asymptotic poincaré compactification and finite-time singularities

ISSN 0202-2893, Gravitation and Cosmology, 2013, Vol. 19, No. 4, pp. 240–245. c© Pleiades Publishing, Ltd., 2013.

Asymptotic Poincare Compactification and Finite-Time Singularities

Spiros Cotsakis*

Research Group of Geometry, Dynamical Systems and Cosmology University of the Aegean,Karlovassi 83 200, Samos, Greece

Received April 2, 2013

Abstract—We provide an extension of the method of asymptotic decompositions of vector fields with finite-time singularities by applying the central extension technique of Poincare to the dominant part of the vectorfield at approach to the singularity. This leads to a bundle of fan-out asymptotic systems whose equilibria atinfinity govern the dynamics of the asymptotic solutions of the original system. We show how this methodcan be useful to describe a single-fluid isotropic universe at the time of maximum expansion, and discusspossible relations of our results to structural stability and non-compact phase spaces.

DOI: 10.1134/S0202289313040099

1. INTRODUCTION

Sufficient conditions for the global existence ofgeneral-relativistic spacetimes imply their physicalduration for an infinite proper time, as shown in [1,2]. The implied singular universes (in the contrapos-itive direction) will necessarily have finite-time sin-gularities, and their classification problem is a highlynon-trivial matter even in the isotropic category, cf.[3–5]. These include, for instance, apart from thestandard singular universes of relativistic cosmology,sudden singularities [6], dark energy ones [7], singularuniverses with interacting fluids [8], universes withhigher-order gravity corrections [9], and brane mod-els [10–12].

A popular method to represent infinity in generalrelativity is Penrose’s conformal method (cf. [13]),wherein the overall structure of a physical spacetimeis conformally changed so that its infinitely remoteregions become the boundary of a new, nonphysicalspacetime. Infinity is then classified according to thebehavior of the various geodesics of the new space-time near its boundary. Until recently, the conformalmethod was the only useful method of studying the“structure of infinity” in relativistic field theories. Infinite dimensions, the corresponding idea to analyzethe global structure of solutions, especially for planarsystems, stereographically is due to Bendixson [14].

The method of asymptotic splittings ([15] and ref-erences therein) offers another asymptotic represen-tation of solutions to the dynamical equations nearinfinity (that is, where some component of the so-lution diverges), and it is possible to describe it inthree main steps as follows. First, we recall some

*E-mail: [email protected]

simple definitions. We shall work on open subsetsor R

n, although our results can be extended withoutdifficulty to any differentiable manifold Mn.

We shall use interchangeably the terms “vectorfield” f : Mn → T Mn and “dynamical system” de-fined by f on Mn, x = f(x), with (·) ≡ d/dt. Also,we will use the terms “integral curve” x(t, x0) of thevector field f with initial condition x(0) = x0, and’solution’ of the associated dynamical system x(t;x0)passing through the point x0, with identical mean-ings. We say that the system x = f(x) (equivalently,the vector field f ) has a finite-time singularity ifthere exists a ts ∈ R and an x0 ∈ Mn such that for allM ∈ R there exists a δ > 0 such that ||x(t;x0)||Lp >M , for |t− ts| < δ. Here x : (0, b) → Mn, x0 = x(t0)for some t0 ∈ (0, b), and || · ||Lp is any Lp norm. Wesay that the vector field has a future (resp. past)singularity if ts > t0 (resp. ts < t0). Note also that t0is an arbitrary point in the domain (0, b) and may betaken to mean “now”. Alternatively, we may set τ =t − ts, τ ∈ (−ts, b − t∗), and consider the solution interms of the new time variable τ , x(τ ;x0), with afinite-time singularity at τ = 0. We see that for avector field to have a finite-time singularity there mustbe at least one integral curve passing through thepoint x0 of Mn such that at least one of its Lp normsdiverges at t = ts, and we write

limt→ts

||x(t;x0)||Lp = ∞, (1.1)

to denote the behavior of the solution on approach tothe finite-time singularity at ts.

