13
Numerical Algorithms � ������ ��� �
On the resolution of singularities of ordinary
di�erential systems
Jukka Tuomela
Institute of Mathematics� Helsinki University of Technology� PL ����� ����� TKK� Finland�
E�mail jukkatuomela�hut�
We show how some di erential geometric ideas help to resolve some singularities of
ordinary di erential systems Hence a singular problem is replaced by a regular one� which
facilitates further analysis of the system The methods employed are constructive and the
regularized systems can also be used for numerical computations
Keywords� singularities� di erential equations� jet spaces
AMS Subject classi�cation� Primary ��F��� ��A��� secondary ��C��
�� Introduction
We continue the investigation of �apparent� singularities of ordinary di�erential
systems which was started in ����� By a di�erential system we mean any system of ODEs
or DAEs� of any order� This large class of systems can be handled in a uniform way
using jet spaces� In particular there is no di�erence between ODEs and DAEs� so the
term DAE is superuous in jet space context� This is perhaps a little surprising since
in the traditional approach to DAEs the di�erence between ODEs and DAEs is always
stressed� see for example ��� Intuitively one might explain this as follows� the traditional
distinction of ODEs and DAEs has more to do with certain representations of objects
rather than objects themselves�
What are these objects then� Usually one thinks of a di�erential equation or system
as a manifold with some vector eld and the solutions as integral curves of this vector
eld� This is inconvenient or too restrictive for several reasons� First transforming a
given system to a rst order system destroys the natural structure of the problem� recall
that highest order derivatives determine the nature of the problem� There is also the
arti cial distinction of autonomous and nonautonomous systems� More important from
the point of view of the present article is the fact that singularities� i�e� zeros� of vector
elds cannot model certain singularities which arise even in very simple systems�
The jet theory provides a richer framework than the use of the tangent bundle�
� J� Tuomela � Singularities of di�erential systems
Di�erential systems are regarded as submanifolds of jet bundles� hence we have quite
many geometric tools available to study them� For example one may inquire if these
systems are singular as manifolds� and if these singularities are generic� The hierarchy
of jet bundles of di�erent order allows a geometric interpretation of di�erentiation �pro�
longation� and elimination �projection� which is quite important as the examples below
demonstrate�
To discuss the solutions it is necessary to use two distinct points of view� one may
consider solutions as integral manifolds of certain distributions or certain sections of some
bundle� Perhaps this sounds a bit confusing� but in fact both notions are important and
useful in their ways� The fundamental objects to be studied are thus de ned geometri�
cally� in particular they are de ned without any reference to any equations� Of course in
actual computations the manipulation of equations is necessary and it is seen that the
tools from ideal theory are very convenient in this task� Also it is quite interesting that
there is a close connection between geometric and algebraic concepts�
Below we will see that the use of jet bundles allows us to regularise some singular
systems� i�e� we can replace the original singular system by a regular one� The examples
are chosen to represent various singular situations where the numerical computation of
the solutions by standard methods would be di�cult or impossible� In each case we
show how to use jets to obtain a system which is convenient for numerical purposes�
The discussion of actual numerical methods� however� is outside the scope of the present
article� see ���� and �����
In spite of the simplicity of the examples� the methods employed can be used to
analyse general systems� in sections ��� and ��� all steps are entirely constructive and no
special property of the initial system is used� Section ��� is a bit di�erent in character
since there the pullback map is explicitly needed� However� the point is that since
the system is imbedded in a �big� jet bundle� this gives more freedom in choosing the
appropriate transformation than the traditional setting allows� Also the nal example is
directly related to problems occurring in applications�
Hence the scope of the tools used is quite large� but its extent is a bit di�cult
to characterize� This is because the word singular�ity� covers such a wide variety of
phenomena and so it does not make much sense to speak about singularities �in general��
Anyway� we think that the geometric point of view of di�erential equations is quite
useful in practice� not only in the resolution of singularities� but also in the analysis and
numerical analysis of di�erential systems�
�� Basic de�nitions
We recall here briey some notions that are needed below� For more information
we refer to ���� �standard di�erential geometry� and ��� �jet spaces�� We will also need
J� Tuomela � Singularities of di�erential systems �
some standard algebraic terminology for which we refer to ��� �ordinary commutative
algebra� and ����� ����� ���� �di�erential algebra��
We will use the convention that the components of the vector are indicated by su�
perscripts and the derivatives �or jet coordinates� by subscripts� All maps and manifolds
are assumed to be smooth� i�e� in nitely di�erentiable� The di�erential �or Jacobian� of
a map f is denoted by df � LetM be a manifold� The tangent �resp� cotangent� space of
M at p �M is denoted by TMp �resp� T �Mp� and the tangent �resp� cotangent� bundle
of M by TM �resp� T �M�� Let M and N be manifolds and let � be a section of T �N
�i�e� one�form�� Given a map f � M �� N we can de ne a section of T �M by
f���Vp� � ��df�Vp��
where Vp � TMp and f�� is called the pull�back of �� Let E � R � Rn and let us
denote the q�th order jet bundle of E by Jq�E�� The coordinates of Jq�E� are denoted by�x� y� y�� � � � � yq�� Let us de ne the one forms
�ij � dyij�� � yijdx i � �� � � � � n j � �� � � � � q ���
Each �ij is then a section of T�Jq�E�� Let us further de ne
Cp ��vp � �TJq�E��p
���ij�vp� � ��Hence C is a n � � dimensional distribution on Jq�E�� called the Cartan distribution��Now let us consider the di�erential system
f�x� y� y�� � � � � yq� � � ���
where f � Jq�E� � R�q���n�� �� Rk� Let Rq � f����� and let us de ne a distribution Don Rq by
Dp � �TRq�p �Cp ���
Now supposing that the system is involutive and that D is one�dimensional we can de nethe solutions as follows�
De�nition �� Solutions of the involutive system ��� are integral manifolds of D�
Since one�dimensional distributions always have integral manifolds� we conclude that
there always exist solutions to our problem�
Note that it is absolutely essential that the system is involutive� otherwise it is
possible that the distributionD is one�dimensional� although �some of� the correspondingintegral manifolds are not really solutions in any reasonable sense� Since the involutivity
is not explicitly needed in the sequel we refer to ����� ���� and ��� for a thorough discussion
� In the special case n � q � �� C is also called the contact distribution
� J� Tuomela � Singularities of di�erential systems
of this important concept and to ���� for a more accessible introduction� Some elementary
examples of applications of jets to di�erential equations can be found in ����
�� Resolution of singularities
Basically we consider a di�erential system �or equation� as a submanifold of a jet
bundle� but in general any manifold M with one�dimensional distribution D on it� may
be interpreted as a �system of� di�erential equation�s�� The word singular�ity� can be
used at least in the following senses�
�� M is not everywhere smooth manifold�
�� dim�Dp� � � at some points of M �
�� Solution curves �or their projections� are not everywhere smooth�
� Classical distinction of general versus singular solution�
All these aspects appear in some ways in the examples that follow� For an extensive
discussion of the last problem we refer to a recent thesis ����� The set of points where
dim�Dp� � � is called the singular set of the distribution� although around regular points
distributions and vector elds are essentially equivalent� the singular sets are more general
that the singularities� i�e� zeros� of vector elds�
We will show below that using jets it is sometimes possible to resolve the singularity
in the sense that the original problem can be replaced by a regular one whose solutions
correspond to the solutions of the original problem�
���� Regularisation by introducing jets
Let us consider the following problem
f�x� y� y�� � x�y��� � �yy� � x � �
From the traditional point of view there is a singularity at x � �� However� putting
E � R � R and R� � f����� � J��E� we see that R� is everywhere smooth� The
distribution D whose integral manifolds are solutions is given by the nullspace of
A �
��y��� � � ��y� �xy� � �y�y� � �
�
Obviously the rank of A is two on R�� hence D is everywhere one�dimensional and the
problem is regular� So in this case simply introducing the jet spaces made the �apparent�
singularity disappear� Note that R� n fy��axisg has two components and projectingthe distribution from one of them to E gives an elementary example of the fact that
J� Tuomela � Singularities of di�erential systems �
singularities of distributions are more general than singularities of vector elds� Indeed�
there is no vector eld around origin which would span the projected distribution�
The integral manifolds of the projected distribution �a family of parabolas� make
a similar pattern than the lines of curvature around an umbilic on an ellipsoid� see ����
vol� �� p� ����� Hence in a simple problem of classical di�erential geometry there arise a
situation which cannot be handled using only tangent bundles and vector elds�
Now the situation in the example is absolutely typical� Let Rq � Jq�E� be involu�tive� let the distribution D on it be de ned by ��� and let Sa be the set of points of Rq
where dim�Dp� � a�
Theorem �� For a generic problem Sa is a smooth manifold of codimension a� � a �or
empty��
Proof� Let m be the dimension of Rq� In convenient coordinates D can be represented
as the nullspace of some �m����mmatrix� The result then follows from the discussion
in ��� p� ���� �
This result has some interesting consequences which are important from practical point of
view� First note that the typical solution does not meet the singular set of the distribution
because this set is of codimension � while classically the solution usually meets the
singular set since in the classical setting the singular set is of codimension �� The above
example illustrates this situation�
The second interesting� and perhaps surprising� fact is that the singular set can
be �large� in a generic situation while it is well�known that a generic vector eld admits
singularities� i�e� zeros� only at isolated points� For example the singular set of the
following equation is a smooth curve�
R� � �y��� � �y��
� � y� � x� � � � �
By the theorem� this property is stable with respect to perturbations�
���� Regularisation by prolongation
Consider again the general system ��� and let Rq � f����� � Jq�E��
De�nition �� The prolongation of f � denoted by ���f�� is a system obtained by putting
together f and its total derivatives� The zero set of ���f� is the prolongation of Rq �
denoted by ���Rq� � Jq���E��
� J� Tuomela � Singularities of di�erential systems
Note that in general the prolongation can be de ned purely geometrically without any
reference to equations� see ���� and ����� However� the above de nition is convenient in
the context of the present paper�
Let us start with a simple problem �Clairaut�s equation�
f�x� y� y�� � �y��� � xy� � y � �
whose solutions are y � cx� c� ��general solution�� and y � �x�� ��singular solution���Again let E � R � R and R� � f����� � J��E�� Obviously R� is smooth� but the
distribution is not one�dimensional at �y��x � �� By the theorem of the previous section
we conclude that the codimension is �wrong� and hence the problem is not generic� Indeed�
considering the equation �y��� � xy� � ay � �� it is easily checked that the codimension
of the singular set is two �for a �� ��� except for the value a � ��� Hence it depends onthe intended application whether it is more appropriate to perturb the coe�cients to get
a generic problem or to insist that a is indeed exactly equal to ���Let us take this latter point of view� Now taking the total derivative of f we obtain
��y� � x�y� and the problem splits into two subsystems
Q� �
��y��� � xy� � y � �
y� � �Q� �
��y��� � xy� � y � �
�y� � x � ���
It is immediately veri ed that both systems are smooth as manifolds and that the codi�
mensions of the singular sets are �right�� The distribution can in the former case readily
be projected back to R�� so we get a regular problem on R��
Again this example is typical� Now f can be considered as a di�erential polynomial
and the set of solutions of R� is characterized by the radical di�erential ideal generated
by f � According to the basic theorem of Ritt� any radical di�erential ideal is a nite
intersection of prime di�erential ideals� In the above example Q� and Q� are the prime
components of the original system� The important point is that we can in practice
