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# Frontiers in Physics

## Original Research ARTICLE

Front. Phys., 15 September 2020 | https://doi.org/10.3389/fphy.2020.00264

# Periodic Solutions of Some Classes of One Dimensional Non-autonomous Equation

• 1Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan
• 2Informetrics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam
• 3Faculty of Mathematics & Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
• 4Department of Mathematics, Cankaya University, Ankara, Turkey
• 5Institute of Space Sciences, Mǎgurele, Romania
• 6Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
• 7Department of Mathematics, College of Arts and Sciences, Prince Sattam Bin Abdulaziz University, Wadi Aldawaser, Saudi Arabia

In this paper, the periodic solutions of a certain one-dimensional differential equation are investigated for the first order cubic non-autonomous equation. The method used here is the bifurcation of periodic solutions from a fine focus z = 0. We aimed to find the maximum number of periodic solutions into which a given solution can bifurcate under perturbation of the coefficients. For classes C3, 8, C4, 3, C7, 5, C7, 6, eight periodic multiplicities have been found. To investigate the multiplicity >9, the formula for the focal value was not available in the literature. We also succeeded in constructing the formula for η10. By implementing our newly developed formula, we are able to get multiplicity ten for classes C7, 3, C9, 1, which is the highest known to date. A perturbation method has been properly established for making the maximal number of limit cycles for each class. Some examples are also presented to show the implementation of the newly developed method. By considering all of these facts, it can be concluded that the presented methods are new, authentic, and novel.

## 1. Introduction

On August 8, 1900, David Hilbert presented a set of mathematical problems [1] to the Second International Congress of Mathematicians in Paris. The sixteenth problem he posed was titled the Problem of the Topology of Algebraic Curves and Surfaces. It is stated in two parts. In the first part, Hilbert suggested a thorough investigation of the relative positions of the separate branches of algebraic curves in nth-order vector fields, which is in the area of real algebraic geometry. In the second part, Hilbert asked for a search for the upper bound of the number of limit cycles and their relative locations in polynomial vector fields of order n. This part of the problem is related to ordinary differential equations and dynamical systems. Generally, this part of the problem is what is usually meant when talking about Hilbert's 16th problem.

Limit cycle theory takes a central role in Hilbert's 16th problem. Studying the number of limit cycles for differential equations is the most difficult part of the problem. The phenomenon of the limit cycle was first discovered and introduced by Poincaré in his four-part article, Integral curves defined by differential equations [25] published between 1881 and 1886.

At that time, Poincaré also noticed the close relationship between the study of limit cycles and the solutions of the global structural problems of a family of integral curves of differential equations. His work was later extended by Bendixson to the well-known Poincaré-Bendixson theorem [6] on the limit set of trajectories of dynamical systems in a bounded region. The driving force behind the study of limit cycle theory was the invention of the triode vacuum tube, which was able to produce stable self-excited oscillations of constant amplitude. It was noted that this kind of oscillation phenomenon could not be described by linear differential equations. At the end of the 1920's Van der Pol [7] developed a differential equation to describe the oscillations of constant amplitude in a triode vacuum tube. Limit cycles are common solutions for all types of dynamical systems. They model systems that exhibit self-sustained oscillations. In other words, these systems oscillate even in the absence of external periodic forcing. For a practical example, consider a specific Holling-Tanner predator-prey model [8]. This model appears to match very well with what happens for many predator-prey species in the natural world, for example, house sparrows and sparrow hawks in Europe, muskrat and mink in Central North America, and white-tailed deer and wolf in Ontario, Canada.

Other examples of self-excited oscillation are the beating of a heart, rhythms in body temperature, hormone secretion, chemical reactions that oscillate spontaneously, and vibrations in bridges and airplane wings. Due to the wide occurrence of limit cycles in science and technology, limit cycle theory has also been extensively studied by physicists, and more recently by chemists, biologists, and economists [916].

We consider the differential equation of the form

where independent variable t and coefficients γ, δ, υ are real-valued functions but z ∈ ℂ. To find the maximum count of periodic solutions we use the complexified form of the equation (1); for more details, see [1720]. Also consider that ∃ β ∈ ℝ such that:

$z(β)=z(0).$

These solutions are periodic, even if γ, δ, and υ are not themselves periodic. Our fundamental focus is to get the maximum number of periodic solutions of any class of the form (1) in which a solution may bifurcate by perturbing the coefficients. Neto [21] states that for Equation (1), until some coefficients are restricted, we are unable to have an upper bound for the number of focal values. The number of periodic solutions depends upon the multiplicity of the solutions z = 0. The multiplicity of z = 0, as solution of (1) is also multiplicity of z = 0; as a zero of the following displacement function

described in complex function theory. For z = 0, the means of computing multiplicity (μ) is explained in Alwash and Llyod [22], but for the sake of ease, we explained it briefly here. We write $z\left(t,0,r\right)=\sum _{i=1}^{\infty }{a}_{i}\left(t\right){r}^{i}$ for 0 ≤ t ≤ β where also r lies in neighborhood of z = 0, and use it in equation (2); for more detail, see [21, 2326]. This provides a differential equation for aμ(t) with some starting conditions a1(0) = 1 and aμ(0) = 0 for i > 1. Therefore

