Topological properties on isochronous centers of polynomial Hamiltonian differential systems

Consider the following complex polynomial Hamiltonian differential systems of degree $n$

where the Hamiltonian function $H(x,y)$ is a polynomial of degree $n+1$ in $\mathbb{C}[x,y]$. Assuming the origin $O$ is a center of Morse type, without loss of generality, $H(x,y)$ can be written as $H(x,y)=(x^{2}+y^{2})/2+h.o.t.$. For a generic level curve $L_{h}$ defined by the algebraic equation $H(x,y)=h$ where $h\in\mathbb{C}$ is sufficiently close to $0$, one can associate a vanishing cycle $\gamma_{h}$ to the critical value $h=0$, which is a 1-dimensional cycle vanishing at $h=0$ in the 1-dimensional homology group $\mathcal{H}_{1}(L_{h},\mathbb{Z})$ and can be characterized by the following purely topological property: modulo orientation and the free homotopy deformation on $L_{h}$, as $h\rightarrow 0$, the cycle $\gamma_{h}$ can be represented by a continuous family of loops on $L_{h}$ of length that tends to zero. This description explains the terminology(see, e.g., [8]). Respectively $T(h)=\oint_{\gamma_{h}}dt$ is called a period function of system (1.5). If $T(h)$ is a nonzero constant independent of $h$ for $h\neq 0$, then the origin is called an isochronous center. This definition coincides with the classical isochronous center when $(x,y)\in\mathbb{R}^{2}$ and $t\in\mathbb{R}$.

One of the most important problems on isochronous centers is to describe the role of the vanishing cycle $\gamma_{h}$ in the 1-dimensional homology group of the compact Riemann surface of $L_{h}$. It is still an open problem until now. In [7], L. Gavrilov has asked the following question for systems (1.5) with only isolated singularities:

In general cases, the above question has a negative answer. Example 3.23 in reference [4] provides a system with

which has an isochronous center at the origin, but the corresponding vanishing cycle is not homologous to zero on the Riemann surface of $L_{h}$. In this counterexample, it is not difficult to see that the critical level curve $L_{0}$ contains at least three different singularities on $\mathbb{C}^{2}$.

What conditions can give a positive answer to Gavrilov’s question? This is also an important and meaningful question, especially it is closely related with the famous Jacobian conjecture on $\mathbb{C}^{2}$, which asserts that the following polynomial map with a constant Jacobian determinant

is a global homeomorphism, where $f=x+h.o.t.$ and $g=y+h.o.t.$ are polynomials in $\mathbb{C}[x,y]$. At present it has been proved only when the degrees of $f$ and $g$ are not too large. Obviously the map $\Phi_{P}$ induces a Hamiltonian system

having an isochronous center of Morse type at the origin with the Hamiltonian function $H(x,y)=(f^{2}+g^{2})/2$.

Also in [7], Proposition 6.1 says that if the vanishing cycle associated to the origin for system (1.15) represents a zero homology cycle on the Riemann surface of a generic level curve, then the map $\Phi_{P}$ is injective, which suffices to guarantee the Jacobian conjecture is true. In addition, he has also proved that(Theorem 4.1 of [7]) Question 1.1 has a positive answer under the conditions that the critical level curve $L_{0}$ contains only a single singular point which is isochronous and $H(x,y)$ is a ‘good’ polynomial having only isolated and simple singularities, where the definition of a good polynomial depends on the Milnor numbers of the complex projective closure $\overline{L_{h}}$ of $L_{h}$ at infinity.

This paper is devoted to look for other conditions to give a positive answer to Question 1.1. Denote by

the period $1$-form of system (1.5). We have the following main theorem.

Applying the above theorem to system (1.15), one can achieve an equivalent description of the Jacobian conjecture.

To prove Theorem 1.2, we will carefully study some real systems induced by complex system (1.5) and the corresponding transformation linearizing an isochronous center. Such systems possess many good properties, such as commutativity, transversality, and so on. Besides, their topological structures near the points at infinity on $L_{h}$ can also provide for us a lot of information for the isochronicity of system (1.5). Letting $H_{n+1}(x,y)$ be the highest degree part of $H(x,y)$, we have the following necessary condition for isochronicity:

In this paper, we will also show an interesting relation between Gavrilov’s question and the following conjecture, which was claimed by X. Jarque and J. Villadelprat in [9], on real systems (1.5), i.e., $(x,y)\in\mathbb{R}^{2},$ and $t\in\mathbb{R}$.

At present, this conjecture is still open and a recent development can be found in [5]. The following theorem indicates that if the Jarque-Villadelprat conjecture is not true, then the Gavrilov’s question must have a negative answer for such real systems.

The paper is organized as follows. We shall first introduce some properties on the commuting real differential systems(or real vector fields) induced by system (1.5) and provide a powerful technique to extend the transformation linearizing an isochronous center. Then we give the detailed proof of the main results and some applications.

Note that if the origin is an isochronous center of Morse type for system (1.5), then there exists an analytic area-preserving transformation(see, e.g, [1, 11, 12])

changing system (1.5) to a linear system

here we say $\Phi$ is area-preserving is equivalent to say its Jacobian determinant $\det(J(\Phi))\equiv 1$, where

Generally speaking, $\Phi$ is only well defined in a small neighborhood of the origin $\mathcal{N}(O)\subseteq\mathbb{C}^{2}$.

By taking advantage of constant Jacobian determinant, one can construct another complex system in $\mathcal{N}(O)$ as follows

which can be also linearized to a linear system

by the same transformation $\Phi$, for the reasons that

and

Consequently, systems (1.5) and (2.15) induce the following four real differential systems(see, e.g. [3]) by taking $(x,y)\in\mathbb{C}^{2}\cong\mathbb{R}^{4}$ but $t\in\mathbb{R}$:

and

where ${\rm i}^{2}=-1$. They can be transformed to the following four real linear systems simultaneously by the same $\Phi$ respectively:

and

Letting $u=u_{1}+\mathrm{i}u_{2}$ and $v=v_{1}+\mathrm{i}v_{2}$ and regarding $\mathbb{C}^{2}\cong\mathbb{R}^{4}=\{(u_{1},u_{2},v_{1},v_{2})\}$, the coefficient matrices of $V_{\ast}$, ${\rm i}V_{\ast}$, $V_{g\ast}$ and ${\rm i}V_{g\ast}$ are respectively

where

Obviously we have

Due to that $\Phi$ is a diffeomorphism, one can get the following important properties for vector fields $V$, ${\rm i}V$, $V_{g}$, and $\ {\rm i}V_{g}$:

1. (1)

   they are commutative pairwise everywhere in $\mathcal{N}(O)$, i.e., as real vector fields, the Lie bracket of any two of them vanishes. So for any two points $p_{1},p_{2}$ in $\mathcal{N}(O)$ except $O$, it takes the same time along any two continuous paths connecting $p_{1}$ and $p_{2}$ consisting of finitely many trajectories of those vector fields.

2. (2)

   their trajectories are transversal pairwise everywhere on $\mathcal{N}(O)-L_{0}$; while on $L_{0}$, $V$(resp. ${\rm i}V$) coincides with ${\rm i}V_{g}$(resp. $-V_{g}$) on one of two branches near $O$ and with $-{\rm i}V_{g}$(resp. $V_{g}$) on the other one;

3. (3)

   the domain in which $V_{g}$ and ${\rm i}V_{g}$ can be well defined is the same to the domain of $\Phi$, but $V$ and ${\rm i}V$ are well defined on the whole complex plane $\mathbb{C}^{2}$;

4. (4)

   near the origin, all of the orbits of system $V$ are closed; on the contrary, system ${\rm i}V$ does not have any closed orbits in $\mathcal{N}(O)$;

5. (5)

   the trajectories of systems $V$ and ${\rm i}V$ are both tangent to $L_{h}$ everywhere, so their restrictions, denoted by $V_{h}$ and ${\rm i}V_{h}$, are two real systems defined well on $L_{h}$.

