The solution of some persistent p : − q resonant center problems

The notion of p : −q resonant center was introduced recently and studied by several authors. In this paper we generalize the notion of a persistent center to a persistent p : −q resonant center and find conditions for existence of a persistent p : −q resonant center for several p : −q resonant systems with quadratic nonlinearities. To prove the sufficiency of the obtained conditions we use either the Darboux theory of integrability or look for a formal first integral of the required form or we use the method based on the blow-up transformation.


Introduction
An essential part of the theory of systems of ODE's is devoted to studying the so-called centerfocus problem of two dimensional analytic systems of ordinary differential equations of the form where u, v are real variables and P(u, v), Q(u, v) are analytic functions whose series expansions start from terms of degree at least two.This is the problem of distinguishing between a center (all trajectories in a neighborhood of the singular point at the origin are ovals) and a focus (all trajectories in a neighborhood of the singular point at the origin are spirals).Most works on the subject are devoted to investigation of polynomial vector fields.By the Poincaré-Lyapunov Corresponding author.Email: matej.mencinger@um.sitheorem [29,34], system (1.1) has a center at the origin if and only if there exists an analytic first integral of the form The theorem says that the qualitative picture of trajectories in a neighborhood of the singular point is related to local integrability of the system: the singular point is a center if and only if there exists an analytic first integral of the form (1.2).
Although the problem of distinguishing between a center and a focus has been studied in many works (see [3,20,35,37,42] and the references therein) it is completely solved only for quadratic systems (P and Q in (1.1) are homogeneous quadratic polynomials [13,28]) and for systems with P and Q being homogeneous cubic polynomials [36].An extensive bibliography about the center problem can be found in [21].
One of the tools to study the problem of distinguishing between a center and a focus is the Poincaré return map, which we compute after introducing polar coordinates in system (1.1).The difficulty in the study of the center problem using this method arises from the complexity in computing the irreducible decomposition of the variety of the ideal generated by the Lyapunov quantities that are the coefficients of the Poincaré first return map.Since it is easier to study complex varieties than real ones we complexify the real system as follows.
Adjoining to this equation its complex conjugate we have the system ẋ = ix + F(x, x), ẋ = −i x + F(x, x).
Consider y := x as a new variable and G = F as a new function.Then, from the latter system we obtain the system of two complex differential equations which we can write in the form ẋ = ix + F(x, y), ẏ = −iy + G(x, y), where x, y are complex variables and F(x, y) and G(x, y) are complex analytic functions whose series expansions start from degree at least two.After the change of time τ = it and rewriting t instead of τ, system (1.3) becomes Following the Poincaré-Lyapunov theorem and [13] we can extend the concept of a center to complex systems of the form (1.4).We say that system (1.4) has center at the origin if it admits a formal first integral of the form In such case we also say that system (1.4) has 1 : −1 resonant center.In [19] the following generalization of the center problem was proposed.Consider differential systems in C 2 with a p : −q resonant elementary singular point, i.e., ẋ = px + P(x, y), where p, q ∈ N and P(x, y) and Q(x, y) are polynomials of the form Determine when the elementary singular point located at the origin is a resonant center where the definition of a resonant center comes from Dulac [13].
In this paper we will use the concept of (weakly) persistent center which was introduced in [7].In [2] the authors generalized the notion of persistent center and weakly persistent center for complex planar differential systems.In [33] these notions were extended to linearizable persistent centers and linearizable weakly persistent centers for complex planar differential systems.
Definition 1.2.The origin O is a (weakly) persistent center of system (1.4) if it is a center of the system ẋ = x + λF(x, y), We now extend the notion of (weakly) persistent center to a (weakly) persistent p : −q resonant center and introduce the following generalization of a p : −q resonant center.
Definition 1.3.The origin O is called a persistent p : −q resonant center (weakly persistent p : −q resonant center) of system (1.5) if it is a p : −q resonant center of the system In [2, Theorem 2.1] the following Theorem was proven for p = q = 1.
To prove the above theorem one just has to rewrite the proof of [2, Proof of Theorem 2.1] and change "center" to "p : −q resonant center".
