Open Journal of Mathematical Analysis
ISSN: 2616-8111 (Online) 2616-8103 (Print)
DOI: 10.30538/psrp-oma2021.0095
Limit cycles of a planar differential system via averaging theory
Department of Mathematics, University of Annaba, Laboratory LMA, P.O. Box 12, Annaba 23000, Algeria.; (H.M & A.M)
\(^{1}\)Corresponding Author: houdaifamelki@gmail.com
Abstract
$$\dot{x}=-y+\varepsilon (1+\sin ^{n}\theta )xP(x,y)$$
$$ \dot{y}=x+\varepsilon (1+\cos ^{m}\theta )yQ(x,y),
$$
where \(P(x,y)\) and \(Q(x,y)\) are polynomials of degree \(n_{1}\) and \(n_{2}\) respectively and \(\varepsilon\) is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of a linear center \(\dot{x}=-y, \dot{y}=x,\) by using the averaging theory of first order.
Keywords:
1. Introduction
One of the main problems in the theory of ordinary differential equations is the study of their limit cycles, their existence, their number and their stability, these properties of limit cycles were studied extensively by mathematicians and physicists, and more recently also by chemists, biologists, economists, etc. A limit cycle of a differential equation is a periodic orbit in the set of all isolated periodic orbits of the differential equation.
The second part of the \(16\)th Hilbert's problem [1] is related to the least upper bound on the number of limit cycles of polynomial vector fields having a fixed degree. The study of differential equations or planar differential systems has been considered by several authors. In [2] the authors studied the limit cycles for a variant of a generalized Riccati equation. Mathieu, in [3] considered the second order differential equation
\begin{equation*} \ddot{x}+b(1+\cos t)x=0, \end{equation*} where b is a real constant. It is called Mathieu equation, which is the simplest mathematical model of an excited system on a parameter. We also recall the Ermakov-Pinney equation which is the Mathieu-Duffing type equation \begin{equation*} \ddot{x}+b(1+\cos t)x-x^\beta=0, \end{equation*} where \(\beta\) is an integer and \(b>0\). The last two equations modeled the dynamics of a system with harmonic parametric excitation and a nonlinear term corresponding to a restoring force, see [4,5].There are several methods exist to study the number of limit cycles that bifurcate from the periodic orbits such as the integrating factor, the abelian integral method, the Poincaré-Melnikov integral method, Poincaré return map and averaging theory. The study of limit cycles for differential equations or planar differential systems by applying the averaging method has been considered by several authors see for instance [6,7,8].
In [9], the authors studied the limit cycles of the second-order differential equations
\begin{equation*} \ddot{x}+\varepsilon(1+\cos^m \theta)Q(x,y)+x=0, \end{equation*} where \(Q(x,y)\) is an arbitrary polynomial of degree \(n\), and for each integer non-negative \(m\).In this paper, our goal is to study the maximum number of limit cycles of a differential planar system bifurcating from the periodic orbits of the linear center
\begin{equation*} \begin{cases} \dot{x}=-y\\ \dot{y}=x, \end{cases} \end{equation*} given byTheorem 1. For all polynomials \(P(x,y)\) and \(Q(x,y)\) have degree \(n_{1}\) and \(n_{2}\) respectively, \(n\) and \(m\) are non-negative integers, then for \(|\varepsilon|>0\) sufficiently small, the maximum number of limit cycles of the differential systems (1) bifurcating from the periodic orbits of the linear center \(\dot{x}=-y, \dot{y}=x\) using averaging theory of first order
- (1) If \(m\) odd and \(n\) odd \begin{equation*} \max \left\{ n_{1},n_{2}\right\}, \end{equation*}
- (2) If \(m\) even and \(n\) even \begin{equation*} \max\left\{\left[ \frac{n_{1}}{2}\right] ,\left[ \frac{n_{2}}{2}\right] \right\}, \end{equation*}
- (3) If \(m\) odd and \(n\) even \begin{equation*} \max \left\{ n_{2},n_{2}+\left[ \frac{n_{1}}{2}\right] -\left[ \frac{n_{2}}{2 }\right] \right\} , \end{equation*}
- (4) If \(m\) even and \(n\) odd \begin{equation*} \max \left\{ n_{1},n_{1}+\left[ \frac{n_{2}}{2}\right] -\left[ \frac{n_{1}}{2 }\right] \right\} , \end{equation*} where \([.]\) denotes the integer part function.
2. The averaging theory of first order
The averaging theory of first for studying periodic orbits was developed in [10,11].Theorem 2. We Consider the differential system
- (i) \(H\) and \(R\) are locally Lipschitz with respect to x,
- (ii) For \(a\in D\) with \(h(a)=0\), there exists a neighborhood \(V\) of a such that \(h(z)\neq0\) for all \(z\in \bar V \setminus\{ a\}\) and \(d_{B}(h,V,0)\neq 0\).
Here we will need some facts from the proof of Theorem 2. Hypothesis (i) assures the existence and uniqueness of the solution of each initial value problem on the interval \(\left[0,T\right]\). Hence, for each \(z\in D\), it is possible to denote by \(x(\cdot,z,\varepsilon)\) the solution of (2) with the initial value \(x(0,z,\varepsilon)=z\).
We consider also the function \(\zeta:D\times(-\varepsilon_{f},\varepsilon_{f})\to\mathbb{R}^n\) defined by
For additional information on the averaging theory, see the books [12,13].
Theorem 3.(Discartes Theorem). Consider the real polynomial \( p(x)=a_{i_{1}}x^{i_{1}}+a_{i_{2}}x^{i_{2}}+......+a_{i_{k}}x^{i_{k}}\) with \( 0\leq i_{1}< i_{2}< ...< i_{k}\) and \(a_{i_{j}}\neq 0\) real constants for \(j\in \{1,2,...,k\}.\) When \(a_{i_{j}}a_{i_{j+1}}< 0,\) we say that \(a_{i_{j}}\) and \( a_{i_{j+1}}\) have a variation of sign. If the number of variations of signs is \(m\), then \(p(x)\) has at most \(m\) positive real roots. Moreover, it is always possible to choose the coefficients of \(p(x)\) in such a way that \(p(x) \) has exactly \(k-1\) positive real roots.
3. Proof Theorem 1
We need the first order averaging theory in to prove of Theorem 1. In order to apply first order averaging method we write system (1), in polar coordinates \((r,\theta)\) where \(x=r\cos(\theta), y=rsin(\theta)\) \(r>0\). If we take \begin{equation*} \left\{ \begin{array}{l} P(x,y)=\displaystyle\sum_{i+j=0}^{n_{1}}a_{ij}x^{i}y^{j}, \\ Q(x,y)=\displaystyle\sum_{i+j=0}^{n_{2}}b_{ij}x^{i}y^{j}, \end{array} \right. \end{equation*} system (1) can be written as follows \begin{equation*} \left\{ \begin{array}{l} \dot{r}=\varepsilon \left(\displaystyle\sum_{i+j=0}^{n_{1}}a_{ij}\left(\cos ^{i+2}\theta \sin ^{j}\theta +\cos ^{i+2}\theta \sin ^{j+n}\theta \right)r^{i+j+1} \right.\\ \;\;\;\;\;\;\;\;\;\;\;\left.+\displaystyle\sum_{i+j=0}^{n_{2}}b_{ij}\left(\cos ^{i}\theta \sin ^{j+2}\theta +\cos ^{i+m}\theta \sin ^{j+2}\theta \right)r^{i+j+1}\right), \\ \dot{\theta}=1+\varepsilon \left(\displaystyle\sum_{i+j=0}^{n_{2}}b_{ij}\left(\cos ^{i+1}\theta \sin ^{j+1}\theta +\cos ^{i+m+1}\theta \sin ^{j+1}\theta \right)r^{i+j} \right.