in general, we will illustrate its use in a number of example problems. We exploitthe following lemma, which encodes convenient sufficient conditions for application of theBanach fixed point theorem.
Lemma 3.6. Let X be a Banach space, x0 ∈ X and suppose that for some r∗ > 0,T : Br∗(x0) → X is a Frechet differentiable mapping. Let Y0 be a positive constant sothat
‖T (x0)− x0‖ ≤ Y0,
and assume that there are positive constants Z1, Z2 so that for all 0 < r ≤ r∗ we have that
‖DT (x)−DT (x0)‖ ≤ Z1 + Z2r,
with x ∈ Br(x0).Then, for any 0 < r ≤ r∗ so that
Z2r2 − (1− Z1)r + Y0 ≤ 0, (32)
there is unique x ∈ Br(x0) so that T (x) = x.
Proof. Note that Br(x0) is a complete metric space. For x ∈ Br(x0) consider
‖T (x)− x0‖ ≤ ‖T (x)− T (x0)‖+ ‖T (x0)− x0‖≤ supz∈Br(x0)
‖DT (z)‖‖x− x0‖+ Y0
≤ (Z1 + Z2r) ‖x− x0‖+ Y0
≤ Z2r2 + Z1r + Y0
≤ r,
by the hypothesis given in Equation (32). Then T maps Br(x0) into itself.Now choose x, y ∈ Br(x0), and consider
‖T (x)− T (y)‖ ≤ supz∈Br(x0)
‖DT (z)‖‖x− y‖
≤ (Z1 + Z2r) ‖x− y‖.
From Equation (32) we have that
Z2r2 + Z1r + Y0 ≤ r,
or
Z2r + Z1 +Y0
r≤ 1.
Now since Z2, Z1, Y0 and r are strictly positive, it follows that
Z2r + Z1 < 1,
so that T is a contraction mapping on Br(x0).
36
3.3.1 A first example: one scalar equation in a single complex variable
Recall that in Section 2.1.2 we studied the analytic linearization problem for a single firstorder analytic differential equation in one complex variable. For the example of the logisticnonlinearity we found that the linearizing change of coordinates has Taylor coefficient givenby
p0 = z0,
where z0 is the equilibrium point,p1 = s,
with s ∈ C an arbitrary scaling, and
pn =−1
λ(n− 1)
n−1∑
k=1
pn−kpk. (33)
for n ≥ 2. Suppose we compute the coefficients pn for 0 ≤ n ≤ N using Equation (33) andthat we choose a numerical radius of validity ν > 0. Then, ideally speaking, we would liketo prove that for some r > 0 we have the bound
sup|σ|≤ν
∣∣PN (σ)− P (σ)∣∣ ≤ r.
In fact, by choosing s appropriately we can arrange that the coefficients pn∞n=0 decayrapidly enough that we simply take ν = 1.
Following the sketch outlined in the previous section, define the mapping Ψ: `1 → `1 by
Ψ(p)n =
z0 if n = 0
s if n = 1−1
λ(n−1)
∑n−1k=1 pn−kpk for n ≥ 2
. (34)
Clearly, p = pn∞n=0 are the Taylor coefficients of the desired parameterization if andonly if p is a fixed point of Ψ. Moreover, if p is a fixed point of Ψ and p ∈ `1 then theparameterization
P (σ) =
∞∑
n=0
pnσn,
is analytic on the unit disk.Let XN
1,1, X∞1,1 be as defined in Section 3.2 (noting that in this case the multi-indices are
one dimensional i.e. m = 1, and that we have also fixed ν = 1). Then Ψ has the projections
Ψ(a) = ΨN (aN ) + T (h, aN ),
where
ΨN (aN )n =
z0 if n = 0
s if n = 1−1
λ(n−1)
∑n−1k=1 a
Nn−ka
Nk for 2 ≤ n ≤ N
0 for n ≥ N + 1
,
and
T (h, aN )n =
0 if 0 ≤ n ≤ N−1
λ(n−1)
∑n−1k=1 an−kak for 2 ≤ n ≤ N
,
37
where a = aN + h. Exploiting the operators C, L, and N defined in Section 3.2, we rewriteT as
T (h, aN )n =−1
λ(n− 1)
(C(aN , aN )n + 2L(aN , h)n + N(h, h)n
)
=−1
λ(n− 1)
(n−1∑
k=1
aNn−kaNk + 2
n−1∑
k=1
aNn−khk +
n−1∑
k=1
hn−khk
),
for n ∈ N.The key is to observe that by computing exactly the coefficient up to order N , we obtain
an exact fixed point of the operator ΨN (exactly in the sense of interval arithmetic, i.e. up tointerval enclosures). These coefficients can be computed exactly (up to interval enclosures)simply iterative evaluating the recursion relations. We denote the sequence of coefficientsso computed by aN ∈ XN
1,1.
