# Limit Cycles of a Class of Perturbed Differential Systems via the First-Order Averaging Method

Authors: Amor Menaceur; Salah Mahmoud Boulaaras; Amar Makhlouf; Karthikeyan Rajagobal; Mohamed Abdalla
Journal: Complexity
 (2021)

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## Abstract

By means of the averaging method of the first order, we introduce the maximum number of limit cycles which can be bifurcated from the periodic orbits of a Hamiltonian system. Besides, the perturbation has been used for a particular class of the polynomial differential systems.


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## Body

## 1. Introduction

As we know that the second part of the 16 Hilbert problem ([1, 2]) wants to find a uniform upper bound for the number of limit cycles of all polynomial differential systems of a given degree, we refer the readers to see [3, 4]. The limit cycles problem and the center problem are fastened on specified classes of systems. For instance, we refer to Kukles systems (see, for example, [5–9]) and Liénard systems given by(1)u˙=v,v˙=−u−fu,where fu is a polynomial in the variable u of degree m. For these systems, in 1977, Lins et al. [10] presented the conjecture that if fu has degree m≥1, then system (1) has at most m/2 limit cycles where [·] denotes the integer part function. They prove this conjecture for m=1,2. The conjecture for m=3 has been proved recently by Chengzi and Llibre in [11].Suppose that polynomialsfu and gu are in the variable u of degrees n and m, respectively; then, Llibre et al. [12] established the following generalized Liénard polynomial differential system:(2)u˙=v,v˙=−gu−fuv,where n+m−1/2 is limit cycles.Llibre and Makhlouf [13] studied the number of limit cycles of the following generalized Liénard polynomial differential system:(3)u˙=−v2p−1,v˙=u2q−1−ϵfuv2n−1,where fu is a polynomial of degree m, p,q, and n are positive integers, and ϵ is a small parameter.They introduced the following theorem.Theorem 1 (see [13]).
Letm be the degree of the polynomial fu, and ϵ≠0; then, the polynomial differential system (3) can have at least m/2 limit cycles.Also, Jianyuan and Shuliang [14] investigated the maximum number of limit cycles of the following polynomial differential system:(4)u˙=−v2p−1,v˙=u2mp−1+ϵpu2mp+mpv2pgu,v−A,where ϵ is a small parameter, A>0, p,m∈ℕ, and gu,v is a polynomial of degree n with g0,0=0.In this manuscript, we discuss the maximum number of limit cycles of the following polynomial differential system:(5)u˙=−v2p−1−ϵpufu,v,v˙=u2q−1−ϵqvfu,v,where fu,v is a polynomial of degree m, ϵ is a small parameter, and p,q∈ℕ. Clearly, system (5) with ϵ=0 is a Hamiltonian system with Hamiltonian(6)Hu,v=12qu2q+12pv2p.More precisely, our main results are the following.Theorem 2.
For the sufficiently smallε, system (5) has at most(7)λ=m2maxp,qlimit cycles bifurcating from the periodic orbits of the center u˙=−v2p−1, v˙=u2q−1, by using the averaging theory of first order.
The proof of Theorem2 is given in Section 3.Theorem 3.
Consider system (5) with q=np, where n is a positive integer; then, for ε sufficiently small, the maximum number of limit cycles of the polynomial differential system (5) bifurcating from the periodic orbits of the center u˙=−v2p−1,v˙=u2np−1 using the averaging theory of first order is(8)a.μ1=12m2m2+3,ifm2≤n−1,b.μ2=nm2−nn−3+22,ifm2≥n.
The proof of Theorem3 is given in Section 4. Also, an example is given with its limit cycles (see Figure 1).Figure 1
Limit cycles for the system in Example1 with ϵ=0.01.


## 2. First-Order Averaging Method

Here, we state the basic outcomes from the averaging theory of first order, which will be used to prove the main outcomes.Theorem 4.
Consider the following two initial-value problems:(9)u˙=ϵRt,u+ϵ2Gt,u,ϵ,u0=u0,(10)v˙=ϵf0v,v0=u0,where u,v and u0∈D which is an open domain of ℝ, t∈0,∞, ϵ∈0,ϵ0, R and G are periodic functions with their period T with its variable t, and f0v is the average function of Rt,v with respect to t, that is,(11)f0v=1T∫0TRt,vdt.Assume that(i)
R, ∂R/∂u, ∂2R/∂u2, G, and ∂G/∂u are well defined, continuous, and bounded by a constant independent by ϵ∈0,ϵ0 in 0,∞×D.(ii)
T is a constant independent of ϵ.(iii)
vt belongs to D on the time scale 1/ϵ. Then, the following statements hold:(a)
On the time scale1/ϵ, we have(12)ut−vt=Oϵ,asϵ⟶0(b)
Ifp is an equilibrium point of the averaged system (10) such that(13)∂f0∂v|v=p≠0,then system (9) has a T-periodic solution ϕt,ϵ⟶p as ϵ⟶0(c)
If (13) is negative, the corresponding periodic solution ϕt,ϵ of equation (9) according to t,u is asymptotically stable for all ϵ sufficiently small; if (13) is positive, then it is unstableFor more information about the averaging theory, see, e. g., [15–17].


## 3. Proof of Theorem2

Thep.q-trigonometrical functions were defined by Liapunov [18]. Let uθ=Csθ and vθ=Snθ be the solution of the following initial value problem:(14)u˙=−v2p−1,v˙=u2q−1,u0=q21pandv0=0.Furthermore, the following properties are satisfied:(a)
The functionsCsθ and Snθ are T–periodic with(15)T=2p−1/2qq−1/2pΓ1/2pΓ1/2qΓ1/2p+1/2q,whereΓ is the gamma function.(b)
Forp=q=1, we have Csθ=cosθ and Snθ=sinθ(c)
pCs2pθ+qSn2qθ=1(d)
LetCsθ and Snθ be the 1.q-trigonometrical functions, when i and j are both even (see [19])(16)∫0TCsiθSnjθdθ=2q−j+1/2Γi+1/2qΓj+1/2Γi+1/2q+j+1/2.We shall need the first-order averaging theory to prove Theorem2. We write system (5) in p,q-polar coordinates r,θ, where u=rpCsθ and v=rqSnθ. In this way, system (5) will become written in the standard form for applying the averaging theory. If we write fu,v=∑i+j=0mai,juivj, then system (5) becomes(17)r˙=−ϵr∑i+j=0mai,jrpi+qjCsiθSnjθ,θ.=r2pq−p−q.Treatingθ as the independent variable, we get from system (17) the following:(18)drdθ=ϵRr,θ,where(19)Rr,θ=−r−2pq+p+q+1∑i+j=0mai,jrpi+qjCsiθSnjθ.By using the notation which is introduced in Section2, we get(20)f0r=−r−2pq+p+q+1T∑i+j=0mai,jrpi+qj∫0TCsiθSnjθdθ,and we write(21)f0r=−r−2pq+p+q+1T∑i+j=0mai,jαi,jrpi+qj,where(22)αi,j=∫0TCsiθSnjθdθ.It is known that(23)αi,j=0,ifiorjis odd,αi,j>0,ifiandjare even.Hence,(24)f0r=−r−2pq+p+q+1T∑s+k=0m/2a2s,2kα2s,2kr2ps+qk.For the simplicity of calculation, letAs,k=a2s,2kα2s,2k; therefore, (24) can be reduced to(25)f0r=−r−2pq+p+q+1T∑s+k=0m/2As,kr2ps+qk.The positive zeros number off0r, as we know, is equal to the following:(26)Kr=∑s+k=0m/2As,kr2ps+qk,and then, to find the real positive roots of Kr, we must find the zeros of a polynomial in the variable t=r2:(27)Ht=∑s+k=0m/2As,ktps+qk.Now, we stretch the polynomial (27) as follows:(28)Ht=A0,0+A1,0tp+A0,1tq+A2,0t2p+A1,1tp+q+A0,2t2q+⋯+Al,0tlp+Al−1,1tl−1p+q+Al−2,2tl−2p+2q+⋯+A2,l−2t2p+l−2q+A1,l−1tp+l−1q+A0,ltql+⋯+Am/2,0tm/2p+Am/2−1,1tm/2−1p+q+Am/2−2,2tm/2−2p+2q+⋯+A2,m/2−2t2p+m/2−2q+A1,m/2−1tp+m/2−1q+A0,m/2tm/2q.So, the degree ofHt is bounded by λ=m/2maxp,q, and we conclude that f0r has at most λ positive root r. Hence, Theorem 2 is proved.


