Complete integrable systems with unconfined singularities
Complete integrable systems with unconfined singularities
V́ıctor Mañosa∗
Departament de Matemàtica Aplicada III,
Control, Dynamics and Applications Group (CoDALab)
Universitat Politècnica de Catalunya
Colom 1, 08222 Terrassa, Spain
victor.manosa@upc.edu
April 10th, 2007
Abstract
We prove that any globally periodic rational discrete system in Kk (where K denotes
either R or C,) has unconfined singularities, zero algebraic entropy and it is complete
integrable (that is, it has as many functionally independent first integrals as the dimen-
sion of the phase space). In fact, for some of these systems the unconfined singularities
are the key to obtain first integrals using the Darboux-type method of integrability.
PACS numbers 02.30.Ik, 02.30.Ks, 05.45.-a, 02.90.+p, 45.05.+x.
Keywords: Singularity confinement, first integrals, globally periodic discrete systems, com-
plete integrable discrete systems, discrete Darboux–type integrability method.
Singularity confinement property in integrable discrete systems was first observed by
Grammaticos, Ramani and Papageorgiou in [1], when studying the propagation of singu-
larities in the lattice KdV equation xi+1j = x
j+1+1/x
j − 1/x
j+1, an soon it was adopted as
a detector of integrability, and a discrete analogous to the Painlevé property (see [2, 3] and
references therein). It is well known that some celebrated discrete dynamical systems (DDS
from now on) like the McMillan mapping and all the discrete Painlevé equations satisfy the
singularity confinement property [1, 4]. In [5, p. 152] the authors write: “Thus singularity
confinement appeared as a necessary condition for discrete integrability. However the suf-
ficiency of the criterion was not unambiguously established”. Indeed, numerical chaos has
been detected in maps satisfying the singularity confinement property [6]. So it is common
knowledge that singularity confinement is not a sufficient condition for integrability, and
some complementary conditions, like the algebraic entropy criterion have been proposed to
ensure sufficiency [7, 8].
Corresponding Author: Phone: 00-34-93-727-8254; Fax: 00-34-93-739-8225.
http://arxiv.org/abs/0704.1326v1
On the other hand a DDS can have a first integral and do not satisfy the singularity
confinement property, as shown in the following example given in [9]: Indeed, consider
the DDS generated by the map F (x, y) = (y, y2/x) which has the first integral given by
I(x, y) = y/x. Recall that a first integral for a map F : dom(F ) ∈ Kk → Kk, is a K–valued
function H defined in U , an open subset of dom(F ) ∈ Kk, satisfying H(F (x)) = H(x) for
all x ∈ U .
The above example shows that singularity confinement is not a necessary condition for
integrability if “integrability” means the existence of a first integral. The first objective
of this letter is to point out that more strong examples can be constructed if there are
considerd globally periodic analytic maps. A map F : U ⊆ Kk → U is globally p–periodic if
F p ≡ Id in U . Global periodicity is a current issue of research see a large list of references
in [10, 11].
Indeed there exist globally periodic maps with unconfined singularities, since global
periodicity forces the singularity to emerge after a complete period. However from [10,
Th.7] it is know that an analytic and injective map F : U ⊆ Kk → U is globally periodic if
and only if it is complete integrable, that is, there exist k functionally independent analytic
first in U). Note that there is a difference between the definition of complete integrable DDS
and the definition of complete integrable continuous DS: For the later case the number of
functionally independent first integrals has to be just k− 1, which is the maximum possible
number; see [13]. This is because the foliation induced by the k−1 functionally independent
first integrals generically have dimension 1 (so this fully determines the orbits of the flow).
Hence, to fully determine the orbits of a DDS, the foliation induced by the first integrals
must have dimension 0, i.e. it has to be reduced to a set of points, so we need an extra first
integral.
In this letter we only want to remark that there exist complete integrable rational
maps with unconfined singularities and zero algebraic entropy (Proposition 1), and that
these unconfined singularities and its pre–images (the forbidden set) in fact play a role in
the construction of first integrals (Proposition 2) for some globally periodic rational maps.