Once we have in hand the unique fixed point aN ∈ XN1,1 of ΨN , what remains is to find
a fixed point of T (h, aN ), where we now think of the second argument aN as being fixed.Define the map T (·, aN ) : X∞1,1 → X∞1,1 by
T (h, aN )n =−1
λ(n− 1)
(C(aN , aN )n + 2L(aN , h)n + N(h, h)n
), (35)
for n ∈ N. One now checks that the Frechet derivative of T (h, aN ) with respect to h, andacting on a vector v ∈ X∞1,1 is given by
[DT (h, aN )v
]n
=−1
λ(n− 1)
(2L(aN , v)n + 2N(h, v)n
), (36)
for n ∈ N.The following lemma facilitates our computer assisted error analysis, and follows directly
from the estimates of Section 3.2.
Lemma 3.7 (The Y and Z bounds). Fix aN ∈ XN1,1 and let T (·, aN ) : X∞1,1 → X∞1,1 be as
defined in Equation (35). Define the constants
Y0 :=1
|λ|
2N∑
n=N+1
1
n− 1
∣∣∣∣∣n−1∑
k=1
aNn−kaNk
∣∣∣∣∣ ,
Z1 :=2
|λ|N
N∑
n=1
∣∣aNn∣∣ ,
and
Z2 :=2
|λ|N.
Then‖T (0, aN )‖X∞1,1 ≤ Y0.
andsup
b∈Br(0)
‖DT (b)‖B(X∞1,1) ≤ Z1 + Z2r.
Proof. The proof is just a matter of applying the bounds derived in Lemmas 3.3 and 3.4 tothe expressions for T and DT given in Equations (35) and (36), and taking into account thefact that m = ν = 1 in the present set up.
Example: validated numerics for the complex logistic equation
38
3.3.2 A second example: system of scalar equations in two complex variablesand stable/unstable manifolds for Lorenz
Recall that the power series coefficients for the Parameterization of a two dimensional sta-ble/unstable manifold for the Lorenz system were found to be solutions of the homologicalequations
[Df(p0)− (mλ1 + nλ2)Id] pmn = smn, (37)
where p0 is the desired fixed point of the Lorenz equations (either the origin or one of “theeyes”)
pmn =
amnbmncmn
,
are the Taylor coefficients of the parameterization of the local manifold, and
smn =
0∑mk=0
∑nl=0 δ
mnkl a(m−k)(n−l)ckl
−∑mk=0
∑nl=0 δ
mnkl a(m−k)(n−l)bkl
=
0(a∗c)mn−(a∗b)mn
,
using the notation of Section 3.2.Suppose now that Df(p0) is diagonalizable, and that the eigenvalues λ1, λ2, λ3 ∈ C,
and the associated eigenvectors are known ξ1, ξ2, ξ3 ∈ R3 are known (at least up to intervalenclosures). Then we will write
Q = [ξ1, ξ2, ξ3],
for the matrix whose columns are the eigenvectors. By the assumption of diagonalizibilityQ is invertible, and we let Q−1 denote its inverse. Letting
Λ =
λ1 0 00 λ2 00 0 λ3
,
denote the diagonal matrix of eigenvalues, and assuming that λ1, λ2 are stable and λ3 isunstable, we have that
Df(p0)− (mλ1 + nλ2)Id = QΛQ−1 − (mλ1 + nλ2)QQ−1
= Q (Λ− (mλ1 + nλ2)Id)Q−1,
from which we obtain that
(Df(p0)− (mλ1 + nλ2)Id)−1
= Q (Λ− (mλ1 + nλ2)Id)−1Q−1.
Here
(Λ − (mλ1 + nλ2)Id)−1 =
(λ1 −mλ1 − nλ2)−1 0 00 (λ2 −mλ1 − nλ2)−1 00 0 (λ3 −mλ1 − nλ2)−1
,
For each (m,n) ∈ N2 define the matrices
Amn =
A11mn A12
mn A13mn
A21mn A22
mn A23mn
A31mn A32
mn A33mn
= Q (Λ− (mλ1 + nλ2)Id)
−1Q−1,
and note that
‖Amn‖ = ‖ (Df(p0)− (mλ1 + nλ2)Id)−1 ‖ ≤ ‖Q‖‖Q−1‖Kmn,
39
where
Kmn := max
(1
|λ1 −mλ1 − nλ2|,
1
|λ2 −mλ1 − nλ2|,
1
|λ3 −mλ1 − nλ2|
)
Here Kmn → 0 as (m + n) → ∞, and none of the denominators is zero by the assumptionthat the eigenvalues are non-resonant (an assumption which holds at all equilibria of theLorenz system, for the classical parameter values).
We now consider the problem in the “Taylor transform” domain. Note that our se-quences have m = 2 dimensional multi-indices and that, by exploiting the free choice ofeigenvector scalings, we able to take ν = 1. More precisely we look for Taylor coeffi-cients in the space `11,2. We start by defining the “diagonal” bounded linear operatorsA12, A13, A22, A23, A32, A33 : `11,2 → `11,2 by
Ajk(a)mn = Ajkmnamn,
for j = 1, 2, 3, and k = 2, 3 for all (m,n) ∈ N2.