The multiplicity (μ) is “μ > 1” if

$a1(β)=1a2(β)=a3(β)=...=aμ-1(β)=0.$

However, aμ(β) ≠ 0. When a1(β) = 1 and aμ(β) = 0, ∀ μ > 1, the origin is center. We can observe from Equation (1) that $\stackrel{·}{{a}_{1}}\left(t\right)={a}_{1}\left(t\right)\upsilon \left(t\right),$ where a1(t) is defined as

$a1(t)=e∫0tυ(s)ds.$

In this way, μ > 1 iff

because a1(t) = 1. We are especially interested in the situation where z = 0 has multiple solutions, so we consider that (4) holds. By using the following transformation

$ξ=ze-∫0tυ(s)ds.$

(1) takes the form

where $\stackrel{^}{\gamma }\left(t\right)=\gamma \left(t\right){e}^{2{\int }_{0}^{t}\upsilon \left(s\right)ds}$ and

We can see that $\stackrel{^}{\gamma }$ and $\stackrel{^}{\delta }$ are periodic if γ, δ, and υ are periodic. By using Lemma (2.6) in Alwash and Llyod [22], consider multiplicity of z = 0 as a periodic solution of (1). If, for equation (1), μ > 1, then the multiplicity of ξ = 0 as a periodic solution of (5) is also μ. So we consider that υ(t) $\stackrel{~}{=}$ 0 in (1). As a result, equation (1) takes the form

where γ and δ may be polynomials (i) in t (ii) in cost and sint (trigonometric functions). The functions ai(t), for i > 1 are calculated by utilizing the relation

with a1(t) = 1. Calculation of these functions is tough because of the integration by parts used in it. Assume that ηi= ai(β); at that point, μ = i if η1 = 1 and ηk = 0 for 2 ≤ ki − 2 but ηi ≠ 0. These ${\eta }_{{i}^{´}s}$ are known as focal values. For i ≤ 8 functions ai(t) and ηi are given in Alwash and Llyod [22]. For i = 9 N, Yasmin calculated a9(t) and η9 in [27]. For i = 10 we have calculated a10(t) and η10 in Nawaz [28], also presented in theorem 2.1 and 2.2.

In section 2, we write formulas with which we can calculate the highest focal value, and we also implement stopping criteria defined in Alwash and Llyod [22]. Some required conditions and the method of perturbation are described in section 3. Section 2 and 3 are mainly concerned with the calculation of focal values, which we will utilize in section 4. In section 4, we consider polynomial coefficients for equation (6) and calculate the focal values. Some examples are given in section 5. In the last section, 6, we make discussions and conclusions.

## 2. Calculation of Focal Values η10

In the following theorem (2.1), some functions a2, a3, ..., a10 are given that are obtained from Equation (7) and are helpful in calculating the periodic solutions.

Theorem 2.1. For the equation (7), conclusive functions a2, a3, ..., a8 are given in Alwash and Llyod [22], and a9, a10 are described below:

and

By using these functions, we obtained the next theorem, 2, which enables us to find the maximum multiplicity in which the integral is like $\int \gamma \left(t\right)\overline{\delta \left(t\right)}dt$; bar “−” shows that integral $\overline{\delta \left(t\right)}={\int }_{0}^{t}\delta \left(t\right)dt$ is definite.

Theorem 2.2. The solution z = 0 of (6) has a multiplicity k, wherever 2 ≤ k ≤ 10 iff ηn = 0 for 2 ≤ nk − 1 and ηn ≠ 0 where

${\eta }_{2}={\int }_{0}^{\beta }\delta$

${\eta }_{3}={\int }_{0}^{\beta }\gamma$

${\eta }_{4}={\int }_{0}^{\beta }\gamma \overline{\delta }$

${\eta }_{5}={\int }_{0}^{\beta }\gamma {\overline{\delta }}^{2}$

${\eta }_{6}={\int }_{0}^{\beta }\gamma {\overline{\delta }}^{3}-\frac{1}{2}{\overline{\gamma }}^{2}\delta$

${\eta }_{7}={\int }_{0}^{\beta }\gamma {\overline{\delta }}^{4}+2\gamma {\overline{\delta }}^{2}\overline{\gamma }$

${\eta }_{8}={\int }_{0}^{\beta }\gamma {\overline{\delta }}^{5}+3\gamma {\overline{\delta }}^{3}\overline{\gamma }+\gamma {\overline{\delta }}^{2}\overline{\overline{\delta }\gamma }-\frac{1}{2}{\overline{\gamma }}^{3}\delta$

