# Existence of Multiple Periodic Solutions for Cubic Nonautonomous Differential Equation

Authors: Saima Akram; Allah Nawaz; Humaira Kalsoom; Muhammad Idrees; Yu-Ming Chu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this article, approaches to estimate the number of periodic solutions of ordinary differential equation are considered. Conditions that allow determination of periodic solutions are discussed. We investigated focal values for first-order differential nonautonomous equation by using the method of bifurcation analysis of periodic solutions from a fine focusZ=0. Keeping in focus the second part of Hilbert’s sixteenth problem particularly, we are interested in detecting the maximum number of periodic solution into which a given solution can bifurcate under perturbation of the coefficients. For some classes like C7,7,C8,5,C8,6,C8,7, eight periodic multiplicities have been observed. The new formulas ξ10 and ϰ10 are constructed. We used our new formulas to find the maximum multiplicity for class C9,2. We have succeeded to determine the maximum multiplicity ten for class C9,2 which is the highest known multiplicity among the available literature to date. Another challenge is to check the applicability of the methods discussed which is achieved by presenting some examples. Overall, the results discussed are new, authentic, and novel in its domain of research.


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## Body

## 1. Introduction

Recently, the bifurcation analysis has attracted the attention of many researchers because it has wild applications in dynamics system and the universal existence of bifurcation in the nature, for example, bifurcation occurs when the small smooth change of parameters leads to qualitative change of its behavior (see [1]), aerodynamic limit cycle oscillation, and nonlinear oscillation in power system; the homoclinic and heteroclinic branches of a limit cycle colliding with one saddle point, two, or more saddle points and multiple biological dynamical systems are also bifurcation.As nature is changing every moment and many changes occurring in nature are periodic like weather, blood flow inside body, circadian rhythm, oceans, and even human behavior, the study of theory of periodic or almost periodic solution is gaining attention. These periodic cycles are often observed to explore the impact of environmental factors in mathematical biology, food supplies, and harvesting. One of the classical mathematics problems is the theoretical calculation of the periodic solution of planar vector field.The question about the number of periodic solution for nonautonomous differential equation continues to attract more interest. Neto in [2] states that for equation (1), we are unable to have upper bound for number of periodic solutions until some coefficients are restricted. Kadry in [3] gives conditions that allow determination for both upper and lower bounds. Our basic focus is to acquire highest periodic solutions of any class of the type (1); our main concern in this paper is this nearby question of bifurcation. Without loss of generality for above said arguments, we are considering the differential equation of the form(1)Z˙=γτZ3+δτZ2+υτZ,where independent variable τ and coefficients γ,δ,υ are real valued functions but Z∈ℂ. To find maximum number of periodic solutions, we use complexified form of equation (1) (for more details, see [2, 4–6]). The above equation is the reminiscent of the equation described in Lloyd [5]:(2)Z˙=p0τZn+p1τZn−1+p2τZn−2+⋯+pnτ,with the assumption that p0τ=1 and p1τ,…,pnτ were periodic functions having same period. It was shown that for n=3 in (2), there are three periodic solutions; also, the equation takes form as Abel’s differential equation of first kind. It is important because of its connection with the well-known Hilbert’s sixteenth problem [7] for differential equations with polynomial coefficients:(3)f˙=ℱf,h,h˙=ℋf,h.Here,ℱ and ℋ are polynomials in f and h. Then, Hilbert’s sixteenth problem is transformed to polynomial equation, where leading coefficient randomly changes its sign.For equation (1), complexified form is used so that we can take the maximum number of periodic solutions for each class using the perturbation method. For this reason, periodic solutions cannot be destroyed by any small perturbation of the coefficients. Consider that for equation (1) there exists β∈ℝ such that(4)Zβ=Z0.These solutions are periodic, even ifγ, δ, and υ are not themselves periodic. Limit cycles bifurcate out of the fine focus Z=0 when the coefficients of γ and δ are slightly perturbed. By this method, the obtained limit cycles are said to be of small amplitude. The number of periodic solutions depends upon the multiplicity of solution Z=0. The multiplicity of Z=0 as a solution of equation (1) is also a multiplicity of the following displacement function:(5)p:r⟶Zβ,0,r−r.ForZ=0, the method to compute multiplicity “μ” is explained in [4]; for the sake of ease, we explained it briefly here. We write Zτ,0,r=∑i=1∞ξiτri for 0≤τ≤β, where r lies in the regions near Z=0, and use it in equation (5). For more details, see [2, 5, 6, 8] equation, which provide differential equation for ξϰτ, having some starting conditions ξ10=1 and ξϰ0=0 for i>1. Therefore,(6)pr=ξ1α−1r+∑i=2∞ξiβri.The multiplicityϰ is “ϰ>1” if(7)ξ1β=1,ξ2β=ξ3β=…=ξϰ−1β=0.However,ξϰβ≠0. When ξ1β=1 and ξϰβ=0, ∀ϰ>1 origin is the center. We can observe from equation (1) that ξ˙1τ=ξ1τυτ, where ξ1τ is defined as(8)ξ1τ=e∫0τυsds.In this way,μ>1 if(9)∫0τυsds=0.Becauseξ1τ=1, we are especially interested about situation when Z=0 has multiple solutions. So, we consider that (9) holds by applying the following conversion:(10)ξ=ze−∫0τυsds.Equation (1) takes the following form:(11)ξ˙=γ^τξ3+δ^τξ3,where γ^τ=γτe2∫0τυsds and(12)δ^τ=δτe2∫0τυsds.We can see thatγ^ and δ^ are periodic if γ,δ, and υ are periodic. By using Lemma 2.6 in [4], we consider multiplicity of Z=0 as periodic solution of (1); if for equation (1), ϰ>1, then the multiplicity of ξ=0 as a periodic solution of (11) is also ϰ. So, we consider that υτ=˜0 in (1). As a result, equation (1) takes the form as(13)Z˙=γτZ3+δτZ2.Hereγ and δ may be polynomials i in τ and ii in cost and sint (trigonometric functions) (for more details, see [4, 9, 10]). The functions ξiτ, for i>1 are calculated by utilizing the present relation:(14)ξ˙i=γ∑j+k+l=ij,k,l≥1ξjξkξl+δ∑j,k≥1j+k=iξjξk,with ξ1τ=1. However, as “i” increases, certain calculations become tremendously complicated because of integration by parts, used in it. Assume that ϰi=ξiβ; at that point, ϰ=i if ϰ1=1 and ϰk=0 for 2≤k≤i−2, but ϰi≠0, and these ϰi′s are known as focal values. For i≤8, functions ξiτ and ϰi are given in [4]; for i=9, Yasmin in [10] had calculated ξ9τ and ϰ9. For i=10, we have calculated ξ10τ and ϰ10 which are given in Section 2. In equation (11), we make restriction for n≤10. We do not give the complete detailed derivation because they form a complicated web.In Section2, we have discussed classical formulas by which we are able to calculate the maximum multiplicity. With the help of classical results by Alwash given in [4], we are able to define some conditions w.r.t stopping criteria for our equation (1), and some suitable perturbations are defined in Section 3. Sections 2 and 3 are mainly concerned for calculation of focal values, which are used in Section 4. In Section 4, calculation of focal values is done for equation (13) having polynomial coefficients. In Section 5, some examples are given, and in Section 6, conclusions about periodic multiplicity are discussed.


