# Limit Cycles for the Class ofD-Dimensional Polynomial Differential Systems

Authors: Zouhair Diab; Amar Makhlouf
Journal: Journal of Applied Mathematics
 (2016)

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## Abstract

We perturb the differential systemx˙1=-x2(1+x1), x˙2=x1(1+x1), and x˙k=0 for k=3,…,d inside the class of all polynomial differential systems of degree n in Rd, and we prove that at most nd-1 limit cycles can be obtained for the perturbed system using the first-order averaging theory.


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## Body

## 1. Introduction

One of the main problems in the theory of differential systems is the study of their periodic orbits, their existence, their number, and their stability. As usual, a limit cycle of a differential system is a periodic orbit isolated in the set of all periodic orbits of the differential system.In [1], the authors studied the differential system(1)x˙=-y+εax+Px,y,y˙=x+εay+Qx,y,where Px,y and Qx,y are arbitrary polynomials of degree n starting with terms of degree 2, a is a real parameter, and ε is small parameter. They proved that for ε≠0 sufficiently small, the maximum number of limit cycles bifurcating from the periodic orbits of the linear center x˙=-y, y˙=x, obtained using the averaging theory of first order, is (n-1)/2 if n is odd and (n-2)/2 if n is even. In the same paper, the authors studied the limit cycles of the differential system(2)x˙=-y+εax+Px,y,z,y˙=x+εay+Qx,y,z,z˙=εbz+Rx,y,z,where Px,y,z, Qx,y,z, and Rx,y,z are arbitrary polynomials of degree n starting with terms of degree 2 and a,b∈R. Then, there exists ε0>0 sufficiently small such that for ε<ε0 there are systems (2) having at least nn-1/2 limit cycles bifurcating from the periodic orbits of the system x˙=-y, y˙=x, and z˙=0.In [2], the authors studied the number of limit cycles of the differential system(3)x˙=-y1+x+εax+Px,y,z,y˙=x1+x+εay+Qx,y,z,z˙=εbz+Rx,y,z,where F(x,y,z), G(x,y,z), and R(x,y,z) are polynomials of degree n starting from terms of degree 2. Then, there exists ε0>0 sufficiently small such that for ε<ε0 there are systems (3) having at least n2 limit cycles bifurcating from the periodic orbits of the system x˙=-y1+x, y˙=x1+x, and z˙=0.In general, to obtain analytically periodic solutions of a differential system is a very difficult work, usually impossible. Here, using the averaging theory of first order, we will study the number of limit cycles of the differential system(4)x˙1=-x21+x1+εax1+P1x1,…,xd,x˙2=x11+x1+εax2+P2x1,…,xd,x˙k=εbkxk+Pkx1,…,xd,k=3,…,d,in Rd, where Pkx1,…,xd for k=1,…,d is a polynomial of degree n starting with terms of degree 2, a,bk∈R, and ε is a small parameter.The problem of studying the limit cycles of system (4) is reduced using the averaging theory of first order to find the zeros of a nonlinear system of d-2 equations with d-2 unknowns. It is known that in general the averaging theory for finding periodic solutions does not provide all the periodic solutions of the system; this is due to two main reasons. First, the averaging theory for studying the periodic solutions of a differential system is based on the so-called displacement function, whose zeros provide periodic solutions of the differential system. This displacement function in general is not global and consequently it cannot control all the periodic solutions of the differential system, only the ones which are in its domain of definition and are hyperbolic. Second, the displacement function is expanded in power series of a small parameter ε, and the averaging theory only controls the zeros of the dominant term of this displacement function. When the dominant term is εk, we talk about the averaging theory of order k. For more details, see, for instance, [3] and the references quoted there. The averaging theory of first order necessary for the results of this paper is summarized in Section 2.Our main result on the limit cycles of the differential system (4) is as follows.Theorem 1.
By applying the first-order averaging theory to the polynomial differential system (4), for ε≠0 sufficiently small at most nd-1 limit cycles bifurcate from the periodic orbits of the differential system x˙1=-x2(1+x1), x˙2=x1(1+x1), x˙3=0, and x˙4=0,…,x˙d=0.Theorem1 is proved in Section 3.