## 4. Proof of Theorem3

Consider the polynomial differential system (5) withq = np. From equation (25), we obtain(29)f0r=−rnp−2p+1+p+1T∑s+k=0m/2As,kr2ps+nk.The zeros positive number off0r is equal to the following:(30)Sr=∑s+k=0m/2As,kr2ps+nk.We write (30) as follows:(31)Sr=A0,0+A1,0r2p+A0,1r2pn+A2,0r4p+A1,1rn+12p+A0,2r4np+A3,0r6p+A2,1rn+22p+A1,2r1+2n2p+A0,3r6npA4,0r8p+A3,1rn+32p+A2,2r2+2n2p+A1,3r1+3n2p+A0,4r8np+⋯+Am/2−2,0rm/2−22p+Am/2−3,1rm/2+n−32p+Am/2−4,2rm/2+2n−42p+⋯+A2,m/2−4r2+m/2−4n2p+A1,m/2−3r1+m/2−3n2p+A0,m/2−2rm/2−22np+Am/2−1,0rm/2−12p+Am/2−2,1rm/2+n−22p+Am/2−3,2rm/2+2n−32p+⋯+A2,m/2−3r2+m/2−3n2p+A1,m/2−2r1+m/2−2n2p+A0,m/2−1rm/2−12np+Am/2,0rm/22p+Am/2−1,1rm/2+n−12p+Am/2−2,2rm/2+2n−22p+⋯+A2,m/2−2r2+m/2−2n2p+A1,m/2−1r1+m/2−1n2p+A0,m/2rm/22np.Let us write (31) as(32)Sr=A0,0+A1,0r2p+A2,0r4p+⋯+Am/2−2,0rm/2−22p+Am/2−1,0rm/2−12p+Am/2,0rm/22p+A0,1r2np+A1,1rn+12p+A2,1rn+22p+⋯+Am/2−3,1rn+m/2−32p+Am/2−2,1rn+m/2−22p+Am/2−1,1rn+m/2−12p+A0,2r4np+A1,2r2n+12p+A2,2r2n+22p+⋯+Am/2−4,2r2n+m/2−42p+Am/2−3,2r2n+m/2−32p+Am/2−2,2r2n+m/2−22p+⋯+A0,m/2−2rm/2−2n2p+A1,m/2−2r1+m/2−2n2p+A2,m/2−2r2+m/2−2n2p+A0,m/2−1rm/2−12np+A1,m/2−1r1+m/2−1n2p+A0,m/2rm/2n.To find the number of positive roots of polynomialsSr, we distinguish two cases.Case 1.
Form/2≤n−1, the number term in polynomial (32) is(33)m2+1+m2+m2−1+⋯+2+1=12m2+2m2+1.
Now, the Descartes theorem of the Appendix will be applied, and the appropriate coefficientsAi,j can be selected for the simple positive zeros number of Sr as at most(34)μ1=12m2+2m2+1−1=12m2m2+3.
Hence, (a) of Theorem3 is proved.Case 2.
Form/2≥n, the number term in polynomial (32) is(35)m2+1+m2+m2−1+⋯+2+1−m2−n+1−m2−n−m2−n−1−⋯−2−1=12m2+2m2+1−m2−n+1m2−n+2=nm2−nn−32,by Descartes theorem of the Appendix, and we can choose the appropriate coefficients Ai,j in order that the simple positive roots number of Sr is at most(36)μ2=nm2−nn−32−1=nm2−nn−3+22.
Hence, (b) of Theorem3 is proved.Example 1.
We consider system (5), where p=1,q=3, and(37)fu,v=2u3+u2v−7.2365u2+5v2−0.5u+0.7605.
In this case,m=3,n=3 and Csθ and Snθ are T-periodic functions with period T=8.4131. From equation (5), we obtain(38)f0r=−1TrA0.0+A1.0r2+A0.1r6.
Upon using (16), we get(39)α0,0=8.4131,α2,0=3.6276andα0,2=2.1033.
So,(40)f0r=−18.4131r6.3982−26.251r2+10.517r6.
This polynomial has two positive real roots,r1=0.5 and r2=1.2. According to statement (a) of Theorem 3, that system has exactly two limit cycles bifurcating from the periodic orbits of the center u˙=−v, u˙=v5, using the averaging theory of first order. Figure 1 shows the limit cycles for Example 1.



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