Prior to state this result we recall some definitions. In the following F will denote a rational
Given F : U ⊆ Kk → U , with F = (F1, . . . , Fk), a rational map, denote by
S(F ) = {x ∈ Kk such that den(Fi) = 0 for some i ∈ {1, . . . , k}},
the singular set of F . A singularity for the discrete system xn+1 = F (xn) is a point
x∗ ∈ S(F ). The set
Λ(F ) = {x ∈ Kk such that there exists n = n(x) ≥ 1 for which Fn(x) ∈ S(x)},
is called the forbidden set of F , and it is conformed by the set of the preimages of the
singular set. If F is globally periodic, then it is bijective on the good set of F , that is
G = Kk \ Λ(F ) (see [11] for instance). Moreover G is an open full measured set ([12]).
A singularity is said to be confined if there exists n0 = n0(x∗) ∈ N such that lim
Fn0(x)
exists and does not belong to Λ(F ). This last conditions is sometimes skipped in the
literature, but if it is not included the “confined” singularity could re–emerge after some
steps, thus really being unconfined.
Rational maps on Kk extend to homogeneous polynomial maps on KP k, acting on
homogeneous coordinates. For instance, the Lyness’ Map F (x, y) = (y, (a + y)/x), associ-
ated to celebrated Lyness’ difference equation xn+2 = (a + xn+1)/xn, extends to KP
Fp[x, y, z] = [xy, az
2 + yz, xz]. Let dn denote the degree of the n–th iterate of the extended
map once all common factors have been removed. According to [7], the algebraic entropy
of F is defined by E(F ) = lim
log (dn)/n.
The first result of the paper is
Proposition 1. Let F : G ⊆ Kk → G be a globally p–periodic periodic rational map. Then
the following statements hold.
(a) F has k functionally independent rational first integrals (complete integrability).
(b) All the singularities are unconfined.
(c) The algebraic entropy of F is zero.
Proof. Statement (a) is a direct consequence of [10, Th.7] whose proof indicates how to
construct k rational first integrals using symmetric polynomials as generating functions.
(b) Let x∗ ∈ S(F ), be a confined singularity of F (that is, there exists n0 ∈ N such that
x{n0,∗} := limx→x∗ F
n0(x∗) exists and x{n0,∗} /∈ Λ(F )). Consider ǫ ≃ 0 ∈ K
k, such that
x∗ + ǫ /∈ Λ(F ) (so that it’s periodic orbit is well defined). Set x{n0,∗,ǫ} := F
n0(x∗ + ǫ). The
global periodicity in Kk \Λ(F ) implies that there exists l ∈ N such that F lp−n0(x{n0,∗,ǫ}) =
F lp(x∗ + ǫ) = x∗ + ǫ, hence
F lp−n0(x{n0,∗,ǫ}) = lim
x∗ + ǫ = x∗.
But on the other hand
F lp−n0(x{n0,∗,ǫ}) = F
lp−n0(x{n0,∗}).
Therefore x{n0,∗} ∈ Λ(F ), which is a contradiction.
(c) Let F̄ denote the extension of F to KP k. F̄ is p–periodic except on the set of pre–
images of [0, . . . , 0] (which is not a point of KP k), hence dn+p = dn for all n ∈ N (where
dn denote the degree of the n–th iterate once all factors have been removed). Therefore
E(F ) = limn→∞ log (dn)/n = 0.
As an example, consider for instance the globally 5–periodic map F (x, y) = (y, (1+y)/x),
associated to the Lyness’ difference equation xn+2 = (1 + xn+1)/xn, which is posses the
unconfined singularity pattern {0, 1∞,∞, 1}. Indeed, consider an initial condition x0 =
(ε, y) with |ε| ≪ 1, and y 6= −1, y 6= 0 and 1 + y + ε 6= 0 (that is, close enough to the
singularity, but neither in the S(F ) nor in Λ(F )). Then x1 = F (x0) = (y, (1 + y)/ε),
x2 = F (x1) = ((1 + y)/ε, (1 + y + ε)/(εy)), x3 = F (x2) = ((1 + y + ε)/(εy), (1 + ε)/y),
and x4 = F (x3) = ((1 + ε)/y, ε), and finally x5 = F (x4) = x0. Therefore the singularity is
unconfined since it propagates indefinitely.