${\eta }_{9}={\int }_{0}^{\beta }\gamma {\overline{\delta }}^{6}-5\gamma {\overline{\delta }}^{4}\overline{\gamma }-2{\overline{\delta }}^{3}\overline{\delta \overline{\gamma }}+20\overline{\delta {\overline{\gamma }}^{2}}+2\overline{\delta \overline{\gamma }}\delta {\overline{\gamma }}^{2}$

and

${\eta }_{10}={\int }_{0}^{\beta }\gamma {\overline{\delta }}^{7}-\frac{1235}{6}\gamma \overline{\gamma }{\overline{\delta }}^{5}-\frac{970}{3}\gamma {\overline{\gamma }}^{2}{\overline{\delta }}^{3}-237\delta {\overline{\delta }}^{2}{\overline{\gamma }}^{3}-24\gamma {\overline{\gamma }}^{2}\delta {\overline{\delta }}^{2}-$$70\overline{\gamma }{\overline{\delta }}^{3}{\gamma }^{2}-21{\overline{\gamma }}^{4}\delta -74\gamma {\overline{\gamma }}^{3}\overline{\delta }+\frac{5}{2}{\overline{\gamma }}^{2}\delta {\overline{\delta }}^{4}+32{\overline{\delta }}^{4}\gamma \overline{\delta \overline{\gamma }}-16\delta {\overline{\delta }}^{4}\overline{\gamma }-15{\overline{\delta }}^{5}{\gamma }^{2}$ $-36\delta \overline{\delta }{\overline{\gamma }}^{2}\overline{\delta \overline{\gamma }}-8\delta {\overline{\delta }}^{4}\gamma \overline{\gamma }.$

In theorem (2.1) some functions a2, a3, ..., a10 are given that are obtained from Equation (7) and are helpful in calculating the periodic solutions. As future work, one can calculate a maximum multiplicity >10 by firstly generalizing theorem 2.1 and 2.2. This should be calculated by substituting the value of i > 10 into Equation (7).

## 3. Conditions for the Center and Method of Perturbation

In this section, we describe some conditions for the center. From theorem 2.2, we find the maximum value μ for different classes of equations. We have to stop calculating multiplicity ηk. We need some conditions that assure that there is no need to proceed further with ηk. For this, we require some conditions that are sufficient for z = 0 as a center. The conditions are given in theorem 3 and corollary 4.

Theorem 3.1. Consider that there are continuous functions f, g defined on I = σ([0, α]) and differentiable function σ with σ(α) = σ(0) such that

$γ(t)=f(σ(t))σ·$
$δ(t)=g(σ(t))σ·,$

then the origin is a center for (6).

Corollary 3.2. Consider that if any δ or γ is identically 0 and the other has mean value zero. The origin is a center.

For more detail, see [17, 19, 20]. After determining the maximum multiplicity μ, we now have to make a series of perturbations of the coefficients, every one of which results in one periodic solution coming out of origin.

For this, suppose the equation of the form given below:

having multiplicity μ = j (suppose). Let U be in the region near 0 in the complex plane containing no periodic solution except z = 0. From theorem (2.4) in Alwash and Llyod [22], the initial point that is contained in U remained fixed concerning a total number of periodic solutions. With the condition that perturbations of the coefficients considered are small enough, our goal is to get η2 = η3 = ... = ηj−2 = 0 but ηj−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 solutions while we get periodic solution ψ(t), where ψ(0) ∈ U as a non-trivial solution. By considering the underlying fact that the complex solutions always appear in conjugate pairs, we can say that ψ is real. Now, let U1 and V1 be the neighborhood of zero and ψ, respectively, such that V1U1U and V1U1 = γ. The periodic solutions around V1 and U1 are preserved when we make a small perturbation in the coefficients. By applying the same procedure as above, our choice is to perturb the coefficients such that ηk = 0 for k = 2, 3, ..., j − 3, but ηj ≠ 0. So that we get μ = j − 2. By applying that procedure, we get two non-trivial real periodic solutions and the zero solution is of multiplicity j − 2. Continuously, in this way, we end up with Equation (8) with μ = 2 and j − 2 being distinct non-trivial (other than zero) real periodic solutions.

## 4. Polynomial Coefficients for Some Classes

Let Ci,j indicate the class of the shape (6) in which the degree of γ is i and δ is j and these are polynomial in “t” only. We consider some classes C7,3, C7,5, C7,6, C3,8, C4,3, C9,1 and will evaluate the maximum multiplicity; for more classes, see [19, 20, 28]. These are described below in the form of theorems as:

Theorem 4.1. Let C7,3 be class of equation of the form

with

where the degree of γ(t) is 7 and δ(t) is 3. Then, μmax(C7,3) ≥ 10.

