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Limit cycles of planar polynomial vector fields - Scholarpedia

# Limit cycles of planar polynomial vector fields

Post-publication activity

Curator: Chengzhi Li

## Definition and Hilbert's 16th problem

### Limit Cycle for a Planar Polynomial Vector Field

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 called stable 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.

Example

Van der Pol's system $\frac{dx}{dt}=y,\ \ \frac{dy}{dt}=-x+\mu (1-x^2)y,\ \mu>0$ has a unique limit cycle, which is stable and hyperbolic.

### Hilbert's and Weak Hilbert's 16th Problem

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 ):

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  and Ilyashenko  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 $F(h,\varepsilon)=\varepsilon M(h)+O(\varepsilon^2),$ where $M(h)=\oint_{L_ h}P_ndy-Q_ndx,$ which is called the first order Melnikov function.

Weak Hilbert's 16th problem posed by V. I. Arnold (1977, 1990) [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 , Varchenko ). During 1994-2002 it has been proved that in generic cases $$Z(3,2)=2$$ (Horozov and Iliev , Zhang and Li , Gavrilov  and Li and Zhang , and in degenerate cases $$Z(3,2)=3$$ ( , , , ,  and ). Also see the second part of the book by Christopher and Li(2007).

### Ye's classification

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) $$\frac{dx}{dt}=-y+dx+lx^2+mxy+ny^2,$$ $$\frac{dy}{dt}=x(1+ax),\,\,\,a\neq 0;$$

(III) $$\frac{dx}{dt}=-y+dx+lx^2+mxy+ny^2,$$ $$\frac{dy}{dt}=x(1+ax+by),\,\,\,b\neq 0,$$

which is called Ye's classification, made by Yanqian Ye in 1960s. Many researches were done by using the classification, see Ye(1986) and the following.

### Some Global Results on Limit Cycles

L. Chen and Y. Ye(1975), X. Yang and Y. Ye(1978) proved that any system in class I has at most one limit cycle. Then Cherkas and Zhilevich ,  and Ryckov  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)  and Cherkas(1986).

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 .

In 1955, Petrovsky and Landis  attempted to prove $$H(2)=3\ .$$ Unfortunately, their proof contained errors. Then in 1979, L. Chen and M. Wang  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  found out some important properties of quadratic systems: a closed orbit is convex; there is a unique singularity in the interior of it; 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  and D. Zhu  respectively proved that a quadratic system with a second or third order weak saddle point does not have a limit cycle. P. Zhang  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)  and Reyn(2007) .

### Some Bifurcation Results of Limit Cycles

In 1952, Bautin  studied the Hopf bifurcation 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 appear.

Zhu(1989) , Joyal and Rousseau(1989)  and Cai and Guo(1990)  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)  proved that in quadratic systems the maximum number of limit cycles that can appear near a homoclinic loop of a nondegenerate Hamiltonian system is 2 under small perturbations with a single parameter, where a Hamiltonian system is called nondegenerate if the perturbed system is Hamiltonian when $$M(h)\equiv 0\ .$$ From their proof the conclusion is also true under small perturbation with arbitrary parameters. Then Iliev  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  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 , Han(1997) . 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 , Zoladek  and Han-Yang  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 saddles.

## Polynomial Systems with Higher Degrees

### Polynomial Equivariant Systems

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: $S=\left( \begin{array}{cc} 1&0\\0&-1 \end{array}\right)=\rm diag (1,-1),$ 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 $S=\left( \begin{array}{cc} \cos\frac{2\pi}{q}&-\sin\frac{2\pi}{q}\\ \sin\frac{2\pi}{q}&\cos\frac{2\pi}{q} \end{array} \right)\equiv Z_{q}.$

Let $F(z,\bar{z})=P_n(\frac{z+\bar{z}}{2},\frac{z- \bar{z}}{2i})+i\,Q_n(\frac{z+\bar{z}}{2},\frac{z-\bar{z}}{2i}).$

Then J. Li and X. Zhao (1989)  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)  and V. I. Arnold(1977) .

