Integrability conditions of a weak saddle in generalized Liénard-like complex polynomial differential systems
- 10.1080/14029251.2020.1819612How to use a DOI?
- Integrability problem; weak saddle; Liénard-like complex polynomial differential systems
We consider the complex differential systemwhere f is the analytic function with aj ∈ ℂ. This system has a weak saddle at the origin and is a generalization of complex Liénard systems. In this work we study its local analytic integrability.
- © 2020 The Authors. Published by Atlantis Press and Taylor & Francis
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1. Introduction and statement of the main results
The center problem for polynomial vector fields in the real plane with an elementary singular point of the form(where h.o.t. means higher order terms) has been the subject of many investigations during these last decades, see for instance [4–6, 10, 11, 14, 16] and the references therein. These type of systems can be embedded by the complex change of variables u = x + iy and v = ū = x − iy into the complex system The next extension of the above system is to consider analytic vector fields in ℂ2 of the form
In this paper we give a simple and self-contained proof of the characterization of the local analytic integrability of a complex analytic differential system in ℂ2 of the form
Systems with linear part of the form (1.3) have a weak saddle at the origin and were studied by several authors, see for instance [12, 13] where the nonlinearities only appear in one equation. The integrability of Liénard systems with a weak saddle were considered in . System (1.3) is a generalization of these Liénard systems with a weak saddle at the origin, already started to be studied in .
Of course we also obtain the characterization of the C∞ integrability of system (1.3) because in the proof we will obtain a formal first integral H that can be C∞ or analytic around the origin. However by results given in  the first integral is always analytic. With the method used in the proof we cannot characterize the existence of a less regular first integral, for instance a Ck first integral.
First we consider the case in which a1 ≠ 0, and after that when a1 = 0, considering both cases in different theorems.
The case in which a1 = 0 with n even is more involved and we can only solve it completely for each fixed degree of f less than or equal to 8 (remains open the problem for a degree greater than 8 which is outside of our current computing facilities).
n odd and aj = 0 for j even,
n ⩾ 4 even and aj = 0 for all j that is not of the form j = (2i + 1)(n − 2) + 2 with i ⩾ 0. Moreover we can give the following sufficient condition of integrability.
Consider system (1.3) with with n > 1 and an ≠ 0. If n ⩾ 4 is even and aj = 0 for all j that is not of the form j = (2i + 1)(n − 2) + 2 with i ⩾ 0, then it is locally integrable at the origin.
System (1.3) with with n > 1 and an ≠ 0 is locally integrable at the origin if and only if one of the conditions (i) or (ii) holds.
Note that system (1.3) has a resonant saddle [1 : −1] at the origin. The classical Liénard system is given byChoosing the variables it can be transformed to This system has been studied by several authors in the last decades, see  and the references therein. Recently in  it was given the characterization of the integrable complex analytic differential systems in ℂ2 of the form This system has a weak saddle at the origin which corresponds with the [1 : −1] resonance case. Our result is then a generalization of this result because we study the Liénard-like system (1.3) that has also a weak saddle at the origin.
2. Proof of Theorem 1.1
We prove Theorem 1.1 for n = 1 and we separate it into the sufficiency and the necessity part. The proof of the general case can be obtained changing a1 by an.
Proof of sufficiency
Doing the affine change of variables
Hence system (2.3) is invariant by the symmetry (u, v, t) → (u, −v, −t). Taking z = v2 and the scaling of time dt = vdτ we get a non-singular point at the origin. The first integral which exists around the origin by the Flow-box theorem can be pulled back to a first integral of the form H(x, y) = xy+h.o.t. of the original system. So, sufficiency is proved.
Proof of necessity
Consider with a1 ≠ 0. As explained in the introduction, to find the saddle quantities we propose a formal first integral of the form
Now we compute the derivative of H along the vector field associated to system (1.3) and we obtain a linear system for each function Hk. Note thatand So, a monomial Rl1,l2xl1yl2 can be written as where
Tl,0 = T0,l = 0 for l ⩾ 1 and T1,1 = 0,
for l ⩾ 1,and (we recall that c1,1 = 0),
for l1 ≠ 2 and l2 ≠ 2 with l1 + l2 ⩾ 3,with the convention that if i = 1 then l2 − 1 ⩾ 2 and if j = 1 then l1 − 1 ⩾ 2.
We first compute H3. In this case we have the linear system
From the second and third equations we get
The linear system for H4 is
Taking into account that c1,2 = −c2,1 we deduce from the third equation that a2 = 0 and thus obtaining a first condition to have formal integrability of system (1.3). Moreover from the second and third equations we getthat is
Now we compute the linear system for H5 and we getwhere
Taking into account that a2 = 0 we get that T2,3 = 3c3,1a1 and T3,2 = 3c1,3a1. Moreover, since we conclude that
Now we prove the theorem by induction. Our induction hypothesis will be that for each n, we haveand if n is even, then
Note that since for i ≠ j we have ci,j = Ti,j/(i − j), we can check that(the case in which i = j is trivially satisfied). In fact, until now we have proven the induction hypothesis for n = 2 and n = 3.