\\ \;\;\;\;\;\;\;\;\;\;\;\left.-\displaystyle\sum_{i+j=0}^{n_{1}}a_{ij}\left(\cos ^{i+1}\theta \sin ^{j+1}\theta +\cos ^{i+1}\theta \sin ^{j+n+1}\theta \right)r^{i+j}\right). \end{array} \right. \end{equation*} Taking \(\theta\) as the new independent variable system, (1) can be written as \begin{align*} \frac{dr}{d\theta } =&\varepsilon \left(\sum_{i+j=0}^{n_{1}}a_{ij}\left(\cos ^{i+2}\theta \sin ^{j}\theta +\cos ^{i+2}\theta \sin ^{j+n}\theta \right)r^{i+j+1} \right.\\ &\left.+\sum_{i+j=0}^{n_{2}}b_{ij}\left(\cos ^{i}\theta \sin ^{j+2}\theta +\cos ^{i+m}\theta \sin ^{j+2}\theta \right)r^{i+j+1}\right)+O(\varepsilon ^{2}) \end{align*}\begin{align*} =&\varepsilon F\left(r,\theta \right)+O(\varepsilon ^{2}), \end{align*} where \begin{eqnarray*} F(r,\theta ) &=&F_{1}(r,\theta )+F_{2}(r,\theta ), \\ F_{1}(r,\theta ) &=&\sum_{i+j=0}^{n_{1}}a_{ij}\left(\cos ^{i+2}\theta \sin ^{j}\theta +\cos ^{i+2}\theta \sin ^{j+n}\theta \right)r^{i+j+1}, \\ F_{2}(r,\theta ) &=&\sum_{i+j=0}^{n_{2}}b_{ij}\left(\cos ^{i}\theta \sin ^{j+2}\theta +\cos ^{i+m}\theta \sin ^{j+2}\theta \right)r^{i+j+1}. \end{eqnarray*} Let \(F_{10}\) be the averaging equation of first order associated with system (1), using the notation introduced in Theorem 2 we compute \(F_{10}\) by integrating \(F_{1}\) with respect to \(\theta\),If \(p\) is a natural number and \(q=0\) we have
\begin{eqnarray*} &&\int_{0}^{2\pi }\sin ^{2l}\theta d\theta =\frac{(2l-1)!!}{2^{l}l!}2\pi ,\\ &&\int_{0}^{2\pi }\sin ^{2l+1}\theta d\theta =0,\\ &&\int_{0}^{2\pi }\cos ^{2l}\theta d\theta =\frac{(2l-1)!!}{2^{l}l!}2\pi, \\&&\int_{0}^{2\pi }\cos ^{2l+1}\theta d\theta =0. \end{eqnarray*} We have also \begin{eqnarray*} \int_{0}^{2\pi }\sin ^{p}\theta \cos ^{2q+1}\theta d\theta &=&0,\\ \int_{0}^{2\pi }\cos ^{p}\theta \sin ^{2q+1}\theta d\theta &=&0. \end{eqnarray*} These last formulas are applicable for arbitrary real \(p\) and non-negative integer \(q\), except the following negative odd integers \(p=-1,-3,...,-(2n+1)\). For more details of these integrals and other, see [14].Now we determine \(\frac{1}{2\pi }\int_{0}^{2\pi }F_{1}(r,\theta )d\theta\), in the following cases
