Dynamics of locally linearizable complex two dimensional cubic Hamiltonian systems

The aim of this paper is to characterize global dynamics of locally linearizable complex two dimensional cubic Hamiltonian systems. By finding invariants, we prove that their associated real phase space \begin{document}$\mathbb R^4$\end{document} is foliated by two dimensional invariant surfaces, which could be either simple connected, or double connected, or triple connected, or quadruple connected. On each of the invariant surfaces all regular orbits are heteroclinic ones, which connect two singularities, either both finite, or one finite and another at infinity, or both at infinity, and all these situations are realizable.

1. Introduction and statement of the main results. The study on the problem of locally analytic linearizability of analytic differential systems at a singularity has a long history, and it can be traced back to Poincaré. Along this direction there have appeared lots of results, see e.g. [4,16] and the references therein. For real planar analytic systems at a singularity having a pair of pure imaginary eigenvalues this problem is equivalent to characterize the existence of isochronous center at the singularity, see for instance the survey paper [4]. In 2015 Llibre and Romanovski [10] extended their study to complex two dimensional polynomial Hamiltonian systems with Hamiltonians being of the form where h is a cubic homogeneous polynomial or consists of cubic and quartic homogeneous polynomials. In the former it was given in [10] the necessary and sufficient conditions for the Hamiltonian system to be linearizable at the origin. In the latter there provided five sufficient conditions for the Hamiltonian system to be linearizable at the origin.
The main aim of this paper is to investigate the global dynamics of locally linearizable complex two dimensional cubic Hamiltonian systems with Hamiltonians of the form (1). Here we do not consider the complex quadratic Hamiltonian systems with Hamiltonians of the form (1) because it will be studied uniformly for general quadratic Hamiltonian system in [17].
We must say that on global dynamics of complex differential systems there are only a few known results, see for instance [1,7,8,19] and the references therein. In fact, even for planar real Hamiltonian systems, there are few differential systems where the global dynamics has been completely characterized, see e.g. [3,5]. Here by finding new invariants we complete the characterization of the global dynamics of the mentioned systems.
Consider complex two dimensional cubic Hamiltonian systems, i.e.
x = x − a 10 x 2 − 2b 01 xy − a 12 y 2 − a 20 x 3 − a 11 x 2 y − 3b 02 xy 2 − a 13 y 3 , y = −y + b 21 x 2 + 2a 10 xy + b 01 y 2 + b 31 x 3 + 3a 20 x 2 y + a 11 xy 2 + b 02 y 3 , with the Hamiltonian x 3 + a 10 x 2 y + b 01 xy 2 + a 12 3 where x, y are complex variables and a ij , b ij are complex coefficients. Llibre and Romanovski [10,Theorem 4] obtained the next result. Note from the proof of [10,Theorem 4] that except in the case (e) all the other cases have the linearization transformation defined only locally. So their global dynamics cannot be characterized by their locally linearized systems. Next we present global dynamics of the Hamiltonian system (2) under each of the five conditions. For doing so, we will use the real form of the complex Hamiltonian system (2). Set Under the real expressions of the complex variables and of complex coefficients, the complex Hamiltonian system (2) can be represented aṡ Correspondingly, the real and imaginary parts of the Hamiltonian (3) are respectively , .
We can check that system (4) is Hamiltonian with the Hamiltonian H R and the first integral H I in involution. Indeed, set ξ = (x 1 , y 2 , y 1 , x 2 ) τ with τ the transpose of a matrix. One haṡ Here ∇ ξ H R denotes the gradient of H R with respect to ξ. As we knew, there are lots of known results on characterization of integrability of differential systems (Hamiltonian or not, see e.g. [11,12,14,15,18]. For integrable real Hamiltonian systems of 2-degrees of freedom, Arnold [2] presented global dynamics of linear Hamiltonian ones with an elliptic-elliptic singularity. Lerman and Umanskiy [9] obtained local topological structure of nonlinear ones near simple singularity. Now we can state our main results of this paper depending on the five conditions of Theorem A. Theorem 1.1. For system (2) under the condition (a) of Theorem A, the following statements hold.
