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Limit cycles of a generalised Mathieu differential system

Data publikacji: 27 Dec 2021
Tom & Zeszyt: AHEAD OF PRINT
Zakres stron: -
Otrzymano: 17 Jun 2021
Przyjęty: 24 Sep 2021
Informacje o czasopiśmie
License
Format
Czasopismo
eISSN
2444-8656
Pierwsze wydanie
01 Jan 2016
Częstotliwość wydawania
2 razy w roku
Języki
Angielski
Introduction and statement of the main results

A limit cycle of a differential system is a periodic orbit having a neighbourhood where it is the unique periodic orbit of the differential system. The notion of limit cycle was introduced in 1881 by Poincaré [10].

The study of the existence and number of limit cycles that a differential system in ℝ2 can exhibit is one of the more difficult problems in the qualitative theory of the differential system in the plane. Thus in 1900, Hilbert [6] presented a list of 23 problems at the International Conference of Mathematicians in Paris; most of these problems were solved partially or completely, but the second part of the 16th problem remains unsolved till today. This problem ask about the existence of an upper bound for the maximal number of limit cycles that polynomial differential systems in ℝ2 of a given degree can exhibit.

A source of producing limit cycles is by perturbing the periodic orbits of a centre; see for instance the papers [3, 11] and the book of Christhoper and Li [5], and the hundreds of references quoted there.

The classical Mathieu's differential equation [9] is x¨+b(1+cosθ)x=0, \ddot x + b(1 + \cos \theta )x = 0, where b is real parameter, and the dots denote second derivative with respect to time t. This equation was first discussed in 1868 by Mathieu while studying the problem of vibrations on an elliptical drumhead. Matthieu's equation has many applications in engineering [12, 13] and also in theoretical and experimental physics [2, 7, 14]. Information on its periodic orbits can be found in [16].

Mathieu's equation can be written as the differential system x˙=y,y˙=b(1+cosθ)x, \dot x = y,\quad \dot y = - b(1 + \cos \theta )x,

In [4], Chen and Llibre studied the limit cycles of the differential system x˙=y,y˙=xε(1+cosmθ)Q(x,y), \dot x = y,\quad \dot y = - x - \varepsilon \left( {1 + \mathop {\cos }\nolimits^m \theta } \right)Q(x,{\kern 1pt} y), where ɛ > 0 is a small parameter and Q(x, y) is an arbitrary polynomial of degree n.

In the present work we study the limit cycles of the following generalisation of differential system (2) as x˙=yε(1+coslθ)P(x,y),y˙=xε(1+cosmθ)Q(x,y), \dot x = y - \varepsilon (1 + \mathop {\cos }\nolimits^l \theta )P(x,{\kern 1pt} y),\quad \dot y = - x - \varepsilon \left( {1 + \mathop {\cos }\nolimits^m \theta } \right)Q(x,{\kern 1pt} y), where ɛ > 0 is a small parameter, l and m are positive integers, and P(x, y) and Q(x, y) are arbitrary polynomials of degree n. More precisely, we study the maximum number of limit cycles which bifurcate from the periodic orbits of the linear centre x˙=y \dot x = y , y˙=x \dot y = - x , when it is perturbed in the form (3).

Our main result is as follows.

Theorem 1

Using the averaging theory of first-order the maximum number of limit cycles of the differential system (3) bifurcating from the periodic solutions of the linear centre x˙=y \dot x = y , y˙=x \dot y = - x is at most:

n if n is even, and l and m are not both even;

n/2 if n,l and m are even;

n if n is odd and l and m are one odd and the other even;

(n − 1)/2 if n is odd, and l and m are even;

n if n, l and m are odd.

Theorem 1 is proved in section 3.

In section 2, we present a summary of the averaging theory of first-order and of the Descartes theorem that we shall need for proving Theorem 1.

The averaging theory of first-order and the Descartes theorem
Averaging theory of first-order

In these subsection we summarise the result stated in Theorems 11.5 of the book of Verhulst [15] on the averaging theory. For a general introduction to the computation of periodic orbits using the averaging theory see book [8].

Consider the periodic differential system dxdθ=X(θ,x)=ε(θ,x)+ε2Φ(θ,x,ε), {{dx} \over {d\theta }} = {\cal X}(\theta ,{\kern 1pt} x) = \varepsilon {\cal F}(\theta ,{\kern 1pt} x) + {\varepsilon ^2}\Phi (\theta ,{\kern 1pt} x,{\kern 1pt} \varepsilon ), where ɛ is a small parameter, x ∈ ℝ, θ ∈ 𝕊1 = ℝ/(2πℤ) and ℱ : 𝕊1 × D → ℝ, Φ : 𝕊1 × D × (−ɛ0, ɛ0) → ℝ are C2 functions, being D an open interval of ℝ and ℱ and Φ are periodic with period 2π in the variable θ.

Now we consider the averaging function f : D → ℝ defined by f(x)=12π02π(θ,x)dθ. f(x) = {1 \over {2\pi }}\int_0^{2\pi } {\cal F}(\theta ,{\kern 1pt} x)d\theta . It is known that if x (θ, x0) is the solution of system (4) such that x (0, x0) = x0, then we have x(2π,x0)x0=εf(x0)+O(ε2). x\left( {2\pi ,{\kern 1pt} {x_0}} \right) - {x_0} = \varepsilon f\left( {{x_0}} \right) + O\left( {{\varepsilon ^2}} \right). So for ɛ > 0 sufficiently small the simple zeros of the averaged function f (x) provide limit cycles of differential equation (4).

The Descartes theorem

In order to study the real zeros of the function f (x) we shall use the Descartes theorem (for a proof see [1]).

Theorem 2

(Descartes Theorem). Consider the real polynomial p(ρ) = ai1ρi1 + a i2ρi2 + ⋯ + ainρin with 0 ≤ i1i2 ≤ ⋯ ≤ in and aij ≠ 0 real constants for j ∈ {1,2,...,n}. When aijaij+1 < 0, we say that aij and aij+1 have a variation of sign. If the number of variations of signs in the polynomial p(ρ) is m, then p(ρ) has at most m positive real roots. Moreover, it is always possible to choose the coefficients of p(ρ) in such a way that p(ρ) has exactly n − 1 positive real roots.

Proof of Theorem 1

Let the polynomials P(x,y)=i+j=0naijxiyj P(x,{\kern 1pt} y) = \sum\nolimits_{i + j = 0}^n {a_{ij}}{x_i}{y_j} and Q(x,y)=i+j=0nbijxiyj Q(x,{\kern 1pt} y) = \sum\nolimits_{i + j = 0}^n {b_{ij}}{x_i}{y_j} be.

We write the differential system (3) in polar coordinates (ρ, θ) defined by x = ρ cos θ and y = ρ sin θ, with ρ > 0, we obtain ρ˙=ε(cosθ(1+cosLθ)P(ρcosθ,ρsinθ)+sinθ(1+cosmθ)Q(ρcosθ,ρsinθ)),θ˙=1+ερ(sinθ(1+cosLθ)P(ρcosθ,ρsinθ)cosθ(1+cosmθ)Q(ρcosθ,ρsinθ)). \matrix{ {\dot \rho } \hfill & { = - \varepsilon \left( {\cos \theta \left( {1 + \mathop {\cos }\nolimits^L \theta } \right)P(\rho \cos \theta ,{\kern 1pt} \rho \sin \theta ) + \sin \theta \left( {1 + \mathop {\cos }\nolimits^m \theta } \right)Q(\rho \cos \theta ,{\kern 1pt} \rho \sin \theta )} \right),} \hfill \cr {\dot \theta } \hfill & { = - 1 + {\varepsilon \over \rho }\left( {\sin \theta \left( {1 + \mathop {\cos }\nolimits^L \theta } \right)P(\rho \cos \theta ,{\kern 1pt} \rho \sin \theta ) - \cos \theta \left( {1 + \mathop {\cos }\nolimits^m \theta } \right)Q(\rho \cos \theta ,{\kern 1pt} \rho \sin \theta )} \right).} \hfill \cr } Taking in the differential system (7) the variable θ as the new independent variable, system (7) reduces to the differential equation dρdθ=ε(cosθ(1+cosLθ)P(ρcosθ,ρsinθ)+sinθ(1+cosmθ)Q(ρcosθ,ρsinθ))+O(ε2)=εi+j=0ncosiθsinjθ(aij(cosθ+cosL+1θ)+bij(sinθ+sinθcosmθ))ρi+j+O(ε2)=ε(θ,x)+O(ε2). \matrix{ {{{d\rho } \over {d\theta }}} \hfill & { = \varepsilon \left( {\cos \theta \left( {1 + \mathop {\cos }\nolimits^L \theta } \right)P(\rho \cos \theta ,{\kern 1pt} \rho \sin \theta ) + \sin \theta \left( {1 + \mathop {\cos }\nolimits^m \theta } \right)Q(\rho \cos \theta ,{\kern 1pt} \rho \sin \theta )} \right) + O\left( {{\varepsilon ^2}} \right)} \hfill \cr {} \hfill & { = \varepsilon \sum\limits_{i + j = 0}^n \mathop {\cos }\nolimits^i \theta \mathop {\sin }\nolimits^j \theta \left( {{a_{ij}}\left( {\cos \theta + \mathop {\cos }\nolimits^{L + 1} \theta } \right) + {b_{ij}}\left( {\sin \theta + \sin \theta \mathop {\cos }\nolimits^m \theta } \right)} \right){\rho ^{i + j}} + O\left( {{\varepsilon ^2}} \right)} \hfill \cr {} \hfill & { = \varepsilon {\cal F}(\theta ,{\kern 1pt} x) + O\left( {{\varepsilon ^2}} \right).} \hfill \cr } Since this differential equation is written in the normal form (4), so we can apply to it the averaging theory of first-order.