- (1) If \(n\) odd and \(n_{1}\) even \begin{eqnarray*} f_{1}(r) &=&\frac{1}{2\pi }\int_{0}^{2\pi }F_{1}(r,\theta )d\theta \\ &=&\frac{1}{2\pi }\int_{0}^{2\pi }\sum_{i+j=0}^{n_{1}}\left[a_{i,j}\left(\sin ^{j}\theta +\sin ^{j+n}\theta \right)\cos ^{i+2}\theta \right]r^{i+j+1}d\theta \end{eqnarray*}\begin{eqnarray*} &=&\frac{1}{2\pi }\int_{0}^{2\pi }\sum_{2q+j=2}^{n_{1}+2}\left[a_{2q-2,j}(\sin ^{j}\theta +\sin ^{j+n}\theta )\cos ^{2q}\theta \right]r^{2q+j-1}d\theta \\ &=&\frac{1}{2\pi }\left[\sum_{2q+2l+1=3}^{n_{1}+1}a_{2q-2,2l+1}r^{2q+2l} \int_{0}^{2\pi }\sin ^{2l+n+1}\theta \cos ^{2q}\theta d\theta \right. \\ &&\left. +\sum_{2l+2q=2}^{n_{1}+2}a_{2q-2,2l}r^{2q+2l-1}\int_{0}^{2\pi }\sin ^{2l}\theta \cos ^{2q}\theta d\theta \right] \\ &=&\sum_{l+q=1}^{n_{1}/2}a_{2q-2,2l+1}r^{2l+2q}\frac{(2q-1)!!}{ (2q+2l+1)(2q+2l-1)...(2l+3)}\frac{(2l+n)!!}{2^{\tfrac{2l+n+1}{2}}(\frac{ 2l+n+1}{2})!} \\ &&+\sum_{l+q=1}^{(n_{1}+2)/2}a_{2q-2,2l}r^{2l+2q-1}\frac{(2q-1)!!}{ (2q+2l)(2q+2l-2)...(2l+2)}\frac{(2l-1)!!}{2^{l}l!}\\ &=&\sum_{l+q=1}^{n_{1}/2}a_{2q-2,2l+1}r^{2q+2l}\frac{ (2l+n)!!(2q-1)!!}{2^{\tfrac{2l+n+1}{2}}(\frac{2l+n+1}{2} )!(2q+2l+1)(2q+2l-1)...(2l+3)} \\ &&+\sum_{l+q=1}^{(n_{1}+2)/2}a_{2q-2,2l}r^{2q+2l-1}\frac{ (2q-1)!!(2l-1)!!}{2^{l+q}l!(q+l)(q+l-1)...(l+1)} \\ &=&\sum_{k=1}^{n_{1}+1}A_{k}r^{k}. \end{eqnarray*}
- (2)If \(n\) odd and \(n_{1}\) odd \begin{eqnarray*} f_{2}(r) &=&\frac{1}{2\pi }\int_{0}^{2\pi }F_{1}(r,\theta )d\theta \\ &=&\frac{1}{2\pi }\int_{0}^{2\pi }\sum_{i+j=0}^{n_{1}}\left[a_{i,j}(\sin ^{j}\theta +\sin^{j+n}\theta )\cos ^{i+2}\theta \right]r^{i+j+1}d\theta \\ &=&\frac{1}{2\pi }\int_{0}^{2\pi }\sum_{2q+j=2}^{n_{1}+2}\left[a_{2q-2,j}(\sin ^{j}\theta +\sin ^{j+n}\theta )\cos ^{2q}\theta \right]r^{2q+j-1}d\theta \\ &=&\frac{1}{2\pi }\left[\sum_{2q+2l+1=3}^{n_{1}+2}a_{2q-2,2l+1}r^{2q+2l} \int_{0}^{2\pi }\sin ^{2l+n+1}\theta \cos ^{2q}\theta d\theta \right. \\ &&\left.+\sum_{2l+2q=2}^{n_{1}+1}a_{2q-2,2l}r^{2q+2l-1}\int_{0}^{2\pi }\sin ^{2l}\theta \cos ^{2q}\theta d\theta \right]\\ &=&\sum_{l+q=1}^{(n_{1}+1)/2}a_{2q-2,2l+1}r^{2q+2l}\frac{ (2l+n)!!(2q-1)!!}{2^{\tfrac{2l+n+1}{2}}(\frac{2l+n+1}{2} )!(2q+2l+1)(2q+2l-1)...(2l+3)} \\ &&+\sum_{l+q=1}^{(n_{1}+1)/2}a_{2q-2,2l}r^{2q+2l-1}\frac{ (2q-1)!!(2l-1)!!}{2^{l+q}l!(q+l)(q+l-1)...(l+1)} \\ &=&\sum_{k=1}^{n_{1}+1}\tilde{A}_{k}r^{k}. \end{eqnarray*}
- (3) If \(n\) even and \(n_{1}\) even \begin{eqnarray*} f_{3}(r) &=&\frac{1}{2\pi }\int_{0}^{2\pi }F_{1}(r,\theta )d\theta \\ &=&\frac{1}{2\pi }\int_{0}^{2\pi }\sum_{i+j=0}^{n_{1}}[a_{i,j}(\sin ^{j}\theta +\sin ^{j+n}\theta )\cos ^{i+2}\theta ]r^{i+j+1}d\theta \end{eqnarray*}\begin{eqnarray*} &=&\frac{1}{2\pi }\int_{0}^{2\pi }\sum_{2q+j=2}^{n_{1}+2}[a_{2q-2,j}(\sin ^{j}\theta +\sin ^{j+n}\theta )\cos ^{2q}\theta ]r^{2q+j-1}d\theta \\ &=&\frac{1}{2\pi }\left[\sum_{2l+2q=2}^{n_{1}+2}a_{2q-2,2l}r^{2q+2l-1}\int_{0}^{2 \pi }(\sin ^{2l}\theta +\sin ^{2l+n}\theta )\cos ^{2q}\theta d\theta \right]\\ &=&\sum_{l+q=1}^{(n_{1}+2)/2}a_{2q-2,2l}r^{2q+2l-1}\frac{(2q-1)!!}{ 2^{q}(q+l)(q+l-1)...(l+1)}\left[\frac{(2l-1)!!}{2^{l}l!}+\frac{(2l+n-1)!!}{2^{ \tfrac{2l+n}{2}}(\frac{2l+n}{2})!}\right] \\ &=&\sum_{\substack{ k=1 \\ k\text{ }odd}}^{n_{1}+1}\bar{A}_{k}r^{k}. \end{eqnarray*}