Now endow(`11,2)3
= `11,2 ⊕ `11,2 ⊕ `11,2, with the product space norm
‖(a, b, c)‖(`11,2)3 := max(‖a‖11,2, ‖b‖11,2, ‖c‖11,2).
Consider the operator Ψ:(`11,2)3 →
(`11,2)3
given by
Ψ(a, b, c)mn =
Ψ(a, b, c)1mn
Ψ(a, b, c)2mn
Ψ(a, b, c)3mn
, (38)
withΨ(a, b, c)j = Aj2(a∗c)−Aj3(a∗b),
for j = 1, 2, 3. Note that a fixed point of Ψ solves the homological equations to all orders,
i.e. if (a, b, c) ∈(`11,2)3
is a fixed point of Ψ, then pmn = (amn, bmn, cmn) are the Taylorcoefficients of the desired parameterization.
Let XN1,2 and X∞1,2 be as defined in Section 3.2. Then Ψ has the decomposition
Ψ(a, b, c) = ΨN (aN , bN , cN ) + T (u, v, w, aN , bN , cN ),
where ΨN : (XN1,2)3 → (XN
1,2)3 and T : (X∞1,2)3⊕ (XN1,2)3 → (X∞1,2)3. Here the components of
ΨN are just the homological equations for 0 ≤ m+ n ≤ N . Suppose that aN , bN , cN ∈ XN1,2
are obtained by solving these homological equation exactly (at least in the sense of intervalarithmetic). Since solving the recursion relations to N -th order gives the fixed point of ΦN ,we need only prove the existence of a fixed point of T (·, ·, ·, aN , bN , cN ) with the XN
1,2 dataconsidered as fixed.
In this case T : (X∞1,2)3 → (X∞1,2)3 is given by
T (u, v, w, aN , bN , cN ) =
T (u, v, w, aN , bN , cN )1
T (u, v, w, aN , bN , cN )2
T (u, v, w, aN , bN , cN )3
, (39)
where, using the notation of Section 3.2, we have
T (u, v, w, aN , bN , cN )j = Aj2(C(aN , cN ) + L(aN , w) + L(cN , u) +N(u,w)
)
−Aj3(C(aN , bN ) + L(aN , v) + L(bN , u) +N(u, v)
).
40
Note that T is Frechet differentiable and that DT : (X∞1,2)3 → (X∞1,2)3 is given by (suppress-ing for a moment the function arguments)
DT =
D1T1 D2T
1 D3T1
D1T2 D2T
2 D3T2
D1T3 D2T
3 D3T3
,
where, for h ∈ X∞1,2 these partial derivatives have the action
D1Tj(u, v, w, aN , bN , cN )h = Aj2
(L(cNh) +N(h,w)
)−Aj3
(L(bN , h) +N(h, v)
)
D2Tj(u, v, w, aN , bN , cN )h = −Aj3
(L(aN , h) +N(u, h),
)
andD3T
j(u, v, w, aN , bN , cN )h = Aj2(L(aNh) +N(u, h)
),
for j = 1, 2, 3.We would like to show that T is a contraction in some small neighborhood of (0, 0, 0) ∈
(X∞1,2)3. Since we adopted the max norm on the product space the norm of the matrix DTis bound by the max of the sums of the row entries. Then, recalling the bounds of Lemma(REFERENCE), we have the following.
Lemma 3.8. Let
KN := ‖Q‖‖Q−1‖ max1≤j≤3
supm+n≥N+1
1
|mλ1 + nλ2 − λj |,
and define
Y0 := 2KN
(2N∑
m+n=N+1
|(aN ∗bN )mn|+ |(aN ∗cN )mn|
),
Z1 := KN
2
∑
1≤m+n≤N|aNmn|+
∑
1≤m+n≤N
(|bNmn|+ |cNmn|
) ,
andZ2 := 4KN .
Then‖T (0, 0, 0)‖(X∞1,2)3 ≤ Y0.
Moreover, for r > 0 let Br(0) denote the ball of radius r about the origin in (X∞1,2)3. Then
sup(a,b,c)∈Br(0)
‖DT (a, b, c)‖B((X∞1,2)3) ≤ Z1 + Z2r.
Proof. First consider, for j = 1, 2, 3 and k = 2, 3, the operators Ajk restricted to X∞1,2. Wehave that
‖Aji‖B(X∞1,2) := sup‖h‖X∞1,2=1
‖Aijh‖X∞1,2
= sup‖h‖1m=1
∞∑
m+n=N+1
|Aijmnhmn|
≤ sup‖h‖11,2=1
∞∑
m+n=N+1
‖Q‖‖Q−1‖|mλ1 + nλ2 − λj |
|hmn|
≤ KN .
41
Then
‖T (0, 0, 0, aN , bN , cN )− 0‖(X∞1,2)3 ≤ max1≤j≤3
‖T (0, 0, 0, aN , bN , cN )j‖X∞1,2
≤ ‖Aj2‖B(X∞1,2)‖C(aN , cN )‖11,2 + ‖Aj3‖B(X∞1,2)‖C(aN , bN )‖11,2
≤ 2KN
(2N∑
m+n=N+1
|(aN ∗bN )mn|+ |(aN ∗cN )mn|
)
= Y0
Now let u, v, w ∈ X∞1,2 all with norm less than or equal to r. Considering the derivativein light of the estimates in Lemma (REFERENCE) one has that
‖DT (u, v, w, aN , bN , cN )‖B((X∞)3) ≤ max1≤j≤3
3∑
k=1
‖DkTj((u, v, w, aN , bN , cN ))‖B(X∞1,2)
≤ max1≤j≤3
3∑
k=1
sup‖h‖X∞1,2=1
‖DkTj((u, v, w, aN , bN , cN ))h‖X∞1,2
≤ KN sup‖h‖X∞1,2=1
(2‖L(aN , h)‖11,2 + ‖L(bN , h)‖11,2 + ‖L(cN , h)‖11,2 + 2‖N(u, h)‖11,2 + ‖N(v, h)‖11,2 + ‖N(w, h)‖11,2
)
≤ KN
2
∑
1≤m+n≤N|aNmn|+
∑
1≤m+n≤N
(|bNmn|+ |cNmn|
)+ 2‖u‖X∞1,2 + ‖v‖X∞1,2 + ‖w‖X∞1,2
≤ Z1 + Z2r
Example: validated numerics for the Lorenz manifolds
3.3.3 The more realistic example: stable/unstable manifolds of equilibriumconfigurations in the four body problem
Let Σ denote the 4× 4 diagonal matrix of eigenvalues and
Q = [ξ1, ξ2, ξ3, ξ4],
denote the matrix of eigenvectors, and Q−1 be its inverse. Define
Σmn = Σ− (mλ1 + nλ2)Id4
=
λ1 − (mλ1 + nλ2) 0 0 00 λ2 − (mλ1 + nλ2) 0 00 0 λ3 − (mλ1 + nλ2) 00 0 0 λ4 − (mλ1 + nλ2)
,
where λ1, λ2 are the stable (or respectively unstable) eigenvalues, and the matrices
Amn =
A11mn A12
mn A13mn A14
mn
A21mn A22
mn A23mn A24
mn
A31mn A32
mn A33mn A34
mn
A41mn A42
mn A43mn A44
mn
= QΣ−1
mnQ−1.
42
with
F 1mn(a, b, c, d, q, r, s, T, U, V )F 2mn(a, b, c, d, q, r, s, T, U, V )F 3mn(a, b, c, d, q, r, s, T, U, V )F 4mn(a, b, c, d, q, r, s, T, U, V )
= QΣ−1
mnQ−1Hmn,
and
Hmn =
0h2mn
0h4mn
,
withh2mn = m1s
2mn +m2s
2mn +m3s
2mn,
andh4mn = m1s
4mn +m2s
4mn +m3s
4mn.
Then for 1 ≤ j ≤ 4 we write
F j(a, b, c, d, q, r, s, T, U, V )mn = Aj2mnh2mn +Aj4mnh
4mn,
where
s2 = −3
2Q−5/200 (a00 − x1) (a∗a+ c∗c)− (a00 − x1)
Q00
(3
2T ∗Q+Q∗T
)+ a∗T,
s2 = −3
2R−5/200 (a00 − x2) (a∗a+ c∗c)− (a00 − x2)
R00
(3
2U ∗R+R∗U
)+ a∗U,
s2 = −3
2S−5/200 (a00 − x3) (a∗a+ c∗c)− (a00 − x3)
S00
(3
2V ∗S + S∗V
)+ a∗V,
s4 = −3
2Q−5/200 (c00 − y1) (a∗a+ c∗c)− (a00 − x1)
Q00
(3
2T ∗Q+Q∗T
)+ c∗T,
s4 = −3
2R−5/200 (c00 − y2) (a∗a+ c∗c)− (a00 − x2)
R00
(3
2U ∗R+R∗U
)+ c∗U,
and
s4 = −3
2S−5/200 (c00 − y3) (a∗a+ c∗c)− (a00 − x3)
S00
(3
2V ∗S + S∗V
)+ c∗V.
Moreover, by definingF 5(a, b, c, d, q, r, s, T, U, V ) =
2(a00 − x1)F 1(a, b, c, d, q, r, s, T, U, V ) + 2(c00− y1)F 3(a, b, c, d, q, r, s, T, U, V ) + a∗a+ c∗c,
F 6(a, b, c, d, q, r, s, T, U, V ) =
2(a00 − x2)F 1(a, b, c, d, q, r, s, T, U, V ) + 2(c00− y2)F 3(a, b, c, d, q, r, s, T, U, V ) + a∗a+ c∗c,
F 7(a, b, c, d, q, r, s, T, U, V ) =
2(a00 − x3)F 1(a, b, c, d, q, r, s, T, U, V ) + 2(c00− y3)F 3(a, b, c, d, q, r, s, T, U, V ) + a∗a+ c∗c,
F 8(a, b, c, d, q, r, s, T, U, V ) =−3
2Q−5/200 F 5(a, b, c, d, q, r, s, T, U, V )− 1
Q00
(−3
2T ∗Q+Q∗T
),
43
F 9(a, b, c, d, q, r, s, T, U, V ) =−3
2R−5/200 F 6(a, b, c, d, q, r, s, T, U, V )− 1
R00
(−3
2U ∗R+R∗U
),
F 10(a, b, c, d, q, r, s, T, U, V ) =−3
2S−5/200 F 7(a, b, c, d, q, r, s, T, U, V )− 1
S00
(−3
2V ∗S + S∗V
),
and
Ψ(a, b, c, d, q, r, s, T, U, V ) =
F 1(a, b, c, d, q, r, s, T, U, V )F 2(a, b, c, d, q, r, s, T, U, V )F 3(a, b, c, d, q, r, s, T, U, V )F 4(a, b, c, d, q, r, s, T, U, V )F 5(a, b, c, d, q, r, s, T, U, V )F 6(a, b, c, d, q, r, s, T, U, V )F 7(a, b, c, d, q, r, s, T, U, V )F 8(a, b, c, d, q, r, s, T, U, V )F 9(a, b, c, d, q, r, s, T, U, V )F 10(a, b, c, d, q, r, s, T, U, V )
,
we have that the sequences of Taylor coefficients for the desired stable parameterization aregiven by the fixed point of Ψ.
Define the product spaces
XN =(XNν,4
)10,
andX∞ =
(X∞ν,4
)10,
and observe that X = XN ⊕X∞.Then, as in the previous examples, Ψ is decomposed as
Ψ(a, b, c, d, q, r, s, T, U, V ) = ΨN (a, b, c, d, q, r, s, T, U, V ) + T (a, b, c, d, q, r, s, T, U, V ).
Let (aN , bN , cN , dN , qN , rN , sN , TN , UN , V N ) = x ∈ XN denote the exact fixed point of ΨN ,computed using the recursion relations. We seek a fixed point of the map T (·, x) : X∞ →X∞.
Now, T is given by
T (H, x) =
T1(H, x)T2(H, x)T3(H, x)T4(H, x)T5(H, x)T6(H, x)T7(H, x)T8(H, x)T9(H, x)T10(H, x)
where the truncated maps now proliferate substantially.Define the linear operators Aj2, Aj4 : X∞ν,4 → X∞ν,4 by
(Ajkh)mn = Ajkmnhmn,
for j = 1, 2, 3, 4 and k = 2, 4, and the components
Tj(H, x) = Aj2T11(H, x) +Aj4T12(H, x),
44
whereT11(H, x) = m1T11(H, x) +m2T11(H, x) +m3T11(H, x),
andT11(H, x) =
−3
2Q−5/200 (a00−x1)
(C(aN , aN ) + C(cN , cN ) + L(aN , h1) + L(cN , h3) + N(h1, h1) + N(h3, h3)
)
− (a00 − x1)
Q00
(3
2(C(TN , QN ) + L(TN , h5) + L(QN , h8)N(h5, h8)
)+
(C(QN , TN )+L(QN , h8)+L(TN , h5)N(h8, h5))+C(aN , TN )+L(aN , h8)+L(TN , h1)+N(h1, h8)
3.3.4 A brief closing example: validated Taylor integrators
45
4 Connecting Orbits and the Method of Projected Bound-aries
We now return to the problem of connecting orbits for differential equations. We are readyto formulate the method of projected boundaries for differential equations.
4.1 Heteroclinic connections between equilibria of vector fields
Let f : RN → RN be a smooth vector field and p, q ∈ RN be hyperbolic equilibria for f .Suppose that P : Dmp → RN and Q : Dmq → RN are parameterizations of the local unstableand stable manifolds at p and q respectively. Just as in the case of maps, we say that thereis a short connection from p to q (relative to P and Q) if the equation
P (θ)−Q(φ) = 0,
has a solution (θ∗, φ∗) ∈ Dmp⊕Dmq . Note however that in the case of differential equationssolutions of this equation cannot be isolated. This is because any infinitesimal flow of anintersection point is again a point of intersection. Then we look for a solution on a fixedsphere in the unstable parameter space.
P Q
0 0
= P (0) Q(0) =
Figure 5: Projected boundaries for flows.
When the images of the local parameterizations P and Q do not intersect then thesituation is a little more involved. Let Φ: RN × R → RN denote the flow generated by f .
46
We define the mapF (θ, φ, T ) = Φ(P (θ), T )−Q(φ),
and seek θ∗, φ∗, T∗ so thatF (θ∗, φ∗, T∗) = 0.
Then the orbit of P (θ∗) is heteroclinic from p to q. An equivalent operator not involvingthe flow Φ is given the integrated form, i.e. consider the operator F : X → X
F(α, β, T, x(t)) :=
(Q(φ(β))− P (θ(α))−
∫ T0f(x(τ)) dτ
P (θ(α)) +∫ t
0f(x(τ)) dτ − x(t)
),
where φ(β) and θ(α) are parameterizations of spheres in the parameter spaces of the unstableand stable parameterizations. Then F maps the space
X := Rn ⊕ C([0, T ],RN ),
to itself.
Discretization of C([0, T ],Rn): The function space must be discretized in order to solvethe problem numerically. In [54, 4] for example the function space is discretized usingpiecewise linear splines. Spectral approximation using Chebyshev series is used in [55, 56].Chebyshev series are “Fourier series for boundary value problems” and are amenable to asequence space a-posteriori analysis as discussed in the Lecture of J.B. van den Berg [1].
Figure 6: An illustration of a “short connection” for the Lorenz system. The green circleon the blue manifold illustrates the “phase condition” used to isolate the connecting orbitsegment.
4.2 Example: Heteroclinic Connections, Patterns, and TravelingWaves
The following application appears in [55], and uses all of the techniques discussed above inorder to solve an interesting pattern formation problem for PDEs. The pattern formationmodel
∂tU = −(1 + ∆)2U + µU− β|∇U|2 −U3 (40)
in the plane, i.e., U = U(t, x) ∈ R, t ≥ 0, x ∈ R2 was introduced in [57]. This equationgeneralizes the Swift-Hohenberg equation [58]. The additional term β|∇U|2, reminiscent
47
Figure 7: A long connection for Lorenz. Now the phase condition is the green circle on thered manifold. The black curve is the solution of the projected boundary value problem. Theboundary value problem is discretized using piecewise linear splines. The pink curves arefound using the conjugacy to the linear dynamics in the stable/unstable parameter spaces.
Figure 8: Long connection for the classic parameter values in Lorenz. The boundary valueproblem is discretized using Chebyshev series.
of the Kuramoto-Sivashinsky equation [51, 52], breaks the up-down symmetry U 7→ −Ufor β 6= 0. The Swift-Hohenberg equation acts as a phenomenological model for patternformation in Rayleigh-Benard convection, with β 6= 0 corresponding to a free boundary atthe top of the convection cell, rather than a fixed one for the symmetric case β = 0 [59].The parameter µ is related to the distance to the onset of convection rolls. For µ < 0 thetrivial equilibrium U ≡ 0 is locally stable, whereas for µ > 0 it is unstable.
48
Depending on the parameter values the dynamics generated by (40) exhibit a variety ofpatterns besides simple convection rolls (a stripe pattern); in particular, hexagonal spot pat-terns are observed. In [60] it is shown that stable hexagonal patterns with small amplitudecan be found for β < 0 only. In [57] the interplay between hexagons and rolls near onset(small µ) is examined using a weakly nonlinear analysis. Introducing a small parameterε > 0, the parameters are scaled as
µ = ε2µ, β = εβ.
In the asymptotic regime ε 1, one can describe the roll and hexagonal patterns viaamplitude equations. The seminal result in [57], based on spatial dynamics and geometricsingular perturbation theory, is that heteroclinic solutions of the system
4B′′1 + cB′1 + µB1 − βB2
2 − 3B31 − 12B1B
22 = 0
B′′2 + cB′2 + µB2 − βB1B2 − 9B31 − 6B2
1B2 = 0(41)
correspond to modulated front solutions, travelling at (rescaled) speed c, which correspondsto an asymptotically small velocity εc in the original spatio-temporal variables. We note thatthe system (41) also shows up in the analysis of solidification fronts describing crystallizationin soft-core fluids.
The variables B1 and B2 represent amplitudes of certain wave modes (slowly varying inthe original variables). We refer to [57] for the details and the rigorous justification of thederivation. The dynamical system (41) has up to seven stationary points:
(B1, B2) = (0, 0) trivial state
(B1, B2) = (Brolls, 0) “positive” rolls
(B1, B2) = (−Brolls, 0) “negative” rolls
(B1, B2) = (B+hex, B
+hex) hexagons
(B1, B2) = (B−hex, B−hex) “false” hexagons
(B1, B2) = (−β/3, B±mm) two “mixed mode” states.
(42)
Here Brolls =√µ/3 and B±hex =
−β±√β2+60µ
30 and B±mm = ± 13
√µ− β2/3. Due to symmetry,
there are two equilibria corresponding to rolls. We refer to [57] for a full discussion ofall states and their stability properties. In this paper we consider the interplay betweenhexagons and rolls (both positive and negative).