In our study we shall use the following formulas for computing the averaged function: 02πcospθsin2qθdθ=(2q1)!!(2q+p)(2q+p2)(p+2)02πcospθdθ=2παp,2q,02πcos2sθdθ=(2s1)!!2ss!2π=2πβ2s,02πcos2s+1θdθ=0,02πcospΘsin2q+1θdθ=0. \matrix{\hfill {\int_0^{2\pi } \mathop {\cos }\nolimits^p \theta \mathop {\sin }\nolimits^{2q} \theta d\theta } & { = {{(2q - 1)!!} \over {(2q + p)(2q + p - 2) \ldots (p + 2)}}\int_0^{2\pi } \mathop {\cos }\nolimits^p \theta d\theta = 2\pi {\alpha _{p,2q}},} \hfill \cr \hfill {\int_0^{2\pi } \mathop {\cos }\nolimits^{2s} \theta d\theta } & { = {{(2s - 1)!!} \over {{2^s}s!}}2\pi = 2\pi {\beta _{2s}},} \hfill \cr \hfill {\int_0^{2\pi } \mathop {\cos }\nolimits^{2s + 1} \theta d\theta } & { = 0,} \hfill \cr \hfill {\int_0^{2\pi } \mathop {\cos }\nolimits^p \Theta \mathop {\sin }\nolimits^{2q + 1} \theta d\theta } & { = 0.} \hfill \cr } where in the first and second formulas p, q and s are positive integers, in the third one s is a non-negative integer, and in the fourth one p is a positive integer and q is a non-negative integer. For more details of these four integrals see [17, pages 152 and 153].

Proof of Theorem 1 when n is even, and l and m are not both even

We consider the following three cases with n even.