- (4) If \(n\) even and \(n_{1}\) odd \begin{eqnarray*} f_{4}(r) &=&\frac{1}{2\pi }\int_{0}^{2\pi }F_{1}(r,\theta )d\theta \\ &=&\frac{1}{2\pi }\int_{0}^{2\pi }\sum_{i+j=0}^{n_{1}}[a_{i,j}(\sin ^{j}\theta +\sin ^{j+n}\theta )\cos ^{i+2}\theta ]r^{i+j+1}d\theta \\ &=&\frac{1}{2\pi }\int_{0}^{2\pi }\sum_{2q+j=2}^{n_{1}+2}[a_{2q-2,j}(\sin ^{j}\theta +\sin ^{j+n}\theta )\cos ^{2q}\theta ]r^{2q+j-1}d\theta \\ &=&\frac{1}{2\pi }\left[\sum_{2l+2q=2}^{n_{1}+1}a_{2q-2,2l}r^{2q+2l-1}\int_{0}^{2 \pi }(\sin ^{2l}\theta +\sin ^{2l+n}\theta )\cos ^{2q}\theta d\theta \right]\\ &=&\sum_{l+q=1}^{(n_{1}+1)/2}a_{2q-2,2l}r^{2q+2l-1}\frac{(2q-1)!!}{ 2^{q}(q+l)(q+l-1)...(l+1)}\left[\frac{(2l-1)!!}{2^{l}l!}+\frac{(2l+n-1)!!}{2^{ \tfrac{2l+n}{2}}(\frac{2l+n}{2})!}\right]\\ &=&\sum_{\substack{ k=1 \\ k\text{ }impair}}^{n_{1}}\bar{A}_{k}r^{k}. \end{eqnarray*}
- (5) If \(m\) odd and \(n_{2}\) even \begin{eqnarray*} f_{5}(r) &=&\frac{1}{2\pi }\int_{0}^{2\pi }F_{2}(r,\theta )d\theta \\ &=&\frac{1}{2\pi }\int_{0}^{2\pi }\sum_{i+j=0}^{n_{2}}\left[b_{i,j}(\cos ^{i}\theta +\cos ^{i+m}\theta )\sin ^{j+2}\theta \right]r^{i+j+1}d\theta \\ &=&\frac{1}{2\pi }\int_{0}^{2\pi }\sum_{i+2q=2}^{n_{2}+2}\left[b_{i,2q-2}(\cos ^{i}\theta +\cos ^{i+m}\theta )\sin ^{2q}\theta \right]r^{i+2q-1}d\theta \\ &=&\frac{1}{2\pi }\left[\sum_{2l+1+2q=3}^{n_{2}+1}b_{2l+1,2q-2}r^{2l+2q} \int_{0}^{2\pi }\cos ^{2l+m+1}\theta \sin ^{2q}\theta d\theta \right. \\ &&\left.+\sum_{2l+2q=2}^{n_{2}+2}b_{2l,2q-2}r^{2l+2q-1}\int_{0}^{2\pi }\cos ^{2l}\theta \sin ^{2q}\theta d\theta \right]\\ &=&\sum_{l+q=1}^{n_{2}/2}b_{2l+1,2q-2}r^{2l+2q}\frac{(2q-1)!!}{ (2q+2l+1)(2q+2l-1)...(2l+3)}\frac{(2l+m)!!}{2^{\tfrac{2l+m+1}{2}}(\frac{ 2l+m+1}{2})!} \\ &&+\sum_{l+q=1}^{(n_{2}+2)/2}b_{2l,2q-2}r^{2l+2q-1}\frac{(2q-1)!!}{ (2q+2l)(2q+2l-2)...(2l+2)}\frac{(2l-1)!!}{2^{l}l!} \end{eqnarray*} \begin{eqnarray*} &=&\sum_{l+q=1}^{n_{2}/2}b_{2l+1,2q-2}r^{2l+2q}\frac{(2l+m)!!(2q-1)!! }{2^{\tfrac{2l+m+1}{2}}(\frac{2l+m+1}{2})!(2q+2l+1)(2q+2l-1)...(2l+3)} \\ &&+\sum_{l+q=1}^{(n_{2}+2)/2}b_{2l,2q-2}r^{2l+2q-1}\frac{ (2l-1)!!(2q-1)!!}{2^{l+q}l!(q+l)(q+l-1)...(l+1)} \\ &=&\sum_{k=1}^{n_{2}+1}B_{k}r^{k}. \end{eqnarray*}