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].

### Analytical property of Abelian integrals

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)  and W. Li(2000)  (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) 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)  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 $$c_3\ ,$$ see Han, Yang, Alina and Gao (2008). Jiang and Han(1999) 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).

### Methods of Studying the Number of Limit Cycles

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  and . 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  proving $$H(3)\geq 13$$ (a different cubic system showing $$H(3)\geq 13$$ was given by C. Li, C. Liu and J. Yang  in the same year, using Abelian integral and ). The Hopf bifurcation of $$Z_5$$ and $$Z_6$$-equivariant planar polynomial vector field of degree 5 was studied in  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  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 form $M(h)=I(h)[\lambda-P(h)].$ 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  for more details. By this detection function method J. Li and Q. Huang(1987) 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, the averaging 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)  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) 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] etc.. From , $$H(4)\geq 20\ .$$ From , $$H(5)\geq 28\ .$$ From , $$H(6)\geq 35\ .$$ From , $$H(7)\geq 50\ .$$ Otrokov  proved that $$H(n) \geq (n^2 + 5n-20-6(-1)^n)/2$$ for $$n \geq 6.$$ Christopher and Lloyd  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  obtained $H(n)\geq \frac{1}{4\ln 2}(n+1)^2\ln (n+1)-\frac{1}{4}\left(\frac{\ln3}{\ln2}-\frac{25}{27}\right)(n+1)^2+n+\frac{2}{3}$ for $$n=3\cdot2^k-1,\,k\geq 1\ .$$

Recently, Han  obtained the following: For any sufficiently small $$\varepsilon>0$$ there exists a positive number $$n^*\ ,$$ depending on $$\varepsilon\ ,$$ such that $H(n)>\left(\frac{1}{2\ln2}-\varepsilon\right)(n+2)^2\ln(n+2)\quad {\rm for}\quad n>n^*.$ Hence, $\lim_{n\rightarrow\infty}\inf \frac{H(n)}{(n+2)^2\ln(n+2)}\geq \frac{1}{2\ln2}.$ That is to say, $$H(n)$$ grows at least as rapidly as $$\frac{1}{2\ln2}(n+2)^2\ln(n+2)\ .$$

## Limit Cycles in Li$${\acute{e}}$$nard Systems

### Existence and Uniqueness of Limit Cycles

In 1926, B. van der Pol  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  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  and . 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 ). 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 $$G\ .$$

### Hopf bifurcation for Polynomial Liénard Systems

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  proved $$\hat H(m, 1)=\left[\frac{m}{2}\right]\ .$$ Han  proved $$\hat H(m, 2)=\left[\frac{2m+1}{3}\right]$$ for all $$m\geq 1\ .$$ Christopher and Lynch  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  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  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 ). The ultimate aim is to establish a general formula for $$\hat H (m,n)$$ as a function of $$m$$ and $$n\ .$$

### Global Results for Polynomial Li$${\rm \acute{e}}$$nard Systems

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 system.

It is easy to see that $$H(1,1)=0\ .$$ In 1977, Lins, de Melo and Pugh  proved that $$H(2,1)=1\ .$$ They also made a conjecture $$H(m,1)=\left[\frac{m}{2}\right]\ .$$ A counterexample was found in  with 4 limit cycles for the case $$m=6\ ,$$ $$n=1\ .$$ In 1988, Coppel  proved that $$H(1,2)=1\ .$$ In 1986, Li  proved $$H(2,2)=1$$ which was then reproved by Dumortier and Li  in 1997 and Luo, Wang, Zhu and Han  in 1997 by using Zhang Zhifen's theorem. In 1996, Dumortier and Li  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  we know $$H(2,3)\geq 5.$$ Recently, Yang, Han and Romanovski  obtained $H(m,3)\geq \left[\frac{3m+14}{4}\right], \ m=3,4,5,6,7,8.$