Limit cycle bifurcations from a nilpotent focus or center of planar systems

In this paper, we study the analytical property of the Poincare return map and the generalized focal values of an analytical planar system with a nilpotent focus or center. Then we use the focal values and the map to study the number of limit cycles of this kind of systems with parameters, and obtain some new results on the lower and upper bounds of the maximal number of limit cycles near the nilpotent focus or center.


Introduction and main result
Consider an analytic system of the forṁ x = y + X(x, y),ẏ = Y (x, y), (1.1) where X, Y = O(|x, y| 2 ) for (x, y) near the origin. The following criterion for the existence of a center or a focus at the origin of (1.1) has been established in [4,15]. Lyapunov [15] also introduced the generalized polar coordinates x = r Cs(θ), y = r n Sn(θ) and the return map to give a way to find focal values in solving the center-focus problem for (1.1), where (Cs(t), Sn(t)) is the solution of the initial probleṁ x = y,ẏ = −x 2n−1 , (x(0), y(0)) = (1, 0).
Then, Stróżyna &Żo ladek [20] proved that this formal normal form can be achieved and it has a center at the origin if b 2j = 0 for all 2j ≥ n − 1.
For the case of n = 2 Liu and Li [11] introduced a different generalized polar coordinates of the form x = r cos θ, y = r 2 sin θ to change (1.1) into the form assuming the origin is a center or a focus. Letr(θ, h) denote the solution of the 2π-periodic satisfyingr(0) = h. Note that the initial value problem is well-defined also for negative h.
In the latter case the origin is called a kth order weak focus of (1.1).
Liu and Li [11] also gave some new methods to compute the focus values v 2 , v 4 , · · · , v 2k , or equivalent values, and studied the problem of limit cycle bifurcations near the origin, finding a new phenomenon: a node can generate a limit cycle when its stability changes.
In this paper we study the problem of limit cycle bifurcations near the origin for the analytic systemẋ where δ = (δ 1 , . . . , δ m ) ∈ D ⊂ R m with D compact, and X, Y = O(|x, y| 2 ) for |x| small and δ ∈ D. Let y = F (x, δ) be the solution of the equation y + X(x, y, δ) = 0. We define the following two functions: then the origin is a center or a focus of (1.9) for all δ ∈ D.
Thus, the Poincaré return map is defined as (2) if n is even, then for all |x 0 | small The above theorem tells us that the function P (x 0 , δ) is analytic in x 0 at x 0 = 0 as n is odd, and not analytic in x 0 at x 0 = 0 as n is even unless the origin is a center (in this case, P is the identity).
Define p n = [1 + (−1) n ]/2. Then the conclusions of the above theorem can be written From the proof of the above theorem we see that v 2k+pn depends on v 1+pn , v 3+pn , · · · , v 2k−1+pn smoothly. Using the theorem we derive the following two statements on limit cycle bifurcations near the origin. (1) If there is an integer k ≥ 1 such that k+1 j=1 |v 2j−1+pn (δ)| > 0 for all δ ∈ D, then there exists a neighborhood U of the origin such that (1.9) has at most k limit cycles in U for all δ ∈ D. (1.14) then for an arbitrary sufficiently small neighborhood of the origin there are some δ ∈ D near δ 0 such that (1.9) has exactly k limit cycles in the neighborhood. all δ ∈ D. Assume that there exist δ 0 ∈ D and an integer k ≥ 1 such that (1.14) is satisfied. If the origin is a center of (1.9) as v 2j−1+pn (δ) = 0, j = 1, · · · , k, then there exists a neighborhood U of the origin such that (1.9) has at most k − 1 limit cycles in U for all δ ∈ D near δ 0 , and also, for an arbitrary sufficiently small neighborhood of the origin there are some δ ∈ D near δ 0 such that (1.9) has exactly k − 1 limit cycles in the neighborhood.
The theorem means that the cyclicity of the system at the point δ 0 is equal to k − 1.
Now, different from [2] and [11]- [14], we give the following new and more reasonable definition.
Definition 1.1. We call v 2k+1+pn (δ) the generalized focal values of order k of (1.9) at the origin.
By Theorem 1.6, we see that a nilpotent focus of order k generates at most k limit cycles under perturbations as long as the perturbations always satisfy (1.11) and (1.12).
The generalized focal values v 1+pn , v 3+pn , · · · , v 2k+1+pn , · · · can be calculated using the normal form of system (1.9). We will give a method how to do it. By Stróżyna anḋ Zo ladek [20] we know that (1.9) has the following analytic normal form: We remark that here f and g in (1.15) may be different from ones given by (1.10). As before, let δ ∈ D ⊂ R m with D compact. Also, suppose for |x| small the function g(x, δ) It is easy to see that the equation G(x, δ) = G(y, δ) for xy < 0 defines a unique analytic Let one of the following conditions be satisfied: Then we have (1) For δ = δ 0 (1.15) has a stable (unstable) focus at the origin.
(2) If further then for an arbitrary sufficiently small neighborhood of the origin there are some δ ∈ D near δ 0 such that (1.15) has at least k limit cycles in the neighborhood.
Here, we should mention that the functions g m and f m−1 depend only on the terms of degree at most m of the expansions of the functions X and Y in (1.9) at the origin.
The Poincaré maps of (1.9) and (1.22) are essentially the same. We can suppose that the Poincaré map of (1.22) is P (x 0 , δ) having the expansion Second, truncating the higher order terms in (1.22) we obtain the following polyno- (1.24) In practice, for given system (1.9) it is not difficult to find the corresponding system (1.24). For (1.24) we can further use Theorem 1.8 to find its focal values at the origin up to any large order. Let P m (x 0 , δ) denote the Poincaré map of (1.24). It has the expansion Here, a problem we would like to solve is the following: For any given The following theorem gives an answer. cients v 1 , v 2 , · · · , v kn in (1.13) depend only on the terms of degree at most (k + 2)n − 2 of the expansions of the functions X and Y at the origin.
Obviously, in the case of n = 1 (the elementary case), the above conclusion is a well-known results.
We organize the paper as follows. In section 2 we first give preliminary lemmas. In section 3 we prove our main results. In section 4 we provide some application examples.

Preliminaries
Consider (1.9). In this section we will always suppose that (1.11) and (1.12) are satisfied.
This means thatr(θ) is also a solution of (2.5). Then the first conclusion follows by the uniqueness of initial problem. The second one follows in the same way. This completes the proof.

Proof of the main results
In this section we prove our main results presented in Theorems 1.4-1.10.
Proof of Theorem 1.5. There are two cases to consider separately.
Finally, noting that l 1 = −k 0 and substituting (3.4) into (3.6) we easily see that This ends the proof.
Proof of Theorem 1.6. For the first part, suppose the conclusion is not true.
The first conclusion follows.
For the second one, by Theorem 1.5, the functiond can be written as where P j (0, δ) = 0. Like in [8] one can show that P j are series convergent in a neighborhood of δ 0 (see also e.g. [18,17]). Further, by (1.14), we can take v 1+pn , v 3+pn , · · · , v 2k−1+pn as free parameters, varying near zero. Precisely, if we change them such that the origin is a stable (unstable) focus of order k of (1.9). If v 2j−1+pn (δ) = 0 for all j ≥ 1 the origin is a center of (1.9).
Proof of Theorem 1.8. Now we consider (1.15), where g satisfies (1.11). Let If f satisfies (1.12), then the origin is a center or focus of (1.15), and where , δ), δ) = G(x, δ) for |x| small. Note that (1.15) is equivalent to the following systeṁ which has the same Poincaré return map P (x 0 , δ) as (1.15). Introducing the change of variables x and t which is equivalent tou The systems (3.10) and (3.11) have the same Poincaré return map, denoted by P 1 (u 0 , δ).
Proof of Theorem 1.9. Let |δ − δ 0 | be small. For n = 2 we have p n = 1. Then the first conclusion is direct from Corollary 3.1 and (1.17)- (1.20). In fact, we have by For the second conclusion, we first keep B 1 (δ) = 0, and vary B 3 (δ), · · · , B 2k−1 (δ) near zero to obtain exactly k − 1 simple limit cycles near the origin. These limit cycles are bifurcated by changing the stability of the focus at the origin k − 1 times. Then we vary B 1 such that 0 < |B 1 | ≪ |B 3 |, and B 1 B 3 < 0. This step produces one more limit cycle bifurcated from the origin by changing the stability of the origin which is a node now by [10]. The theorem is proved for the case of n = 2.
This finishes the proof.
Proof of Theorem 1.10. Consider (1.22). Without loss of generality, we can assume X m+1 = 0 in (1.22). Otherwise, it needs only to introduce a change of variables v = y + X m+1 (x, y). In this case, we can write (1.22) into the forṁ For the sake of convenience below, we rewrite the functions g, f and ϕ j as follows: where f j , g j and ϕ jl , j ≥ 0, l ≥ 1, are polynomials in x of degree at most n − 1, and ϕ j0 , j ≥ 0, are polynomials in x with degree at most n − 2.
Hence, by Lemma 2.3, (1.23) and (3.26) we come to the following conclusion: For any j ≥ 1, v j (δ) depends only onR k andS k with 0 ≤ k ≤ j − 1. Similarly, for j ≥ 1 and 0 ≤ l ≤ n − 1 or jn ≤ l + jn ≤ (j + 1)n − 1,S l+jn andR l+jn depend only on the coefficients of degree l of the polynomials g j , f j , xϕ j0 and ϕ i,j−i with i = 0, · · · , j − 1 in x. In other words, for jn + 1 ≤ u ≤ (j + 1)n,S u−1 andR u−1 depend only on the coefficients of degree u − 1 − jn of the polynomials g j , f j , xϕ j0 and ϕ i,j−i Thus, for all k ∈ N [in,(i+1)n−1] ,S k andR k depend only on g i , f i , xϕ i0 and ϕ l,i−l with l = 0, · · · , i − 1. And for k ∈ N [jn,u−1] ,S k andR k depend only on the coefficients of degree k − jn of the polynomials g j , f j , xϕ j0 and ϕ l,j−l with l = 0, · · · , j − 1 in x.
We claim that if j ≥ 0, m ≥ (j + 1)n, then for jn + 1 ≤ u ≤ (j + 1)n, v u (δ) depends only on the functions g i , f i , with i = 0, · · · , j − 1 and the coefficients of degree at most u − 1 − jn of the polynomials g j , f j in x.
In fact, by the above discussion, we need only to prove ϕ 00 = 0 in the case j = 0 and ϕ ls = 0 for l + s ≤ j and 0 ≤ l ≤ j − 1 in the case j > 0. This can be shown easily since By (3.20) again, the above claim can be restated that if j ≥ 0, m ≥ (j + 1)n, then for jn + 1 ≤ u ≤ (j + 1)n, v u (δ) depends only on the coefficients of degree at most 2n + u − 2 of g and the coefficients of degree at most n + u − 2 of f in x. Thus, for any integers k and m satisfying k ≥ 1 and m ≥ (k + 1)n, by taking j = 0, · · · , k we know that for all 1 ≤ u ≤ (k + 1)n, v u (δ) depends only on the coefficients of degree at most 2n + u − 2 of g and the coefficients of degree at most n + u − 2 of f in x.
In this case, for all 1 ≤ u ≤ (k + 1)n, v u (δ) depends only on g m and f m−1 in (3.19). Then the conclusion of Theorem 1.10 follows.

Application examples
In this section we give some application examples based on the examples given in [2].
Moreover, there can be 3 limit cycles near the origin. See Theorem 4.1 in [2] and its proof.
Then there exists a neighborhood V of the origin such that the system (4.1) has at most 3 limit cycles in V .
(4.2) By Theorem 4.2 in [2] and its proof if A 2 < 2 then the origin of (4.2) is always a focus with |v 2 | + |v 4 | + |v 6 | + |v 8 | > 0. Moreover, there are systems inside (4.2) with at least 3 limit cycles around the origin. Then by Theorem 1.6 again we have Proposition 4.2. Let A, B and C be bounded parameters with A 2 < 2. Then there exists a neighborhood V of the origin such that the system (4.2) has at most 3 limit cycles in V .
(2) If b 0 = 0, there are systems inside (4.3) which have at least k limit cycles near the origin.