One of the most interesting problems in the theoryof singularities of vector fields is to find the structureof the set of points x0 in Mn such that, when evolved

240

ASYMPTOTIC POINCARE COMPACTIFICATION 241

through the dynamical system defined by the vectorfield, the integral curve of f passing through a point inthat set satisfies the property (1.1). The evolutionary,geometric nature of this set especially near singular-ities of the field, is of prime interest in this paper. Inthe method of asymptotic splittings, we imagine thaton approach to a finite time singularity, f decomposesinto a dominant part f (0) and another, subdominantpart (consisting possibly of more than one additivecomponents f (μ), μ = 1, · · · , k), as follows:

f = f (0) + f (1) + · · · + f (k). (1.2)

This asymptotic decomposition is highly non-unique.Then, at the first step, for each particular decomposi-tion we drop all terms after f (0) and replace the exactequation x = f(x) with the asymptotic equation

x = f (0)(x). (1.3)

Repeating this for all possible decompositions of f ,we end up with a bundle of asymptotic dynamicalsystems to work with. At Step two, we look forscale-invariant solutions (called dominant balances)in each asymptotic system. Each such system maycontain many possible dominant balances resulting invarious possible dominant behaviors near the singu-larity. Balancing each asymptotic system requires acareful asymptotic analysis of the subdominant partsin the various vector field decompositions. At the laststep, we check the overall consistency of the approx-imation scheme and build asymptotic solutions foreach acceptable balance in a term-by-term iterationprocedure that ends up with a formal series expansionrepresentation of the solutions. When the wholeprocedure is completed, we have finally constructedformal series developments of particular or generalsolutions of the original set of equations, valid in a lo-cal neighborhood of the finite-time singularity. Fromsuch expansions we can deduce all possible dominantmodes of approach of the field to the singularity, de-cide on the generality of the constructed solutions andalso determine the size and part of the space of initialdata that led to such a solution.

However, this method has two shortcomings:1. It does not give any information about the

qualitative behavior of individual orbits.2. It does not distinguish between the behavior of

those solution components tending to +∞ from thoseones tending to −∞.

The qualitative method of central projections andcompactifications due to Poincare is usually pre-sented [16–19] as being suitable only for homoge-neous vector fields and singularities at infinity, that is,when we have a homogeneous vector field divergingas t → ∞. The purpose of this paper is to emphasizethe fact that this method, when considered from the

viewpoint of asymptotic splittings, is really suitablefor any weight-homogeneous vector field that blowsup at a finite time (or at infinity). In this way, we maysuitably adapt the ingenious method of Poincare’sto study the finite-time singularities that arise incosmology and relativistic field theory, wherein ahealthy dose of problems having precisely this kindof vector field and asymptotic structure appear withphase spaces of any finite number of dimensions. It ishoped that this will provide a useful alternative to thestereographic techniques of Bendixson and Penrosementioned above.

In this paper, we revisit the method of centralprojections and use it as a suitable complement anddevelopment of the method of asymptotic splittings,in order to decide about the qualitative behaviorsof the asymptotic orbits that result from the vari-ous possible asymptotic decompositions. In the nextsection, we give the details of the method of centralprojection for one-dimensional systems and describethe state of maximum expansion for a universe witha single fluid in general relativity, a problem knownnot to admit a complete description with standardmethods. In Section 3, we describe the amalgamationof asymptotic splittings and central projections in twodimensions, and in the last section we discuss variousissues that lie ahead and result from the approachtaken in this paper.

2. THE ONE-DIMENSIONAL CASE

Let us consider the one-dimensional system x =f(x) and assume that there is a finite time ts suchthat the solution x has a singularity, x(ts) = +∞,or x(ts) = −∞. In general relativity (which is theprincipal area of application we have in mind for thetechniques discussed here), one is accustomed toview such singularities as following from other plau-sible physical and geometric hypotheses—the so-called singularity theorems [21–23]. We imagine theone-dimensional phase space Γ = {x : x ∈ R} of thesystem as sitting at the (X, 1) (cotangent) line of an(X,Z) plane, and we let x be any particular state ofthe system on this line (so that x = X in this case).The Poincare central projection prc is a map from Γto the upper semi-circle S1