entirely work with algebraic ideals and consequently use Gr�obner basis techniques� This
is explained in ���� p� ��� to which we refer for further details� but the idea is as
follows� First we check if the given system is prime� if not we factor it and proceed with
each subsystem separately� Then we check the involutivity and see if by di�erentiating
and eliminating any new equations can be obtained� If there are new equations� the
system may not be prime anymore� so we have to factor again� take subsystems and
di�erentiate� Hence we obtain a tree structure where the original system is at the root
and prime�involutive components are at the leaves�
Of course the actual implementation of the algorithm is quite complicated� and
in practice it may well be best to do each of the subproblems separately� and try by
inspection to simplify the task� For example one may be interested only in one branch
of the tree� so it would be a waste of time to compute the whole tree�
J� Tuomela � Singularities of di�erential systems �
Going back to the Clairaut�s equation� one could say that from the point of view
of numerical anaslysis the original problem was ill�posed� because it was not generic�
However� the resulting systems Q� and Q� are numerically stable� Hence to treat nu�
merically non�generic situation it may be necessary rst to use symbolic computation to
produce a well�posed problem before actual numerical treatment of the problem� Why
should one be interested in a non�generic problem� One reason is that in some cases it
is the �singular solution� which is really the solution which one wants to compute� see ���
p� ���� for an example� In the present example if a �� ��� no solution of the perturbedsystem is close to the singular solution of the unperturbed system�
Let us then consider another example where prolongation resolves the singularity�
f�x� y� y�� � �y��� � �y� � y� � y� � � � � ���
This de nes a curve in the �y� y���plane
with a double point at p � ��� ��� We
denote this curve by K and hence R� �
f����� � R�K� The set of singular points�i�e� the points at which df vanishes� is
S � R�fpg� Now the geometry ofR� sug�
gests that through each point of S there
could pass two smooth solutions� see the
picture on the right�
00.5
1
1.5
2
x
-1
-0.5
0
y
0.5
0.75
1
1.25
1.5
y1
00.5
1
1.5
2
x
0.
0.
1
1
1
In E this would mean that at each point of R� f�g� there are two solutions whichmeet tangentially� Now if two curves intersect tangentially� one expects that in general
their second derivatives would not coincide �note that R� intersects itself transversally�
and hence we would obtain a regular situation if the information on second derivatives
were also available�
To this end we prolong f and get the system
���f� �
��y��� � �y� � y� � y� � � � �
�y�y� � �y� � �y�y� � �yy� � �
� J� Tuomela � Singularities of di�erential systems
Let us denote the zero set of these
equations in �y� y�� y���space by ���K� so���R�� � R � ���K� � J��E�� Let � �
���K� �� K be the projection induced by
the projection �y� y�� y�� �� �y� y��� Now
we would like � to be bijective in a neigh�
borhood of p � K� except that the ber of pshould consist of two points� However� the
ber of p is the entire dotted line ��� �� y��
in the picture� We would like to eliminate
this line somehow�
-1 -0.5 0 0.5y
0 0.5 11.52y1
-1
0
1
y2
-1 -0.5 0 0 5
0 0.5 11.52
Note that the components of ���f� can be interpreted as elements of the ring
Q�y� y� � y��� Let I be the ideal generated by ���f�� We are interested in the solution setsand therefore we computed the prime components of the radical of I� denoted by pI ��This gave
pI � I� � I�� where I� � �y� y� � �� and�
I� ���y��
� � �y� � y� � y� � � � �y�y� � �y� � �y�y� � �yy� ��y�y � ��y� � ��y � ���y� � ��y� � �y � ��y��� � ��y� � ��y � ��y���
�
So like in the previous example� the system decomposed into � subsystems� However� in
this case the subsystem corresponding to I� is not interesting as a di�erential equation�Now I� describes a curve in a three�dimensional space so we would expect that it werea zero set of two equations� instead the ideal obtained has four generators� From the
ideal theoretic point of view we cannot simply drop two generators� the ideal would
change and also the corresponding solution sets� However� we need only the �piece� of
the variety in the neighborhood of the singular set and so we can proceed as follows�
First the fourth generator gives the exceptional ber� ����p� � f��� ����g� Next wecheck that the gradients of the second and fourth generator are linearly independent in