Proof. By using theorem 2.2, we calculate

Thus, multiplicity of z = 0 is μ = 2 if η2 ≠ 0, and multiplicity μ = 3 if η2 = 0 but η3 ≠ 0. If η2 = η3 = 0, then from (10) and (11), we take

and

Now, by using (12) and (13), γ(t) and δ(t) take the form of:

$γ(t)=b(2t−1)+c[(2t)2−43]+d[(2t)3−2]+e[(2t)4−165]+h[(2t)7−16],$
$δ(t)=l[(2t)3-2].$

and η4 is a constant multiple of “l ” given as:

$η4=-l(-3920h-224e+90c+105b)1575.$

If η4 = 0 then either l = 0 or

If l = 0, then δ(t) = 0 and η3 = 0 shows the mean value of γ(t) is zero. So by corollary 3.2, the origin is a center. Suppose l ≠ 0. If (14) holds, then η5 is calculated as:

$η5=-8l2(264e-35c+165b)675675.$

If η5 = 0, then either l = 0 or 264e − 35c + 165b = 0. But l ≠ 0(taken above), so we substitute

and calculate η6 as:

$η6=cl(200984c+6416388l2+534699b)81723972330.$

If η6 = 0, then either c = 0 or

because we already take l ≠ 0. If c = 0 then by using (15), γ(t), and δ(t) take the following form:

$γ(t)=[8t3-2][b(t4-t)+d],$
$δ(t)=l[8t3-2].$

Let σ(t) = 2t4 − 2t; then $\stackrel{·}{\sigma }\left(t\right)=8{t}^{3}-2$. Also, σ(0) = σ(1). So γ(t) and δ(t) are as follows:

$γ(t)=[b(t4-t)+d]σ·,$
$δ(t)=lσ·.$

By using theorem 3.1 the origin is a center having f(σ)= [b(t4t) + d] and g(σ) = l. Thus, suppose c ≠ 0. By using (16), we have η7 as follows:

$η7=491l2(12l2+b)(2060488754705b+22506362768324l2-757960558278d)1336288792941284910.$

Recall that l ≠ 0 (considered above). If η7 = 0 then either b = −12l2 or

If (17) holds, then we find

$η8=491l(307857d-901484l2)ρ170323661397720454173105679513834037327588400000.$

Here

Now if η8 = 0 then either

or because l ≠ 0, ρ ≠ 0. If l ≠ 0, 307857d − 901484l2 ≠ 0 but (18) holds, then we have $d={p}_{i}{l}^{2}.$ For i = 1, 2, with p1 = 340.28404100, p2 = −377.123069100, and in each case η9 is a multiple of l7, and l ≠ 0 (taken above). If (19) holds, then we compute η9 as:

$η9=32l5(37026759569911l-1736569072760400)999670687490475.$

If ${\varkappa }_{9}=0$ then, as l ≠ 0 considered above gives that l5 ≠ 0, we takes value of l as:

If (20) holds, then we calculate ${\varkappa }_{10}$ as:

$ϰ10=-5981909850866458913426128067910330261654972183986994762723753707656668378300214360133880839216718016650781737345237712655039420363082187505657708544000000000000000049738789970173287324409068649857544943273129235904875486944604484243602120619367493715976441567349747422942013452414709681196758182064297581227921.$

Here, ${\varkappa }_{10}$ is equal to a constant number that is non-zero. Thus, we conclude that the multiplicity of class C7,3 is 10, i.e., μmax(C7,3) ≥ 10.

Theorem 4.2. For equation

With

Choose ϵj for 1 ≤ j ≤ 8 to be non-zero and small as compared to ϵj−1. Then (21) has eight distinct non-trivial real periodic solutions.

Proof. If we substitute ϵp = 0, ∀ p = 1, 2, ..., 8, and coefficients are as given in Equations (22) and (23). So, the multiplicity of the origin $\varkappa$ is 10. Now, choose ϵ1 ≠ 0 and ϵp = 0 for 2 ≤ p ≤ 8; then it can be easily seen that ${\varkappa }_{9}$ is a constant multiple of ϵ1, but So, the multiplicity reduces by one and $\varkappa =9.$ For that reason, one periodic solution bifurcates out of the origin. Now, set ϵ1 ≠ 0, ϵ2 ≠ 0 and ϵp = 0 for 3 ≤ p ≤ 8; then we have ${\varkappa }_{p}=0$ for p = 2, 3, ..., 7. But ${\varkappa }_{8}$ results in a form of ϵ2 with some constant multiple. So, $\varkappa =8.$ Now, set ϵ1 ≠ 0, ϵ2 ≠ 0, ϵ3 ≠ 0 and ϵp = 0 for 4 ≤ p ≤ 8; then we have ${\varkappa }_{p}=0$ for p = 2, 3, ..., 6. But ${\varkappa }_{7}$ results in a form of ϵ3 with some constant multiple. If ϵ2 is sufficient small, then there are two non-trivial real periodic solutions. Further, moving in the present way, we have eight real periodic non-trivial solutions.