First we observe that it follows from the induction hypotheses that
Indeed, taking the notationby the induction hypotheses (i.e., a2 = 0 and al+1−j = 0 for l + 1 − j even) and taking into account that in the definition of Ti,j given in (i)–(iii) we get that i + j < n, we readily obtain:
Hence, if l is odd thenand if l is even (since by assumptions al = 0)
Furthermore, proceeding in the same manner, taking the notation p = l1 + l2, we get
In particular, if n is odd the determinant of the above linear system is different from zero and hence the system is compatible and determined and so we can determine all the coefficients ci,j with i + j = n in the form
In particular, for either i = 1 or j = 1 we get (see the definition of Tl1,1)
On the other hand, for m = n even, the determinant of the corresponding linear system is zero, so we have that all the coefficients ci,j where i + j = n with (i, j) ≠ (n/2, n/2) are completely determined and we have the extra condition
Note that the conditions with i + j = n with (i, j) ≠ (n/2, n/2) follow exactly as in the case n odd and so we obtain that
Now take m = 2n. Then proceeding as above we have the condition
So, if n is odd we get the identity 0 = 0 and if n is even we get the identityas we wanted to show. This shows that when a1 ≠ 0 system (1.3) has a formal first integral when f is odd. Hence we have proved the necessity, completing the proof of Theorem 1.1.
3. Proof of Theorem 1.2
We first prove the sufficiency of Theorem 1.2. System (1.3) under the assumptions (i) of Theorem 1.2 takes the form (2.1) with a1 ≠ 0, or with a1 = 0. Both cases are included in Theorem 1.1. Hence the proof given in Theorem 1.1 is also valid for this case.
ak = 0 for k ≠ 4m with m ∈ ℕ,
ai = 0 for all i ⩽ 8 except for i = 6.
In fact case (a) is valid for any degree of f so we present the general proof of this case. Under the assumptions of case (a) system (1.3) takes the form
Doing the affine change of variables in (2.2), system (3.1) becomes system (2.3). Now proceeding as in the proof of the sufficiency part of Theorem 1.1 and we conclude that there exists a first integral of the form H(x, y) = xy + h.o.t. of the original system.
In the case (b) system (1.3) with f of degree less than or equal to 8 becomes
System (3.2) has the analytic first integralthat is well-defined around the origin. Hence we have an integrable saddle at the origin.
Now we prove the necessity of Theorem 1.2. As in the proof of Theorem 1.1. to find the saddle quantities we propose a formal first integral of the form in (2.4)–(2.5). We first compute H3. In this case we have the linear system (2.6) with a1 = 0. Since the determinant of that linear system is compatible and determined, we get that ci,j = 0 for i + j = 3. The linear system for H4 is the one given in (2.7) with a1 = 0. From the third equation we get that a2 = 0. So, a condition to have a formal first integral in this case is a2 = 0. This proves that a necessary condition for system (1.3) with a1 = 0 to have a formal first integral is a2 = 0. The linear system for H5 is
From here we have c5,0 = c4,1 = c1,4 = c0,5 = 0, c3,2 = −a3 and c2,3 = a3. The linear system for H6 gives the following result c6,0 = c5,1 = c3,3 = c1,5 = c0,6 = 0, c4,2 = −a4/2 and c2,4 = a4/2. Solving the different linear systems for H7, H8, H9, H10 and H11 we do not find any extra necessary condition. The linear system for H12 gives the condition . However to go further with this method is very difficult and we do not see how to prove by induction the conditions for any degree of f.
So, now we have fixed the degree of f less than or equal to 8 (due to the fact that the machine does not allow us to go further). In order to compute the necessity in this case we use the following method. Using the change of variables X = x + iy, Y = x − iy and the scaling of time t ↦ −t/i system (1.3) takes the form
Taking polar coordinates X = r cosθ and Y = r sinθ system (3.3) becomes
4. Proof of Theorem 1.3
Now we propose the change of variablesDoing this change we obtain that becomes Note that
Now we claim that
Proceeding in the same manner we have that for ℓ oddwhere in the last step we have used the induction hypothesis. So, and claim (4.5) is proved.
Note that proceeding as above,where is a polynomial. Hence, by equation (4.2) we have
Analogously we havewhere is a polynomial. Hence, by equation (4.3) we get
Next we make a rescaling of time of the form dτ = (xn−2 + yn−2)dt. In this way the above system becomeswhere the dot means derivative in the new time τ. The above system does not have a singular point at the origin. Hence since it is polynomial in the variables (X, Y) it has an analytic first integral around the origin by the Flow-box theorem. This first integral can be pulled back to a first integral of the form H(x, y) = xy + h.o.t. of the original system. This completes the proof of the theorem.
The authors would like to thank the reviewers comments and suggestions that significantly improved the initial version of the paper. The first author is partially supported by a MINECO/ FEDER grant number MTM2017-84383-P and an AGAUR (Generalitat de Catalunya) grant number 2017SGR 1276. The second author is supported by FCT/Portugal through UID/MAT/04459/2013.
Cite this article
TY - JOUR AU - Jaume Giné AU - Claudia Valls PY - 2020 DA - 2020/09/04 TI - Integrability conditions of a weak saddle in generalized Liénard-like complex polynomial differential systems JO - Journal of Nonlinear Mathematical Physics SP - 664 EP - 678 VL - 27 IS - 4 SN - 1776-0852 UR - https://doi.org/10.1080/14029251.2020.1819612 DO - 10.1080/14029251.2020.1819612 ID - Giné2020 ER -