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Maoan Han et al. (2010), Scholarpedia, 5(8):9648.
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The authors' remark. In this article we would like to provide a comprehensive review on the study of limit cycles of planar polynomial vector fields, closely related to Hilbert's 16th problem. In particular, we reviewed the progress of research on the quadratic systems, higher-degree polynomial systems, as well as Liénard systems. There are so many researches in this area that it is impossible to give a complete review for the study of limit cycles.
This article is written with the purpose to mainly include contributions of Chinese scholars.
Let \(P_n\) and \(Q_n\) be polynomials in \((x,y)\) satisfying
\(\max\{\deg(P_n),\deg(Q_n)\}=n\ .\) Then the equations \((E_n)\)
\[\frac{dx}{dt}=P_n(x,y),\ \ \frac{dy}{dt}=Q_n(x,y)
define a planar polynomial system which corresponds to a polynomial
vector field on the plane. Any nontrivial periodic solution of the
system determines a closed curve on the phase plane, called a
periodic orbit.
A periodic orbit is called a limit cycle if there is a neighborhood
of it such that it is the \(\alpha-\)limit set or \(\omega-\)limit set
of all points in the neighborhood. Furthermore, the limit cycle is
if it is the
\(\omega-\)limit set of all points in the
neighborhood. It is called unstable if not stable.
Using the solutions near a limit cycle one can define a return map
or Poincar\({\rm \acute{e}}\) map which can be used to define
multiplicity of the limit cycle. A limit cycle is called simple or
hyperbolic if it has multiplicity one.
\frac{dx}{dt}=y,\ \ \frac{dy}{dt}=-x+\mu (1-x^2)y,\ \mu&0
has a unique limit cycle, which is stable and hyperbolic.
At the Second International Congress of Mathematicians, Paris, in
1900, D. Hilbert posed his 23 mathematical problems. The second part
of his 16th problem can be stated as follows (see [53]):
For a given integer \(n\geq 2\ ,\) what is the maximum number of limit
cycles of system \((E_n)\) for all possible \(P_n\) and \(Q_n\ ?\) And how
about the possible relative positions of the limit cycles?
This problem is still open, but it has been proved by \({\rm
\acute{E}}\)calle [27] and Ilyashenko [60] that the number of limit
cycle is finite for a given system \((E_n)\ .\)
Note that the problem is trivial for \(n=1\ :\) a linear system may have
periodic orbits but has no limit cycle. Let \(H(n)\) denote the
maximum number for \(n\geq 2\ ,\) called Hilbert number. Is \(H(n)\)
finite? It is also open ever for \(n=2\ .\)
Now consider a near-Hamiltonian polynomial system of the form
\frac{dx}{dt}=\frac{\partial H_m}{\partial y}+\varepsilon
P_n(x,y),\quad \frac{dy}{dt}=-\frac{\partial H_m}{\partial
x}+\varepsilon Q_n(x,y),
where \(H_m=H_m(x,y)\) is a polynomial in \(x,y\) of degree \(m\) such
that the equations \(H_m(x,y)=h\) define a family of closed curves
\(L_h\) for \(h\) in an interval. We take a segment \(\sigma\ ,\)
transversal to each oval \(L_h\ ,\) and choose the values of the
function \(H\) itself to parameterize \(\sigma\ ,\) and denote by
\(\gamma(h,\varepsilon)\) a piece of the orbit of the perturbed system
between the starting point \(h\) on \(\sigma\) and the next
intersection point \(P(h,\varepsilon)\) with \(\sigma\ .\) The ``next
intersection is possible for sufficiently small \(\varepsilon\ ,\)
since \(\gamma(h,\varepsilon)\) is close to \(L_h\ .\) Then the
displacement function \(F(h,\varepsilon)=P(h,\varepsilon)-h\) can be
expressed as
F(h,\varepsilon)=\varepsilon M(h)+O(\varepsilon^2),
M(h)=\oint_{L_
h}P_ndy-Q_ndx,
which is called the first order Melnikov function.
Weak Hilbert's 16th problem posed by V. I. Arnold () [1,2]
is the following.