Corollary 4.1. For an equation

if γ(t) and δ(t) are as given in theorem 4.1, Equation (24) has ten real periodic solutions if γ and υ are small enough.

Proof. If γ = 0 and υ = 0, μ = 2 then (24) has eight real periodic solutions. If γ ≠ 0 but is small enough, then μ = 1 and by using the same arguments as in the above theorem, there are nine distinct periodic solutions other than 0; z = 0 is another solution. Thus, we have ten real periodic solutions.

Theorem 4.3. For class C7,5 consider δ(t) = i + n(2t + 1)5 and

$γ(t)=a+b(2t+1)+c(2t+1)2+f(2t+1)5+h(2t+1)7.$

Then μmax(C7,5) ≥ 8.

Proof. By using theorem 2.2, we have

$η2=1823n+i,$
$η3=410h+1823f+133c+2b+a.$

Thus, the multiplicity of z = 0 is μ = 2 if η2 ≠ 0. And multiplicity μ = 3 if η2 = 0 but η3 ≠ 0. If η2 = η3 = 0, then we calculate η4 as:

$η4=-n(-114470h+1079c+372b)189.$

If η4 = 0 then either n = 0 or

If n = 0, then η3 = 0 shows i = 0; hence, δ(t) = 0 and η2 = 0 implies that the mean value of γ(t) is zero. By corollary 3.2, the origin is a center. Suppose n ≠ 0. By using (25), we compute η5 as:

$η5=-8n2(886519816c+499588553b)2109395925.$

If η5 = 0 then

because we already take n ≠ 0. If (26) holds, then η6 is:

$η6=2bn(-379667599239958655624n2+5813410092109719b)815558371657762539807.$

If η6 = 0 then either b = 0 or

because n ≠ 0. If b = 0, then by using (26), (25), γ(t) and δ(t) take the form:

$γ(t)=f[(2t+1)5-1823],$
$δ(t)=n[(2t+1)5-1823].$

Let $\sigma \left(t\right)=\frac{1}{2}{\left(2t+1\right)}^{6}-182t$; then $\stackrel{·}{\sigma }\left(t\right)=3{\left(2t+1\right)}^{5}-182$. Also, σ(0) = σ(1). So, we can write it as:

$γ(t)=13fσ·,δ(t)=13nσ·.$

Then, by theorem 3.1, the origin is a center with f(σ)= $\frac{1}{3}f$ and $g\left(\sigma \right)=\frac{1}{3}n.$ So we take b ≠ 0. If (27) holds, then η7 is as follows:

Now, if η7 = 0, recalling that n ≠ 0 then

With holding (28), we have

$η8=2300397726692597332894199628622545916981609410130707279936768793823482030814697509595313095537997395261781753311944606875n7.$

Which is a constant multiple of n7 and is non-zero because n ≠ 0 (taken above). Thus we conclude that the multiplicity of class C7,5 is 8, i.e., μmax(C7,5) ≥ 8.

Theorem 4.4. For equation

Let

$δ(t)=-1823n+ϵ6+n(2t+1)5,$
$γ(t)=-v1+ϵ5+b(2t+1)+c(2t+1)2+f(2t+1)5+h(2t+1)7.$

Proof. With

$v1=-410h-1823f-133c-2b,b=3796675992399586556245813410092109719n2+ϵ2,c=-19470134860245482491747529020318381984429n2-499588553886519816ϵ2+ϵ3,f=9888560913986218905316863901375259954255683216749379832543161535450n2+ϵ1,$

and

$h=-5769075464309404974283565331028214n2-281257136387664ϵ2+1079114470ϵ3+ϵ4.$

If ϵj(1 ≤ j ≤ 6), σ1 and σ2 are chosen to be non-zero and also

$|σ2|≪|σ1|≪|ϵ6|≪|ϵ5|≪...≪|ϵ1|.$

Then (29) has eight distinct real periodic solutions other than zero.

Theorem 4.5. Let δ(t) = j + p(t − 1)6 and

$γ(t)=a+c(t-1)2+d(t-1)3+g(t-1)6+h(t-1)7.$

For class C7,6 of the form (9), then μmax(C7,6) ≥ 8.