(I) If b 01 = 0, the associated real four dimensional phase space R 4 is foliated by real 2-dimensional invariant surfaces, in which one is the plane y = 0, another one saying S 0 is diffeomorphic to the y-plane, all the others saying S are diffeomorphic to the y-plane minus the origin, and they approach the infinity of the x-plane when y → 0.
-System (2) restricted to y = 0 is an unstable linear star node.
-System (2) restricted to the y-plane is equivalent to a stable linear star node. -System (2) restricted to each S has its orbits being topological lines and positively going to the infinity of the x-plane. (II) If b 01 = 0, system (2) has exactly two singularities S * a1 and S * a2 , and the associated real four dimensional phase space R 4 is foliated by real 2-dimensional invariant surfaces, in which there are two invariant planes y = 0 and b 01 y = 1, two invariant surfaces L a1 and L a2 passing respectively S * a1 and S * a2 , and all the other invariant surfaces L a approaching both y = 0 and b 01 y = 1 at their infinity.
-System (2) restricted to y = 0 is an unstable linear star node.
-System (2) restricted to b 01 y = 1 is a stable linear star node.
-All orbits, except two, on L a1 positively approach S * a1 and negatively go to infinity of b 01 y = 1.
-All orbits, except two, on L a2 negatively approach S * a2 and positively go to infinity of y = 0.
-All orbits, except two, on L a are heterolinic to infinity of both y = 0 and b 01 y = 1.
-The two exceptional orbits mentioned above are also heteroclinic with some special connection described in the proof of this theorem.
Under the condition (b) of Theorem A, system (4) has the same dynamics as those in Theorem 1.1, which can be seen by exchanging the coordinates (x 1 , x 2 ) with (y 1 , y 2 ). Theorem 1.2. For system (2) under the condition (c) of Theorem A, the following statements hold.
(I) If b 01 = 0, system (2) has the same dynamics as those in statement (I) of Theorem 1.1. (II) If b 01 = 0, system (2) has exactly three singularities S * c1 , S * c2 and S * c3 , and the associated real four dimensional phase space R 4 is foliated by real two dimensional invariant surfaces, in which there are three invariant planes y = 0, 2b 01 y = 3 and b 01 y = 3, one invariant surface L c1 passing both S * c1 and S * c3 , one invariant surface L c2 passing S * c2 , and all the other invariant surfaces L c having three boundaries approaching respectively y = 0, 2b 01 y = 3 and b 01 y = 3 at their infinities.
-System (2) restricted to any one of y = 0, 2b 01 y = 3 and b 01 y = 3 is linear, and has a star node, which is unstable, stable and unstable. -L c1 is the global stable manifold of both S * c1 and S * c3 , and on it all orbits, except two, negatively approach infinity of 2b 01 y = 3.
-L c2 is the global unstable manifold of S * c2 , and on it all orbits, except two, positively approach infinity of either y = 0 or of b 01 y = 3.
-All orbits on L c are heterolinic, and except four of them, negatively go to infinity of 2b 01 y = 3 and positively approach infinity of either y = 0 or b 01 y = 3. -The two or four exceptional orbits mentioned in the last three items have one end approaching infinity of one of y = 0, 2b 01 y = 3 and b 01 y = 3, and the other end tending to infinity of R 4 with finite slope.
Under the condition (d) of Theorem A, system (4) has the same dynamics as those in Theorem 1.2, which can be obtained by exchanging the coordinates (x 1 , x 2 ) with (y 1 , y 2 ).
Under the condition (e) of Theorem A, as shown in [10] by Llibre and Romanovski system (2) can be globally linearized via the symplectic transformation Correspondingly, system (4) is reduced tȯ where This means that the two first integrals H R and H I of system (4) are functionally independent, and so the system is integrable in the Liouvillian sense. Let X a be the vector field associated to system (4). Some calculations show that This verifies that F 1 = 0 is always an invariant plane of system (5), and so is F 2 = 0 provided that b 01 = 0, i.e. b 2 1 + b 2 2 = 0. Here we have used the fact that F 2 can be written as According to the Darboux theory of integrability [13,20], F 1 and F 2 are called Darboux polynomials of system (4), and K 1 and K 2 are factors of F 1 and F 2 , respectively.