+ sending each phase pointto a point on the circle such that

prc : Γ → S1+ : x �→ θ, with x = cot θ. (2.1)

In this way, the two possible infinities of x at the twoends of Γ are bijectively separated (note that this can-not happen with a stereographic map like Penrose’s),

x = +∞ → θ = 0, (2.2)

x = −∞ → θ = π, (2.3)

GRAVITATION AND COSMOLOGY Vol. 19 No. 4 2013

242 COTSAKIS

the new phase space, the so-called Poincare’s circle,S1

+ is now compact, θ ∈ [0, π], and the original sys-tem x = f(x) reads

θ = − sin2 θf(cot θ) ≡ g(θ). (2.4)

So past singularities are met as θ ↓ 0, while futureones are at θ ↑ π. Let us suppose for the sake ofillustration that we are interested in the asymptoticproperties of the solution near a past singularity.In the next step of the Poincare central projectionmethod, we are after a singular asymptotic systemon the compactified phase space S1

+. According to themethod of asymptotic splittings, the field f is asymp-totically decomposed into two parts, a dominant onef (0), and another subdominant one, f (sub), i.e., f =f (0) + f (sub), in a highly non-unique manner. Wetake any particular decomposition for which f (0) =axM , M ∈ Q, and asymptotically as θ ↓ 0 we find thatg(θ) = −aθ2−M , that is, we arrive at the singularasymptotic system

θ = −aθ2−M . (2.5)

As in the method of asymptotic splittings, we needto repeat the whole procedure below for all possibledecompositions. Notice that this system is only validlocally around the singularity at θ = 0, not every-where on the compactified phase space S1

+. The laststep is now to obtain a singularity-free system validlocally around the singularity. For this purpose, onechanges the time t to a new one, τ , given by d/dt =θ1−Md/dτ , to arrive at the complete asymptoticsystem defined on the Poincare circle,

θ = −aθ, (2.6)

and valid near θ = 0. We conclude that the systemhas a past attractor when a > 0, otherwise all orbitsare repelled near the singularity.

To assess the merits of this method, we revisitthe evolution of the single-fluid FL models in generalrelativity with a linear equation of state p = (γ − 1)ρ,studied in [20], pp. 58–60. Introducing the densityparameter Ω defined in terms of the fluid density ρand the Hubble rate H by Ω = ρ/3H2, the evolutionof the system is governed by the equation

Ω = −(3γ − 2)(1 − Ω)Ω, (2.7)

and it is well known that for closed models, after afinite time interval, at the instant of maximum expan-sion, Ω blows up to +∞, thus making the descrip-tion of the whole evolution incomplete ([20], p. 60).To describe this, we choose the decomposition hav-ing f (0) = aΩ2, where a = 3γ − 2 and M = 2. Theresulting asymptotic system is θ = −(3γ − 2), and

taking θ = 0 at t = tmax, we find that θ = −(3γ −2)(t− tmax). The central projection gives Ω = cot θ =cot(−(3γ − 2)(t − tmax)), near the point of maximumexpansion.

3. DIMENSION 2

For homogeneous vector fields, the various reduc-tions involved in central projections are described inRefs. [16–19] and will not be repeated here. Below,as in the method of asymptotic splittings, we focuson weight-homogeneous vector fields and finite-timesingularities exclusively. Any C1 function h is calledweight-homogeneous if we can find a d ∈ R and anonzero vector w ∈ R

n such that for all numbers t wehave

h(tw1x1, tw2x2, · · · , twnxn)

= tdh(x1, x2, · · · , xn). (3.1)

We then speak of the weight vector w = (wi) andthe weighted degree deg(h,w) = d of the functionh. For instance, the function f1 = 2xy is weight-homogeneous with weight vector w = (−1,−1) andweighted degree d = −2, and f2 = 4y2 − x2 is alsoweight-homogeneous with the same w and d. Whenw = (1, · · · , 1), we say that h is a homogeneousfunction.