Case 1: m and l are odd. Then we compute the averaged function (5), and we obtain f(ρ)=12π02π(ρ,Θ)dθ=12π02πi+j=0n(aijcosi+1θsinjθ+aijcosi+l+1θsinjθ+bijcosiθsinj+1θ+bijcosi+mθsinj+1θ)ρi+jdθ=12π02πi+2q=2n+1((ai,2q1cosi+1θ+ai,2q1cosi+l+1θ)sin2q1θ+(bi,2q1cosiθ+bi,2q1cosi+mθ)sin2qθ)ρi+2q1+i+2q=2n((ai,2qcosi+1(θ)+ai,2qcosi+l+1θ)sin2q(θ)+(bi,2qcosiθ+bi,2qcosi+mθ)sin2q+1θ)ρi+2qdθ=12π[2s+1+2q=3n+102πb2s+1,2q1cos2s+1+mθsin2qθρ2s+2qdθ+2s+2q=2n02πb2s,2q1cos2sθsin2qθρ2s+2q1dθ+2s+1+2q=3n+102π(a2s+1,2qcos2s+2θsin2q(θ)ρ2s+1+2qdθ+2s+2q=2n02πa2s,2qcos2s+l+1θsin2q(θ)ρ2s+2qdθ]+s+q=1n/2b2s+1,2q1α2s+1+m,2qρ2s+2q+s+q=1n/2b2s,2q1α2s,2qρ2s+2q1+s+q=1n/2a2s+1,2qα2s+2,2qρ2s+1+2q+s+q=1n/2a2s,2qα2s+l+1,2qρ2s+2q=k=1n+1Akρk. \matrix{ {f(\rho )} \hfill & { = {1 \over {2\pi }}\int_0^{2\pi } {\cal F}(\rho ,\Theta )d\theta } \hfill \cr {} \hfill & { = {1 \over {2\pi }}\int_0^{2\pi } \sum\limits_{i + j = 0}^n \left( {{a_{ij}}\mathop {\cos }\nolimits^{i + 1} \theta \mathop {\sin }\nolimits^j \theta + {a_{ij}}\mathop {\cos }\nolimits^{i + l + 1} \theta \mathop {\sin }\nolimits^j \theta } \right.} \hfill \cr {} \hfill & {\quad \left. { + {b_{ij}}\mathop {\cos }\nolimits^i \theta \mathop {\sin }\nolimits^{j + 1} \theta + {b_{ij}}\mathop {\cos }\nolimits^{i + m} \theta \mathop {\sin }\nolimits^{j + 1} \theta } \right){\rho ^{i + j}}d\theta } \hfill \cr {} \hfill & { = {1 \over {2\pi }}\int_0^{2\pi } \sum\limits_{i + 2q = 2}^{n + 1} \left( {\left( {{a_{i,2q - 1}}\mathop {\cos }\nolimits^{i + 1} \theta + {a_{i,2q - 1}}\mathop {\cos }\nolimits^{i + l + 1} \theta } \right)\mathop {\sin }\nolimits^{2q - 1} \theta } \right.\;} \hfill \cr {} \hfill & {\quad \left. { + \left( {{b_{i,2q - 1}}\mathop {\cos }\nolimits^i \theta + {b_{i,2q - 1}}\mathop {\cos }\nolimits^{i + m} \theta } \right)\mathop {\sin }\nolimits^{2q} \theta } \right){\rho ^{i + 2q - 1}}} \hfill \cr {} \hfill & {\quad + \sum\limits_{i + 2q = 2}^n \left( {\left( {{a_{i,2q}}\mathop {\cos }\nolimits^{i + 1} (\theta ) + {a_{i,2q}}\mathop {\cos }\nolimits^{i + l + 1} \theta } \right)\mathop {\sin }\nolimits^{2q} (\theta )} \right.\;} \hfill \cr {} \hfill & {\quad \left. { + \left( {{b_{i,2q}}\mathop {\cos }\nolimits^i \theta + {b_{i,2q}}\mathop {\cos }\nolimits^{i + m} \theta } \right)\mathop {\sin }\nolimits^{2q + 1} \theta } \right){\rho ^{i + 2q}}d\theta } \hfill \cr {} \hfill & { = {1 \over {2\pi }}\left[ {\sum\limits_{2s + 1 + 2q = 3}^{n + 1} \int_0^{2\pi } {b_{2s + 1,2q - 1}}\mathop {\cos }\nolimits^{2s + 1 + m} \theta \mathop {\sin }\nolimits^{2q} \theta {\rho ^{2s + 2q}}d\theta } \right.\;} \hfill \cr {} \hfill & {\quad + \sum\limits_{2s + 2q = 2}^n \int_0^{2\pi } {b_{2s,2q - 1}}\mathop {\cos }\nolimits^{2s} \theta \mathop {\sin }\nolimits^{2q} \theta {\rho ^{2s + 2q - 1}}d\theta } \hfill \cr {} \hfill & {\quad + \sum\limits_{2s + 1 + 2q = 3}^{n + 1} \int_0^{2\pi } ({a_{2s + 1,2q}}\mathop {\cos }\nolimits^{2s + 2} \theta \mathop {\sin }\nolimits^{2q} (\theta ){\rho ^{2s + 1 + 2q}}d\theta } \hfill \cr {} \hfill & {\quad + \left. {\sum\limits_{2s + 2q = 2}^n \int_0^{2\pi } {a_{2s,2q}}\mathop {\cos }\nolimits^{2s + l + 1} \theta \mathop {\sin }\nolimits^{2q} (\theta ){\rho ^{2s + 2q}}d\theta } \right]} \hfill \cr {} \hfill & {\quad + \sum\limits_{s + q = 1}^{n/2} {b_{2s + 1,2q - 1}}{\alpha _{2s + 1 + m,2q}}{\rho ^{2s + 2q}} + \sum\limits_{s + q = 1}^{n/2} {b_{2s,2q - 1}}{\alpha _{2s,2q}}{\rho ^{2s + 2q - 1}}} \hfill \cr {} \hfill & {\quad + \sum\limits_{s + q = 1}^{n/2} {a_{2s + 1,2q}}{\alpha _{2s + 2,2q}}{\rho ^{2s + 1 + 2q}} + \sum\limits_{s + q = 1}^{n/2} {a_{2s,2q}}{\alpha _{2s + l + 1,2q}}{\rho ^{2s + 2q}}} \hfill \cr {} \hfill & { = \sum\limits_{k = 1}^{n + 1} {A_k}{\rho ^k}.} \hfill \cr }

