# On Behavior of the Periodic Orbits of a Hamiltonian System of Bifurcation of Limit Cycles

Authors: Amor Menaceur; Mohamed Abdalla; Sahar Ahmed Idris; Ibrahim Mekawy
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In light of the previous recent studies by Jaume Llibre et al. that dealt with the finite cycles of generalized differential Kukles polynomial systems using the first- and second-order mean theorem such as (Nonlinear Anal., 74, 1261–1271, 2011) and (J. Dyn. Control Syst., vol. 21, 189–192, 2015), in this work, we provide upper bounds for the maximum number of limit cycles bifurcating from the periodic orbits of Hamiltonian system using the averaging theory of first order.


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## Body

## 1. Introduction

Among the many interesting problems in the qualitative theory of planar polynomial differential systems is the study of their limit cycles (see [1, 2]). In particular, concerning Kukles differential system of the form,(1)x˙=−y,y˙=fx,y,has a long history, where fx,y is a polynomial with real coefficients of degree n. Since it was first introduced in Kukles 1944, many researchers have concentrated on its maximum number of limit cycles and their location. See, for example, [3–5].In [6], Llibre and Mereu studied the maximum number of limit cycles using the averaging theory as follows:(2)x˙=y,y˙=−x−∑k⩾1εkfkx+gkxy+hkxy2+dky3,where, for every k, the polynomials fkx,gkx, and hkx have degree n1,n2, and n3, respectively. d0k≠0 is a real number and ε is a small parameter.Also, Makhlouf and Menaceur [7] studied the maximum number for the more generalized polynomial Kukles differential systems in the form(3)x˙=y,y˙=−x−∑k⩾1εkfkx+gkxy+hkxy2+gkxy3.The number of limit cycles bifurcating from the centerx˙=−y2p−1 and y˙=x2q−1, where p,q are positive integers, for the following two kinds of polynomial differential systems,(4)x˙=−y2p−1,y˙=x2q−1−εfxy2n−1,x˙=−y2p−1−εpxfx,y,y˙=x2q−1−εqyfx,y,were investigated in the works [8, 9], respectively. In the current study, we discuss the maximum number of limit cycles of the following differential system:(5)x˙=−y2p−1,y˙=x2q−1−εfx+gxy2n−1+hxy2n+lxy2n+1,where p,q, and n are positive integers, the polynomials fx,gx,hx, and lx have degree n1,n2,n3, and n4, respectively, and ε is a small positive parameter. Clearly, system (5) with ϵ=0 is an Hamiltonian system with(6)Hx,y=12qx2q+12py2p.Our main theorems are given as follows.Theorem 1.
For the sufficiently smallε, system (5), using averaging theory of first order, has at most(7)maxn22p,n42p+q,
The limit cycles bifurcating from the periodic orbits of the center arex˙=−y2p−1 and y˙=x2q−1, where . denotes the integer part function.The proof of Theorem1 is given in Section 3.Theorem 2.
Consider system (5) with q=lp, l is a positive integer, and ε sufficiently small; let Hni,l denote the maximum number of limit cycles of the polynomial differential system (5) bifurcating from the periodic orbits of the center x˙=−y2p−1andy˙=x2lp−1 using the averaging theory of first order; then,(8)aHni,l=n22+n42+1,ifn22<l,bHni,l=n42+l+1,ifl≤n22≤l+n42,cHni,l=n22,ifl+n42<n22.The proof of Theorem2 is given Section 4.


## 2. First-Order Averaging Method

The averaging theory is an interesting method to research the limit cycles. Here, some specific function, associated to the initial system, is stated.Theorem 3.
The two initial value problems are as follows:(9)x˙=εRt,x+ε2Gt,x,ε,x0=x0,(10)y˙=εf0y,y0=x0,where x,y and x0∈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 f0y is the average function of Rt,y with respect to t, i.e.,(11)f0y=1T∫0TRt,ydt.
Assume that(i)
R, ∂R/∂x, ∂2R/∂x2, G, and ∂G/∂x 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)
yt belongs to D on the time scale 1/ε. Then, the following statements hold:(i)
On the time scale1/ε, we have(12)xt−yt=Oε,asε⟶0.(ii)
Ifp is an equilibrium point of the averaged system (10), such that(13)∂f0∂y|y=p≠0,then system (9) has a T-periodic solution ϕt,ε⟶p as ε⟶0.(iii)
If (11) is a negative, therefore, the corresponding periodic solution ϕt,ε of equation (9) according to t,x is asymptotically stable, for all ε sufficiently small; if (11) is a positive, then it is unstable.For more information about the averaging theory, see [10–12].


## 3. Proof of Theorem1

Here, we need to transform system (5) to the canonical from (9). Doing the change of p,q-polar coordinates x=rpCsθ and y=rqSnθ (see Appendix) and taking θ as an independent variable, then system (5) can be written as(14)r˙=−εr−q+1Snθ2p−1frpCsθ+grpCsθSnθ2n−1rq2n−1+hrpCsθSnθ2nr2qn+lrpCsθSnθ2n+1rq2n+1,θ.=rpq−p−q−εpr−qCsθfrpCsθ+grpCsθSnθ2n−1rq2n−1+hrpCsθSnθ2nr2qn+lrpCsθSnθ2n+1rq2n+1.If we write(15)fx=∑k=0n1akxk,gx=∑k=0n2bkxk,hx=∑k=0n3ckxk,lx=∑k=0n4dkxk,then system (14) becomes(16)r˙=−εr−q+1∑k=0n1akCsθkSnθ2p−1rpk+∑k=0n2bkCsθkSnθ2p+n−1rpk+q2n−1+∑k=0n3ckCsθkSnθ2p+n−1rpk+2qn+∑k=0n4dkCsθkSnθ2p+nrpk+q2n+1,θ.=rpq−p−q−εpr−q∑k=0n1akCsθk+1rpk+∑k=0n2bkCsθk+1Snθ2n−1rpk+q2n−1+∑k=0n3ckCsθk+1Snθ2nrpk+2qn+∑k=0n4dkCsθk+1Snθ2n+1rpk+q2n+1.where θ is the independent variable we get from system (16). From(17)drdθ=εRr,θ+Oε2,where(18)Rr,θ=−r−pq+p+1∑k=0n1akCsθkSnθ2p−1rpk+∑k=0n2bkCsθkSnθ2p+n−1rpk+q2n−1+∑k=0n3ckCsθkSnθ2p+n−1rpk+2qn+∑k=0n4dkCsθkSnθ2p+nrpk+q2n+1.According to the notation introduced in Section2, we have(19)f0r=−r−pq+p+1T∑k=0n1akrpk∫0TCsθkSnθ2p−1dθ+∑k=0n2bkrpk+q2n−1∫0TCsθkSnθ2p+n−1dθ+∑k=0n3ckrpk+2qn∫0TCsθkSnθ2p+n−1dθ+∑k=0n4dkrpk+q2n+1∫0TCsθkSnθ2p+ndθ,and we write(20)f0r=−r−pq+p+1T∑k=0n1akIk,2p−1rpk+∑k=0n2bkIk,2p+n−1rpk+q2n−1+∑k=0n3ckIk,2p+n−1rpk+2qn+∑k=0n4dkIk,2p+nrpk+q2n+1,where(21)Ii,j=∫0TCsiθSnjθdθ.It is known that(22)Ii,j=0,ifiorjis odd,Ii,j>0,ifiandjare even.Hence,(23)f0r=−rq2n−p−1+p+1T∑k=0kevenn2bkIk,2p+n−1rpk+∑k=0kevenn4dkIk,2p+nrpk+2q,we obtain(24)f0r=−rq2n−p−1+p+1T∑s=0n2/2b2sI2s,2p+n−1r2ps+∑s=0n4/2d2sI2s,2p+nr2ps+q.For the simplicity of calculation, letBs=b2sI2s,2p+n−1 and Ds=d2sI2s,2p+n; therefore, (24) can be reduced to(25)f0r=−rq2n−p−1+p+1T∑s=0n2/2Bsr2ps+∑s=0n4/2Dsr2ps+q.As we all know, the number of positive roots off0r is equal to that of(26)Nr=∑s=0n2/2Bsr2ps+∑s=0n4/2Dsr2ps+q.Then, to find the real positive roots ofNr, we must find the zeros of a polynomial in the variable ρ=r2:(27)Mρ=∑s=0n2/2Bsρps+∑s=0n4/2Dsρps+q.So, the degree ofMρ is bounded by μ=maxn2/2p,n4/2p+q, we conclude that f0r has at most μ positive root r. Hence, Theorem 1 is proved.