In this paper we seek for systems having p : −q resonant center within systems of the form (1.7), where P and Q are quadratic polynomials and (p, q) is either (1,2), (1,3), (1,4) or (2,3).Such systems are written as where x, y, a ij , b ji ∈ C. To find necessary conditions we use the approach described in the next section.Then, using several methods we prove the existence of a first integral of the form (1.6).

Preliminaries
To determine if system (1.5) has a resonant center at the origin, by Definition 1.1 we look for a formal first integral of the form (1.6) satisfying the identity Similar as in case of a regular center the (formal) series for Ψ reduces to Ψ = g q,p (x q y p ) 2 + g 2q,2p (x q y p ) 3 + g 3q,3p (x q y p ) 4 + • • • , where g kq,kp is called the k-th saddle quantity (or k-th focus quantity [35]).Saddle quantities are polynomials in the coefficients a ij , b ji of system (1.5).We see that by Definition 1.1 system (1.5) has a resonant center at the origin if and only if Thus, to obtain conditions for resonant center at the origin of system (1.5) we have to find the set of all parameters (a, b) where all polynomials g kq,kp vanish, i.e. we need to find the variety of the ideal g kq,kp : k = 1, 2, . . . 1 .
If we restrict our attention to the systems (1.7), then, for any fixed λ and µ we can easily compute g kq,kp = g kq,kp (λ, µ, a, b) and obtain saddle quantities which can be considered as polynomials in λ and µ.Furthermore, the coefficient g (m,n) kq,kp (a, b) in the term with λ m µ n plays an important role in the analysis of the persistent resonant centers.We call it the k (m,n) -th persistent saddle quantity.If the origin is a center of system (1.7) for all λ, µ ∈ C, then it is by Definition 1.3 a persistent center of system (1.5).
Because of Theorem 1.4 we can for (1.7) always assume that λµ = 0. We now define the following sets of polynomials Therefore, in order to find necessary conditions for the existence of a persistent p : −q resonant center for system (1.5) we have to find first few saddle quantities and then to compute the variety of the ideal generated by these saddle quantities.
Note that the variety of the ideal C p is always easier to obtain than the (regular) center variety V( g kq,kp (a, b) : k ∈ N ) since the saddle quantities, g (m,n) kq,kp (a, b) are split compared to (regular) saddle quantities g kq,kp (a, b).Also note that if system has persistent p : −q resonant center, then it also has p : −q resonant center which will be useful fact in the next section where for some cases the (regular) p : −q resonant center problem has been solved, already.
In the following section we present the main results of the paper.We find necessary and sufficient conditions for some persistent p : −q resonant quadratic systems.For proving the sufficiency of the obtained conditions we mainly use the Darboux theory of integrability which is one of the main methods for proving the existence of first integrals for polynomial systems of differential equations on C 2 (or R 2 ).We recall briefly some results related to this theory.We consider systems where x, y ∈ C, P and Q are polynomials without constant terms that have no nonconstant common factor, and m = max(deg(P), deg(Q)).By the definition a Darboux factor of system (2.1) is a polynomial f (x, y) such that where K(x, y) is a polynomial of degree at most n − 1 (K(x, y) is called the cofactor).The polynomial f defines an invariant algebraic curve f = 0 of system (2.1).A simple computation shows that if there are Darboux factors then the equation admits the Darboux integrating factor 2) The definition of Darboux integrating factor is consistent with the classical definition of an integrating factor.For proving the sufficiency of the conditions concerning the existence of Darboux integrating factor we several times refer to following theorem, or more precisely to part (ii) of the following theorem.
has a local (reciprocal) integrating factor of the form (2.2), with f i analytic in x and y and nonzero α i , then the system is • integrable if q is irrational; • integrable or orbitally normalizable if q is a nonzero rational.