Since f (ρ) is a polynomial generated by a linear combination of the monomials {ρ, ρ2, . . ., ρn+1}. Using Descartes theorem it follows that the polynomial f (ρ) have at most n simple positive zeros, and consequently from subsection 2.1 we get using the averaging theory of first-order that for ɛ > 0 sufficiently small the differential system (3) has at most n limit cycles bifurcating from the periodic solutions of the linear centre (3) with ɛ = 0.

Case 2: m is odd and l is even. Working as in the case 1 we obtain f(ρ)=12π02π(ρ,θ)dθ=s+q=1n/2b2s+1,2q1α2s+1+m,2qρ2s+2q+s+q=1n/2b2s,2q1α2s,2qρ2s+2q1+s+q=1n/2a2s+1,2qα2s+2,2qρ2s+1+2q+s+q=1n/2a2s+1,2qα2s+l+2,2qρ2s+1+2q=k=1n+1A˜kρk. \matrix{ {f(\rho )} \hfill & { = {1 \over {2\pi }}\int_0^{2\pi } {\cal F}(\rho ,\theta )d\theta } \hfill \cr {} \hfill & { = \sum\limits_{s + q = 1}^{n/2} {b_{2s + 1,2q - 1}}{\alpha _{2s + 1 + m,2q}}{\rho ^{2s + 2q}} + \sum\limits_{s + q = 1}^{n/2} {b_{2s,2q - 1}}{\alpha _{2s,2q}}{\rho ^{2s + 2q - 1}}} \hfill \cr {} \hfill & {\quad + \sum\limits_{s + q = 1}^{n/2} {a_{2s + 1,2q}}{\alpha _{2s + 2,2q}}{\rho ^{2s + 1 + 2q}} + \sum\limits_{s + q = 1}^{n/2} {a_{2s + 1,2q}}{\alpha _{2s + l + 2,2q}}{\rho ^{2s + 1 + 2q}} = \sum\limits_{k = 1}^{n + 1} {{\tilde A}_k}{\rho ^k}.} \hfill \cr }

Again, f (ρ) is a polynomial generated by the monomials {ρ, ρ2, ⋯, ρn+1}, and as in case 1 by Descartes theorem it follows that for ɛ > 0 sufficiently small, the differential system (3) has at most n limit cycles bifurcating from the periodic solutions of the linear centre (3) with ɛ = 0.

Case 3: m is even and l is odd. Then f(ρ)=12π02π(ρ,θ)dθ=s+q=1n/2b2s,2q1α2s+m,2qρ2s+2q1+s+q=1n/2b2s,2q1α2s,2qρ2s+2q1+s+q=1n/2a2s+1,2qα2s+2,2qρ2s+1+2q+s+q=1n/2a2s,2qα2s+l+1,2qρ2s+2q=k=1n+1A^kρk. \matrix{ {f(\rho )} \hfill & { = {1 \over {2\pi }}\int_0^{2\pi } {\cal F}(\rho ,\theta )d\theta } \hfill \cr {} \hfill & { = \sum\limits_{s + q = 1}^{n/2} {b_{2s,2q - 1}}{\alpha _{2s + m,2q}}{\rho ^{2s + 2q - 1}} + \sum\limits_{s + q = 1}^{n/2} {b_{2s,2q - 1}}{\alpha _{2s,2q}}{\rho ^{2s + 2q - 1}}} \hfill \cr {} \hfill & {\quad + \sum\limits_{s + q = 1}^{n/2} {a_{2s + 1,2q}}{\alpha _{2s + 2,2q}}{\rho ^{2s + 1 + 2q}} + \sum\limits_{s + q = 1}^{n/2} {a_{2s,2q}}{\alpha _{2s + l + 1,2q}}{\rho ^{2s + 2q}} = \sum\limits_{k = 1}^{n + 1} {{\hat A}_k}{\rho ^k}.} \hfill \cr }

As in the previous two cases, we conclude that for ɛ > 0 sufficiently small the differential system (3) has at most n limit cycles bifurcating from the periodic solutions of the linear centre (3) with ɛ = 0.

Proof of Theorem 1 when n, l and m are even

Working as in case 1, we compute the averaged function and we get f(ρ)=12π02π(ρ,θ)dt=s+q=1n/2b2s,2q1α2s+m,2qρ2s+2q1+s+q=1n/2b2s,2q1α2s,2qρ2s+2q1+s+q=1n/2a2s+1,2qα2s+2,2qρ2s+1+2q+s+q=1n/2a2s+1,2qα2s+l+2,2qρ2s+1+2q=k=1(n/2)+1B˜kρ2k1. \matrix{ {f(\rho )} \hfill & { = {1 \over {2\pi }}\int_0^{2\pi } {\cal F}(\rho ,\theta )dt} \hfill \cr {} \hfill & { = \sum\limits_{s + q = 1}^{n/2} {b_{2s,2q - 1}}{\alpha _{2s + m,2q}}{\rho ^{2s + 2q - 1}} + \sum\limits_{s + q = 1}^{n/2} {b_{2s,2q - 1}}{\alpha _{2s,2q}}{\rho ^{2s + 2q - 1}}} \hfill \cr {} \hfill & {\quad + \sum\limits_{s + q = 1}^{n/2} {a_{2s + 1,2q}}{\alpha _{2s + 2,2q}}{\rho ^{2s + 1 + 2q}} + \sum\limits_{s + q = 1}^{n/2} {a_{2s + 1,2q}}{\alpha _{2s + l + 2,2q}}{\rho ^{2s + 1 + 2q}} = \sum\limits_{k = 1}^{(n/2) + 1} {{\tilde B}_k}{\rho ^{2k - 1}}.} \hfill \cr }

Since f (ρ) is a polynomial generated by a linear combination of the monomials {ρ, ρ3, . . ., ρn+1}. Using Descartes theorem it follows that the polynomial f (ρ) have at most n/2 simple positive zeros, and consequently from subsection 2.1 we get using the averaging theory of first-order that for ɛ > 0 sufficiently small, the differential system (3) has at most n/2 limit cycles bifurcating from the periodic solutions of the linear centre (3) with ɛ = 0.

Proof of Theorem 1 if n, l and m are odd

Again working as in case 1, we compute the averaged function and we obtain f(ρ)=12π02π(ρ,θ)dθ=s+q=1(n1)/2b2s+1,2q1α2s+1+m,2qρ2s+2q+s+q=1(n+1)/2b2s,2q1α2s,2qρ2s+2q1+s+q=1(n1)/2a2s+1,2qα2s+2,2qρ2s+1+2q+s+q=1(n+1)/2a2s,2qα2s+l+1,2qρ2s+2q=k=1n+1Ckρk. \matrix{ {f(\rho )} \hfill & { = {1 \over {2\pi }}\int_0^{2\pi } {\cal F}(\rho ,\theta )d\theta } \hfill \cr {} \hfill & { = \sum\limits_{s + q = 1}^{(n - 1)/2} {b_{2s + 1,2q - 1}}{\alpha _{2s + 1 + m,2q}}{\rho ^{2s + 2q}} + \sum\limits_{s + q = 1}^{(n + 1)/2} {b_{2s,2q - 1}}{\alpha _{2s,2q}}{\rho ^{2s + 2q - 1}}} \hfill \cr {} \hfill & {\quad + \sum\limits_{s + q = 1}^{(n - 1)/2} {a_{2s + 1,2q}}{\alpha _{2s + 2,2q}}{\rho ^{2s + 1 + 2q}} + \sum\limits_{s + q = 1}^{(n + 1)/2} {a_{2s,2q}}{\alpha _{2s + l + 1,2q}}{\rho ^{2s + 2q}} = \sum\limits_{k = 1}^{n + 1} {C_k}{\rho ^k}.} \hfill \cr }

Since the polynomial f (ρ) has the monomials {ρ, ρ2, . . . , ρn+1}. Using Descartes theorem it follows that the polynomial f (ρ) has at most n simple zeros. Therefore for ɛ > 0 sufficiently small the differential system (3) has at most n limit cycles bifurcating from the periodic solutions of the linear centre (3) with ɛ = 0.

Proof of Theorem 1 if n is odd and l and m are one odd and the other even

We distinguish two cases with n odd.

Case 1: m is odd and l is even. Computing the averaged function we get f(ρ)=12π02π(ρ,θ)dθ=s+q=1(n1)/2b2s+1,2q1α2s+1+m,2qρ2s+2q+s+q=1(n+1)/2b2s,2q1α2s,2qρ2s+2q1+s+q=1(n1)/2a2s+1,2qα2s+2,2qρ2s+1+2q+s+q=1(n1)/2a2s+1,2qα2s+l+2,2qρ2s+1+2q=k=1nC˜kρk. \matrix{ {f(\rho )} \hfill & { = {1 \over {2\pi }}\int_0^{2\pi } {\cal F}(\rho ,\theta )d\theta } \hfill \cr {} \hfill & { = \sum\limits_{s + q = 1}^{(n - 1)/2} {b_{2s + 1,2q - 1}}{\alpha _{2s + 1 + m,2q}}{\rho ^{2s + 2q}} + \sum\limits_{s + q = 1}^{(n + 1)/2} {b_{2s,2q - 1}}{\alpha _{2s,2q}}{\rho ^{2s + 2q - 1}}} \hfill \cr {} \hfill & {\quad + \sum\limits_{s + q = 1}^{(n - 1)/2} {a_{2s + 1,2q}}{\alpha _{2s + 2,2q}}{\rho ^{2s + 1 + 2q}} + \sum\limits_{s + q = 1}^{(n - 1)/2} {a_{2s + 1,2q}}{\alpha _{2s + l + 2,2q}}{\rho ^{2s + 1 + 2q}} = \sum\limits_{k = 1}^n {{\tilde C}_k}{\rho ^k}.} \hfill \cr }

Case 2: m is even and l is odd. Then the averaged function f(ρ)=12π02π(ρ,θ)dθ=s+q=1(n+1)/2b2s,2q1α2s+m,2qρ2s+2q1+s+q=1(n+1)/2b2s,2q1α2s,2qρ2s+2q1+s+q=1(n1)/2a2s+1,2qα2s+2,2qρ2s+1+2q+s+q=1(n1)/2a2s,2qα2s+l+1,2qρ2s+2q=k=1nC^kρk. \matrix{ {f(\rho )} \hfill & { = {1 \over {2\pi }}\int_0^{2\pi } {\cal F}(\rho ,\theta )d\theta } \hfill \cr {} \hfill & { = \sum\limits_{s + q = 1}^{(n + 1)/2} {b_{2s,2q - 1}}{\alpha _{2s + m,2q}}{\rho ^{2s + 2q - 1}} + \sum\limits_{s + q = 1}^{(n + 1)/2} {b_{2s,2q - 1}}{\alpha _{2s,2q}}{\rho ^{2s + 2q - 1}}} \hfill \cr {} \hfill & {\quad + \sum\limits_{s + q = 1}^{(n - 1)/2} {a_{2s + 1,2q}}{\alpha _{2s + 2,2q}}{\rho ^{2s + 1 + 2q}} + \sum\limits_{s + q = 1}^{(n - 1)/2} {a_{2s,2q}}{\alpha _{2s + l + 1,2q}}{\rho ^{2s + 2q}} = \sum\limits_{k = 1}^n {{\hat C}_k}{\rho ^k}.} \hfill \cr }

Now the polynomials f (ρ) in cases 1 and 2 are generated by the monomials {ρ, ρ2, . . . , ρn}. Using Descartes theorem it follows that the polynomial f (ρ) has at most n − 1 simple zeros. Therefore for ɛ > 0 sufficiently small the differential system (3) has at most n − 1 limit cycles bifurcating from the periodic solutions of the linear centre (3) with ɛ = 0.

