Local bifurcation of limit cycles and center problem for a class of quintic nilpotent systems

For a class of fifth degree nilpotent system, the shortened expressions of the first eight quasi-Lyapunov constants are presented. It is shown that the origin is a center if and only if the first eight quasi-Lyapunov constants are zeros. Under a small perturbation, the conclusion that eight limit cycles can be created from the eight-order weakened focus is vigorously proved. It is different from the usual Hopf bifurcation of limit cycles created from an elementary critical point. Mathematical Subject Classification: 34C07; 37G10.


Introduction and statement of the main results
Two main open problems in the qualitative theory of planar analytic differential systems are characterizing the local phase portrait at an isolated critical point and the determination and distribution of limit cycles. Recall that a critical point is said to be of focus-center type if it is either a focus or a center. In what follows, this problem is called the focus-center problem or the monodromy problem, which is usually done by the blow-up procedure. Of course, if the linear part of the critical point is non-degenerate (i.e., its determinant does not vanish) the characterization is well known. The problem has also been solved when the linear part is degenerate but not identically null, see [1][2][3].
On the other hand, once we know that a critical point is of focus-center type, one comes across another classical problem, usually called the center problem or the stability problem, that is of distinguishing a center from a focus. The Poincaré-Lyapunov theory was developed to solve this problem in the case where the critical point is non-degenerate, see [4,5]. From a theoretical viewpoint, the study of this problem for a concrete family of differential equations goes through the calculation of the so-called Lyapunov constants, which gives the necessary conditions for center, see [6,7]. To completely solve the stability problem of polynomial systems of a fixed degree, although the Hilbert basis theorem asserts that the number of needed Lyapunov constants is finite, which is the number is still open.
Probably the most studied degenerated critical points are the nilpotent critical points. For these points, zero is a double eigenvalue of the differential matrix, but it is not identically zero. Nevertheless, given an analytic system with a nilpotent monodromic critical point it is not an easy task to know if it is a center or a focus. Analytic systems having a nilpotent critical point at the origin were studied by Andreev [1] in order to obtain their local phase portraits. However, Andreev's results do not distinguish between a focus and a center. Takens [8] provided a normal form for nilpotent center of foci. Moussu [3] found the C ∞ normal form for analytic nilpotent centers. Berthier and Moussu in [9] studied the reversibility of the nilpotent centers. Teixeira and Yang [10] analysed the relationship between reversibility and the center-focus problem for systemṡ where X(x, y) and Y(x, y) are real analytic functions without constant and linear terms, defined in a neighborhood of the origin.
It is well known that the dynamical behavior of a dynamical system depends on its parameters. As these parameters are varied, changes may occur in the qualitative structure of the solutions for certain parameter values. These changes are called bifurcations and the parameter values are called a bifurcation set. For a given family of polynomial differential equations usually the number of Lyapunov constants needed to solve the center-focus problem is also related with the so-called cyclicity of the point, i.e., the number of limit cycles that appear from it by small perturbations of the coefficients of the given differential equation inside the family considered (see [11] for cases where this relation does not exist). A classical way to produce limit cycles is by perturbing a system which has a center, in such a way that limit cycles bifurcate in the perturbed system from some of the periodic orbits of the period annulus of the center of the unperturbed system.
For a planar dynamical system, if the origin is an elementary critical point and the linearized system at the origin has a simple pair of pure imaginary eigenvalues ±iω, ω > 0, then, under a small perturbation of the parameters, a small amplitude limit cycle can be created in a small neighborhood of the origin. This local change of the phase portraits is called Hopf bifurcations.
If the origin is not an elementary critical point, when the parameters are changed, what happens in a small neighborhood of the origin? This bifurcation phenomena is called the bifurcation of multiple critical point. To the best of our knowledge, there are essentially three different ways, the normal form theory [6], the Poincaré return map [12] and Lyapunov functions [13], of studying the center-focus problem of nilpotent critical points, see for instance [3,14,15]. On the other hand, the three tools mentioned above have been also used to generate limit cycles from the critical point, see for instance [15][16][17], respectively.
In [18,19] it is proved that any analytic nilpotent center is limit of an analytic linear type center, i.e., given any nilpotent center of a system X 0 , there always exists a oneparametric perturbation X μ which has a center for any μ ≠ 0 such that for μ 0 we have that X μ X 0 . Therefore, any nilpotent center can be detected using the same methods that for a nondegenerate center, for instance the Poincaré-Liapunov method.
Here we are glad to highlight the work of Liu and Li [20], where a new definition of the focal value, quasi-Lyapunov constant, are given for the three-order nilpotent critical point. Meanwhile, the equivalence of quasi-Lyapuonv constant with focal value is proved. A linear recursive formula to compute quasi-Lyapunov constants is also presented. Afterward, they proved that if the three-order nilpotent origin is a m-order weakened focus, then, by a small perturbation for the unperturbed system, there exist m limit cycles in a neighborhood of the origin. At the same time, the origin becomes an elementary critical point and two complex singular points.
The aim of this article is to use the integral factor method introduced in [20], in order to compute what will be called quasi-Lyapunov constants (see Section 2) for a three-order nilpotent critical point in the following quintic system: (1:3) In addition, by applying them we give the center condition and lower bound for the cyclicity of the origin, i.e., N(5) ≥ 8.
Our main results are summarized in the following two theorems: Consider the following perturbed system of (1.3) dx dt = δx + y + a 50 x 5 + a 41 x 4 y + a 32 x 3 y 2 + a 14 xy 4 + a 05 y 5 , (1:5) When 0 <δ ≪ 1, in a small neighborhood of the origin, system (1.5) has exact three critical points: O(0, 0), (x 1 , y 1 ) and (x 2 , y 2 ), where O(0, 0) is an elementary critical point, another two critical points are complex with when δ 0, three critical points coincide to become the three-order nilpotent criti- Theorem 1.2. Assume that the origin of system (1.3) is an eight-order weakened nilpotent focus. Then, under a small perturbation of system (1.3), for a small parameter δ, in a neighborhood of the origin of system (1.5), there exist 8 limit cycles enclosing the elementary node O(0, 0). Theorems 1.1 and 1.2 will be proved in Sections 3 and 4, respectively.