More precisely, (i) if all f i (0, 0) = 0, then the system is integrable; (ii) if at most one f i (0, 0) vanishes and the corresponding Darboux factor has one of forms f i (x, y) = x + o(x, y) and f i (x, y) = y + o(x, y), then the system is integrable; (iii) if exactly two factors f 1 (x, y) = x + o(x, y) and f 2 (x, y) = y + o(x, y) vanish at the origin, then the system is integrable, except when the two coefficients α 1 and α 2 are both integers greater than 1, in which case it is orbitally normalizable;

Main results
In this section we consider the problem of persistent p : −q resonant center of system (1.8) for the following values of p and q: According to Definition 1.3 we look for a systems with resonant center within the family for all λ, µ ∈ C. a) Case p = 1, q = 2.
Theorem 3.1.System (1.8) has a persistent 1 : −2 resonant center at the origin if and only if one of the following four conditions holds: In order to obtain conditions listed above we compute first four saddle quantities g 2,1 , . . ., g 8,4 of system (3.1) and obtain and so on.Therefore, we have . In a similar way we also obtain C 2 , C 3 and C 4 and it turns out that C Hence, using the routine minAssGTZ [9] of computer algebra system Singular [26] we compute the decomposition of the variety of ideal C p 3 and obtain four components listed in Theorem 3.1.For the sufficiency of these conditions we use [18,19], where authors solved the resonant center problem for system (1.8) with (p, q) = (1, 2).They found 20 conditions for a resonant center and among them there are also the above listed four conditions corresponding to the persistent resonant centers.Since in [19] the authors showed that in each of the 20 cases there is an analytic first integral of the form (1.6), the proof of this theorem is completed.Proof.The computation of obtained conditions goes in a similar way as in previous case.The sufficiency is ensured using [12] where 1 : −3 resonant center problem for system (1.8) was solved.Proof.The computation of saddle quantities and the corresponding ideals is similar as in previous two cases.The above five persistent resonant center cases of system (1.8) are listed among 55 conditions (proven to be necessary and sufficient) for the existence of a 1 : −4 resonant center in [17].Consequently, we have five necessary and sufficient conditions for persistent 1 : −4 resonant centers.The 2 : −3 resonant center problem (1.5) was solved only when (1.5) is cubic Lotka-Voltera system, see [10,22].In the quadratic case the problem is still open.Here, we do not give a complete list of all 2 : −3 resonant center conditions for (1.8), but we present three systems with resonant center, which are also persistent resonant center.The proof of Theorem 3.5 is given in Section 5.

Proof of sufficiency of Theorem 3.4
To prove the sufficiency of the conditions, we apply the Darboux theory of integrability to construct the Darboux integrating factor in all cases unless in case 3, where we look for a formal first integral of the form Ψ(x, y) = ∑ ∞ k=1 f k (x)y k .Below is the case-by-case analysis.Case 1.In this case system (3.1) has the form: We find two invariant lines l 1 = y and l 2 = y − 5 b 01 µ , which help us to construct the Darboux integrating factor M = l −4/5 1 l −(5a 01 λ+6b 01 µ)/5b 01 µ 2 for b 01 = 0 and µ = 0.By Theorem 2.1 there exists a first integral of the form (1.6) with p = 1 and q = 5.Remark 4.1.By Theorem 1.4 we can conclude, that origin is a center of system also for µ = 0.If b 01 = 0 this case coincides with the Case 3 of this theorem for a 10 = 0. Note that in this case the rational functions f k (x) become polynomial.
Case 2. In this case system (3.1) is written as: and it has invariant line l 1 = x and two invariant curves which allows us to construct a Darboux integrating factor M = l 4 1 (l 2 l 3 ) −1 .To prove the existence of a first integral of the form (1.6) with p = 1 and q = 5 we refer to Theorem 2.1.

Case 3.
In this case we find only one invariant curve f 1 = y, which is not enough to construct Darboux first integral or Darboux integrating factor.Note that conditions in this case are b 2,−1 = 0, b 10 = 0 and b 01 = 0, the corresponding system is ẋ = x + λ(a 10 x 2 + a 01 xy + a −12 y 2 ), ẏ = −5y.