For given integers \(m\) and \(n\) find the maximum \(Z(m,n)\) of the
numbers of isolated zeros of the Abelian integrals \(M(h)\) for all
possible \(P_n\ ,\) \(Q_n\) and \(H_m\ .\)
It is easy to see that \(Z(2,n)=[\frac{n-1}{2}]\) since \(M(h)\) is a
polynomial in this case. In 1984 it has been proved \(Z(m,n)&\infty\)
(Khovansky [64], Varchenko [108]). During
it has been
proved that in generic cases \(Z(3,2)=2\) (Horozov and Iliev [55],
Zhang and Li [124], Gavrilov [30] and Li and Zhang [71], and in
degenerate cases \(Z(3,2)=3\) ( [31], [56], [57], [127], [128] and
[11]). Also see the second part of the book by Christopher and
Li(2007)[14].
Any quadratic system with a center or focus can be transformed into
one of the following systems
(I) \(\frac{dx}{dt}=-y+dx+lx^2+mxy+ny^2,\) \(\frac{dy}{dt}=x;\)
(II) \(\frac{dx}{dt}=-y+dx+lx^2+mxy+ny^2,\)
\(\frac{dy}{dt}=x(1+ax),\,\,\,a\neq 0;\)
\(\frac{dx}{dt}=-y+dx+lx^2+mxy+ny^2,\)
\(\frac{dy}{dt}=x(1+ax+by),\,\,\,b\neq 0,\)
which is called Ye's classification, made by Yanqian Ye in
1960s. Many researches were done by using the classification, see
Ye(] and the following.
L. Chen and Y. Ye(1975)[8], X. Yang and Y. Ye(] proved that
any system in class I has at most one limit cycle. Then Cherkas and
Zhilevich [13], [6] and Ryckov [93] proved that any system in class
III with \(a=0\) has at most one limit cycle. Hence, a quadratic
system with a straight line solution has at most one limit cycle.
There are some studies on the non-existence of limit cycles around a
third order focus of quadratic systems between 1976 and 1986. The
conclusion was completely proved in 1986 by Li(1986) [65] and
Cherkas(1986)[12].
Many people have their contribution to the uniqueness of limit
cycles, showing that any quadratic system has at most one limit
cycle around a second order weak focus and it
is hyperbolic if
exists, including S. Cai, Z. Wang, W. Chen, M. Han, P. Zhang, W. Li,
S. Gao, X. Wang, X. Huang and J. Reyn, investigating various
particular cases under different conditions. The final proof for
general quadratic systems was given by Zhang(1999) in [120].
In 1955, Petrovsky and Landis [90] attempted to prove \(H(2)=3\ .\)
Unfortunately, their proof contained errors. Then in 1979, L. Chen
and M. Wang [7] and S. Shi (1980) found respectively examples of
quadratic systems having 4 limit cycles, giving \(H(2)\geq 4\ .\) It is
widely conjectured that \(H(2)=4\ .\)
In 1959 T. Tung [99] found out some important properties of
quadratic systems: a cl there is a unique
singularity i two closed orbits are similarly
(resp. oppositely) oriented if their interiors have (resp. do not
have) common points. Hence, the distribution of limit cycles of
quadratic systems have only one or two nests. Around 2000 P. Zhang
[121,122] proved that in two nests case at least one nest contains a
unique limit cycle. Therefore, (2,2)-distribution of limit cycles
for a quadratic system is impossible.
P. Zhang and S. Cai [123] and D. Zhu [129] respectively proved that
a quadratic system with a second or third order weak saddle point
does not have a limit cycle. P. Zhang [119] further proved that a
quadratic system with a first order saddle point has at most one
limit cycle. The global geometry of quadratic systems is also an
interesting topic, for studies in this aspect see D. Schlomiuk
[94,95,96], for example.
There are many more other results on the non-existence, uniqueness
of limit cycles, existence of two limit cycles and global phase
analysis. See Ye(1986) [112] and Reyn(2007) [91].
In 1952, Bautin [3] studied the
for quadratic
systems and proved that there are at most 3 limit cycles bifurcated
from a weak focus or center of such a system, and 3 limit cycles can
Zhu(1989) [129],
Joyal and Rousseau(1989) [63] and Cai and
Guo(1990) [5] respectively proved that in quadratic systems there
are at most 3 limit cycles bifurcated from an isolated homoclinic
loop. However, it is still open if the number 3 can be achieved.