Proof of Theorem 1 if n is odd, and l and m are even

Computing the averaged function we obtain f(ρ)=12π02π(ρ,θ)dθ=s+q=1(n+1)/2b2s,2q1α2s+m,2qρ2s+2q1+s+q=1(n+1)/2b2s,2q1α2s,2qρ2s+2q1+s+q=1(n1)/2a2s+1,2qα2s+2,2qρ2s+1+2q+s+q=1(n1)/2a2s+1,2qα2s+l+2,2qρ2s+1+2q=k=1(n+1)/2Dkρ2k1+k=1(n1)/2D^kρ2k+1. \matrix{ {f(\rho )} \hfill & { = {1 \over {2\pi }}\int_0^{2\pi } {\cal F}(\rho ,\theta )d\theta } \hfill \cr {} \hfill & { = \sum\limits_{s + q = 1}^{(n + 1)/2} {b_{2s,2q - 1}}{\alpha _{2s + m,2q}}{\rho ^{2s + 2q - 1}} + \sum\limits_{s + q = 1}^{(n + 1)/2} {b_{2s,2q - 1}}{\alpha _{2s,2q}}{\rho ^{2s + 2q - 1}}} \hfill \cr {} \hfill & {\quad + \sum\limits_{s + q = 1}^{(n - 1)/2} {a_{2s + 1,2q}}{\alpha _{2s + 2,2q}}{\rho ^{2s + 1 + 2q}} + \sum\limits_{s + q = 1}^{(n - 1)/2} {a_{2s + 1,2q}}{\alpha _{2s + l + 2,2q}}{\rho ^{2s + 1 + 2q}}} \hfill \cr {} \hfill & { = \sum\limits_{k = 1}^{(n + 1)/2} {D_k}{\rho ^{2k - 1}} + \sum\limits_{k = 1}^{(n - 1)/2} {{\hat D}_k}{\rho ^{2k + 1}}.} \hfill \cr }

The polynomial f (ρ) is generated by the monomials {ρ, ρ3, . . . , ρn}. Using Descartes theorem it follows that the polynomial f (ρ) has at most (n − 1)/2 simple zeros. Therefore for ɛ > 0 sufficiently small the differential system (3) has at most (n − 1)/2 limit cycles bifurcating from the periodic solutions of the linear centre (3) with ɛ = 0.

Proof of Theorem 1 if n, l and m are odd

Again working as in case 1, we compute the averaged function and we obtain f(ρ)=12π02π(ρ,θ)dθ=s+q=1(n1)/2b2s+1,2q1α2s+1+m,2qρ2s+2q+s+q=1(n+1)/2b2s,2q1α2s,2qρ2s+2q1+s+q=1(n1)/2a2s+1,2qα2s+2,2qρ2s+1+2q+s+q=1(n+1)/2a2s,2qα2s+l+1,2qρ2s+2q=k=1n+1Ckρk. \matrix{ {f(\rho )} \hfill & { = {1 \over {2\pi }}\int_0^{2\pi } {\cal F}(\rho ,\theta )d\theta } \hfill \cr {} \hfill & { = \sum\limits_{s + q = 1}^{(n - 1)/2} {b_{2s + 1,2q - 1}}{\alpha _{2s + 1 + m,2q}}{\rho ^{2s + 2q}} + \sum\limits_{s + q = 1}^{(n + 1)/2} {b_{2s,2q - 1}}{\alpha _{2s,2q}}{\rho ^{2s + 2q - 1}}} \hfill \cr {} \hfill & {\quad + \sum\limits_{s + q = 1}^{(n - 1)/2} {a_{2s + 1,2q}}{\alpha _{2s + 2,2q}}{\rho ^{2s + 1 + 2q}} + \sum\limits_{s + q = 1}^{(n + 1)/2} {a_{2s,2q}}{\alpha _{2s + l + 1,2q}}{\rho ^{2s + 2q}} = \sum\limits_{k = 1}^{n + 1} {C_k}{\rho ^k}.} \hfill \cr }

Since the polynomial f (ρ) has the monomials {ρ, ρ2, ⋯, ρn+1}. Using Descartes theorem it follows that the polynomial f (ρ) has at most n simple zeros. Therefore for ɛ > 0 sufficiently small the differential system (3) has at most n limit cycles bifurcating from the periodic solutions of the linear centre (3) with ɛ = 0.

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