We look for a formal first integral in the form Ψ(x, y) = ∑ ∞ k=1 f k (x)y k .The functions f k are determined recursively by the differential equation For k = 1, 2, 3, 4 (setting the integration constant equal to 1) we obtain Suppose by induction that f k (x) = p 5k (x) (1+a 10 λx) 5k , where p 5k (x) denotes a polynomial of degree at most 5k and k = 1, . . ., n − 1.In order to complete this task we solve the differential equation using the induction assumption about the form of f n−1 and f n−2 .
The general solution of linear differential equation of the form In this case g(x) = 5n x(1+a 10 λx) and h(x) = p 5n−4 (x) x(1+a 10 λx) 5n−3 , yielding e g(x)dx = x 5n (1+a 10 λx) 5n and Rewriting e − g(x)dx h(x) as and integrating, yields for some a 0 , a 1 , . . ., a 5n−1 .Therefore, using (4.3) and choosing integration constant as C = 1 we obtain the solution of (4.1) where p 5n denotes a polynomial of degree at most 5n.Therefore, it exists analytic first integral of the form Ψ(x, y) = ∑ ∞ k=1 f k (x)y k whose power series expansion is of the form , for a 10 = 0 and µ = 0. Thus, for a 10 = 0 and µ = 0 system (4.4) has a first integral of the form (1.6) with p = 1 and q = 5 according to Theorem 2.1.
In case µ = 0 we again refer to Theorem 1.4 and we can conclude, that origin is a center of system also for µ = 0. On the other hand if µ = 0, system (4.4) becomes subcase of system in Case 3 of this theorem, which is already proven to be integrable.
In case a 10 = 0 the corresponding system (4.4) is We look for a formal first integral in the form Ψ(x, y) = ∑ ∞ k=5 f k (y)x k .The functions f k (y) are determined recursively by the differential equation where f 3 (y) = f 4 (y) = 0. We seek for polynomial solutions of (4.5).We claim that a particular solution to (4.5) for k ≥ 5 is a linear polynomial.For k = 5 this is trivial to check, since (4. . Inductively, if f 5 (y), f 6 (y), . . ., f k−2 (y) and f k−1 (y) are linear polynomials, then clearly (4.5) takes the form where A k and B k are some constants.Note that this is a linear ODE of first order, whose nonhomegeneous part is being a linear polynomial A k + B k y.This clearly yields a particular solution for f k (y) in form of a linear polynomial.In particular, it is trivial to check that for k ≥ 7 the solution to (4.5) takes the form Now, setting C 5 = 1, C 6 = 0 and C k = 0 for k ≥ 7 proves the existence of an analytic first integral of the form Ψ(x, y) = ∑ ∞ k=5 f k (y)x k which is of the form (1.6) with p = 1 and q = 5.After the blow-up transformation (see [15] where this transformation is shown that it is useful to obtain the sufficiency) we obtain the following system We now look for the first integral of the system (5.2) of the form We compute Ψ = ∂Ψ(z,y) ∂z P (z, y) + ∂Ψ(z,y) ∂y Q (z, y) and for each k ≥ 5 set the coefficient of power y k to zero.Setting f 4 (z) = 0 this yields for k ≥ 5 the following recurrence differential equation for f k (z) and f k−1 (z) For k = 5, 6, 7, 8, 9 we find where p 4 (z), q 5 (z) and r 6 (z) are some polynomials of degree at most 4, 5 and 6, respectively.So, we assume that where R k−3 (z) = ∑ k−3 j=0 ρ j z j denotes a polynomial of degree at most k − 3. We prove this by induction.We have to solve the following differential equation where Suppose Then f k (z) = ∑ k−3 j=0 (j + 3) ρ j z j+2 , and It is very important to see that the coefficient to the highest power n in expression (5.4) vanishes Also, note that the lowest power of expression (5.4) is obviously z 2 .This implies that differential equation (5.3) becomes where W n−3 (z) is a polynomial of degree at most n − 3. From differential equation (5.5) according to (4.3) we have A direct integration yields e g(z)dz = z 3n 5 , z k+2− 3n 5 dz = 5w k z k+3 5(k+3)−3n , and finally For C = 0 we obtain f n (z) = z 3 Q n−3 (z) , where Q n−3 (z) is a polynomial of degree n − 3, which completes the proof.