Horozov and Iliev(1994) [54] proved that in quadratic systems the
maximum number of limit cycles that can appear near a homoclinic
loop of a nondegenerate
is 2 under small
perturbations with a single parameter, where a Hamiltonian system is
called nondegenerate if the perturbed system is
\(M(h)\equiv 0\ .\) From their proof the conclusion is also true under
small perturbation with arbitrary parameters. Then Iliev [56] proved
that the maximum number of limit cycles near a homoclinic loop of a
degenerate Hamiltonian system is 2 under small perturbation with a
single parameter by using up to the fourth order Melnikov functions.
Han, Ye and Zhu [47] further proved that this conclusion is also
true for the case of degenerate Hamiltonian systems under small
perturbation with arbitrary parameters. Thus, in quadratic systems
the maximum number of limit cycles near
a homoclinic loop of any
Hamiltonian system is 2 under arbitrary small perturbations. There
are also some studies on the number of limit cycles near a
homoclinic loop of an integrable non-Hamiltonian system under
quadratic perturbations, see He and Li [52], Han(1997) [32]. The
results obtained positively suggest that 2 is also the maximum
number for the integrable case.
There may appear a polycycle or heteroclinic loop with two or three
elementary saddles in quadratic systems. The results obtained by
Dumortier, Roussarie and Rousseau [24], Zoladek [130] and Han-Yang
[45] show that at most (resp, three) two limit cycles can be
bifurcated from a polycycle with two (resp., three) saddles. Also,
in the both cases, two limit cycles can appear.
It seems very difficult even impossible to find an example of
quadratic systems having three limit cycles near a polycycle with 3
We consider polynomial plane systems with some symmetry. For
convenience, we write the system \(E_n\) into the vector form
\frac{du}{dt}=V(u),
where \(u=(x,y)\) and \(V(u)=(P_n(x,y),Q_n(x,y))\ .\)
Let \(S\ :\) \( {R}^{2}\rightarrow {R}^{2}\)
be an invertible differentiable map. We say that \(S\)
is a symmetry of the above system or the system is \(S-\)
equivariant if
DS(u)\cdot V(u)=V(S(u)),\quad u\in {R}^{2}.
It is easy to see that the system is
\(S-\) equivariant if and only
if it is invariant under the transformation \(S\ .\)
If \(S\) is linear, then the above becomes
SV(u)=V(S(u)),\quad u\in {R}^{2}.
For example, if \(S\) is the reflection:
\begin{array}{cc}
\end{array}\right)=\rm diag (1,-1),
then the \(S-\) equivariance means
P_n(x,y)=P_n(x,-y),\quad Q_n(x,y)=-Q_n(x,-y).
In this case, the flow is symmetric with respect to the \(x\)-axis.
Now let \(S\) be a rotation by angle \(2\pi/q\) for an integer \(q\geq
2\ .\) Then
\begin{array}{cc}
\cos\frac{2\pi}{q}&-\sin\frac{2\pi}{q}\\
\sin\frac{2\pi}{q}&\cos\frac{2\pi}{q}
\end{array}
\right)\equiv Z_{q}.
F(z,\bar{z})=P_n(\frac{z+\bar{z}}{2},\frac{z-
\bar{z}}{2i})+i\,Q_n(\frac{z+\bar{z}}{2},\frac{z-\bar{z}}{2i}).
Then J. Li and X. Zhao (1989) [84] proved that the system
\(\frac{du}{dt}=V(u)\) is \(Z_{q}\)-equivariant if and only if the
function \(F\)
has the form
F(z,\bar{z})=\sum_{l\geq
1}p\,_{l}(|z|^{2})\bar{z}\,^{lq-1}+\sum_{l\geq
0}h\,_{l}(|z|^{2})z\,^{lq+1},
where \(p_{l}\) and \(h_{l}\) are complex polynomials. For the
discussion of \(Z_{q}\)-equivariant, also see F. Takens(1974) [98] and
V. I. Arnold(1977) [1].
In particular, for a polynomial system
of degree \(5\ ,\) it is
\(Z_2\)-equivariant if and only if \(V(-u)=-V(u)\ ;\) it is
\(Z_q\)-equivariant \((3\leq q\leq 6)\) if and only if the function \(F\)
has the following form
\begin{array}{ccl}
(1)&q=3,&F(z,\bar{z})=(A_{0}+A_{1}|z|^{2}+A_{2}|z|^{4})z+(A_{3}+A_{4}|z|^{2})\bar{z}^{2}+A_{5}z^{4}+A_{6}\bar{z}^{5},\\
(2)&q=4,&F(z,\bar{z})=(A_{0}+A_{1}|z|^{2}+A_{2}|z|^{4})z+(A_{3}+A_{4}|z|^{2})\bar{z}^{3}+A_{5}{z}^{5},\\
(3)&q=5,&F(z,\bar{z})=(A_{0}+A_{1}|z|^{2}+A_{2}|z|^{4})z+A_{3}\bar{z}^{4},\\
(4)&q=6,&F(z,\bar{z})=(A_{0}+A_{1}|z|^{2}+A_{2}|z|^{4})z+A_{3}\bar{z}^{5}.
\end{array}
There have been many studies on the limit cycles of
\(Z_q\)-equivariant polynomial systems, see [73,78,74,75,80,81,82].
As before, let \[M(h)=\oint_{L_ h}P_ndy-Q_ndx,
\] where \(L_h\) denotes a
closed orbit
defined by \(H_m(x,y)=h\)
for \(h\in (h_1,h_2)\ .\) Obviously, \(M\) is
analytic at each \(h\in (h_1,h_2)\ .\) There have been
some studies on
the analytical property of \(M\) at the endpoints \(h_1,h_2.\) Let
\(h_0\in \{h_1,h_2\},\) and \(L_{h_0}\) denote the limit of \(L_h\) as
\(h\rightarrow h_0\ .\) If \(L_{h_0}\) is an elementary center, it was
proved in M. Han(2000) [35] and W. Li(2000) [85] (Theorem 3.9 in
Chapter 4) that \(M\) is analytic at \(h=h_0\ .\) Thus, for \(h\) near \(h_0\)
\(M\) has the following expansion
\[M(h)=\sum_{i\geq 0}b_i(h-h_0)^{i+1}.\]
A program was established in Han, Yang and Yu (2009)[46] for
computing the coefficients \(b_i\) in the above expansion. If
\(L_{h_0}\) is a homoclinic loop passing through a hyperbolic saddle,
it was proved in Roussarie(1986) [92] that \(M\) has the expansion
\[M(h)=\sum_{i\geq
0}[c_{2i}(h-h_0)^{i}+c_{2i+1}(h-h_0)^{i}\ln|h-h_0|].\]
For the explicit formulas of the coefficients \(c_0\ ,\) \(c_1\ ,\) \(c_2\) and \(c_3\ ,\)
Han, Yang, Alina and Gao (2008)[44].
Han(1999)[62] proved that the above expansion remains true if
\(L_{h_0}\) is a heteroclinic loop passing
through two or more hyperbolic saddles. Recently, Han etc.
[38,48,118] considered the cases of \(L_0\) being a nilpotent center
or homoclinic loop passing through a cusp or nilpotent saddles, and
obtained expansions of \(M\ .\) All of those expansions have an
important application: one can use the coefficients appeared in the
expansions to produce zeros of \(M\) near both endpoints and hence to
study the number of limit cycles in Hopf bifurcation and homoclinic
bifurcation etc.. For example, one can give an explicit condition
which ensures the existence of 7 limit cycles near a double
homoclinic loop, see Han and Zhang(2006)[49].
There are three main aspects in studying the number of limit cycles.
The first is Hopf bifurcation. There have been many results in this
aspect. The main technique is to compute focus values or Liapunov
constants, see for example [28] and [113]. By this method, P. Yu and
M. Han [115,116, 114] studied Hopf bifurcations of \(Z_2\)-equivariant
cubic systems, proving \(H(3)\geq 12\ .\) Then \(Z_2\)-equivariant cubic
systems were further studied by J. Li and Y. Liu in [79] proving
\(H(3)\geq 13\) (a different cubic system showing \(H(3)\geq 13\) was
given by C. Li, C. Liu and J. Yang [66] in the same year, using
Abelian integral and [21]). The Hopf bifurcation of \(Z_5\) and
\(Z_6\)-equivariant planar polynomial vector field of degree 5 was
studied in [79] in detail, and obtained 25 and 24 limit cycles
respectively. Another method is to compute the first coefficients
appearing in the expansion of the first order Melnikov function at a
center, say, it is proved in [46] that a center of Hamiltonian
quadratic system generates at most 5 limit cycles under
perturbations of cubic polynomials.
The second aspect is limit cycle bifurcation from a family of
periodic orbits for near-Hamiltonian or near-integrable systems. The
main idea is to estimate the number of zeros of the Melnikov
function \(M(h)\ .\) In many cases the function can be written into the
\[M(h)=I(h)[\lambda-P(h)].\]
Then in order to analyze the zeros of \(M(h)\) one needs to study the
geometrical property of the curve \(\lambda=P(h)\) on the \(\lambda-P\)
plane. The function \(P(h)\) is called detection function. This
method was first introduced by J. Li etc. in 1990s, and is very
valid in many applications to polynomial systems. See [73] for
more details. By this detection function method J. Li and Q.
Huang(1987)[77] obtained \(H(3)\geq 11\ .\)
There are many papers concerning the weak Hilbert's 16th
problem, by using the Picard-Fuchs equations, the Argument Principle,
method, the Picard-Lefschetz formula and by using some techniques, see for
example [9,10,17,11,22,25,26,30,55,58,59,67,68,
69,70,71,127,128] etc..
The third aspect is limit cycle bifurcation near a homoclinic loop
or a poly-cycle for near-Hamiltonian or near-integrable systems. One
way is to use the expansion of the Melnikov function at the loop,
discussing the number of limit cycles near the loop. The method
was originated by Roussarie(1986) [92] for the case of homoclinic loop
and developed in [62,44,40,109]
for the cases of double homoclinic loops and
poly-cycles. The other way is to produce limit cycles by changing
the stability of a homoclinic loop originated by Han(1997)[33] and
developed with many applications to polynomial systems in [47,37,45,39,49,51,50,41,42,103,102,29,105,106,107,104,43]
From [39], \(H(4)\geq 20\ .\) From [103], \(H(5)\geq 28\ .\) From [101], \(H(6)\geq
35\ .\) From [83], \(H(7)\geq
50\ .\) Otrokov [89] proved that \(H(n) \geq (n^2 + 5n-20-6(-1)^n)/2\) for \(n \geq
Christopher and Lloyd [15] proved that
H(n)\geq \frac{1}{2\ln 2}(n+1)^2\ln
(n+1)-\frac{35}{24}(n+1)^2+3n+\frac{4}{3}
for \(n=2^k-1,\,k\geq 2\ .\) Then
Li, Chan and Chung [76]
H(n)\geq \frac{1}{4\ln 2}(n+1)^2\ln
(n+1)-\frac{1}{4}\left(\frac{\ln3}{\ln2}-\frac{25}{27}\right)(n+1)^2+n+\frac{2}{3}
for \(n=3\cdot2^k-1,\,k\geq 1\ .\)
Recently, Han [36] obtained the following: For any sufficiently
small \(\varepsilon&0\) there exists a positive number \(n^*\ ,\)
depending on \(\varepsilon\ ,\) such that
H(n)&\left(\frac{1}{2\ln2}-\varepsilon\right)(n+2)^2\ln(n+2)\quad
{\rm for}\quad n&n^*.
\lim_{n\rightarrow\infty}\inf
\frac{H(n)}{(n+2)^2\ln(n+2)}\geq \frac{1}{2\ln2}.