Denote by $\varphi(\cdot,t)$(resp. ${\rm i}\varphi,\ \varphi_{g},\ {\rm i}\varphi_{g}$, $\varphi_{\ast},\ {\rm i}\varphi_{\ast},\ \varphi_{g\ast},$ and ${\rm i}\varphi_{g\ast}$) the flow map induced by $V$(resp. ${\rm i}V$, $V_{g},\ {\rm i}V_{g}$, $V_{\ast}$, ${\rm i}V_{\ast}$, $V_{g\ast},$ and ${\rm i}V_{g\ast}$), i.e., for any given point $p$, $\varphi(p,t)$ takes the value at time $t$ of the solution of equations $V$ with initial value $p$ at $t=0$. The commutativity between those systems means that each one of the flow maps above preserves the orbits of any other system in the domain of $\Phi$. We shall take advantage of this observation to extend the domain of $\Phi$ to a bigger one than $\mathcal{N}(O)$. Without loss of generality, we assume $\mathcal{N}(O)$ is a sufficiently small and homeomorphic to a open ball $\{(x,y)\in\mathbb{C}^{2}:\ \left|x\right|^{2}+\left|y\right|^{2}<\epsilon\}$ centered at $O$ with radius $\epsilon$. Denote by $H$ the following map

For a closed orbit $\sigma$ of $V$ in $\mathcal{N}(O)$ such that $H(\sigma)\neq 0$, and two sufficiently small number $t_{1},t_{2}$, the space

is a real 3-dimension sub-manifold of $\mathcal{N}(O)$ and transversal to ${\rm i}V$ at every point. Then $\Phi(\Gamma_{\sigma})$ is also a real 3-dimension sub-manifold of $\Phi(\mathcal{N}(O))$ and transversal to ${\rm i}V_{\ast}$ at every point.

Along the trajectories of ${\rm i}V$ passing through $\Gamma_{\sigma}$ in $\mathcal{N}(O)$, the transformation $\Phi$ can be expressed by the flow map ${\rm i}\varphi$ as follows: for any point $p\in\Gamma_{\sigma},$ and any sufficiently small $t$, we have

Clearly the vector fields ${\rm i}V$ and ${\rm i}V_{\ast}$ are well defined globally, the above equation can be extended to a larger interval $I$ for time $t\in\mathbb{R}$ such that ${\rm i}\varphi(p,t)\not\in\mathcal{N}(O)$, if ${\rm i}V$ satisfies the following two conditions:

1. C1.

   the trajectories of ${\rm i}V$ could not return into the domain where $\Phi$ has already been defined well;

2. C2.

   there is no point $P$ at infinity such that ${\rm i}\varphi(p,t)$ tends to $P$ as $t$ tends a finite moment $t_{0}$ for some a point $p_{0}\in\Gamma_{\sigma}$.

If the trajectories of ${\rm i}V$ from a open subset of $\Gamma_{\sigma}$ go to a point at infinity when $t$ tends $\infty$, then the interval $I$ for those points can be $[0,+\infty)$ or $(-\infty,0]$. While if C1 holds but C2 not, then $I$ can only be $[0,t_{0})$ or $(t_{0},0]$ at such a point $p_{0}$(see Figure 1).

Noticing that the vector fields $V$ and $V_{\ast}$ are also well defined globally, we can also perform the above operation along the trajectories of $V$ if it satisfies the conditions C1 and C2.

In a word, one can extend the transformation $\Phi$ to an open domain $\mathcal{D}$ as big as possible according to the above operation along the trajectories of $V$ and ${\rm i}V$. Although $\mathcal{D}$ may be much bigger than $\mathcal{N}(O)$, we have $\mathcal{D}\cap L_{h}$ is still homeomorphic to $\mathcal{N}(O)\cap L_{h}$ for any $h$ sufficiently close to $0$.

To prove the main results, we still need to know some information about the points at infinity on $L_{h}$. It is better to deal with it in the projective space $\mathbb{CP}^{2}$. Assume the projective closure $\overline{L_{h}}$ are defined by the following homogeneous equations

where $H_{k}$ represents the homogeneous part of degree $k$ of $H(x,y)$. For a generic value $h$, the set of singularities on $\overline{L_{h}}$, denoted by $\Sigma_{h}$, consists of only some points at infinity on $L_{h}$. The Riemann surface of $L_{h}$ coincides with the resolution of $\overline{L_{h}}$ by a birational map. Generally speaking, The algebraic curve $\overline{L_{h}}$ may have more than one connected branches near a point $P\in\Sigma_{h}$. The number of such branches is equal to the number of essentially different Puiseux expressions associated to $P$(see, e.g., [10]).