- (6) If \(m\) odd and \(n_{2}\) odd \begin{eqnarray*} f_{6}(r) &=&\frac{1}{2\pi }\int_{0}^{2\pi }F_{2}(r,\theta )d\theta \\ &=&\frac{1}{2\pi }\int_{0}^{2\pi }\sum_{i+j=0}^{n_{2}}\left[b_{i,j}(\cos ^{i}\theta +\cos ^{i+m}\theta )\sin ^{j+2}\theta \right]r^{i+j+1}d\theta \\ &=&\frac{1}{2\pi }\int_{0}^{2\pi }\sum_{i+2q=2}^{n_{2}+2}\left[b_{i,2q-2}(\cos ^{i}\theta +\cos ^{i+m}\theta )\sin ^{2q}\theta \right]r^{i+2q-1}d\theta \\ &=&\frac{1}{2\pi }\left[\sum_{2l+1+2q=3}^{n_{2}+2}b_{2l+1,2q-2}r^{2l+2q} \int_{0}^{2\pi }\cos ^{2l+m+1}\theta \sin ^{2q}\theta d\theta \right. \\ &&\left.+ \sum_{2l+2q=2}^{n_{2}+1}b_{2l,2q-2}r^{2l+2q-1}\int_{0}^{2\pi }\cos ^{2l}\theta sin ^{2q}\theta d\theta \right] \\ &=&\sum_{l+q=1}^{(n_{2}+1)/2}b_{2l+1,2q-2}r^{2l+2q}\frac{ (2l+m)!!(2q-1)!!}{2^{\tfrac{2l+m+1}{2}}(\frac{2l+m+1}{2} )!(2q+2l+1)(2q+2l-1)...(2l+3)} \\ &&+\sum_{l+q=1}^{(n_{2}+1)/2}b_{2l,2q-2}r^{2l+2q-1}\frac{ (2l-1)!!(2q-1)!!}{2^{l+q}l!(q+l)(q+l-1)...(l+1)} \\ &=&\sum_{k=1}^{n_{2}+1}\tilde{B}_{k}r^{k}. \end{eqnarray*}
- (7) If \(m\) even and \(n_{2}\) even \begin{eqnarray*} f_{7}(r) &=&\frac{1}{2\pi }\int_{0}^{2\pi }F_{2}(r,\theta )d\theta \\ &=&\frac{1}{2\pi }\int_{0}^{2\pi }\sum_{i+j=0}^{n_{2}}\left[b_{i,j}(\cos ^{i}\theta +\cos ^{i+m}\theta )\sin ^{j+2}\theta \right]r^{i+j+1}d\theta \\ &=&\frac{1}{2\pi }\int_{0}^{2\pi }\sum_{i+2q=2}^{n_{2}+2}\left[b_{i,2q-2}(\cos ^{i}\theta +\cos ^{i+m}\theta )\sin ^{2q}\theta \right]r^{i+2q-1}d\theta \\ &=&\frac{1}{2\pi }\left[\sum_{2l+2q=2}^{n_{2}+2}b_{2l,2q-2}r^{2l+2q-1}\int_{0}^{2 \pi }(\cos ^{2l}\theta +\cos ^{2l+m}\theta )\sin ^{2q}\theta d\theta \right] \\ &=&\sum_{l+q=1}^{(n_{2}+2)/2}b_{2l,2q-2}r^{2l+2q-1}\frac{(2q-1)!!}{ 2^{q}(q+l)(q+l-1)...(l+1)}\left[\frac{(2l-1)!!}{2^{l}l!}+\frac{(2l+m-1)!!}{2^{ \tfrac{2l+m}{2}}(\frac{2l+m}{2})!}\right]\\ &=&\sum_{\substack{ k=1 \\ k\text{ }odd}}^{n_{2}+1}\bar{B}_{k}r^{k}. \end{eqnarray*}
- (8) If \(m\) even and \(n_{2}\) odd \begin{eqnarray*} f_{8}(r)&=&\frac{1}{2\pi }\int_{0}^{2\pi }F_{2}(r,\theta )d\theta \end{eqnarray*}\begin{eqnarray*} &=&\frac{1}{2\pi }\int_{0}^{2\pi }\sum_{i+j=0}^{n_{2}}\left[ b_{i,j}(\cos ^{i}\theta +\cos ^{i+m}\theta )\sin ^{j+2}\theta \right]r^{i+j+1}d\theta \\ &=&\frac{1}{2\pi }\int_{0}^{2\pi }\sum_{i+2q=2}^{n_{2}+2}\left[b_{i,2q-2}(\cos ^{i}\theta +\cos ^{i+m}\theta )\sin ^{2q}\theta \right]r^{i+2q-1}d\theta \\ &=&\frac{1}{2\pi }\left[\sum_{2l+2q=2}^{n_{2}+1}b_{2l,2q-2}r^{2l+2q-1}\int_{0}^{2 \pi }(\cos ^{2l}\theta +\cos ^{2l+m}\theta )\sin ^{2q}\theta d\theta \right] \\ &=&\sum_{l+q=1}^{(n_{2}+1)/2}b_{2l,2q-2}r^{2l+2q-1}\frac{(2q-1)!!}{ 2^{q}(q+l)(q+l-1)...(l+1)}\left[\frac{(2l-1)!!}{2^{l}l!}+\frac{(2l+m-1)!!}{2^{ \tfrac{2l+m}{2}}(\frac{2l+m}{2})!}\right]\\ &=&\sum_{\substack {k=1 \\ k\text{ }odd}}^{n_{2}}\bar{B}_{k}r^{k}. \end{eqnarray*}