That is to say,
\(H(n)\) grows at least as rapidly as
\(\frac{1}{2\ln2}(n+2)^2\ln(n+2)\ .\)
In 1926, B. van der Pol [100] investigated the triode vacuum tube
and found a phenomenon of stable self-excited oscillations of
constant amplitude, producing a second order differential equation
of the form \[x''+\mu(x^2-1)x'+x=0.\] The stable self-excited
oscillations correspond to a stable limit cycle. Then in 1928
Li\({\acute{e}}\)nard [86] studied the following equation
\[x''+f(x)x'+g(x)=0,\]
where \(f\) is even and \(g\) odd. The equation is equivalent to a plane
system of the form
\[\dot{x}=y, \dot{y}=-f(x)y-g(x)\]
or of the form
\[\dot{x}=y-F(x), \dot{y}=-g(x), F(x)=\int_0^xfdx\]
which is called a Li\({\acute{e}}\)nard system. A more general
system of the form \[\dot{x}=h(y)-F(x), \dot{y}=-g(x),
F(x)=\int_0^xfdx\] is called a generalized Li\({ \acute{e}}\)nard
system, where \(f\ ,\) \(g\) and \(h\) are continuous functions. There are
many studies on the nonexistence and uniqueness of limit cycles and
the existence of one or more limit cycles. For details, see [112]
and [126]. There are different methods for proving the uniqueness of
limit cycles. One of them is to compare the integrals of the
divergence of the system along two limit cycles, which was
originated by Zhang Zhifen in 1958 (see [125]). She proved the
following which is called Zhang Zhifen's theorem:
Let \(G\) denote a rectangle containing the origin. If \(xg(x)&0\ ,\)
\(yh(y)&0\) and the functions \(f(x)/g(x)\) and
\(h(y)\) are strictly
monotone in their variable for all \((x,y) \in G\) with \(x\neq0\ ,\)
\(y\neq0\ ,\) then the generalized Li\({\rm \acute{e}}\)nard system has at
most one limit cycle in \(G\ .\)
Consider the Liénard system \[\dot x = y,\ \dot y =
-f_m(x)y-g_n(x),\]
where \(f_m\) and \(g_n\) are polynomials in \(x\) of degree \(m\) and \(n\) respectively.
Let \({H}(m, n)\) denote the maximum number of limit cycles for all possible \(f_m\) and \(g_n\ ,\) and let
\(\hat{H}(m, n)\) denote the maximum number of small-amplitude limit
cycles of the above system that can be bifurcated from a focus. We
have obviously \({H}(m, n)\geq \hat{H}(m, n)\ .\)
In 1984, Blows and Lloyd [4] proved \(\hat H(m,
1)=\left[\frac{m}{2}\right]\ .\) Han [34] proved \(\hat H(m,
2)=\left[\frac{2m+1}{3}\right]\) for all \(m\geq 1\ .\) Christopher and
Lynch [16] obtained \(\hat H(m, 2)=\hat H(2,
m)=\left[\frac{2m+1}{3}\right]\) for all \(m\geq 1\ ,\) and \(\hat H(m,
3)=\hat H(3, m)=2\left[\frac{3m+6}{8}\right]\) for all \(1&m\leq 50\ .\)
The authors [16] also gave a table of \(\hat{H}(m, n)\) for some
specific values of \(m\) and \(n\ .\) The table was complemented in Yu and Han [117] for some more
values of \(m\) and \(n\ .\) If \(f\) is even and \(g\) is odd with \(n=3\ ,\)
then \(\hat H(m,3)=m\) for all \(m\geq 2\) (see [61]).
The ultimate aim is to establish a
general formula for \(\hat H (m,n)\) as a function of \(m\) and \(n\ .\)
We have seen that the number \(\hat{H}(m, n)\) gives a lower bound of
\({H}(m, n)\ .\) Finding \({H}(m, n)\) is the Hilbert 16th problem for the
Li\'enard System. We call \({H}(m, n)\) the Hilbert's number for the
It is easy to see that \(H(1,1)=0\ .\) In 1977, Lins, de Melo and Pugh
[87] proved that \(H(2,1)=1\ .\) They also made a conjecture \(
H(m,1)=\left[\frac{m}{2}\right]\ .\) A counterexample was found in [23]
with 4 limit cycles for the case \(m=6\ ,\) \(n=1\ .\) In 1988, Coppel [18]
proved that \(H(1,2)=1\ .\) In 1986, Li [72] proved \(H(2,2)=1\) which was
then reproved by Dumortier and Li [19] in 1997 and Luo, Wang, Zhu
and Han [88] in 1997 by using Zhang Zhifen's theorem. In 1996,
Dumortier and Li [20] proved that \(H(1,3)=1\ .\) What is the Hilbert's
number for all other cases is still open. Then an interesting
problem is to find a better lower bound of it than \(\hat{H}(m, n)\ .\)
It is conjectured that \(H(3,1)=1\ .\) From Dumortier and Li [21] we
know \(H(2,3)\geq 5.\) Recently, Yang, Han and Romanovski [110]
\[H(m,3)\geq \left[\frac{3m+14}{4}\right], \ m=3,4,5,6,7,8.
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