Bifurcation of critical periods of a quartic system

For the polynomial system ẋ = ix + xx̄(ax2 + bxx̄ + cx̄2) the study of critical period bifurcations is performed. Using calculations with algorithms of computational commutative algebra it is shown that at most two critical periods can bifurcate from any nonlinear center of the system.


Introduction
Consider a system of ordinary differential equations on R 2 of the forṁ where u and v are real unknown functions and P and Q are polynomials without constant and linear terms. The singularity at the origin of system (1.1) is either a center or a focus. In a neighborhood of a center the so-called period function T (r) gives the least period of the periodic solution passing through the point with coordinates (u, v) = (r, 0) inside the period annulus of the center. If T (r) is constant in a neighbourhood of the origin, then the center at the origin is called isochronous. For a center that is not isochronous any value r > 0 for which T (r) = 0 is called a critical period. The problem of critical period bifurcations is aimed on estimation of the number of critical periods that can arise near the center under small perturbations. It was investigated for the first time by Chicone and Jacobs [2] in 1989 for quadratic systems and some Hamiltonian systems. After that, many studies were devoted to the problem (see, e.g. [1,5,7,10,12,15,16,17,18,19,20,21] and references given there). One of difficulties in investigations of this problem is that before studying the critical periods bifurcation for a polynomial system one should resolve the center problem for the system, that is, find all systems in the family with a center at the origin.
Studies of the center problem are usually simpler if one considers the problem in the complex setting. To perform a complexification we can make the substitution x = u + iv obtaining from (1.1) the complex differential equatioṅ a jk x j+1xk . (1.2) Adjoining to (1.2) its complex conjugate and consideringā jk as a new parameter b kj andx as a distinct unknown function y we obtain the systeṁ a jk x j+1 y k = ix + P (x, y), This system is called the complexification of (1.1) and it is equivalent to (1.2) when y =x and b kj =ā jk . By Poincaré-Lyapunov theorem system (1.1) has a center at the origin if and only if it admits in a neighbourhood of the origin an analytic first integral of the form which is equivalent to the existence of a first integral of the form for system (1.2). Thus, extending the notion of center from real systems to systems (1.3) it is said that complex system (1.3) has a center at the origin if in a neighbourhood of the origin it admits an analytic first integral of the form (1.4) Since to each a jk in the first equation of (1.3) corresponds the parameter b kj in the second equation of (1.3), system (1.3) has 2 parameters. We denote the ordered 2 -tuple of the parameters of (1.3) by (a, b); that is, (a, b) = (a p1q1 , . . . , a p q , b q ,p , . . . , b q1p1 ), (1.5) and we use the notation C[a, b] for the ring of polynomials in the variables a p1q1 , a p2q2 , . . . , b q1p1 over C.
Recently, García, Llibre and Maza [6] studied limit cycle bifurcations near a center or a focus at the origin of the quintic system written in the complex form as the equationẋ which, in order to use the notation similar to the one in (1.2), we write as the complex equationẋ In this paper we study critical period bifurcations from the center at the origin of system (1.6). We first describe a way to compute the period function of system (1.2) using the normal form of its complexification (1.3). Then we prove that at most three critical periods can bifurcate from any nonlinear center of the system.

Preliminaries
To study critical period bifurcations of system (1.6) we have to compute a series expansion of the period function T (r) of the system. One possibility is to pass to polar coordinates. This way is geometrically and theoretically straightforward, however it is not computationally efficient since one needs to compute integrals of trigonometric polynomials, and this is a difficult task in the case of polynomials of high degree.
Another possible computational approach relies on calculations of Poincaré-Dulac normal form of the complexification (1.3). We briefly remind it following to [14] and [5].
As it is well-known after a change of coordinates (2. 2) The normal form (2.2) is not uniquely defined since the so-called resonant coefficient h and by Y K the ideal generated by the first K pairs of the coefficients, The normal form of a particular system (a * , b * ) with the fixed parameters is linear when all the coefficients of the normal form evaluated at (a * , b * ) are equal to zero, Y that is, when the point (a * , b * ) belongs to the variety of the ideal Y. The variety V(I) of a polynomial ideal I is the set of common zeros of all polynomials of the ideal. The variety V L := V(Y) is called the linearizability variety of system (1.3).
As it is well known system (1.1) has an isochronous center at the origin if and only if the system is linearizable. Thus, the real systems (1.1), which parameters after the complexification are in V L , have isochronous centers at the origin.
For system (1.3) one can find a function (1.4) such that where g kk is a polynomial in the coefficients of system (1.3). The polynomial g kk is called the k-th focus quantity. Clearly, system (1.3) with fixed coefficients (a * , b * ) has a center at the origin if and only if g kk ≡ 0 for all k ∈ N. We call the ideal the Bautin ideal of system (1.3). The variety of B, V C = V(B), is called the center variety. We will also use the ideal generated by the first K focus quantities, which we denote Let us denote It is easy to see that the origin is a center for (1.3) if and only if G ≡ 0, in which case H has purely imaginary coefficients and the distinguished normalizing transformation converges. We also define When (1.3) is the complexification of a real system one can recover the real system by replacing every occurrence of y 2 byȳ 1 in each equation of (2.2). In such case, performing the transformation y 1 = re iϕ we obtain from (2.2) the equations forṙ andφ as follows:ṙ We write the function H as The integration of the second equation in (2.4) gives the least period of the periodic solution of (??)assing through the point with coordinates (r, 0) as for some coefficients p 2k . The center at the origin of system (1.6) corresponding to a parameter a * is isochronous if and only if p 2k (a * ,ā * ) = 0 for k ≥ 1.
It is easy to see that p 2k are polynomials in the parameters a,ā of system (1.2). We can extend the polynomial functions p 2k (a,ā) to the set of parameters (a, b) setting in (2.4) y 2 instead ofȳ 1 . Then instead of (2.5) we obtain the function which coincides with the period function (2.5) when b =ā.
We call the polynomial p 2k (a, b) in (2.6) the k-th isochronicity quantity. Using (2.5) and the formula for the inversion of series the first three polynomials p 2k are computed as: Since values of the isochronicity quantity p 2k are of interest only on the center variety, we should work with the equivalence class [p 2k ] of p 2k in the coordinate ring C[V C ] of the center variety, which can be viewed as the set of equivalence classes of polynomials C[a, b] by V C . That is, for polynomials f, g ∈ C[a, b], We denote and for K ∈ N, The ideal P is called the isochronicity ideal.
Finally, we remind that given a Noetherian ring R and an ordered set 3. An upper bound for critical periods bifurcating from centers of system (1.6) Along with system (1.6) we consider its complexificatioṅ Our study is based on the following theorem which is an immediate corollary of [5, Theorem 5. is m, and (c) a primary decomposition of P K + √ B can be written R ∩ N where R is the intersection of the ideals in the decomposition that are prime and N is the intersection of the remaining ideals in the decomposition. Then for any system of family (1.2) corresponding to (a * ,ā * ) ∈ V C \ V(N ), at most m − 1 critical periods bifurcate from a center at the origin.
Thus, to estimate the number of bifurcating critical periods of system (3.1) we have to know the center and linearizability varieties of the system.
First we note that it follows from Corollary 3.4.6 in [14] that for system (3.1) the focus quantities g 2k+1,2k+1 are zero polynomials. Using the results of [9] we can easily prove the following statement. where B 14 = g 2,2 , g 4,4 , g 6,6 , g 8,8 , g 10,10 , g 12,12 , g 14,14 , and it consists of four components defined by the following prime ideals: Proof. Using the algorithm in [14, Chapter 3] and a Mathematica code similar to the one given in [14, Fig. 6.1 of Appendix] we computed the focus quantities g 2,2 , g 4,4 , . . . , g 14,14 (since the expressions are long, we do not present them here, but one can easily compute them using any available computer algebra system). Then, using the routine minAssGTZ, which is based on the algorithm of [8], of the computer algebra system Singular [3] we found that the minimal associate primes of B 14 are the prime ideals I 1 , . . . , I 4 in the statement of the theorem.