Recall from the Darboux theory of integrability [13,20] that for a polynomial vector field Z defined in R n or C n with coordinates z, a polynomial F ∈ C[z] is a Darboux polynomial of the vector field Z if there exists a polynomial K ∈ C[z] such that where K is cofactor of F . If the vector field Z has p Darboux polynomials F j with cofactors K j , j = 1, . . . , p satisfying with σ j ∈ C constants not all vanishing, then F σ1 1 . . . F σp p is a first integral of the vector field Z, which is called a Darboux first integral.
(I) b 01 = 0, i.e. b 1 = b 2 = 0. By (6) and (7) the two first integrals H R and H I of system (5) have their level sets intersecting transversally everywhere except on the invariant plane F 1 = 0. So, it follows from the Liouville-Arnold integrability theorem (see e.g. [8,19]) that the phase space R 4 of system (5) when b 01 = 0 is foliated by two dimensional invariant manifolds.
(II) b 01 = 0, i.e. b 2 1 = b 2 2 = 0. Again by (6) and (7) the level sets of the two first integrals H R and H I of system (5) intersect transversally everywhere outside the invariant planes F 1 = 0 and F 2 = 0. Note from the expression (8) of F 2 with the Liouville-Arnold integrability theorem that the phase space R 4 of system (5) when b 01 = 0 is foliated by two dimensional invariant manifolds. Now we have another Darboux polynomial So again by the Darboux theory of integrability [13,20] system (5) has the rational first integral is also a Darboux polynomial with the cofactor K 4 = 2b 1 y 1 − 2b 2 y 2 − 1. Moreover, F 4 can be written in .
Note that the third and fourth equations of system (5) are independent of x 1 , x 2 , and they form a 2-dimensional system with exactly two singularities which are respectively stable and unstable nodes. Easy calculations verify that F 4 = 0 is an invariant elliptic cylinder and contains the two singularities and two heteroclinic orbits connecting these two singularities. By computing the dynamics of this two dimensional system at the infinity of the (y 1 , y 2 )-plane we get the phase portrait as shown in Fig. 1 A B Figure 1. Phase portrait of the last two equations of system (5) Lifting the invariant orbits and singularities of the 2-dimensional system to the phase space R 4 of system (5), we get the two invariant planes F 1 = 0 and F 2 = 0 lifted from S 1 and S 2 respectively, and two families of 3-dimensional invariant cylinders, which are heteroclinic to F 1 = 0 and F 2 = 0. System (5) restricted to F 1 = 0 is linear and has the origin as an unstable star node. System (5) restricted to F 2 = 0 is also linear and has a unique singularity, which is a stable star node.
We can check that system (5) has exactly two singularities which are both saddles and have eigenvalues (1, 1, −1, −1), where Since F 1 = 0 passes through S * a1 , by the dynamics on F 1 = 0 it follows that system (5) has a 2-dimensional stable analytic manifold tangent to the (y 1 , y 2 )plane. Note that (H R , H I )| S * a1 = (0, 0), and (H R , H I ) = (0, 0) has a branch with the expression given by where A 1i and B 1i are polynomials of degrees two and one, respectively. Denote this branch by L a1 . Then L a1 is the global stable analytic manifold, and it approaches the infinity of F 2 = 0 when F 2 → 0. On L a1 all orbits except two ones positively tend to S * a1 and negatively go to the infinity of the invariant plane F 2 = 0. The exceptional two ones have one positively approaching S * a1 and negatively tending to the infinity of (x 1 , x 2 )-plane when y 2 1 + y 2 2 → ∞; and another one negatively approaching the infinity of the invariant plane F 2 = 0 and positively going to the infinity of (x 1 , x 2 )-plane when y 2 1 + y 2 2 → ∞.