A weight-homogeneous vector field f =(f1, · · · , fn) is one for which we can find a a commonweight vector w for the fi’s. When the components fi,i = 1, · · · , n, have weighted degrees di = deg(fi, w),i = 1, · · · , n,, we say that the vector field f hasweighted degree deg(f,w) = d with d = (di). Forexample, the vector field f = (4y − x2/2, xy + x3)has the weight vector w = (−1,−2) and weighteddegree d = (−2,−3).

An important subclass of weight-homogeneousvector fields are scale-invariant vector fields, forwhich their weight vectors and degrees are related by

d = w − 1, (3.2)

with 1 here meaning the vector having 1 in everyslot (hence, di = wi − 1). Thus the degree d of ascale-invariant vector field measures its failure frombeing exactly homogeneous. An obvious propertyof any scale-invariant field f with weight vector pis that the “scale-invariant” system x = f admits asolution of the form x = atp, a ∈ C

n (in this nota-tion, xi = ait

pi in components), when the algebraicsystem a · p = f(a) has a nontrivial solution (that is,some component of a is nonzero). In this case, thesolution x = atp is invariant under the scaling t → λt,x → λpx.

In the method of asymptotic splittings, we lookfor weight-homogeneous decompositions, that is,



vector field decompositions of the form (1.2) such thatthe following conditions hold:

1. all vector field “components” f (j) in the sum areweight-homogeneous with the same weight vector w;

2. their weighted degrees are given by d(j) = w −j − 1, and d(0) > d(1) > · · · > d(k);

3. the most nonlinear, additive component f (0) isscale-invariant.

For the sake of illustration, we shall confine ourattention to planar systems of the form

x = P (x, y), y = Q(x, y), (3.3)

and we shall assume that this system has a finite-timesingularity at time ts, as in the definition (1.1) (ev-erything we do generalizes easily to any finite dimen-sion). As in the 1-dimensional case of the previoussection, since the original system (3.3) is incomplete,we are eventually after a complete, “asymptotic” sys-tem defined on the 2-dimensional sphere S2. Takingthe phanar phase space (x, y) to be the Z = 1 planeof a new 3-space with coordinates X, Y , Z, theembedding

prc : R2 → S2

+ : (x, y) �→ (X,Y,Z) : x = X/Z,

y = Y/Z, X2 + Y 2 + Z2 = 1, (3.4)

interprets any blow-up in the solution vector (x, y),x, y → ∞, as the limit Z → 0. (We note that anysign in the x, y’s is taken care of by the signs of thenew X,Y variables, and so we can restrict ourselvesto the positive semi-sphere, that is, consider only thedirection Z ↓ 0.) Thus infinity in x, y is now (at) theZ = 0 circle of infinity, X2 + Y 2 = 1. This is thefirst step of the so-called Poincare compactificationfor the system (3.3) wherein the embedding prc is thecentral projection transform, and sends a planarphase point (x, y) to a point of the semi-sphere (the“Poincare sphere”) S2

+. Under the map (3.4), thesystem (3.3) becomes the singular system [16–19]

X = Z((1 − X2)P − XY Q

)

Y = Z(−XY P + (1 − Y 2)Q

)

Z = −Z2(XP + Y Q). (3.5)

We next aim at obtaining asymptotically, througha series of transforms, first a reduction of the sys-tem (3.5) that will be complete and valid on S2

+, andsecondly another reduction valid precisely only alongthe circle of infinity. We suppose we have used themethod of asymptotic splittings and ended up with thespectrum of all possible asymptotic decompositionsof the form (1.2) of the vector field (P (x, y), Q(x, y)).We shall work with a particular weight-homogeneousdecomposition that has a dominant part given by

f (0) = (P (0), Q(0)), (3.6)

and we suppose that a = (ai), i = 1, 2, is theweighted degree of f (0). We then define

M = max{|ai|, i = 1, 2}, (3.7)

and using this, we can introduce the time transformt → τ(t) (monotone when Z > 0),

d

dt= Z1−M d

dτ, (3.8)

so that the singular system (3.5) becomes the follow-ing system for the dominant part of the vector field(P,Q) ((′) ≡ d/dτ ):