## 2. Limit Cycles via Averaging Theory

Roughly speaking, we can say that the averaging theory gives a quantitative relation between periodic solutions of a nonautonomous periodic differential system and the solutions of its averaged differential system, which is autonomous. The next result provides a first-order approximation inε for the limit cycles of a periodic differential system; for a proof, see Theorem 2.6.1 of [4] and Theorem 11.5 of [5].Theorem 2.
One considers the following two initial-value problems:(5)x˙=εft,x+ε2gt,x,ε,x0=x0,(6)y˙=εf0y,y0=x0,where x, y, x0∈D, and D is an open subset of Rn, t∈[0,∞), |ε|≤ε0, t∈[0,∞), |ε|≤ε0, f and g are periodic of period T in the variable t, and f0(y) is the averaged function of f(t,x) with respect to t; that is, (7)f0y=1T∫0Tft,ydt.Assume that(i)
f, its Jacobian ∂f/∂x, its Hessian ∂2f/∂x2, g, and its Jacobian ∂g/∂x are defined, continuous, and bounded by a constant independent of ε in [0,∞)×D and |ε|≤ε0;(ii)
T is a constant independent of |ε|;(iii)
y(t) belongs to D on the time interval [0,1/|ε|].
Then, the following statements hold:(a)
On the time scale1/|ε|, we have that x(t)-y(t)=O(ε) as ε→0.(b)
Ifp is a singular point of the averaged system (6) such that the determinant of the Jacobian matrix ∂f0/∂y|y=p is not zero, then there exists a limit cycle φ(t,ε) of period T for system (5) such that φ(0,ε)→p as ε→0.(c)
The stability or instability of the limit cycleφ(t,ε) is given by the stability or instability of the singular point p of the averaged system (6) when p is a hyperbolic singular point.To prove Theorem1, we need the following three lemmas which are proved in [6].Before doing the proof of Theorem1, we recall the Bézout theorem which will be used later on; for a proof of this result, see [7].Theorem 3 (Bézout theorem).
Letqj be polynomials in the variables (x1,…,xd) of degree dj for j=1,…,d. Consider the following polynomial system: q1(x1,…,xd)=0,…,qd(x1,…,xd)=0, where (x1,…,xd)∈Rd. If the number of solutions of this system is finite, then it is bounded by d1⋯dd.Lemma 4.
Fori,j∈N, one defines (8)Ii,j=12π∫02πcosi⁡θsinj⁡θ1+cos⁡θdθ.Then, Ii,j=0 if and only if j is even. For i,j∈N with j even, one has (9)
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.Lemma 5.
The following equalities hold. Fork∈N, one has (10)Ek=12π∫02πcosk⁡θdθ=0ifkisodd,Ckk/22-kifkiseven;I0,0=12π∫02π11+rcos⁡θdθ=11-r2.Lemma 6.
Fori∈N, one has (11)Ii,0=12π∫02πcosi⁡θ1+rcos⁡θdθ=-1iri1-r2+∑l=1l≡imod2i-1l-12l-ii-li-l2r-l.