## 2. Calculation of the Focal Valuesϰ10

For equation (14), the functions ξ2,ξ3,…,ξ8 are given in [4], whereas ξ9,ξ10 are present in following theorem.Theorem 1.
For equation (14), conclusive functions ξ2,ξ3,…,ξ8 are given in [4]. The function ξ9 is calculated in [11] and we succeeded to calculate ξ10 which are given below:(15)ξ9=δ¯8+7δ¯6γ¯+δ¯6γ¯+6δ¯5δ¯γ¯+2δ¯5γ¯δ¯+5δ¯4δ¯2γ¯+3δ¯4γ¯γ¯+3δ¯3γ¯δ¯2+5δ¯4γγ¯¯+392δ¯4γ¯2−2δ¯3δγ¯2¯+24δ¯3δ¯γ¯γ¯+6δ¯3γγ¯¯δ¯−10δ¯3γδ¯γ¯¯+12δ¯γ¯δ¯3γ¯+4γ¯δδ¯3γ¯¯+4δ¯3γ¯δ¯3+436δ¯2γ¯3+4γ¯3δδ¯¯+4δ¯2γ¯γδ¯2¯−10δδ¯γ¯γδ¯2¯¯+152γ¯2δ¯2γ¯+2δ¯2δ¯2γ¯−2δ¯4γ¯+8γ¯δδ¯3¯+2δ¯δ¯2γδ¯γ¯¯+26δ¯γδ¯2γ¯δ¯+6δ¯2γ¯γ¯−6δ¯2γγ¯¯+12δ¯2δ¯γγ¯¯+16δ¯2γ¯δδ¯γ¯−16δ¯3γδ¯γ¯¯+9δ¯2δ¯γ2¯+9δ¯γ2¯γ¯−δγ¯3¯δ¯+358γ¯4−6δ¯γ¯δγ¯2¯+6γ¯δγ¯2δ¯+33δ¯γ¯2δ¯γ¯−24δγ¯2δ¯γ¯¯+6δ¯2γ¯γ¯γ¯−4δγ¯δγ¯2¯¯.ξ10=δ¯9−232δ¯7γ¯−12356δ¯5γ¯γ¯+3δ¯5γ¯γ¯+111γ¯δ¯4δ¯γ¯−444γ¯δδ¯3δ¯γ¯¯+20γ¯δ¯δ¯4γ¯−12γ¯δδ¯4γ¯¯+2143γ¯δ¯3δ¯2γ¯+3γδ¯7−36γ¯2δδ¯δγ¯¯¯−48δ¯γ¯2δδ¯γ¯¯−16γδ¯γ¯δ¯2γ¯¯−8δ¯δγ¯δγ¯2¯¯+8δ¯2δ¯2γδγ¯¯−160γ¯δδ¯2δ¯2γ¯¯+152γ¯2δ¯3γ¯−1272δ¯4γδ¯γ¯¯+30γ¯δ¯2δ¯3γ¯−68γ¯δδ¯δ¯3γ¯¯+24γ¯δ¯2γδγ¯¯+10159γ¯3δ¯3−237δδ¯2γ¯3¯+8γ¯δ¯7−112γ¯δ¯2δγ¯2¯+26γ¯δδ¯δγ¯2¯¯+3192γ¯2δ¯2δ¯γ¯−174δδ¯γ¯2δ¯γ¯¯−90γ¯γδ¯δ¯2γ¯¯+9γ¯δ¯3γγ¯¯+40γ¯δ¯γγ¯δ¯2¯−24γ¯δγγ¯δ¯2¯+3γ¯δ¯2γδ¯γ¯¯−154γ¯γδ¯2δ¯γ¯−24γγ¯2δ¯2δ¯+24γγ¯2δ¯δ¯2γ¯+42γ¯δδ¯γ¯2¯−5γδ¯6¯−32γ¯δγ¯3¯−21δγ¯4¯+δδ¯γ¯δγ¯2¯¯−154γ¯2δγ¯2¯+1694γ¯4δ¯+70γ¯δ¯δ¯γ¯2−2γδ¯δ¯4γ¯¯+10γ¯3δ¯γ¯+92δ¯4δ¯3γ¯−74γγ¯3δ¯¯+8δ¯δδ¯γ¯3¯+7δ¯6δ¯γ¯+12δ¯4δ¯γ¯+2δ¯δ¯6γ¯−16γ¯δδ¯4¯−70γ¯δ¯3γ2¯+6δ¯5δ¯2γ¯−6γδ¯δ¯4γ¯¯+2δ¯3δ¯3γ¯+10δ¯γγ¯δ¯4¯+26γ¯2δ¯5−52δ¯4δγ¯2¯+52δδ¯4γ¯2¯+732γ¯δ¯4δ¯γ¯+16δ¯δ¯3γ¯δ¯+9δ¯2γγ¯δ¯3¯−45γ¯δ¯5+12γ¯δ¯δ¯2γ¯−21δ¯2γδ¯3γ¯¯+8δ¯γ¯δδ¯3γ¯¯−1603γδ¯3δ¯2γ¯¯+19γγ¯2δ¯3γ¯+32δ¯4γδγ¯¯¯−20δ¯γ¯δ¯δδ¯2γ¯¯+24δ¯γ¯2δ¯2γ¯+43δ¯3δ¯2γ¯−3130γδ¯5¯−9703γγ¯2δ¯3¯+12δ¯δγ¯2¯δγ¯¯+132δ¯2γδγ2¯¯+3δ¯2δ¯2γδ¯γ¯¯+42δ¯2δ¯2γ¯δ¯γ¯−20δ¯δ¯3γδ¯γ¯¯−12γδ¯δ¯2γ¯¯−12δ¯δ¯2γ¯γ¯+12δ¯3δ¯2γ¯γ¯+32δ2γ¯δ¯δ¯2γ¯¯−32δ¯δ¯3γδγ¯¯¯+14δ¯3δ¯γ¯2−28γδ¯δ¯γ¯2−32δ¯2δγ¯3¯−15δ¯5γ2¯−8δδ¯4γγ¯¯−24γ¯2δδ¯2γ¯¯+343γ¯δ¯3δ¯2γ¯+2δ¯δ¯γ¯δ¯3γ¯+δ¯δ¯2γ¯2.By using theseξi functions, we get Theorem 2 that enables us to find the maximum multiplicity in which integral is like ∫γτδτ¯dτ and bar “−” shows integral δτ¯=∫0τδτdτ, which is definite.Theorem 2.
The solutionZ=0 of (13) has a multiplicity k, wherever 2≤k≤10 if ϰn=0 for 2≤n≤k−1 and ϰn≠0 where(16)ϰ2=∫0βδ,ϰ3=∫0βγ,ϰ4=∫0βγδ¯,ϰ5=∫0βγδ¯2,ϰ6=∫0βγδ¯3−12δγ¯¯δ,ϰ7=∫0βγδ¯4+2γδ¯2γ¯,ϰ8=∫0βγδ¯5+3γδ¯3γ¯+γδ¯2δ¯γ¯−12γ¯3δ,ϰ9=∫0βγδ¯6−5γδ¯4γ¯−2δ¯3δγ¯¯+20δγ¯2¯+2δγ¯¯δγ¯2,ϰ10=∫0βγδ¯7−12356γγ¯δ¯5−9703γγ¯2δ¯3−237δδ¯2γ¯3−24γγ¯2δδ¯2−70γ¯δ¯3γ2−21γ¯4δ−74γγ¯3δ¯+52γ¯2δδ¯4+32δ¯4γδγ¯¯−16δδ¯4γ¯−15δ¯5γ2−36δδ¯γ¯2δγ¯¯−8δδ¯4γγ¯.