INTERMEDIATE SYSTEM:

X ′ = ZM((1 − X2)P (0) − XY Q(0)

)

Y ′ = ZM(−XY P (0) + (1 − Y 2)Q(0)

)

Z ′ = −ZM+1(XP (0) + Y Q(0)

). (3.9)

There is an important difference from the homoge-neous case treated in [16–19], in that we consideronly the dominant part of the vector field asymptot-ically, and this will be different for each particularasymptotic decomposition of the field. The “sub-dominant” part is truly subdominant asymptoticallybecause of the validity of the subdominance condi-tion [15]

limt→0

f sub(atp)tp−1

= 0. (3.10)

In a sense, the standard presentation of the Poincaremethod corresponds to the all-terms-dominant de-composition presently (the only possible one whenthe field is homogeneous). We can further make thissystem a complete asymptotic system valid on S2

+ byintroducing the sphere vector field (PS2

+, QS2

+)(X,Y,

Z) defined by the forms

PS2+(X,Y,Z) = ZMP (0)

(X

Z,Y

Z

), (3.11)

QS2+(X,Y,Z) = ZMQ(0)

(X

Z,Y

Z

). (3.12)

Then the above intermediate system becomes valid onS2

+ without singularities, namely,

COMPLETE ASYMPTOTIC SYSTEM ON S2+:

X ′ = (1 − X2)PS2+− XY QS2

+

Y ′ = −XY PS2+

+ (1 − Y 2)QS2+

Z ′ = −Z(XPS2

++ Y QS2

+

). (3.13)


244 COTSAKIS

We note that in this system any additive termsproportional to Z are asymptotically negligible at ap-proach to the singularity, and so this system asymp-totically becomes a system valid on the (X,Y )-planealong the circle of infinity, S1 : X2 + Y 2 = 1, that is,we have

ASYMPTOTIC SYSTEM ON THE CIRCLE OF IN-FINITY:

X ′ = −Y (XQS1 − Y PS1) (3.14)

Y ′ = X(XQS1 − Y PS1),

on the circle X2 + Y 2 = 1. (3.15)

We have introduced here the circle vector field(PS1 , QS1)(X,Y ) given by

PS1(X,Y ) = PS2+(X,Y, 0), (3.16)

QS1(X,Y ) = QS2+(X,Y, 0). (3.17)

Some remarks about the system (3.14), (3.15)defined on the Poincare circle of infinity are in order.First, through the series of transforms defined abovefor any weight-homogeneous vector field f with afinite-time singularity, we have found that infinity inthe original variables x, y is now an invariant manifold(circle), with the dynamics near infinity determined bythe constrained 2-dimensional system (3.14)–(3.15),effectively 1-dimensional, on the equator. Secondly,for any given system of the form (3.3), there cor-respond precisely the same number of systems ofthe form (3.14)–(3.15) as the number of admissi-ble asymptotic decompositions of the original vectorfield (3.3). We note that an asymptotic splittingis admissible provided we come up eventually witha valid asymptotic solution in the form of a formalseries. In particular, we have to find the Kowalevskayaexponents of the various dominant balances of thegiven decomposition and check that each of themleads to a consistent asymptotic scheme as in [15].

The equilibria of (3.14), (3.15) correspond to“equilibria at infinity” of the flow of (3) on thePoincare sphere S2

+, and they are precisely the pointswhere

XQS1 − Y PS1 = 0 (3.18)

X2 + Y 2 = 1, (3.19)

where (PS1 , QS1)(X,Y ) is the circle vector field in-troduced above. Therefore the study of the flow of theoriginal system (3.3) near its finite-time singularitiesis equivalent to that of the behavior of orbits of theasymptotic system (3.14), (3.15) near its equilibriagiven by (3.18), (3.19). Because of the circle condi-tion (3.19), these equilibria are of the form (X,Y, 0)and come in as antipodal pairs distributed along thecircle of infinity, hence it is only necessary to study

the flow along equilibria having X > 0, and secondly,those having Y > 0 (in particular, they cannot bothbe zero). The flow will be topologically equivalent(with direction reversed) at the antipodal points if M ,defined in (3.7), is odd (even) as in the homogeneouscase, cf. [17, 19].