We proved that the formal first integral of (5.2) is of the form Setting R k−3 (z) = ∑ k−3 j=0 ρ j z j and applying the inverse blow-up transformation z → x y , y → y yields  (5.7) We look for the first integral of the form Again we compute Ψ = ∂Ψ(z,y) ∂z P (z, y) + ∂Ψ(z,y) ∂y Q (z, y) and for each k ≥ 5 we set the coefficient of power y k to zero.For k ≥ 5 this yields the following recurrence differential equation for For k = 5, 6, 7, 8, 9 we find where p 5 (z), q 6 (z) and r 7 (z) are polynomials of degree at most 5, 6 and 7, respectively.Now we assume that where R k−2 (z) = ∑ k−2 j=0 ρ j z j denotes polynomial of degree at most k − 2. We prove this by induction.We have to solve the following differential equation where W n−2 (z) is a polynomial of degree at most n − 2. This is the case, since the term (µb 01 z − λ(a −12 + a 01 z + a 10 z 2 )) • f n−1 (z) contains the highest power in the above expression, deg(µb 01 z − λ(a −12 + a 01 z + a 10 z 2 )) = 2 and deg( which is of the form (4.2) and the corresponding solution is of the form (4.3).From differential equation (5.9) and (4.2) it follows A direct integration yields e g(z)dz = z 3n 5 , , and finally For C = 0 we finally obtain f n (z) = Q n−2 (z) , where Q n−2 (z) is a polynomial of degree n − 2, which completes the proof.We proved that the formal first integral of (5.7) is of the form which is a formal integral of system (5.6) of the form (1.6) with p = 2 and q = 3.After computing Ψ = ∂Ψ(z,y) ∂z P (z, y) + ∂Ψ(z,y) ∂y Q (z, y) we set the coefficient to power y k for each k ≥ 5 to zero.This yields for k ≥ 5 the following recurrence differential equation for f k (z) and f k−1 (z) (k − 1) µb 10 z f k−1 (z) − 3k f k (z) + 5z f k (z) + λa 01 z − µb 10 z 2 f k−1 (z) = 0.
For k = 5, 6, 7, 8, 9 we find f 5 (z) = z 3 , f 6 (z) = z 3 (λa 01 − µb 10 z), f 7 (z) = z 3 p 2 (z), f 8 (z) = z 3 q 3 (z), f 9 (z) = z 3 r 4 (z), where p 2 (z), q 3 (z) and r 4 (z) are polynomials of degree at most 2, 3, and 4, respectively.We can assume that f k (z) = z 3 R k−5 (z) , where R k−5 (z) = ∑ k−5 j=0 ρ j z j denotes a polynomial of degree at most k − 5. Again we use induction to prove this assumption.To this end we solve the following differential equation Now we can see that the highest power in expression (5.13) is n − 3 and the lowest power of expression (5.13) is obviously z 2 .This implies that differential equation (5.12) becomes where W n−5 (z) is some polynomial of degree at most n − 5. From differential equation (5.14) using (4.2) it follows For C = 0 we finally obtain f n (z) = z 3 Q n−5 (z) , where Q n−5 (z) is a polynomial of degree n − 5, which completes the proof by induction.
We proved that the formal first integral of (5.11) is of the form Similar as in previous two cases we set R k−5 (z) = ∑ k−5 j=0 ρ j z j and apply inverse blow-up transformation z → x y , y → y to obtain Ψ (x, y) = Ψ x y , y = x 3 y 2 + ψ 3,3 x 3 y 3 + ψ 4,2 x 4 y 2 + h.o.t., which is a formal integral of (5.10) of the required form.

(
iv) if (iii) is satisfied and there exists a Darboux change of one coordinate transforming one of the equations into the normal form ẋ = xh(u) or ẏ = −qyh(u), where h(u) = 1 + O(u) and u = x c y d is the resonant monomial as in Case II or Case III [4, Theorem 4.3], then the system is normalizable.