But the Lyness’ equation is complete integrable since it has the following two functionally
independent first integrals [10]:
H(x, y) =
xy4 + p3(x)y
3 + p2(x)y
2 + p1(x)y + p0(x)
I(x, y) =
(1 + x)(1 + y)(1 + x+ y)
Where p0(x) = x
3 + 2x2 + x, p1(x) = x
4 + 2x3 + 3x2 + 3x + 1, p2(x) = x
3 + 5x2 + 3x + 2,
p3(x) = x
3+x2+2x+1. The extension of F to CP 2 is given by F̄ [x, y, z] = [xy, z(y+z), xz],
which is again 5–periodic, hence dn = dn+5 for all n ∈ N, and the algebraic entropy is
E(F ) = limn→∞ log (dn)/n = 0.
More examples of systems with complete integrability, zero algebraic entropy and un-
confined singularities, together with the complete set of first integrals can be found in [10].
The second objective of this letter is to notice that the unconfined singularities can even
play an essential role in order to construct a Darbouxian–type first integral of some DDS,
since they can help to obtain a closed set of functions for their associated maps. This is the
case of some rational globally periodic difference equations, for instance the ones given by
xn+2 =
1 + xn+1
, xn+3 =
1 + xn+1 + xn+2
, and xn+3 =
−1 + xn+1 − xn+2
, (1)
To show this role we apply the Darboux–type method of integrability for DDS (developed
in [14] and [15, Appendix]) to find first integrals for maps.
Set F : G ⊆ Kk → Kk. Recall that a set of functions R = {Ri}i∈{1,...,m} is closed under
F if for all i ∈ {1, . . . ,m}, there exist functions Ki and constants αi,j, such that
Ri(F ) = Ki
j 6= 1. Each function Ki is called the cofactor of Ri. Very briefly, the method
works as follows: If there exist a closed set of functions for F , say R = {Ri}i∈{1,...,m}, it can
be tested if the function H(x) =
i (x) gives a first integral for some values βi, just
imposing H(F ) = H.
In this letter, we will use the unconfined singularities of the maps associated to equations
in (1) and its pre–images to generate closed set of functions.
Proposition 2. Condider the maps F1(x, y) =(y, (1 + y)/x), F2(x, y, z) =(y, z, (1 + y + z)/x),
and F3(x, y, z) =(y, z, (−1 + y − z)/x) associated to equations in (1) respectively. The fol-
lowing statements hold:
(i) The globally 5–periodic map F1 has the closed set of functions R1 = {x, y, 1 + y, 1+ x+
y, 1 + x}, which describe Λ(F1), and generates the first integral
I1(x, y) =
(1 + x)(1 + y)(1 + x+ y)
(ii) The globally 8–periodic map F2 has the closed set of functions R2 = {x, y, z, 1 + y +
z, 1 + x+ y + z + xz, 1 + x+ y}, which describe Λ(F2), and generates the first integral
I2(x, y, z) =
(1 + y + z)(1 + x+ y)(1 + x+ y + z + xz)
(iii) The map F3 has the closed set of functions R3 = {x, y, z,−1 + y − z, 1 − x − y + z +
xz,−1 + x− z − xy − xz + y2 − yz, 1− x+ y + z + xz,−1 + x− y}, which describe Λ(F3),
and generates the first integral
I3(x, y, z) = (−1 + y − z)(1 − x− y + z + xz)(1− x+ y + z + xz)(−1 + x− z − xy − xz +
y2 − yz)(x− y − 1)/(x2y2z2).
Proof. We only proof statement (ii) since statements (i) and (iii) can be obtained in the
same way. Indeed, observe that {x = 0} is the singular set of F2. We start the process of
characterizing the pre–images of the singular set by setting R1 = x as a “candidate” to be
a factor of a possible first integral. R1(F2) = y, so {y = 0} is a pre–image of the singular
set {R1 = 0}. Set R2 = y, then R2(F2) = z in this way we can keep track of the candidates
to be factors of I4. In summary:
R1 := x ⇒ R1(F2) = y,
R2 := y ⇒ R2(F2) = z,
R3 := z ⇒ R3(F2) = (1 + y + z)/x = (1 + y + z)/R1,
R4 := 1 + y + z ⇒ R4(F2) = (1 + x+ y + z + xz)/x = (1 + x+ y + z + xz)/R1,
R5 := 1 + x+ y + z + xz ⇒ R5(F2) = (1 + y + z)(1 + x+ y)/x = R4(1 + x+ y)/R1,
R6 := 1 + x+ y ⇒ R6(F2) = 1 + y + z = R4.