To study the flow near such equilibria, we followa “fan-out” technique, described in [17, 19] for allterms in the dominant case. Suppose we have anequilibrium, a solution of (3.18), (3.19) with Y > 0.We define the fan-out map

ξ =X

Y, ζ =

Z

Y(3.20)

which spreads any small neighborhood of that equi-librium on the circle back out onto the tangentplane to the Poincare sphere at Y = +1. Then thegeometric dynamics determined by the asymptoticsystem (3.14)–(3.15) near this equilibrium is trans-formed into a dynamics described by the fan-outsystem near its corresponding equilibrium in termsof the new variables ξ, ζ :

FAN-OUT SYSTEM FOR Y > 0:

ξ = ζMPS1

(ξ

ζ,1ζ

)− ξζMQS1

(ξ

ζ,1ζ

), (3.21)

ζ = ζM+1QS1

(ξ

ζ,1ζ

). (3.22)

This is valid in a neighborhood of the point (0,+1, 0).Similarly, there is a completely analogous fan-outreduction for any equilibrium having X > 0. The finalresult is

FAN-OUT SYSTEM FOR X > 0:

η = λMQS1

(1λ

,η

λ

)− ηλMPS1

(1λ

,η

λ

), (3.23)

λ = −λM+1QS1

(1λ

,η

λ

), (3.24)

where the fan-out map in this case is η = Y/X, λ =Z/X. Note that in both fan-out systems we useasymptotic decompositions of the field.

We conclude that the dynamics of the originalsystem near its singularities is determined by thenature of the equilibria of the corresponding fan-outsystems given above, and these can be studied usingstandard dynamical systems methods. In particular,if these turn out to be hyperbolic, then the situationwill result in a relatively easy problem, whereas whenwe have non-hyperbolic equilibria, the dynamics inphase space “at infinity” will be more involved, andthe resulting global phase portraits will look morecomplicated.



4. DISCUSSIONWe have described a combination of the tech-

niques of asymptotic splittings and central extensionsand provided an asymptotic form of the Poincarecompactification suitable for analysis of finite-timesingularities. This applies to weight-homogeneousvector fields and their decompositions near a timewhen (some component of) the solution blows up.This description will be useful in treating problems inmathematical cosmology and other fields where suchsituations naturally arise.

Although we have described the asymptotics of thePoincare central extension method in dimensions 1and 2, there is nothing to prevent these resultsfrom being valid in any finite dimension. The 3-dimensional case of a homogeneous vector field witha singularity as t → ∞ is described in [17], and itis straightforward to formulate the results of thepresent paper in this context. In practice, however,we do not expect to deal with problems in dimensionshigher than four, except in rare circumstances. Foran abstract formulation in general dimensions onmanifolds, we refer to [24].

The study of infinity in the context of the presentpaper opens the way for related structural stabil-ity studies and applications to mathematical cos-mology where such issues have especially importantphysical and geometric interpretations, for instance,a characterization of instabilities through the pos-sible nonhyperbolicity of equilibria at infinity. Al-though, Peixoto’s theorem for structurally stable vec-tor fields concerns compact manifolds, there is asuitable extension of structural stability to the non-compact case (for a modern introduction to theseresults, see [17]).

Such results are of great importance in the presentcontext, for although having the field blowing up onlyas t → ∞ leads possibly to compact phase spaces,and so stability may be natural for such vector fieldswhen they are further compactified on the sphere, thepresence of finite-time singularities and the relatedpossibility for orbits to escape to infinity in a finitetime governed only by the dominant part of the vectorfield, opens real possibilities for the influence of non-compact phase spaces. This is eventually related tothe possible validity of the index theorem under theprecise conditions of the problem analyzed here, andit is perhaps interesting that known results in thisdirection, see [25], are not directly relevant to theissues considered in this paper.

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asymptotic poincaré compactification and finite-time singularities

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