Preliminaries
In this section, we summarize some definitions and results about the center-focus problem of three-order nilpotent critical points of the planar dynamical systems that we shall use later on. For more details and proofs about these results see [20].
In canonical coordinates the Lyapunov system with the origin as a nilpotent critical point can be written in the form: (2:1) Suppose that the function y = y(x) satisfies X(x, y) = 0, y(0) = 0. Lyapunov proved (see for instance [23]) that the origin of system (2.1) is a monodromic critical point (i. e., a center or a focus) if and only if where n is a positive integer. The monodromy problem in the case of a nilpotent singular point was also solved in [24].
be the unique solution of the function equation X(x, f(x)) = 0, f(0) = 0 at a neighborhood of the origin. If there are an integer m and a nonzero real number a, such that we say that the origin is a high-order singular point of system (2.1) with the multiplicity m.
It is different from the center-focus problem for the elementary critical points, we know from Lemma 2.2 that when k > 1 for the first non-zero ν k (-2π), k is an even integer.
(1) For any positive integer m, ν 2m (-2π) is called the m-th focal value of system (2.5) in the origin.
We say that f m is equivalent to g m , denoted by f m~gm . If f 1 = g 1 and for all positive integers m, f m~gm , we say that the function sequences {f m } and {g m } are equivalent, denoted by {f m }~{g m }.
We know from Lemma 2.2 and Definition 2.2 that for the sequence {ν k (-2π)}, k ≥ 2, we have ν 2k+1 (-2π)~0, k = 1, 2, .... We next state the results concerning with bifurcation of limit cycles of system (2.5). Consider the perturbed system of (2.5) where X(x, y), Y(x, y) are given by (2.6). Clearly, when 0 < |δ| ≪ 1, in a neighborhood of the origin, there exist one elementary node at the origin and two complex critical points of system (2.16) at (x 1 , y 1 ) and (x 2 , y 2 ), where (2:17) When δ 0, one elementary node and two complex critical points coincide to become a three-order critical point. Let We have that (2:23) Consider the system (2:24) where g = {g 1 , g 2 , ..., g m-1 } is (m-1)-dimensional parameter vector. Let m−1 } be a point at the parameter space. Suppose that for ∥g -g 0 ∥ ≪ 1, the functions of the right hand of system (2.24) are power series of x, y with a non-zero convergence radius and have continuous partial derivatives with respect to g. In addition, (2:25) For an integer k, letting ν 2k (-2π, g) be the k-order focal value of the origin of system (2.24) δ = 0 .
We give the following key results, which define the quasi-Lyapunov constants and provide a way of computing them.
Theorem 2.2. For system (2.5), one can construct successively a formal series i.e., one can construct successively the terms with the coefficients c ab satisfying a ≠ 0 of the formal series where for all k, M k (x, y) is a k-homogeneous polynomial of x, y and sμ = 0. Now, (2.35) can be written by It is easy to see that (2.36) is linear with respect to the function M, so that we can easily find the following recursive formulae for the calculation of c ab and ω m (s, μ). we can obtain a solution group of {c ab } of (2.41), thus, we have (2:42) Clearly, the recursive formulae presented by Theorem 2.4 is linear with respect to all c ab . Accordingly, it is convenient to realize the computations of quasi-Lyapunov constants by using computer algebraic system like Mathematica.

Proof of Theorem 1.1
Now we start the preparation of the proof of Theorem 1.1. Obviously, the origin of system (1.3) is a three-order nilpotent critical point which is a center or a focus. Straightforward computation by using the recursive formulae shown in Theorem 2.4 and computer algebraic system Mathematica gives the following result. For detailed recursive formulae, please see Appendix.
the vector field defined by system (3.4) is symmetrical with respect to the origin. Therefore, we have All the above discussion allows to finish the proof of Theorem 1.1.

Proof of Theorem 1.2
We proceed to show that 8 limit cycles can be bifurcated in this instance. We found that the highest possible order for a weakened focus at the origin is eight. First of all, we need to find the conditions under which the nilpotent origin of system (1.3) is a eight-order weakened focus. From the fact λ 1 = λ 2 = λ 3 = λ 4 = λ 5 = λ 6 = λ 7 = 0, λ 8 ≠ 0, the following statement holds.
Theorem 4.1. The origin is a weakened focus of maximum order eight for system (1.3). It is of order eight if and only if one of the following four sets of conditions holds: