Period annulus of the harmonic oscillator with zero cyclicity under perturbations with a homogeneous polynomial field

In this work we prove, using averaging theory at any order in the small perturbation parameter, that the period annulus of the harmonic oscillator has cyclicity zero (no limit cycles bifurcate) when it is perturbed by any fixed homogeneous polynomial field.


Introduction and main result
Consider an arbitrary polynomial planar vector field ẋ = −y + ε P(x, y, ε), ẏ = x + ε Q(x, y, ε), (1.1) where P, Q ∈ R{ε}[x, y] are polynomials in the state variables x and y with coefficients depending analytically on the small perturbation parameter ε ∈ R.Here the dot denotes, as usual, derivative with respect to the time independent variable.The unperturbed system (1.1) with ε = 0 is the harmonic oscillator which has a period annulus P given by the punctured phase plane P = R 2 \{(0, 0)}.Limit cycle bifurcations for the vector fields (1.1) can be produced either from the open set P or from its boundary ∂P = {(0, 0)} ∪ L ∞ , where L ∞ is the line at infinity (equator of the Poincaré compactification).In this paper we do not pay attention to the Hopf bifurcations at the origin neither to the bifurcations at infinity (see Remark 1 of [5] for a simple example of limit cycle bifurcation at L ∞ ).
Let X ε be the vector field associated to system (1.1).We denote by Cycl(X ε , P ) the cyclicity of P under the perturbations (1.1) with |ε| 1, that is, the maximum number of limit cycles of (1.1) bifurcating from the circles that foliates P.
Corresponding author.Email: garcia@matematica.udl.catEssentially there are two methods for finding limit cycles of (1.1) bifurcating from P which are averaging methods or Melnikov functions method.It is worth to emphasize that in [1] it is proved the equivalence between both methods.
In [5] the global upper bound [k(n − 1)/2] on Cycl(X ε , P ) is given where n is defined as n = max{deg(P), deg(Q)} and k is the order of the first Melnikov function associated to (1.1) which is not identically zero.Also in [5] some values of Cycl(X ε , P ) have been obtained for the values 1 ≤ k ≤ 6 showing in most cases that the above upper bound is sharp.As far as we know, the bifurcation of limit cycles from P was first analyzed (with the alternative method based on the inverse integrating factor) in [3] under the assumption that (P, Q) has arbitrary degree n ≥ 1 and it is independent of ε.
In this work we will compute Cycl(X ε , P ) for any value of n and k but restricted to the special kind of deformations (1.1) having the perturbation field (P, Q) independent of ε and homogeneous in x and y.Thus we will analyze the perturbations of the form where the nonlinearities P n and Q n are arbitrary homogeneous polynomials in x and y and its coefficients are the components of the parameter vector λ, which does not depend on ε.Our main result is that no limit cycles bifurcate from P under deformations (1.2), which we restate as follows.
Theorem 1.1.The period annulus of the harmonic oscillator has cyclicity zero when it is perturbed by any fixed homogeneous polynomial field.

Proof of Theorem 1.1
Introducing polar coordinates x = r cos θ, y = r sin θ, family (1.2) is written as where A and B are homogeneous trigonometric polynomials of degree n + 1 with coefficients λ given by Here the perturbative parameter ε ∈ I with I ⊂ R a small interval containing the origin.Therefore, for |ε| sufficiently small, we can write system (2.1) into the analytic differential equation dr dθ Since, F (θ, r; λ, 0) ≡ 0 it follows that equation (2.2) is written in the standard form of the averaging theory with period 2π.The method of averaging is a classical tool that allows to study the dynamics of the periodic nonlinear differential systems.The reader can consult for example the book [7] or, for recent advances, the papers [2] and [6].
From the analyticity of (2.2) and the fact F (θ, r; λ, 0) ≡ 0 it follows that the solution r(θ; z, λ, ε) of (2.1) with initial condition r(0; z, λ, ε) = z ∈ R + can be expanded into the convergent power series in ε as r(θ; z, λ, ε) = z + ∑ j≥1 r j (θ, z, λ) ε j where r j (θ, z, λ) are real analytic functions such that r j (0, z, λ) = 0. Therefore, from the results in [4] it follows that the recursive expressions of r j (θ; z, λ) for j ≥ 1 are given by (2.4) We can assume without loss of generality that the function r(•; z, λ, ε) is defined on the interval [0, 2π] provided that ε is close enough to 0. So we can define the displacement map Clearly, the isolated positive zeros z 0 ∈ R + of d(•, λ, ε) are initial conditions for the 2π-periodic solutions of the differential equation (2.2) and they are in one-to-one correspondence with the limit cycles of system (1.2) bifurcating from the circle x 2 + y 2 = z 2 0 .The displacement map d is analytic at ε = 0, hence it can be expressed via the following convergent series expansion We call the coefficient functions f i (z; λ) the averaged functions (they are also called Melnikov functions) which are clearly given by We say that a branch of limit cycles bifurcates from the point z 0 ∈ R + if there is a function z * (λ, ε) (may be only defined in a half-neighborhood of zero) such that z * (λ, 0) = z 0 and d(z * (λ, ε), λ, ε) ≡ 0. It is well known, see [7] for example, that under these conditions it follows that z 0 must be a zero of the function f (•; λ) where is the first subindex such that f (z; λ) ≡ 0.
Let N be the set of non-negative integers.We recall that given the pair (i, j) ∈ N 2 , the function θ 0 sin i (τ) cos j (τ) dτ is a trigonometric polynomial (that is, a function in R[sin(θ), cos(θ)]) plus an eventual linear term αθ where α = 0 only in case that both i and j are even numbers.More generally, when (i, j, k) ∈ N 3 , the function We claim that where ] is a trigonometric polynomial with polynomial coefficients in R[θ, λ].We will prove the claim by induction over k.From (2.2) we have F 1 (θ, z; λ) = z n A(θ; λ), hence from the first equation in (2.4) we get where R 1 (θ, λ) = θ 0 A(τ; λ) dτ.Therefore the claim is true for k = 1.Assume now by induction hypothesis that r j (θ, z, λ) = R j (θ, λ)z j(n−1)+1 where by definition R j ∈ R[θ, λ][sin(θ), cos(θ)] for all 1 ≤ j ≤ k − 1.Then, since all integer subindex j p appearing in (2.4) satisfy 1 ≤ j p ≤ ≤ k − 1, it follows that (2.9) Therefore, using (2.3), (2.8) and (2.9) we rewrite (2.4) like (2.7) with Once the claim (2.7) is proved we get that R k (2π, λ) ∈ R[λ] for all k ∈ N and therefore, from (2.6), that the averaged functions are f k (z; λ) = P k (λ)z k(n−1)+1 where P k ∈ R[λ] for all k ∈ N. Hence it is clear that the only finite point from where 2π-periodic orbit bifurcation can occur in the differential equation (2.2) is just from the initial condition z 0 = 0 which corresponds to the singularity located at the origin of the vector field (1.2).So no periodic orbit bifurcation appear in the period annulus and the theorem is proved.

Some remarks
Theorem 1.1 is not true if the perturbation field (P n , Q n ) is not homogeneous, see for example [3].One of the most simple counterexamples is given by the van der Pol differential equation ẍ + x = ε(1 − x 2 ) ẋ which is a perturbation of the harmonic oscillator with associated vector field ẋ = y, ẏ = −x + ε(1 − x 2 )y.Computations show that the first averaged function is f 1 (z) = 1 4 πz(z 2 − 4) so that from the circle x 2 + y 2 = z 2 0 = 4 the van der Pol limit cycle bifurcates.
the set of trigonometric polynomials with real polynomial coefficients in θ.The fact that R[θ][sin(θ), cos(θ)] is closed under sums, products and quadratures will be key in what follows.