## 3. Conditions for Center and Method of Perturbation

In this section, we describe some conditions for center. From Theorem2, we find maximum value ϰ for different classes of nonautonomous equation of the form as (1). Stopping criteria are defined for calculating maximum multiplicity ϰk. We need some conditions that assure that there is no need to proceed further (ϰk). For this, suitable conditions that are sufficient for Z=0 as a center are given below in the form of theorems and corollaries.Theorem 3.
Consider that there are continuous functionsf, g defined on I=σ0,α and differentiable function σ with σα=σ0 such that(17)γτ=fστσ˙,δτ=gστσ˙.Then, the origin is a center for (13).Theorem 4.
The origin is a center for equation (13) if γ is a constant multiple of δ and ∫0βδτdt=0.Corollary 1.
If anyδ or γ is identically 0 and other has mean value zero, then the origin is a center.After determining the maximum multiplicityϰ, we have to make series of perturbation of the coefficients, every one of which results in one periodic solution to come out of origin. For more details, see [4, 5, 12].For this, suppose equation of the form given below:(18)Z˙=γτZ3+δτZ2,having multiplicity ϰ=j (suppose). Let U be in the regions near 0 in the complex plane containing no periodic solutions except Z=0. From Theorem 2.4 in [4], the initial point which is contained in U remained fixed with respect to total number of periodic solutions with the restriction that perturbation of the coefficients considered remained small enough. Our goal is to get ϰ2=ϰ3=…=ϰj−2=0 but ϰj−1≠0 by perturbing and making suitable choices of γ and δ, if possible. Obviously, the most effective solutions in U and ψ are zero solution while we get periodic solutions ψτ where ψ0∈U as nontrivial solutions. By considering the underlying fact that the complex solutions always appear in conjugate pair, we can say that ψ is real. Let U1 and V1 be the neighborhood of zero and ψ, respectively, such that V1∪U1⊂U and V1∩U1=ϕ. The periodic solutions around V1 and U1 are preserved when we take small perturbation in the coefficients. By applying the same procedure as above, our choice is to perturb the coefficients such that ϰk=0 for k=2,3,…,j−3 but ϰj≠0. So, we get ϰ=j−2. By applying that procedure, we get two nontrivial real periodic solutions with zero solution having multiplicity j−2. In this way, we end up with equation (18) having ϰ=2 and j−2 distinct nontrivial (other than zero) real periodic solutions.


## 4. Polynomial Coefficients for Some Classes

For the polynomial “τ,” consider Cp,q indicates the class of equation of the form (13) with degree p and q for γ and δ, respectively. We have evaluated the maximum multiplicity denoted by the symbol “μ” for some classes like C7,7,C8,5,C8,6,C8,7, and C9,2 that are presented below in the form of theorems. For more classes with maximum multiplicity, see [9].Theorem 5.
LetC9,2 be class of equation of form (13), with(19)γτ=a+bτ+cτ2+dτ3+fτ5+jτ9.δτ=k+mτ2.
Then, we concludeμmaxC9,2≥10.Proof.
Using Theorem2, we take(20)ϰ2=k+13m,ϰ3=a+12b+13c+14d+16f+110j.
Thus, multiplicity ofZ=0 is ϰ=2 if ϰ2≠0. And multiplicity is ϰ=3 if ϰ2=0 but ϰ3≠0. If ϰ2=ϰ3=0, then by using value of “k” and “a,” γτ and δτ are as follows:(21)γτ=bτ−12+cτ2−13+dτ3−14+fτ5−16+jτ9−110,(22)δτ=mτ2−13.
And we computeϰ4 as given below:(23)ϰ4=−m−7938j−7150f−3861d+6006b2162160.
Ifϰ4=0, then either m=0 or(24)j=−71507938f−38617938d+60067938b.
Ifm=0, then δτ=0 and ϰ3=0 gives that mean value of γτ is zero. Thus, origin is a center from Corollary 1. So, consider m≠0. Now, if (24) holds, then we calculate(25)ϰ5=m2−4070f−2727d+1302b200037600.
Ifϰ5=0, we consider m≠0 which implies that(26)f=−27274070d+13024070b.
And by using (26), we take ϰ6 as(27)ϰ6=−md+4b43475961b+47732590m2−7030731d4535399964595500.
Ifϰ6=0, we consider m≠0 and either d=−4b or(28)b=−4773259043475961m2+703073143475961d.
Ifd=−4b, then (26) gives f=3b, and (24) gives j=0. By using values of d,f,j, equations (21) and (22) take the following form:(29)γτ=cτ2−13+b3τ3−3ττ2−13,δτ=mτ2−13.
Letστ=τ3−τ; then, σ˙τ=3τ2−1. Also, σ0=σ1. So, we write above equations as(30)γτ=13c+b3τ3−3τσ˙,δτ=m3σ˙.
Thus, from Theorem3, origin is the center with(31)fσ=13c+b3τ3−3τ,and gσ=m/3. So, we take d≠−4b. If (28) holds, then we compute ϰ7 as(32)ϰ7=−129007m2−8m2+3d244401908873421d−51374543768976m2+343350184670592c5750115960595749963075535680.
Ifϰ7=0, recalling that m≠0, then either d=8/3m2 or(33)d=513745143768976244401908873421m2−343350184670592244401908873421c.
Ifd=8/3m2, then(34)γτ=13c+m2−2τ3+2τσ˙,δτ=m3σ˙.
From Theorem3, origin is the center with fσ=1/3c+m2−2τ3+2τ and gσ=m/3. So, consider d≠8/3m2. By using (33), we calculate ϰ8 as(35)ϰ8=5609m269232c+108205m2ϖ464011623410213337535103836105743901025559739592193301950,where(36)ϖ=4403049224583165170231259327919284c2−4081167090232676699163689650491745cm2−79907221802386910423708643546694695176m4.
Now, ifϰ8=0, then either ϖ=0 or(37)c=−108205269232m2,because m≠0. If (37) holds but ϖ≠0,m≠0, then we compute ϰ9 as(38)ϰ9=−m57692367147318m+6418199848808833796688520035034624.
If equation (37)≠0, l≠0, but ϖ=0 holds, then c=vim2/3 for i=1,2 with v1=45.009923140 and v2=−17.20305420. If ϰ9=0 recalling that m≠0, we take value of m as(39)m=−6418199848808837692367147318.
If (39) holds, then we calculate ϰ10 as(40)ϰ10=−23619930325536057260679648648108179733182380751902439747138622675536890453034093572801545113943979704942365450213502975589695413549237735086533209175509154447014600254621183889944351436363320478791521135226094109654616738193865483843471924056732079968557936379195349112486049662164194337751040.
Hereϰ10 is equal to constant number which is nonzero. Hence, we conclude that multiplicity of class C9,2 is 10, i.e., μmaxC9,2≥10.Theorem 6.
Consider the equation given below:(41)Z˙=γτZ3+δτZ2,with(42)γτ=108205807696−6418199848808837692367147318+ε12+112446022613981467302957807ε2−118170412478235571ε3−36646105ε4−30439ε5−110ε6+ε7+−23−6418199848808837692367147318+ε12−1850833373734481467302957807ε2+234357714491987ε3+ε4t+−108205269232−6418199848808837692367147318+ε12+ε2t2+83−641819848083769236147318+ε12−11445061568648146730297807ε2+ε3t3+−64181984808837692367147318+ε12+2864912505632982713481ε2+6512035ε4−197126007318823714ε3+ε5t5+−509145851520188014636417ε2+8384554345961ε3+52111ε4−35753969ε5+ε6t9.δτ=64181998488088323077101441954−13ε1+ε8+−6418199848808837692367147318+ε1t2,
Chooseεp for 1≤p≤8 to be nonzero and small as compared to εp−1. Then, there exist eight real nontrivial periodic solutions for equation (41).Proof.
If we substituteεr=0, ∀r=1,2,…,8 and coefficients are as written above, then the multiplicity of the origin is 10. Choose ε1≠0 and εr=0 for 2≤r≤8; it can be easily seen that ϰ9 is constant multiple of ϵ1 but ϰ2=ϰ3=…=ϰ7=ϰ8=0, so the multiplicity reduces by one, and ϰ=9. For that reason one periodic solution bifurcates out of the origin. Now, set ϵ1≠0, ϵ2≠0 and ϵr=0 for 3≤r≤8, and we have ϰr=0 for r=2,3,…,7 but ϰ8 results in form of ϵ2 with some constant multiple, so ϰ=8. Now, set ε1≠0, ε2≠0,ε3≠0 and εr=0 for 4≤r≤8; then, we have ϰr=0 for r=2,3,…,6 but ϰ7 results in form of ϵ3 with some constant multiple. If ϵ2 is sufficiently small, then there are two nontrivial real periodic solutions. Further continuing the same procedure, we own eight real periodic nontrivial solutions.Corollary 2.
For an equation(43)Z˙=γτZ3+δτZ2+τ+υ.
Ifγτ and δτ are as given in Theorem 6 and τ and υ are small enough, equation (43) has ten real periodic solutions.Proof.
Given thatτ=0, υ=0, and ϰ=2, then (43) has eight real periodic solutions. If τ≠0 but small enough, then ϰ=1; using the same arguments as above, we have nine periodic solutions. These are distinct and other than 0, Z=0 is another such solution, and thus we take ten real periodic solutions.Theorem 7.
ConsiderC8,j as the class of equations of form (13) having degree 8 for γτ and j for δτ; here j=5,6,7. Then, we cease with the results μmaxC8,j≥8, for all j=5,6,7, respectively.Proof.
We first consider the class in which degree ofγτ is 7,i.e., C8,7. The classes in which degree of γτ is less than 7 are the special cases. Let(44)γτ=a+bτ+cτ2+dτ3+eτ4+fτ5+gτ6+hτ7+iτ8δτ=k+lτ+mτ2+nτ3+oτ4+pτ5+qτ6+rτ7.
By using Theorem2, we calculate(45)ϰ2=k+l2+m3+n4+o5+p6+q7+r8,ϰ3=a+b2+c3+d4+e5+f6+g7+h8+i9.
Ifϰ2=ϰ3=0, we calculate ϰ4 as(46)ϰ4=−1360bm+1360cl+1240dl−1240bn+1210el−1560cn+1560dm−1210bo+51008fl−1360co+1360em−51008bp+51008gl+51512fm+1900en−900do−51512cp−51008bq+71440hl+1280gm+1560fn−71440br−1560dp−1280cq51092gp+84725io−5608fr+56048hp+12880hq−12880gr+11680iq+727540ir.
Fromϰ4, we cannot substitute any value to proceed further. As a result, we make restriction of the coefficients for class C8,7 like c=d=f=g=l=m=n=o=p=q=0 in the system of equation (44).(1)
Let(47)γτ=a+bτ+eτ4+hτ7+iτ8,δτ=k+rτ7.Utilizing Theorem2, we get(48)ϰ2=k+r8,ϰ3=a+b2+e5+h8+i9.Thus, multiplicity ofZ=0 is ϰ=2 if ϰ2≠0. And multiplicity ϰ=3 if ϰ2=0 but ϰ3≠0. If ϰ2=ϰ3=0, then we calculate ϰ4 as(49)ϰ4=−7r−104i+612e+1989b2864160.Ifϰ4=0, then either r=0 or(50)i=612104e+1989104b.Ifr=0, then ϰ2=0 gives k=0; hence, δτ=0. If ϰ2=0, it means that the mean value of γτ is zero. By Corollary 1, origin is a center. Suppose that r≠0. By using (50), we compute(51)ϰ5=−r2−768e+31213b980179200.Ifϰ5=0, then(52)e=31213768b.We have already takenr≠0. If (52) holds, then(53)ϰ6=343br2809107r2+142686496b836532043776000.Ifϰ6=0, then either b=0 or(54)b=−2809107142686496br2,becauser≠0. If b=0, then from (50) and (52), e=0,i=0, respectively. By using these values, γτ and δτ take the form as(55)γτ=kτ7−18,δτ=rτ7−18.Letστ=τ8−τ; then, σ˙τ=8τ7−1; also, σ0=σ1. So, we can write as(56)γτ=18kσ˙,δτ=18rσ˙.Utilizing Theorem3, origin is the center with fσ=1/8k and gσ=1/8r. So, we take b≠0. If (54) holds, then ϰ7 is given by(57)ϰ7=−749407323r4−31704452399633263173r2+5552181576645742592h137618755815002650025303511924736000.Now ifϰ7=0, recalling that r≠0, then(58)h=317044523996332631735552181576645742592r2.Holding (58), we have(59)ϰ8=1085106792375126997264524872221165313173475371176413089188756096613789685106805706496326217236480000r7.That is, constant multiple ofr7 and r is also nonzero. Thus, we conclude that multiplicity of class C8,7 is 8, i.e., μmaxC8,7≥8.(2)
For the classC8,6, the degree of δτ=6 and γτ=8; we suppose that c=d=f=h=l=m=n=o=p=r=0 in system of equation (44). Let(60)γτ=a+bτ+eτ4+gτ6+iτ8,δτ=k+qτ6.By using Theorem2, we calculate(61)ϰ2=k+q7,ϰ3=a+b2+e5+g7+i9.Thus, multiplicity ofZ=0 is ϰ=2 if ϰ2≠0. And multiplicity ϰ=3 if ϰ2=0 but ϰ3≠0. If ϰ2=ϰ3=0, then(62)a=−b2−e5−g7−i9,(63)k=−q7.By using these values from (62) and (63), we have(64)γτ=bτ−12+eτ4−15+gτ6−17+iτ8−19,(65)δτ=qτ6−17.We calculateϰ4 as(66)ϰ4=−q−3i+6e+25b5040.Ifϰ4=0, then either q=0 or(67)i=63e+253b.Ifq=0, then (63) gives k=0, so δτ=0. For ϰ2=0, the mean value of γτ is zero. So, by Corollary 1, origin is the center. Consider that q≠0; if (50) holds, then ϰ5 is(68)ϰ5=−q2−44399616e+1080979445b35418224041200.Ifϰ5=0, then either q=0 or e=1080979445/44399616b. But as we have already seen that q≠0, by using value of e, we have(69)ϰ6=bq984408064224939q2+20917403154855088b1850072878986734223360.Now if we takeϰ6=0, then as q≠0, either b=0 or(70)b=−98440806422493920917403154855088q2.Ifb=0, then equations (64) and (65) take the form as below:(71)γτ=gτ6−17,δτ=qτ6−17.Definingστ=τ7−τ, we see σ⋅τ=7τ6−1; also, σ0=σ1. So, it can be written as(72)γτ=g7σ˙,δτ=q7σ˙.By Theorem3, having fσ=g/7 and gσ=q/7, the origin is a center. So, we take that b≠0. By using (70), we have ϰ7 as(73)ϰ7=−132769951q4−131733796145486983522308400541863q2+36113870596748585190134519660544g.143608906065580564710660947856402888834825584640.Ifϰ7=0, recalling that q≠0 (considered above), then(74)g=13173379614548698352230840054186336113870596748585190134519660544q2.Holding (74), we find(75)ϰ8=4236266672814893139248947441614198027669758565038839578999348891763171305362071787861985267233073102761553912581534515200q7.Asq≠0, we cannot proceed further. We conclude that multiplicity of class C8,6 is 8, i.e., μmaxC8,6≥8.(3)
For the classC8,5, the degree of γτ=8 and δτ=5; we suppose that c=d=g=h=l=m=n=o=q=r=0 in system (44). By using Theorem 2 in this case, we calculate(76)ϰ2=k+p6,ϰ3=a+b2+e5+f6+i9.Thus, multiplicity ofZ=0 is ϰ=2 if ϰ2≠0. And multiplicity ϰ=3 if ϰ2=0 but ϰ3≠0. Suppose ϰ2=ϰ3=0; then, by using value of aandj, δτ and γτ are given as follows:(77)γτ=bτ−12+eτ4−15+fτ5−16+iτ8−19,(78)δτ=pτ5−16.We calculateϰ4 as(79)ϰ4=−p−176i+120e+825b166320.Ifϰ4=0, either p=0 or(80)i=120176e+825176b.Ifp=0, then ϰ2=0, and k=0. As a result, δτ=0. Also, ϰ3=0 reflects that the mean value γτ=0. From Corollary 1, origin is the center. Hence, we substitute that p≠0. If (80) holds, we have(81)ϰ5=−5p2−1976e+42075b7763019264.Now ifϰ5=0, either p=0 or −1976e+42075b=0. But we have already taken p≠0. So, we take(82)e=420751976b.We haveϰ6 as(83)ϰ6=bp93192188p2+958055593b375931952547840.Ifϰ6=0, then either p≠0, b=0 or 93192188p2+958055593b=0. If b=0, then from (82), e=o. By using these values in (80), we see i=0. So, equations (77) and (78) can be written as(84)γτ=fτ5−16,δτ=pτ5−16.
Definingστ=τ6−τ, we see σ⋅τ=6t5−1; also, σ0=σ1. From Theorem 3, the origin is a center along fσ=f/6 and gσ=p/6. We substitute b≠0. By using (82), we proceed for ϰ7 as(85)ϰ7=−1226213p4−167427531234665540327p2+44481776203542888210f1696485821872625651060876535652800.
Ifϰ7=0, recalling that p≠0 (considered above), then(86)f=16742753123466554032744481776203542888210p2.
If (86) holds, we proceed towards ϰ8 as(87)ϰ8=303851190335731455169925842743585940987721491717862570679063691665355228636923726082688000p7.
ϰ8 is a constant multiple of p7; as q≠0, we cannot proceed further. Hence, we conclude that multiplicity of class C8,5 is 8, i.e., μmaxC8,5≥8.Theorem 8.
For classC7,7 both γτ and δτ have degree 7 with the variable 2τ. Then, for equation(88)Z˙=γτZ3+δτZ2,μmaxC7,7≥8.Proof.
Let(89)γτ=a+b2τ+f2τ5+g2τ6+h2τ7,δτ=j+q2τ7.
By Theorem2, we have(90)ϰ2=16q+j,(91)ϰ3=a+b+647g+163f+16h.
Thus, multiplicity ofω=0 is ϰ=2 if ϰ2≠0 and ϰ=3 if ϰ2=0 but ϰ3≠0. If ϰ2=ϰ3=0, from (90) and (91), we substitute j=−16q and a=−b−64/7g−16/3f−16h. Now δτ and γτ take the form as(92)γτ=b2τ−1+f2τ5−163+g2τ6−647+h2τ7−16,(93)δτ=q2τ7−16.
Substitutingjanda, we calculate ϰ4 as(94)ϰ4=−8q336g+400f+147b945.
Ifϰ4=0, then either q=0 or(95)g=−400336f−147336b.
Ifq=0, from (93), δτ=0 and ϰ3=0 reflect that γτ has mean value zero. Using Corollary 1, the origin is a center. We suppose q≠0 and (95) holds; then,(96)ϰ5=−512q2−34f+1323b580635.
Ifϰ5=0, then as taken above, q≠0 implies that(97)f=132334b.
Now by using (97), we calculate ϰ6:(98)ϰ6=28bq18243730176q2+234430835b41989265175.
Ifϰ6=0, either b=0 or(99)b=−18243730176234430835q2,as q≠0. Now if b=0, f=0 gives g=0. As a result, equations (92) and (97) take the form:(100)γτ=h2τ7−16,δτ=p2τ7−16.
Considerστ=2τ8−256t; then, σ˙τ=162τ7−16; also, σ0=σ1. We write γτ and δτ as(101)γτ=16hσ˙,δτ=16pσ˙.
By Theorem3, origin is a center having fσ=16h and gσ=16p. Therefore, we conclude that b≠0. By taking , ϰ7 is computed as(102)ϰ7=−8543709116235776q435792018807075552q2+32995285488045h223319838020134861120561875.
Ifϰ7=0, recalling that q≠0 (taken above), we have(103)h=−3579201880707555232995285488045q2.
If (103) holds, we calculate(104)ϰ8=1075351124332333325077765752450609230924134481921713318730986483405512695232955553868915625q7,which is a constant multiple of q7 and also nonzero as q≠0. Hence, we conclude that the multiplicity is 8, i.e., μmaxC7,7≥8.Theorem 9.
For equation(105)Z˙=γτZ3+δτZ2+ϰ1Z+ϰ2,consider(106)γτ=−v1+ε5+b2τ+f2τ5+g2τ6+h2τ7,δτ=−16q+ε6+128qt7.Here,(107)v1−b−647g−163f−16h,b=−18243730176234430835q2+ε2,g=630548924208173274965q2−12719272ε2−2521ε3+ε4,f=−120682275114243985324195q2+132334ε2+ε3,h=−3579201880707555232995285488045q2+ε1.Ifεp1≤p≤6, ϰ1 and ϰ2 are chosen to be nonzero and(108)ϰ2≪ϰ1≪ε6≪ε5≪…≪ε1.Then, (105) has eight distinct nontrivial real periodic solutions.Proof.
If we substituteεp for 1≤p≤6, the proof is similar to Theorem 6. Hence, it is omitted.


## 5. Examples

In this section, we are presenting some examples which are helpful to show the applicability of the methods presented.Example 1.
Consider the differential equation:(109)dzdt=etz3+sintz2.
Hereγt,δt are transcendental functions, but we use the power series representation by neglecting the terms “tn” for n>5, so we have γt=et=a+bt+ct2/2!+dt3/3!+et4/4!+ft5/5! and δt=sint=ht−it3/3!+jt5/5!. Here a,b,c,d,e,f,h,i,j are constants having same value equal to 1. The solution of the above equation is given below.Solution.
We substitutei=e=0, for suitable restriction of the coefficients. By using Theorem 2, we calculate(110)η2=12h+1720j,η3=a+12b+16c+124d+1720f.
Thus, multiplicity ofz=0 is μ=2 if η2≠0. And multiplicity μ=3 if η2=0 but η3≠0. If η2=η3=0, then we take h=−2/720j, and(111)a=−12b−16c−124d−1720f.
By using these values,η4 is calculated as(112)η4=−j5f+192d+768c+1800b43545600.
Ifη4=0, then either j=0 or(113)f=−1925d−7685c−18005b.
Ifj=0, then h=0; this shows that δt=0, and η3=0 reflects that mean value of γt is zero. By Corollary 1, the origin is a center. Suppose j≠0; if (113) holds, we calculate η5 as(114)η5=j25071d+16384c56043187200000.
Ifη5=0, then j≠0taken above; we substitute d=−16384/5071c and calculate η6 as(115)η6=cj56201893j2+115870113000c13964771180634962688000.
Ifη6=0, then either c=0 or(116)c=−56201893115870113000j2,because j≠0 (already taken above). If c=0, then d=0; substituting these values in (113) gives f=−1800/5b, and (111) results in a=0. By using these values, γt and δt take the following form:(117)γt=ft55!−2t720,δt=jt55!−2t720.
Letσt=t6/5!−t2/720; then, σ˙t=6t5/5!−2t/720. Also, σ0=σ1. So, we can write γt=1/6!fσ˙ and δt=1/6!jσ˙. From Theorem 3, origin is a center having fσ=1/6!f and gσ=1/6!j. Suppose that c≠0; by using (116), we have η7 as follows:(118)η7=−11083j4−125099092877941j2+1849231385825760000b21873119144875211817269842176000000000.
Recalling thatj≠0 (considered above), if η7=0, then we put(119)b=1250990928779411849231385825760000j2,and calculate η8 as(120)η8=83017743756733094728041979679218732377679448920984181770497759873277952000000000000j7,which is a constant multiple of j7, and j≠0. Thus, we conclude that (109) has eight periodic solutions.Example 2.
Consider the differential equation:(121)dzdt=γtz3+δtz2,with γt= equation of circle =ax2+by2+cx+dy+e and δt= point circle = circle with (center (0, 0) and radius=0=gx2+hy2; we then calculate the periodic solutions as given below.Solution.
By using Theorem2, we calculate(122)η2=13g+13h,η3=13a+13b+12c+12d+e.
Thus, multiplicity ofz=0 is μ=2 if η2≠0. And multiplicity μ=3 if η2=0 but η3≠0. If η2=η3=0, then we put g=−h, and e=−1/3a−1/3b−1/2c−1/2d. By using these values, we get(123)η4=−h−5b−4d+5a+4c720.
Ifη4=0, then either(124)b=−45d+55a+45c,or h=0. We get δt=0 for h=0, and η3=0 shows that mean value of γt is zero. So, by Corollary 1, the origin is a center. Suppose that h≠0. Now by using (124), we calculate η5=0 as(125)η5=h273d+1280a+1097c1701000.
Ifη5=0, recalling that h≠0, we put(126)d=−128073a−109773c,and we calculate(127)η6=h128a+117c163845a+131076c+1825h23988436760.
Ifη6=0, then either a=−117/128c or(128)a=−131076163845c−1825163845h2,because h≠0. If a=−117/128c holds but (128) does not hold, we calculate η7 as(129)η7=ch2−12845056h2+24891165c100429391462400.
Now, if (128) holds but a=−117/128c does not hold, we calculate η7 as(130)η7=−h2−1248682406400ch2+6393449027394c2+97998898375h43656745734764122000.
From (130), we cannot substitute any value to proceed further, and ifη7=0 for equation (129), we substitute c=12845056/24891165h2 and calculate η8=0. Hence, in both the cases, we succeeded to calculate η7 as nonzero. Thus, we conclude that (121) has seven periodic solutions.


## 6. Conclusion and Discussion

In this paper, we presented the maximum number of periodic orbits which are usually called limit cycles. Limit cycles are closed paths that are isolated from set of all periodic orbits. To find the maximum number of limit cycles, the perturbation technique has been adopted. We adopt a systematic procedure to define coefficients of higher-order polynomials. We furnished the maximum number of periodic solutions for some classesC7,7,C8,5,C8,6,C8,7. They are found with 8 periodic solutions at most. To find out more periodic solutions, keeping the motivation of second part of Hilbert’s sixteenth problem in our mind, we developed formulas for ξ10 and ϰ10. With the help of newly developed formulas, we are able to calculate multiplicity 10 for class C9,2. This multiplicity is the highest known multiplicity in the literature to date. We also verified the results by carrying out some examples given in Section 5, which shows that presented methods are new, applicable, and authentic.



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