the neighborhood of these points� therefore the second and fourth generator give the
required smooth curves�
It would be nice if this could be done algorithmically� i�e� given a �radical� ideal
and a regular point b of the corresponding variety M � compute a representation of M
in a neighborhood of b as a zero set of as many equations as the codimension of the
manifold� Perhaps the solution to this problem is well�known� however� we did not nd
it in the literature�
� The computations were performed with SINGULAR� see ���� Because I� is known a priori� the same result can be obtained by computing the radical of the saturation
of I with respect to I� � ie I� �pI I�
�
J� Tuomela � Singularities of di�erential systems �
Finally let us note that no point of ���K� projects to ���� �� � K� This is relatedto the fact that if the solution is projected from J��E� to E � then the projected curveis not smooth at �x����� However� this is not problematic in our intended application�since we need to prolong only in a neighborhood S� Hence if we follow numerically a
particular solution� we can do the computations in J��E� until we get too close to S� thenwe pass S in J��E� and then switch back to J��E� again�
Now is the situation described in the previous example generic� In some sense it
is not� because by Sard�s theorem generic inverse images are smooth manifolds� How�
ever� considering R� as an immersion� the situation is seen to be generic since the self�
intersection is transverse� To prove that in this case the prolongation produces an imbed�
ding we must introduce some terminology�
De�nition �� Let Rq be a subbundle of Jq�E�� A section y of E is a solution of Rq � if
its q�jet is a section of Rq �
Of course it is rather unfortunate that we have to introduce another concept of solution�
but this is really necessary� We will also need another kind of prolongation�
De�nition �� Let Rq be a subbundle of Jq�E�� Suppose that q�jets of its solutions
foliate it� The sectional prolongation of Rq � �s��Rq� � Jq���E�� is the subset de ned byq � ��jets of the solutions of Rq �
Theorem �� Let E � R � R and R� � J��E� be an immersion with transverse self�intersections� Suppose that �local� solutions exist outside the singular set and that the
singular set is not itself �an image of� a solution� Then �s��R�� is an imbedding in J��E��
Proof� Let g � R� � R� be an immersion� Let S be the curve of self�intersection and
choose domains i� i � �� �� such that g is a di�eomorphism on i� � � � � and �� g� �� � g� �� � S� Now g� i� are submanifolds of J��E� and they have �local�solutions which foliate g� i�� Then it is straightforward to verify that �s��g� i�� are
disjoint if the intersection is transverse� �
Working in this way geometrically we do not have the problems of the �dotted line� of the
example� However� in general the map g is not known in practice� so in applications one
must work with equations� Intuitively it is quite plausible that in general the prolongation
�improves� the situation� indeed� the dimension of the manifold remains the same in the
prolongation while the dimension of the ambient space increases� Hence one expects that
after su�cient number of prolongations one obtains an imbedding because eventually the
prolonged manifold can intersect itself transversally only by not intersecting at all�
�� J� Tuomela � Singularities of di�erential systems
Anyway� the theorem shows that the situation of the example is typical� and that
the �dotted line� is related only to algebraic representation of the problem� The analogy
of prolongation in this context and blow up in algebraic geometry �see for example
���� is rather obvious� and it is quite striking that such basically trivial operation� i�e�
di�erentiating given equations� gives such powerful results�
���� Regularisation by pull�back
We will nally treat two examples where a convenient pull�back produces a regular
problem� The rst example demonstrates the exibility of the jet spaces� one can con�
sider transformations of jet coordinates separately� allowing more possibilities than the
traditional approach� The solutions are obtained by pulling back the one�forms which
de ne the relevant distribution and the original solutions are recovered with the same
map which de ne the pullback� The second example is directly related to applications�
similar situations arise in systems which have constraint singularities ����
First consider the problem
f�x� y� y�� � y�y��� � � � �
whose solutions are given by y� � �� �x � c��� Let R� � f����� as usual� In this case
introducing jet spaces does not directly allow one to pass the singularity� Indeed following
the solutions in R� one would never reach the singular point� one could compute only
one branch of the solution at a time�
Now it is well�known that this type of singular curves are usually obtained by
projecting a smooth curve on a fold� So the idea is to introduce a new manifold with a
fold in such a way that the original solutions can be recovered from this new setting by
projection�
In this simple case the solution is rather immediate� Consider the map � �
�x� y� z� �� �x� y� ��z� and let M � ����R��� Taking the closure of M � i�e� adding the
x�axis� we get a smooth manifold !M which is evidently the zero set of g�x� y� z� � y�z��
Then taking the pull�back of � � dy � y�dx we get ��� � dy � �zdx� Now recall that
we are not interested in one forms as such� only in distributions de ned by them� Hence
we can multiply forms by non�zero functions and in particular we can replace ��� by
� � zdy � dx� This can be smoothly extended to !M �
J� Tuomela � Singularities of di�erential systems ��
So the original manifold is replaced by!M and the distribution !D on !M is the
nullspace of
A �
�� � ��z�� z �
�
Clearly !D is one�dimensional on !M � so the
problem is regular� and the original solu�
tion curves can be obtained with projec�
tion �x� y� z� �� �x� y�� In the gure on the
right there are some solution curves as well
as their projections�
01
2
3x
0 0.250.50.751
y
-2
-1
0
1
z
01
2
3x
0 0.250.50.751
As our nal example consider the following problem which is taken from ����
y�� � y�� �x� ����ex � � � �y�� � y�� �x� � ���ex � �x � ��y��� � �y��� � �x� ����� � �x� � ���� � �
Of course we could denote by y� and treat it in the same way as other variables� How�
ever� the form of the system resembles the form of mechanical systems with holonomic
constraints and Lagrange multipliers� and in those cases� like in the present one� is not
really needed in computations� Let E � R�R�� since the system is linear in derivatives�
we do not need to work with J��E�� because the distribution in J��E� can be projectedto E � Indeed� di�erentiating the last equation and combining it with the other two weobtain �
B � � y�
� � y�
y� y� �
CA�By��y��
CA �
�B �x� ����ex � ��x� � ���ex � �x�x� � x��� ���
CA ���
Now the distribution is given by the nullspace of
A �
��y�� � �
�y�� � �
�
The linear system ��� can be solved� except on the x�axis� and computing further the
nullspace of A gives a one�dimensional distribution D on E nfx�axisg� In this simple caseone could compute D symbolically� however� this is not necessary� Let us de ne
f�x� y� � �y��� � �y��� � �x� ����� � �x� � ����
and let M � f����� � E � Evidently M is smooth except at the vertex p � ����� �� ���
The distribution D obtained above restricts further to M n fpg�
�� J� Tuomela � Singularities of di�erential systems
Now in ��� it was observed that there is
at least one smooth solution going through
p� To compute this solution numerically it
would be nice to have a smooth problem
whose solutions would give the original so�
lutions� To this end we introduce the map
��x� y� �
�� ���x�
pjyj��
jyj y�� x � ���
�x��
pjyj��
jyj y�� x � ���
and de ne !M � ����M�� In the picture!M is shown above andM below� Evidently
� is bijective� except at the vertex� where
the ber is the unit circle� Note also that �
restricted to !M is continuous� Again this
process is analogous to a blow�up�
Intuitively one may understand the form
of � as follows� rotate !M in di�erent di�
rections for x ��� and x � ��� while
shrinking the exceptional ber to a point�
Then the solution curves of !M andM cor�
respond to each other after a half turn�0 0.250.50.75 1
-0.500.5
-0.5
0
0.5
0 0 250 50 75 1
-0.500.5
0 0.250.50.75 1
-101
-1
0
1
0 0 250 50 75 1
-101
The distribution !D on !M can be de ned by a pull�back as in the previous example or
simply by transporting tangent vectors� It is rather straightforward to work numerically
with !M as a manifold� However� the numerical computation of the distribution near the
exceptional ber may require some care�
In both examples the map � was needed analytically� However� the choice of the
form of the map is not very critical� Also � is required only locally near the singular set�
which further facilitates the choice of the map�
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