By the results of [9] if the parameters (a, b) of system (3.1) are from one of the varieties V(I 1 ), . . . , V(I 4 ), then the corresponding systems have a center. This means that (3.2) holds.
Note, that taking into account that V(B) is a complex variety, from (3.2) we obtain that the radical of B coincides with the radical of B 14 , that is, To find the linearizability variety of system (3.1) and the isochronicity quantities p 2k we have computed the normal form of system (3.1) up to the order 17 and found four first non-zero pairs of the resonant coefficients Y Then, using (2.7) for the calculation of p 4 and computing the series expansions (2.6) in order to find p 8 , p 12 and p 16 we obtain the first four non-zero reduced isochronicity quantities (by the reduced quantities we mean the polynomials obtained in such way that in formulas (2.7) and their extensions to any p 2k only terms containing the highest order coefficients of the normal form are taking into account; it is sufficient to work with the reduced quantities since the other terms of p 2k are in the ideal p 2 , . . . , p 2k−2 ) of system (3.1) as follows: Proof. Using the routine minAssGTZ of Singular we found that the minimal associate primes of ideals Y 8 and B 14 , P 8 are the same. Namely, they are the ideals: By the results of [13] systems with the coefficients from the varieties of these ideals are linearizable. This proves (3.3).
We now can estimate the number of critical periods near a center at the origin of system (1.6). To this end, with the routine radical of the computer algebra system Singular we compute the radical of the Bautin ideal B = B 14 denoted R 14 , that is, (one can also compute R 14 using the routine intersect of Singular and the ideals I 1 − I 4 given in the statement of Proposition 3.2 since it is follows from the proof of Proposition 3.2 that R 14 = ∩ 4 k=1 I k ). Then with the reduce of Singular we check that for k = 2, 3, 4 the remainder of the division of the polynomial p 4k by a Groebner basis of the ideal p 4 , . . . , p 4(k−1) , R 14 is nonzero. That means, that (3.4) holds, which, in turn, yields that the Bautin depth of P 8 in C[V C ] is 4.
Then, with the routine primdecGTZ [4,8] of Singular we have computed the primary decomposition of the ideal Q = P 8 , R 14 and found that Q = ∩ 13 k=1 Q k , where Q 1 , . . . , Q 6 are prime ideals given in the statement of Proposition 3.2, Q 7 , . . . , Q 13 are some ideals defined by many polynomials (for these reason we do not present them here, however the interested reader can easily compute Q and the primary decomposition Q = ∩ 13 k=1 Q k with an appropriate computer algebra system using the ideals P 8 and I 1 − I 4 presented above) whose associate primes are:

Thus,
√ Q k = Q k for k = 1, . . . , 6 and √ Q k = Q k for k = 7, . . . , 13; that is, the ideals R and N from the statement of Theorem 3.1 are R = ∩ 6 k=1 Q k and N = ∩ 13 k=7 Q k . To find systems (1.6) whose coefficients are in the variety of the ideal N we perform as follows. Let T s = Q s+6 for s = 1, . . . , 7. Using the intersect of Singular we compute the ideal T = ∩ 7 k=1 T k and find that T   Substituting the values from (3.6) into polynomials of the ideal T R given in (3.5) we find that also A 13 = B 13 = 0.
It means that the only system of the form (1.6) whose parameters are in the variety of the ideal N is the linear system (1.2), that is, the systemẋ = ix. Thus, by Theorem 3.1 at most 3 critical periods bifurcate from non-linear isochronous centers of system (1.6).