While the weakly nonlinear analysis in [57] provided a vast reduction in complexityfrom (40) to (41), one outstanding issue remained: the heteroclinic solutions are difficult toanalyse rigorously due to the nonlinear nature of the equations (41). In the limit c → ∞various connections could be found through a further asymptotic reduction [57], but for finitewave speeds the analysis of (41) was out of reach. In the present paper we introduce v acomputer-assisted, rigorous method for finding heteroclinic solutions for c = 0, i.e. standingwaves. In particular we find connections between hexagons and rolls with zero propagationspeed, i.e., the two patterns coexist.
The system (41) is gradient-like for c 6= 0, while it is Hamiltonian for c = 0. Hence, forhexagons and rolls to coexist their free energy must be equal, a situation that occurs when
µ = 7+3√
630 β2. We prove the following theorem, which settles the conjecture in [57] about
the coexistence of hexagons and rolls:
49
−20 −15 −10 −5 0 5 10 15 20−0.5
0
0.5
−20 −15 −10 −5 0 5 10 15 20−0.5
0
0.5
Figure 9: At the bottom are graphs of B1 (red) and B2 (blue) representing heteroclinicsolutions of (41) that connect the hexagon state to the positive rolls (on the left) and
negative rolls (and the right). The parameter values are c = 0, µ = 7+3√
630 and β = 1,
corresponding to the assumptions in Theorem 4.1. At the top we illustrate the correspondingstationary patterns of (40). We note that the two phase transitions from rolls to hexagonshave distinctive features. On the left, the stripes (“positive” rolls) undergo pearling, whichgradually leads to separation into spots (hexagons). On the right, the stripes (“negative”rolls) develop transverse waves, which break up into a block structure that then transformsinto hexagonal spots. The figures of the patterns at the top were made using ε = 1
3 . Forsmaller ε the transition between the two states is more gradual.
Theorem 4.1. For parameter values µ = 7+3√
630 β2 and c = 0 there exists a heteroclinic
orbit of (41) between the hexagons and positive rolls, see (42), as well as a heteroclinic orbitbetween the hexagons and negative rolls.
The heteroclinic solutions are depicted in Figure 9, together with the correspondingpatterns of the PDE (40). These orbits thus represent two types of stationary domain wallsbetween hexagons and rolls (spots and stripes). While each heteroclinic connection existson a parabola in the (β, µ) parameter plane, a parameter scaling reduces this to a singleconnecting orbit.
The problem is solved using a scheme very similar to the one illustrated above for theLorenz system. So, an approximate solution unum, obtained through a numerical calculation.We then construct an operator which has as its fixed points the heteroclinic solutions, andwe set out to prove that this operator is a contraction mapping on a small ball around unum
in an appropriate Banach space. The ball should be small enough for the estimates to besufficiently strong to prove contraction, but large enough to include both unum (the centerof the ball) and the solution (the fixed point). Qualitatively, considering the numericalapproximations of solutions depicted as graphs in Figure 9, we can choose the radius of theball so small that the solution is guaranteed to lie within the thickness of the lines.
Note: the rigorous justification in [57] of the link between the PDE (40) and the ODE
50
-20 -15 -10 -5 0 5 10 15 20-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Figure 10: Left: co-existing hexagonal and trivial patterns for β = 1 and µ = − 2135 . Right:
corresponding connecting orbit with u and v components in red and blue, respectively.
system (41) focuses on the case of nonzero propagation speed. The case of stationary frontsis only briefly mentioned in [57], since in that case the ODEs can be found more directlyby a reduction to a spatial center manifold. In both the zero and nonzero speed setting,the rigorous derivation of (41) relies on a (weakly) nonlinear analysis to justify that onemay ignore higher order terms in ε. Reciprocally, a heteroclinic solution of (41) is onlyguaranteed to survive in the full PDE system for small ε > 0, if the orbit is a transversalintersection of stable and unstable manifolds. This is clearly not the case in (41) for c = 0,since the system is Hamiltonian. On the other hand, one should be able to exploit the“robustness” of the solution, which is a by-product of the contraction argument, to justifyrigorously that existence of the front extends to the full system for small ε > 0. However,since stationary coexistence (i.e. a pinned front) is a co-dimension one phenomenon, thisneeds to be viewed in the context of embedding the c = 0 case into a one parameter familyof heteroclinic orbits with wave speed c = c(γ). In that setting transversality is recovered.
Finally, to find a connection for c = 0 between the hexagons and the trivial state (theappropriate parameter value for coexistence is µ = − 2
135 β2), one needs to deal with reso-
nances between eigenvalues in the spectrum of the trivial state. The resonance problem istreated in [40], and there the interested reader can find the computer assisted proof of theconnection between the hexagons and the trivial state.
4.3 Heteroclinic connections between periodic orbits of vector fields
The method of projected boundaries is extended to connecting orbits between periodic orbitssimply by replacing the parameterizations of the local stable/unstable manifolds of the fixedpoints by the parameterizations of the local stable/unstable manifolds of the periodic orbits.