. We can show that (H R , H I ) = (v 21 , v 22 ) has two branches F 2 = 0 and L a2 with the latter the 2-dimensional global unstable analytic manifolds of S * a2 , which approaches the infinity of the invariant plane F 2 = 0 when F 2 → 0. Moreover, on L a2 all orbits except two ones negatively tend to S * a2 and positively go to infinity of the invariant plane F 1 = 0. The exceptional two ones have one negatively approaching S * a2 and positively tending to the infinity of (x 1 , x 2 )plane when y 2 1 + y 2 2 → ∞; and another one positively approaching the infinity of the invariant plane F 1 = 0 and negatively going to the infinity of (x 1 , x 2 )-plane when y 2 1 + y 2 2 → ∞. For ∈ {(0, 0), (v 21 , v 22 )}, we get that the level set (H R , H I ) = , denoted by L a , satisfies the following expressions with A 1 , B 1 polynomials of degree six in y 1 , y 2 , whose expressions will not be presented here, because they are tedious and could be obtained by direct calculations. Some further calculations show that L a is defined in the full (y 1 , y 2 )-plane except at S 1 and S 2 , where the 2-dimensional invariant manifold L a approximates the infinity of F 1 = 0 and F 2 = 0, respectively. Moreover, on L a all orbits are topological lines, and are heteroclinic to the infinity of the (x 1 , x 2 )-plane, either along F 1 = 0, or along F 2 = 0, or during the process y 2 1 + y 2 2 → ∞. This proves statement (II) and consequently completes the proof of the theorem.
Hence F c1 is always a Darboux polynomial of system (9), and so are F c2 and F c3 provided that b 1 and b 2 do not simultaneously vanish.
Moreover, we can check that the gradients of H Rc (ξ) and H Ic (ξ) are linearly independent in R 4 if and only if ξ does not belong to the invariant sets {F c1 = 0} ∪ {F c2 = 0} ∪ {F c3 = 0}.
(I) b 01 = 0. Applying similar calculations as in the proof of Theorem 1.1 we get that system (9) has the same dynamics as those in statement (I) of Theorem 1.1. (II) b 01 = 0, i.e. b 2 1 + b 2 2 = 0. By the above calculations it follows that F c2 and F c3 are also Darboux polynomials of system (9). In addition, we can rewrite F c2 and F c3 in the form Hence, F c1 = 0, F c2 = 0 and F c3 = 0 are three invariant planes, which contain respectively exactly one of the three singularities, saying S * c1 , S * c2 and S * c3 , of system (9). Furthermore, we can check that is also a Darboux polynomial of system (9). The invariant hyperplane F c4 = 0 passes through the three singularities S * c1 , S * c2 and S * c3 . On the invariant plane F c1 = 0 system (9) is reduced tȯ On the invariant plane F c2 = 0 system (9) is reduced tȯ On the invariant plane F c3 = 0 system (9) is reduced tȯ is a 2dimensional surface, denoted by L c , whose expression is where . Note that the last two equations of system (9) are independent of x 1 , x 2 , and have three singularities , which are stable, unstable and stable nodes, respectively. Further studying the dynamics of these two equations in the (y 1 , y 2 )-plane using the Poincaré compactification [6] we get the phase portrait of the last two equations of system (9) in the Poincaré disc as shown in Fig. 2.
A C B Figure 2. Phase portrait of the last two equations of system (9) Lifting the orbits in Fig. 2 we get invariant planes and 3-dimensional invariant surfaces of system (9). The three singularities S c1 , S c2 and S c3 are lifted to the three invariant planes F c1 = 0, F c2 = 0 and F c3 = 0, all the other regular orbits which are heteroclinic are lifted to 3-dimensional invariant surfaces of system (9).

4.
Conclusions. Complex 2-dimensional polynomial Hamiltonian systems whose dynamics were characterized are very few. Even for complex planar quadratic and cubic Hamiltonian systems, it is also open on the characterization of their global dynamics. As a first step to understand the global dynamics of this kind of systems, here we focus on some special systems, for instance the locally linearizable complex 2-dimensional Hamiltonian systems. By finding new invariants together with qualitative theory of dynamical systems we completely characterize the global dynamics of the mentioned systems.