- (a) If \(m\) odd and \(n\) odd
- (a.1) n\(_{1}\) even et n\(_{2}\) even \begin{equation*} F_{10}(r)=\sum_{k=1}^{n_{1}+1}A_{k}r^{k}+\sum_{k=1}^{n_{2}+1}B_{k}r^{k}, \end{equation*}
- (a.2) n\(_{1}\) odd et n\(_{2}\) odd \begin{equation*} F_{10}(r)=\sum_{k=1}^{n_{1}+1}\tilde{A}_{k}r^{k}+\sum_{k=1}^{n_{2}+1}\tilde{B }_{k}r^{k}, \end{equation*}
- (a.3) n\(_{1}\) odd et n\(_{2}\) even \begin{equation*} F_{10}(r)=\sum_{k=1}^{n_{1}+1}\tilde{A}_{k}r^{k}+ \sum_{k=1}^{n_{2}+1}B_{k}r^{k}, \end{equation*}
- (a.4) n\(_{1}\) even et n\(_{2}\) odd \begin{equation*} F_{10}(r)=\sum_{k=1}^{n_{1}+1}A_{k}r^{k}+\sum_{k=1}^{n_{2}+1}\tilde{B} _{k}r^{k}. \end{equation*}
- (b) If \(m\) even and \(n\) even.
- (b.1) n\(_{1}\) even et n\(_{2}\) even \begin{equation*} F_{10}(r)=\sum_{\substack{ k=1 \\ k\text{ }odd}}^{n_{1}+1}\bar{A} _{k}r^{k}+\sum_{\substack{ k=1 \\ k\text{ }odd}}^{n_{2}+1}\bar{B}_{k}r^{k}, \end{equation*}
- (b.2) n\(_{1}\) odd et n\(_{2}\) odd \begin{equation*} F_{10}(r)=\sum_{\substack{ k=1 \\ k\text{ }odd}}^{n_{1}}\bar{A} _{k}r^{k}+\sum_{\substack{ k=1 \\ k\text{ }odd}}^{n_{2}}\bar{B}_{k}r^{k}, \end{equation*}
- (b.3) n\(_{1}\) odd et n\(_{2}\) even \begin{equation*} F_{10}(r)=\sum_{\substack{ k=1 \\ k\text{ }odd}}^{n_{1}}\bar{A} _{k}r^{k}+\sum_{\substack{ k=1 \\ k\text{ }odd}}^{n_{2}+1}\bar{B}_{k}r^{k}, \end{equation*}
- (b.4) n\(_{1}\) even et n\(_{2}\) odd \begin{equation*} F_{10}(r)=\sum_{\substack{ k=1 \\ k\text{ }odd}}^{n_{1}+1}\bar{A} _{k}r^{k}+\sum_{\substack{ k=1 \\ k\text{ }odd}}^{n_{2}}\bar{B}_{k}r^{k}. \end{equation*}
- (c) If \(m\) odd and \(n\) even.
- (c.1) n\(_{1}\) even et n\(_{2}\) even \begin{equation*} F_{10}(r)=\sum_{\substack{ k=1 \\ k\text{ }odd}}^{n_{1}+1}\bar{A} _{k}r^{k}+\sum_{k=1}^{n_{2}+1}B_{k}r^{k}, \end{equation*}
- (c.2) n\(_{1}\) odd et n\(_{2}\) odd \begin{equation*} F_{10}(r)=\sum_{\substack{ k=1 \\ k\text{ }odd}}^{n_{1}}\bar{A} _{k}r^{k}+\sum_{k=1}^{n_{2}+1}\tilde{B}_{k}r^{k}, \end{equation*}
- (c.3) n\(_{1}\) odd et n\(_{2}\) even \begin{equation*} F_{10}(r)=\sum_{\substack{ k=1 \\ k\text{ }odd}}^{n_{1}}\bar{A} _{k}r^{k}+\sum_{k=1}^{n_{2}+1}B_{k}r^{k}, \end{equation*}
- (c.4) n\(_{1}\) even et n\(_{2}\) odd \begin{equation*} F_{10}(r)=\sum_{\substack{ k=1 \\ k\text{ }odd}}^{n_{1}+1}\bar{A} _{k}r^{k}+\sum_{k=1}^{n_{2}+1}\tilde{B}_{k}r^{k}. \end{equation*}
- (d) If \(n\) odd and \(m\) even.
- (d.1) n\(_{1}\) even et n\(_{2}\) even \begin{equation*} F_{10}(r)=\sum_{k=1}^{n_{1}+1}A_{k}r^{k}+\sum_{\substack{ k=1 \\ k\text{ } odd}}^{n_{2}+1}\bar{B}_{k}r^{k}, \end{equation*}
- (d.2) n\(_{1}\) odd et n\(_{2}\) odd \begin{equation*} F_{10}(r)=\sum_{k=1}^{n_{1}+1}\tilde{A}_{k}r^{k}+\sum_{\substack{ k=1 \\ k \text{ }odd}}^{n_{2}}\bar{B}_{k}r^{k}, \end{equation*}
- (d.3) n\(_{1}\) odd et n\(_{2}\) even \begin{equation*} F_{10}(r)=\sum_{k=1}^{n_{1}+1}\tilde{A}_{k}r^{k}+\sum_{\substack{ k=1 \\ k \text{ }odd}}^{n_{2}+1}\bar{B}_{k}r^{k}, \end{equation*}
- (d.4) n\(_{1}\) even et n\(_{2}\) odd \begin{equation*} F_{10}(r)=\sum_{k=1}^{n_{1}+1}A_{k}r^{k}+\sum_{\substack{ k=1 \\ k\text{ } odd}}^{n_{2}}\bar{B}_{k}r^{k}. \end{equation*}
4. Example
We consider the system5. Conclusion
In the present paper, by using the averaging theory of the first order we show that the maximum number of the limit cycles bifurcating from linear center \( \dot{x}=-y, \dot{y}=x,\) for a generalized planar differential system.Author Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.Conflicts of Interest
"The authors declare no conflict of interest."References
- Hilbert, D. (1900). Mathematische probleme, lecture, second Internat Congr Math (Paris, 1990), Nachr. Ges. Wiss. G" ttingen Math. Phys. KL., 253-297. English Translation, Bulletin of the American Mathematical Society, 8(1902), 437-479. [Google Scholor]
- Llibre, J., & Valls, C. (2017). Limit cycles for a variant of a generalized Riccati equation. Applied Mathematics Letters, 68, 76-79. [Google Scholor]
- Mathieu, É. (1968). Mémoire sur le movement vibratoire d'une membrane de forme elliptique. Journal de Mathématiques Pures et Appliquées, 13, 137-203. [Google Scholor]
- Torres, P. J. (2006). Non-trivial periodic solutions of a non-linear Hill's equation with positively homogeneous term. Nonlinear Analysis: Theory, Methods & Applications, 65(4), 841-844. [Google Scholor]
- Zamora, M. (2017). A note on the periodic solutions of a Mathieu-Duffing type equations. Mathematische Nachrichten, 290(7), 1113-1118. [Google Scholor]
- Llibre, J., & Mereu, A. C. (2011). Limit cycles for generalized Kukles polynomial differential systems. Nonlinear Analysis: Theory, Methods & Applications, 74(4), 1261-1271. [Google Scholor]
- Boulfoul, A., Makhlouf, A., & Mellahi, N. (2019). On the limit cycles for a class of generalized kukles differential systems. Journal of Applied Analysis & Computation, 9(3), 864-883. [Google Scholor]
- García, B., Llibre, J., & Del Río, J. S. P. (2014). Limit cycles of generalized Liénard polynomial differential systems via averaging theory. Chaos, Solitons & Fractals, 62, 1-9. [Google Scholor]
- Chen, T., & Llibre, J. (2019). Limit cycles of a second-order differential equation. Applied Mathematics Letters, 88, 111-117. [Google Scholor]
- Buica, A., & Llibre, J. (2004). Averaging methods for finding periodic orbits via Brouwer degree. Bulletin des Sciences Mathematiques, 128(1), 7-22. [Google Scholor]
- Llibre, J., Mereu, A. C., & Teixeira, M. A. (2010, March). Limit cycles of the generalized polynomial Liénard differential equations. In Mathematical Proceedings of the Cambridge Philosophical Society (Vol. 148, No. 2, pp. 363-383). Cambridge University Press. [Google Scholor]
- Sanders, J. A., Verhulst, F., & Murdock, J. (2007). Averaging Methods in Nonlinear Dynamical Systems(Vol. 59). New York: Springer. [Google Scholor]
- Wiggins, S. (1990). Nonlinear Differential Equations and Dynamical Systems. Universitex, Berlin, Springer-Verlag. [Google Scholor]
- Gradshteyn, I. S., & Ryzhik, I. M. (2014). Table of Integrals, Series, and Products. Academic press. [Google Scholor]