## 4. Proof of Theorem2

Consider the polynomial differential system (5) with q=lp; from equation (25) we obtain(28)f0r=−rlp2n−p−1+p+1T∑s=0n2/2Bsr2ps+∑s=0n4/2Dsr2ps+l.As we all know, the number of positive roots off0r is equal to that of(29)Gr=B0+B1r2p+B2r4p+⋯+Bn2/2r2pn2/2+D0r2pl+D1r2pl+1+D2r2pl+2+⋯+Dn4/2r2pl+n4/2.To find the number of positive roots of polynomialsGr, we distinguish 3 cases.(33)f0r=−r6n−4TB0+B1r2+B2r4+D0r6+D1r8+D2r10,where Bs=b2sI2s,2 and Ds=d2sI2s,4. Using (A.3) of the Appendix, we obtain(34)I0,2=2.1033,I2,2=0.60460,I4,2=0.32339,I0,4=0.63098,I2,4=0.15115,I4,4=6.9298×10−2.Case 1.
Forn2/2<l, the number terms in polynomial (29) is n2/2+n4/2+2. Now, we shall apply the Descartes theorem of the Appendix, we can choose the appropriate coefficients Bi and Dj so that the simple positive roots’ number of Gr is at most n2/2+n4/2+1. Hence, (a) of Theorem 2 is proved.Case 2.
Forl≤n2/2≤l+n4/2, the number terms in polynomial (29) is(30)n22+n42+2−n22−l+1=n42+l+1.
By Descartes Theorem, we can choose the appropriate coefficientsBi and Dj so that the simple positive roots’ number of Gr is at most n4/2+l. Hence, (b) of Theorem 2 is proved.Case 3.
Forl+n4/2<n2/2, the number terms in polynomial (29) is n2/2+1; by Descartes Theorem, we can choose the appropriate coefficients Bi and Dj so that the simple positive roots number of Gr is at most n2/2. Hence, (c) of Theorem 2 is proved.Example 1.
We consider system (5), with p=1,q=3,n=1, and(31)fx=∑k=02akxk,gx=∑k=04bkxk,hx=∑k=02ckxk,lx=∑k=04dkxk,where(32)a0=1,a1=2.3,a2=4.7,b0=−0.5,b1=1.1,b2=6.3,b3=2.5,b4=−15.32,c0=2.2,c1=−6.4,c2=7.3,d0=4.65,d1=3.4,d2=−5.24,d3=6.4,d4=1.13.
In this case,Csθ and Snθ are T-periodic function with period T=8.4131. From equation (28), we obtainSo,(35)f0r=−r2T−1.0517+3.8083r2−4.9553r4+2.9320r6−0.79257r8+7.8473×10−2r10.This polynomial has four positive real roots:r1=0.6,r2=0.8,r3=1.1,r4=1.3, and r5=2. According to statement (a) of Theorem 2, the system has exactly 5 limit cycles bifurcating from the periodic orbits of the center x˙=−y and y˙=x5, using the averaging theory of first order.


## 5. Conclusion

In this work, by using averaging theory of the first order, we have proved upper bounds for the maximum number of limit cycles bifurcating from the periodic orbits of the Hamiltonian system. In addition, in the next work, a new condition with a new method will be used to prove our main result in this study.



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