Rewriting the homogeneous part $H_{n+1}(x,y)$ of degree $n+1$ as follows:

where $\alpha_{i},\ \beta_{i}\in\mathbb{C}$ such that $\alpha_{i}:\beta_{i}\neq\alpha_{j}:\beta_{j}$ if $i\neq j,$ the projective coordinate of a point $P^{i}\in\Sigma_{h}$ can be represented by $[\beta_{i}:\alpha_{i}:0]$. Up to a projective change of coordinates, we can always assume its projective coordinate is $[1:0:0]$. Then it is convenient to adopt a pair of new affine coordinates $(X,Y)$, where

and the Puiseux expressions near $P^{i}$ are totally determined by the Puiseux expressions of equation

near the origin. According to the classical theory of Puiseux(see, e.g., ([6])), each branch of an algebraic curve near a singularity can be parameterized by a Puiseux series of the following form.

In general the coefficients $\{c_{i}\}$ may depend on $h$ on different level curve $L_{h}$, so sometimes we replace $\rho(s)$ with $\rho(s,h)$ to emphasize it. Taking the Puiseux parameterization $x=s^{-\mathsf{p}},\ y=s^{\mathsf{q}-\mathsf{p}}\rho(s,h)$ into system (1.5), we obtain a complex 1-dimension ordinary differential equation

on a branch of $\overline{L_{h}}$ near $P^{i}$, where $\lambda\not=0,\ \mathsf{k}\in\mathbb{Z}.$ Then the real systems $V_{h}$ and ${\rm i}V_{h}$ are changed to the following forms respectively under this parameterization:

and

Their topological structures can be classified into the following four classes according to the value of $\mathsf{k}$ near $s=0$:

• •

  $\mathsf{k}>1$. The orbits of real system (3.4)(or system (3.5)) form $2(\mathsf{k}-1)$ petals in a sufficiently small neighborhood of $s=0$, any one of them is tangent to a separatrix of the petals at $s=0$.

• •

  $\mathsf{k}=1$. If $\lambda$ is a pure imaginary number, the point $s=0$ is of center-focus type; while for $\lambda\in\mathbb{R}$, it is a node, and for other numbers, it is a focus.

• •

  $\mathsf{k}=0$. The point $s=0$ is not a singularity for real systems (3.4) and (3.5).

• •

  $\mathsf{k}<0$. The system (3.4)(or system (3.5)) has a saddle structure in a sufficiently small neighborhood of $0$ except $s=0$.

The Puiseux parameterizations can be determined completely by the so-called Newton polygon of the singularity. Given an irreducible polynomial $F(X,Y)=\sum_{k,l}b_{kl}X^{k}Y^{l}$ with $F(0,0)=0$, denote by $\Lambda(F)$ the carrier of $F(X,Y)$, i.e. $\Lambda(F)=\{(k,l)\in\mathbb{Z}^{2}\ |\ b_{kl}\neq 0\}.$ Assuming that $Q_{1},Q_{2}\in\mathbb{R}^{2}$, let

be the straight line segment from $Q_{1}$ to $Q_{2}$. Consider the convex subset $A$ on $\mathbb{R}^{2}$ consisting of those $(X,Y)\in\mathbb{R}^{2}$ such that $X\geq X_{0}$ and $Y\geq Y_{0}$ for some $(X_{0},Y_{0})\in[Q_{1},Q_{2}]$ where $Q_{1},Q_{2}\in\Lambda(F)$.

In this section, we first prove the following important lemmas. It is not difficult to see the vanishing cycle $\gamma_{h}$ can be represented by a given closed orbit of system $V_{h}$ near the origin(we still denote this orbit by $\gamma_{h}$).

By this lemma, we have the following immdiately.

Below we shall show that, under the assumption of Theorem 1.2, in the second case of the above lemma, the number $\mathsf{k}$ must be equal to $1$.

This lemma tells us the period $1$-form $\omega$ of system (1.5) has at least one pole at a point $P^{i}$ at infinity on a branch of $\overline{L_{h}}$ near $P^{i}$. The following lemma will show that the multiplicity of such a point $P^{i}$ can not be too low, i.e., we have