Consider again f : RN → RN a smooth vector field and suppose that γ0 : [0, T0] → RNand γ1 : [0, T1]→ RN are a pair of hyperbolic periodic orbits for f . We write
d
dθγ0(θ) = f [γ0(θ)] for all θ ∈ [0, T0],
andd
dφγ1(φ) = f [γ1(φ)] for all φ ∈ [0, T1],
subject to the constraints
γ0(0) = γ0(T0) and γ1(0) = γ1(T1).
51
Suppose in addition that P : [0, 2T0] × Dm0 → RN and Q : [0, 2T1] × Dm1 → RN are pa-rameterizations of the local unstable and stable manifolds of γ0 and γ1 respectively, i.e. wesuppose that
P (θ, v) ∈Wuloc(γ0), for all θ ∈ [0, 2T0], v ∈ Dm0 ,
andQ(φ,w) ∈W s
loc(γ1), for all φ ∈ [0, 2T1], w ∈ Dm1 .
Finally, assume that(m0 + 1) + (m1 + 1) = N + 1,
i.e. that the dimension of the unstable manifold of γ0 plus the dimension of the stablemanifold of γ1 is one more than the dimension of the phase space. This dimension countallows for the possibility of a transverse and isolated heteroclinic orbit from γ0 to γ1.
Just as in the case of maps and equilibria of differential equations, we say that there isa short connection from γ0 to γ1 (relative to P and Q) if the equation
P (θ, v)−Q(φ,w) = 0,
has a solution(θ∗, v∗, φ∗, w∗) ∈ [0, 2T0]× [0, 2T1]×Dm0 ×Dm1 .
Note however that in the case of differential equations solutions of this equation cannot beisolated. This is because any infinitesimal flow of an intersection point is again a point ofintersection. Then we look for a solution on a fixed sphere in the unstable parameter space.
Suppose instead that the images of the local parameterizations P and Q do not intersect.Let θ(α), φ(β) be parameterizations of m0,m1 dimensional spheres, and Φ: RN × R→ RNdenote the flow generated by f . Define the map
F (α, β, v, w, T ) = Φ(P (θ(α), v), T )−Q(φ(β), w),
and seek α∗, β∗, v∗, w∗, T∗ so that
F (α∗, β∗, v∗, w∗, T∗) = 0.
Then the orbit of P (θ(α∗)) is heteroclinic from p to q. Note that F : RN → RN is a balancedsystem of equations. Figure 11 illustrates the geometric idea behind the mapping F .
In some situations it is desirable to study an operator which does not explicitly involvethe flow operator, and just as in the case of connections between equilibria we have theintegrated form of the operator F : X → X
F(α, β, v, w, T, x(t)) :=
(Q(φ(β), w)− P (θ(α), v)−
∫ T0f(x(τ)) dτ
P (θ(α), v) +∫ t
0f(x(τ)) dτ − x(t)
),
where φ(β) and θ(α) are parameterizations of spheres in the parameter spaces of the unstableand stable parameterizations. Then F maps the space
X := Rn0+n1 ⊕ C([0, T ],RN ),
to itself.
Discretization of C([0, T ],Rn): The function space must be discretized in order to solvethe problem numerically and again splines [54, 4] and Chebyshev series [55, 56] are goodoptions. Sequence space a-posteriori analysis as discussed in the Lecture of J.B. van denBerg [1] can be used to design computer assisted proofs.
52
(↵) ()
1
0
(↵) ()
P
Wuloc(0)
W sloc(1)
Q
P ((↵)) Q(())
[P (), T] = Q(())
Figure 11: The method of projected boundaries for flows. We fix a boundary sphere on thestable/unstable manifolds and look for a time of flight which takes us from one sphere toanother.
4.3.1 Examples
The method described above is useful for computing heteroclinic and homoclinic connectingorbits for differential equations. For example in Figure 12 we see a homolcinic connectionorbit for a periodic orbit near the attractor in the Lorenz system. Figure 13 illustratessix different periodic orbits (in blue) and a number of heteroclinic and homoclinic orbitsbetween them. The periodic orbits and their connections fill out the attractor fairly well.The connections are illustrated by a connectivity graph in Figure 14. The A-B symbols codethe number of left/right winds around the two “eyes” of the Lorenz attractor. Lastly inFigure 15 we illustrate a homoclinic connecting orbit for the system of ten ODEs associatedwith the finite dimensional projection of the Kurramoto-Shivisinsky equations.
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Figure 12: Homoclinic connection for a periodic orbit in the attractor of the Lorenz systems.
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Figure 13: Heteroclinic and homoclinic connection between six different periodic orbits inthe attractor of the Lorenz systems.
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AB
ABBB AAB
ABBAABB
AAAB
Figure 14: Directed graph depiction of the connections.
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