## 3. Proof of Theorem1

Doing the change to polar coordinatesx1=rcos⁡θ, x2=rsin⁡θ, system (4) becomes (12)r˙=εar+∑i=2n∑i1+⋯+id=iri1+i2x3i3⋯xdidai1,i2,…,idi,1cosi1+1⁡θsini2⁡θ+ai1,i2,…,idi,2cosi1⁡θsini2+1⁡θθ˙=1+rcos⁡θ+εr∑i=2n∑i1+⋯+id=iri1+i2x3i3⋯xdidai1,i2,…,idi,2cosi1+1⁡θsini2⁡θ-ai1,i2,…,idi,1cosi1⁡θsini2+1⁡θx˙k=εbkxk+∑i=2n∑i1+⋯+id=iai1,i2,…,idi,kri1+i2x3i3⋯xdidcosi1⁡θsini2⁡θ,where k=3,…,d. Taking θ as the new independent variable instead of t, this differential system can be written as (13)drdθ=εFθ,r,x3,…,xd+Oε2,dxkdθ=εGkθ,r,x3,…,xd+Oε2,for k=3,…,d, where (14)Fθ,r,x3,…,xd=arD0,0+∑i=2n∑i1+⋯+id=iri1+i2x3i3⋯xdidai1,i2,…,idi,1Di1+1,i2+ai1,i2,…,idi,2Di1,i2+1,Gkθ,r,x3,…,xd=bkxdD0,0+∑i=2n∑i1+⋯+id=iai1,i2,…,idi,kri1+i2x3i3⋯xdidDi1,i2,with (15)Di1,i2=cosi1⁡θsini2⁡θ1+rcos⁡θ.Now, using the notation introduced in Lemma 4 and applying the first-order averaging method, we must find the zeros of the system (16)fr,x3,…,xd=0,gkr,x3,…,xd=0,fork=3,…,d,where (17)fr,x3,…,xd=12π∫02πFθ,r,x3,…,xddθ=arI0,0+∑i=2n∑i1+⋯+id=iri1+i2x3i3⋯xdidai1,i2,…,idi,1Ii1+1,i2+ai1,i2,…,idi,2Ii1,i2+1,gkr,x3,…,xd=12π∫02πGkθ,r,x3,…,xddθ=bkxkI0,0+∑i=2n∑i1+⋯+id=iai1,i2,…,idi,kri1+i2x3i3⋯xdidIi1,i2,for k=3,…,d, and (18)Ii,j=12π∫02πDi1,i2dθ=12π∫02πcosi1⁡θsini2⁡θ1+cos⁡θdθ.Theorem 7.
Lett=1-r2 and k=3,…,d. The function tgk(t,x3,…,xd) is a polynomial of degree n in the variables t and xk, while rtf(t,x3,…,xd) is a polynomial of degree n+1. Moreover, rtf(t,x3,…,xd)=(t-1)Q(t,x3,…,xd), where Q is a polynomial in the variables t and x3,x4,…,xd of the degree at most n.Proof.
The functiongk is a linear combination of xkI0,0 and ri1+i2x3i3⋯xdidIi1,i2, where 2≤i1+i2+⋯+id≤n.
Lemma4 claims that (19)rmIm,0=-1m1-r2-1+Xmr,where Xm is an even polynomial of the degree m-1 if m is odd and of degree m-2 otherwise. Using the variable t=1-r2, we conclude that (20)rmIm,0=-1m+X^mtt,where X^mt is an odd polynomial of degree m or m-1. Since rmIm,0 vanishes at r=0, the functions rkIk,0 for k=2,…,m span the space of functions of the form [A+X^(t)]/t vanishing at t=1 with deg⁡X^(t)=m or m-1, respectively. Lemma 4 implies that any function ri+jIi,j is of the form (21)ϕi,j=Yi,jt+X^i+jtt,where Yi,jt is an even polynomial in t of the degree j (j is necessarily even by Lemma 4) and X^i+jt is a polynomial in t of the degree i+j or i+j-1. We conclude that the functions ri1+i2Ii1,i2, where 2-i3+⋯+id≤i1+i2≤n-i3+⋯+id, generate the space of functions Z(t)/t, where deg⁡Z≤n-i3+⋯+id (and, in addition, Z(1)=0). Therefore, {Pk(t,x3,…,xd)/t,deg⁡Pk≤n}, k=3,…,d.
In a similar way,f is a linear combination of r and terms ri1+i2x3i3⋯xdidIi1+1,i2 and ri1+i2x3i3⋯xdidIi1,i2+1, where 2≤i1+i2+⋯+id≤n. We conclude that the functions ri1+i2Ii1+1,i2 and ri1+i2Ii1,i2+1, where 2-i3+⋯+id≤i1+i2≤n-i3+⋯+id, generate the space of functions Z(t)/rt, where deg⁡Z≤n+1-i3+⋯+id. We have f(0,x3,…,xd)=0 which implies Z(1)=0. Therefore, (t-1)Q(t,x3,…,xd)/rt,deg⁡Q≤n. So the polynomials Q(t,x3,…,xd) and tgk(t,x3,…,xd) in the variables t,x3,…,xd have at most degree n. Hence, by the Bézout theorem, the maximum number of solutions of tgk(t,x3,…,xd)=0 for k=3,…,d and Q(t,x3,…,xd)=0 is at most nd-1 for 0<t<1.Thus, from Theorems2 and 3, it follows that the maximum number of limit cycles bifurcating from the differential system (4) is nd-1 obtained using the averaging theory of first order. This completes the proof of Theorem 1.


## 4. An Application of Theorem1

In system (4), we consider the case n even and (22)P1x1,…,xd=∑i=2na0,0,…,ii,1xdi+a1,0,…,0,12,1x1xd,P2x1,…,xd=0,Pkx1,…,xd=∑i=2,ievennai,0,…,0i,kx1i+a0,i,…,0i,kx2ifork=3,…,d.Computing the averaged functions and taking t=1-r2, we have (23)r1-r2fr,x3,x4,…,xd=ar2+a1,0,…,0,12,1xd-∑i=2na0,0,…,ii,1xdi1-1-r2=1-ta1+t+a1,0,…,0,12,1xd-∑i=2na0,0,…,ii,1xdi=1-ta1+t-Q¯xd,where Q¯(xd) is an arbitrary polynomial in xd of degree n such that Q¯(0)=0. At the same time, the averaged function corresponding to Pkx1,…,xd satisfies (24)1-r2gkr,x3,x4,…,xd=bkxk+1-r2∑i=2ievennriai,0,0,…,0i,kIi,0+a0,i,0,…,0i,kI0,i,for k=3,…,d. It is easy to obtain the following relations: (25)
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.Looking at the second term of the first relation and at the first term of the second relation, we obtain that riIi,0 and riI0,i for even 2≤i≤n are independent. In particular, using Lemmas 4, 5, and 6, we obtain (26)1-r2gkr,x3,x4,…,xd=bkxk+g1,kr+g2,kr,fork=3,…,d,where (27)g1,k=∑i=2ievennai,0,0,…,0i,k-1-r2∑m=0mevenn-1rm∑i=m+1ievennai,0,…,0i,k2-mmm2,g2,k=∑m=0mevennAm,krm+1-r2∑m=0mevenn-1Bm,krm,where (28)Am,k=∑i=0ievenna0,i,0,…,0i,kdi-m,i,Bm,k=∑i=0ievenna0,i,0,…,0i,kdi-m,i∑l>0levennei-m,l,di-m,i=-1i-m/2i2i-m2,ei-m,l=-1l-12l-i-mi-m-li-m-l2.Writing t=1-r2, the polynomials gs,k(r)=Ps,k(t) satisfy the conditions gs,k(0)=Ps,k(1)=0 for s=1,2 and k=3,…,d. Then, we can define a polynomial of degree n in t: (29)P¯kt=P1,kt+P2,kt=t-1P~kt.Due to the independence of riIi,0 and riI0,i and the arbitrariness of the coefficients ai,0,0,…,0i,k and a0,i,0,…,0i,k, the polynomial P¯k(t) is an arbitrary polynomial such that P¯k(1)=0. In fact, it is obvious that g1,k and g2,k have n/2 parameters, respectively, where n/2 coefficients a0,i,0,…,0i,k allow choosing the first term of g2,k arbitrarily except for the term with m=0, implying that the even terms of Pk¯(t) are arbitrary except for the constant term, while the other n/2 coefficients ai,0,0,…,0i,k allow choosing the second term in g1,k arbitrarily, implying that the odd terms of Pk¯(t) are arbitrary. Therefore, the polynomial Pk¯(t) of the degree n satisfies Pk¯(1)=0 and has n arbitrary coefficients. The number of solutions of f(r,x3,x4,…,xd)=0, gk(r,x3,x4,…,xd)=0, for k=3,…,d, is equal to the number of the intersection points of the curves(30)l1:a1+t-Q¯xd=0.lk-1:bkxk+P¯kt=0,fork=3,…,d.Hence, by the Bézout theorem, the maximum number of the common solutions of system (30) is at most nd-1 for 0<t<1. We can find nd-1 intersection points on f(r,x3,…,xd)=0 with gk(r,x3,…,xd)=0 for k=3,…,d, r∈(r0,1), 0<r0≪1, which (using the averaging theory; see Theorem 2) give rise to nd-1 limit cycles bifurcating from periodic orbits of the system x˙1=-x2(1+x1), x˙2=x1(1+x1), x˙3=0, and x˙4=0,…,x˙d=0.



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*Source: 1868027-2016-03-07.xml*