From this computations we can observe that R2 = {Ri}i=1,...,6 is a closed set under F2.
Hence a natural candidate to be a first integral is
I(x, y) = xαyβzδ(1 + y + z)γ(1 + x+ y)σ(1 + x+ y + z + xz)τ
Imposing I(F2) = I, we get that I is a first integral if α = −τ , β = −τ , δ = −τ , γ = τ , and
σ = τ . Taking τ = 1, we obtain I2.
A complete set of first integrals for the above maps can be found in [10].
As a corollary of both the method and Proposition 2 we re–obtain the recently discovered
second first integral of the third–order Lyness’ equations (also named Todd’s equation). This
“second” invariant was already obtained independently in [10] and [16], with other methods.
The knowledge of this second first integral has allowed some progress in the study of the
dynamics of the third order Lyness’ equation [17].
Proposition 3. The set of functions R = {x, y, z, 1+ y+ z, 1+x+ y, a+x+ y+ z+xz} is
closed under the map Fa(x, y, z) = (y, z, (a + y + z)/x) with a ∈ R, which is associated to
the third order Lyness’ equation xn+3 = (a+ xn+1 + xn+2)/xn. And gives the first integral
Ha(x, y, z) =
(1 + y + z)(1 + x+ y)(a+ x+ y + z + xz)
Proof. Taking into account that from Proposition 2 (ii) when a = 1, I2 is a first integral for
F{a=1}(x, y, z), it seem that a natural candidate to be a first integrals could be
Hα,β,γ(x, y, z) =
(α+ y + z)(β + x+ y)(γ + x+ y + z + xz)
for some constants α, β and γ. Observe that
R1 := x ⇒ R1(Fa) = y,
R2 := y ⇒ R2(Fa) = z,
R3 := z ⇒ R3(Fa) = (a+ y + z)/x = K3/R1, where K3 = a+ y + z,
at this point we stop the pursuit of the pre–images of the singularities because they grow
indefinitely, and this way doesn’t seem to be a good way to obtain a family of functions
closed under Fa. But we can keep track of the rest of factors in Hα,β,γ.
R4 := α+ y + z ⇒ R4(Fa) = (a+ αx+ y + z + xz)/x,
R5 := β + x+ y ⇒ R5(Fa) = β + y + z,
R6 := γ + x+ y + z + xz ⇒ R6(Fa) = (γ + y + z)x+ (a+ y + z)(1 + y)/x.
Observe that if we take α = 1, β = 1, and γ = a, we obtain R4(Fa) = R6/R1, R5(Fa) = R4
and R6(Fa) = K3(R5/R1). Therefore {Ri}i=1,...,6 is closed under Fa, furthermore Ha =
(R4R5R6)/(R1R2R3) is such that Ha(Fa) = Ha
In conclusion, singularity confinement is a feature which is present in many integrable
discrete systems but the existence of complete integrable discrete systems with unconfined
singularities evidences that is not a necessary condition for integrability (at least when
“integrability” means existence of at least an invariant of motion, a first integral). However
it is true that globally periodic systems are themselves “singular” in the sense that they are
sparse, typically non–generic when significant classes of DDS (like the rational ones) are
considered.
Thus, the large number of integrable examples satisfying the the singularity confinement
property together with the result in [9, p.1207] (where an extended, an not usual, notion
of the singularity confinement property must be introduced in order to avoid the periodic
singularity propagation phenomenon reported in this letter -see the definition of periodic
singularities in p. 1204-) evidences that singularity confinement still can be considered
as a good heuristic indicator of “integrability” and that perhaps there exists an interesting
geometric interpretation linking both properties. However, although some alternative direc-
tions have been started (see [18] for instance), still a lot of research must to be done in order
to understand the role of singularities of discrete systems, their structure and properties in
relation with the integrability issues.
Acknowledgements. The author is partially supported by CICYT through grant DPI2005-
08-668-C03-01. CoDALab group is partially supported by the Government of Catalonia
through the SGR program. The author express, as always, his deep gratitude to A. Cima
and A. Gasull for their friendship, kind criticism, and always good advice.
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