Proof. By using theorem 2.2, we have

Thus, the multiplicity of z = 0 is μ = 2 if η2 ≠ 0, and multiplicity μ = 3 if η2 = 0 but η3 ≠ 0. If η2 = η3 = 0, by using values of a & j, , δ(t) and γ(t) are as follows:

$γ(t)=c[(t-1)2-13]+d[(t-1)3+14]+g[(t-1)6-17]+h[(t-1)7-18],$
$δ(t)=p[(t-1)6-17].$

Also, we calculate η4 as:

$η4=p(792c-486d+77h)221760.$

If η4 = 0 then either p = 0 or

If p = 0, then δ(t) = 0 and η3 = 0 implies that mean value γ(t) = 0. From corollary 3.2, the origin is a center, so consider p ≠ 0. If (32) holds, then we have η5 as:

$η5=-p2(484c-51d)54428220.$

If η5 = 0, then as p ≠ 0 (considered above) implies

And by using (33), we calculate η6 as:

$η6=cp(51382814976p2+520996995995c)115437830545837800.$

If η6 = 0, then either c = 0 or

because p ≠ 0. If c = 0 then γ(t) and δ(t) are:

$γ(t)=g[(t-1)6-17],δ(t)=p[(t-1)6-17].$

Let σ(t) = (t − 1)7t, then $\stackrel{·}{\sigma }\left(t\right)=7{\left(t-1\right)}^{6}-1$. Also, σ(0) = σ(1). So it takes new form as:

$γ(t)=g7σ·,δ(t)=p7σ·.$

By theorem 3.1, having f(σ)= $\frac{g}{7}$ and $g\left(\sigma \right)=\frac{p}{7}$, the origin is a center, so take c ≠ 0. Using (34), we have η7 as:

$η7=-41979424p4(4135364653107809477799p2+748144295421365642240g)536575324409227144872262909825473125.$

If η7 = 0, recalling that p ≠ 0 (η5), then

If (35) holds, then we find η8 as:

$η8=679565224952546531953909700744690288107705057731767297065597743067007681874695840512233681534101600000p7.$

which is a constant multiple of p5, and p is also non-zero (as shown above). Thus, we conclude that the multiplicity of class C7,6 is 8, i.e., μmax(C7,6) ≥ 8.

Theorem 4.6. Let C9,1 be a class of equation of the form (9), with

$γ(t)=c+dt+et2+ft3+kt8+lt9.$
$δ(t)=m+nt.$

We then see that μmax(C9,1) ≥ 10.

Proof. Using theorem 2.2, we take

Thus, the multiplicity of z = 0 is μ = 2 if η2 ≠ 0, and the multiplicity μ = 3 if η2 = 0 but η3 ≠ 0. If η2 = η3 = 0, then by using values of “k” and “a, ” γ(t) and δ(t) are as follows:

Also, we compute η4, given below as:

$η4=n(108l+112k+99f+66e)23760.$

If η4 = 0 then either n = 0 or

If n = 0, then δ(t) = 0 and η3 = 0 gives that the mean value of γ(t) is zero. Thus, the origin is a center from corollary 3.2, so consider n ≠ 0. Now, if (38) holds, then η5 is as below:

$η5=-n2(56k+297f+198e)7207200.$

If η5 = 0, then as we already take n ≠ 0 it implies

and by using (39), η6 is:

$η6=-n(2e+3f)(32e-57n2+29f)836559360.$

If η6 = 0, then as we already consider n ≠ 0 either $f=-\frac{2}{3}e$ or

If $f=-\frac{2}{3}e$ then (36) and (37) are of the following form:

$γ(t)=(t-12)[d+e(-23t2+23t-13)],$
$δ(t)=n(t-12).$

Let σ(t) = t2t; then $\stackrel{·}{\sigma }\left(t\right)=2t-1$. Also, σ(0) = σ(1). So we can write

$γ(t)=12[d+c(-23t2+23t)]σ·,$
$δ(t)=n2σ·.$

Thus, from theorem 3.1, the origin is a center with

$f(σ)=12[d+c(-23t2+23t)],$

and $g\left(\sigma \right)=\frac{n}{2},$ so we take $f\ne -\frac{2}{3}e.$ Holding (40), we compute η7 as:

$η7=19n2(3n2+f)(-348307f+1697231n2+1445136d)254514115215360.$

If η7 = 0, recalling that n ≠ 0, then either f = −3n2 or

If f = −3n2, then

$γ(t)=12[d+n2(-3t2+3t)]σ·,$
$δ(t)=n2σ·.$

From theorem (3.1), the origin is a center with f(σ)= $\frac{1}{2}\left[d+{n}^{2}\left(-3{t}^{2}+3t\right)\right]$ and $g\left(\sigma \right)=\frac{n}{2},$ so consider f ≠ −3n2. Using (41), we calculate η8 as:

$η8=23n(1122d+2129n2)ζ1029474284594079894479022764851200.$

where ζ = 273879615326996052d2 + 1713735341555455508dn2 − 132695961322089231627n4. Now, if η8 = 0 then either ζ = 0 or

because n ≠ 0. If Equation (42) ≠ 0, n ≠ 0, but ζ = 0, then $b={r}_{i}{n}^{2}$ for i = 1, 2, where r1 = 38.208145460, r2 = −50.722660920. If (42) holds, but ζ ≠ 0, n ≠ 0, then we compute η9

$η9=-n5(5168n+449059)2749402656.$

If η9 = 0, then we substitute $n=-\frac{449059}{5168}$ and calculate η10 as:

$η10=-2775260563886524309086510149628730887409296815148671775886791647382029788119627136595362915957451747785441280$

That is a non-zero constant number. Thus, we conclude that the multiplicity of class C9,1 is 10, i.e., μmax(C9,1) ≥ 10.

Theorem 4.7. Choose n, k, l with nl ≠ 0. In the equation

Let

$δ(t)=44905910336-12ϵ1+ϵ8+(-4490595168+ϵ1)t,$

and

$γ(t)=u1+dt+et2+ft3+kt8+lt9.$

With

and

$l=314618154179684ϵ2+1045576ϵ3+5518ϵ4-2827ϵ5+ϵ6.$

If ϵk(1 ≤ k ≤ 8), σ1 and σ2 are chosen to be non-zero such that

$|σ2|≪|σ1|≪|ϵ8|≪|ϵ7|≪...≪|ϵ1|.$

then (43) has ten real distinct non-trivial periodic solutions.

Proof. The proof is similar to that in theorem in (4.2), so it is omitted.

It is pertinent to mention that μmax(C4,3) ≥ 5 given in Yasmin and Ashraf [20] but we succeeded in increasing its multiplicity from 5 to 8, i.e., μmax(C4,3) ≥ 8 by using variable (t) instead of (2t-1).

Theorem 4.8. Let

$γ(t)=e+ft+gt2+ht3+it4,$
$δ(t)=a+dt3.$

for the class C4,3 of form (9). Then we prove that μmax(C4,3) ≥ 8.

Proof. From theorem 2.2, we calculate solutions as:

$η2=a+d4.$
$η3=e+f2+g3+h4+i5.$

Thus, the multiplicity of z = 0 is μ = 2 if η2 ≠ 0, and multiplicity μ = 3 if η2 = 0 but η3 ≠ 0. If η2 = η3 = 0, then

By using (44) and (45), γ(t) and δ(t) are as given below:

Also we calculate η4 as:

$η4=-d(-28i+45g+105f)25200.$

If η4 = 0 then either d = 0 or

If d = 0, then δ(t) = 0, and also η3 = 0 gives that δ(t) has mean value 0. From corollary 3.2, the origin is a center. Consider d ≠ 0. By using (48), we compute η5 as:

$η5=-d2(199g+1617f)60540480.$

If η5 = 0 as d ≠ 0, then

By using (49), we compute η6 as:

$η6=gd(78600753d2+13597688g)2050291017815040.$

If η6 = 0, then either g = 0 or

because we already take d ≠ 0. If g = 0, then (46) and (47) can be written as:

$γ(t)=h(t3-14),$
$δ(t)=d(t3-14).$

Consider σ(t) = t4t; then $\stackrel{·}{\sigma }\left(t\right)=4{t}^{3}-1$. Also σ(0) = σ(1), so it can written as:

$γ(t)=h4σ·(t),$
$δ(t)=d4σ·(t).$

By theorem 3.1, the origin is a center with $f\left(\sigma \right)=\frac{1}{4}h$ and $g\left(\sigma \right)=\frac{1}{4}d$. So take g ≠ 0. If (50) holds, then η7 is:

$η7=-491d4(-704698056497d2+61560932862h)102299877970079928320.$

If η7 = 0, recalling that d ≠ 0, then we take

If (51) holds, then we compute η8 as:

$η8=46697914451605811263464917723384478186490400805647418695680d7.$

where η8 is a constant multiple of d7 and d ≠ 0 (taken above). Thus, we conclude that the multiplicity of class C4,3 is 8, i.e., μmax(C4,3) ≥ 8.

Theorem 4.9. Let C3,8 be a class of equation of the form (9) with

$γ(t)=bt+dt3,$
$δ(t)=ft+ht3+jt5+lt7+mt8.$

Then μmax(C3,8) ≥ 8 when j = 1.

Proof. First we suppose that j ≠ 1. From theorem 2.2, we see:

Thus the multiplicity of z = 0 is μ = 2 if η2 ≠ 0, and multiplicity μ = 3 if η2 = 0 but η3 ≠ 0. If η2 = η3 = 0, then from (52) and (53) we take:

and

Now by using (54) and (55), we calculate η4 as:

$η4=-d(1400m+1287l+858j)1235520.$

If η4 = 0 then either d = 0 or

If d = 0, then from (55), γ(t) = 0 and η3 = 0 gives that the mean value of δ(t) is zero. So by corollary 3.2, the origin is a center. Thus, suppose d ≠ 0. If (56) holds, then η5 is:

$η5=-d(2j+3l)(30744j+59850h+8041l)683726400000.$

If η5 = 0 then either

or

because we already take d ≠ 0. If (57) holds, then γ(t) and δ(t) are of the form:

$γ(t)=d(t3-t2),$
$δ(t)=(t3-t2)[h+j(35600t2+88525t3700t5+23t4+31400t3+43t2-32800t+25)].$

Let $\sigma \left(t\right)=\frac{{t}^{4}}{4}-\frac{{t}^{2}}{4}$, then $\stackrel{·}{\sigma }\left(t\right)={t}^{3}-\frac{t}{2}$. Also σ(0) = σ(1). γ(t) and δ(t) are thus written as:

$γ(t)=dσ(t)·,$
$δ(t)=[h+j(35600t2+88525t3700t5+23t4+31400t3+43t2-d32800t+25)]σ·(t).$

So by theorem 3.1, the origin is a center with f(σ)= d and

$g(σ)=[h+j(35600t2+88525t3700t5+23t4+31400t3+43t2-32800t+25)].$

Therefore, we suppose that 3l + 2j ≠ 0. Holding (58), we compute η6, which is a constant multiple of ξ, as:

$η6=-d(3l+2j)ξ.$

where

$ξ=1532329720524j2+1273229285332jl+1690381942750100d+25542706569l2.$

Now η6 = 0 only if ξ = 0, because we have already discussed the possibility of d = 0, (3l + 2j) = 0; in each case, the origin is a center. For ξ = 0, we substitute

$d=-153232639720524169038161950100j2-127322239285332169038942750100jl-25542700956569169038161942750100l2.$

and obtain

and

We cannot draw any conclusion looking at η7 and η8. Thus, for simplification, we substitute j = 1. Then, η7 and η8 becomes

$η7={d(3l+2)(−20052663449157741455869750437+5254614244207520979577091004807l−26054076800585569433880148475302841l2+24863637539682162604430011l3).$

and

$η8={−d(3l+2)(−6049125746474233569752271066018358646532627l+704985851569387782175507850879295207088l2−36773517798842421434494904030251207123117l3+7260790045236054007143552578113438l4).$

For multiplicity >6, we have to prove that the cubic in η7 and quartic in η8 have no common zero. For this, we suppose that both have a common zero. Then we get a linear relation in j and l, j + kl = 0(say), and for this value of j, η8 is a constant multiple of dl5 ≠ 0. Thus, we conclude that the multiplicity of class C3,8 is 8, i.e., μmax(C3,8) ≥ 8 with j = 1.

Theorem 4.10. Suppose l = α be a real root of the equation

${-20052663449157741455869750437+5254614244207520979577091004807l-26054076800585569433880148475302841l2+24863637539682162604430011l3=0.$

Choose

Such that6| ≪ |ϵ5| ≪ |ϵ4| ≪ |ϵ3| ≪ |ϵ2| ≪ |ϵ1|. Then equation $\stackrel{·}{z}=\gamma \left(t\right){z}^{3}+\delta \left(t\right){z}^{2},$ has six real periodic non-trivial solutions. Where

$γ(t)=bt+dt3.$
$δ(t)=ft+ht3+jt5+lt7+mt8.$

with j = 1.

Proof. If we put ϵj for j as; 1 ≤ j ≤ 6, instead of 1 ≤ j ≤ 8 then the proof is similar to that for theorem (4.2), so it is omitted.

## 5. Examples

The following examples demonstrate the applicability of our main results.

Example: Consider the differential equation:

Here γ(t), δ(t) are transcendental functions, but we use the power series representations by neglecting the terms ‘tn’ for n > 4. Like γ(t) = et = a + bt + ct2 + dt3 + et4, δ(t) = cos t = jkt2 + lt4, with a = b = j = 1, $c=k=\frac{1}{2!},d=\frac{1}{3!},e=\frac{1}{4!},$ and then calculate the periodic solutions.

SOLUTION: We substitute k = 0, and by using theorem 2.2, we calculate:

Thus, the multiplicity of z = 0 is μ = 2 if η2 ≠ 0, and multiplicity μ = 3 if η2 = 0 but η3 ≠ 0. If η2 = η3 = 0, then we take $j=-\frac{1}{120}l,$ and $a=-\frac{1}{2}b-\frac{1}{6}c-\frac{1}{24}d-\frac{1}{120}e.$ By using these values, η4 is calculated as:

$η4=-l(14d+105c+360b)1814400.$

If η4 = 0 then either l = 0 or

If l = 0, then δ(t) = 0 and η3 = 0 shows that the mean value of γ(t) is zero. So by corollary 3.2, the origin is a center. Suppose l ≠ 0. If (62) holds, we have

$η5=-l2(28c+325b)6054048000.$

If η5 = 0, then l ≠ 0(taken above), and we substitute $c=-\frac{325}{28}b,$ and calculate η6 as:

$η6=bl(-21078407l2+10315069600b)1261504744980480000.$

If η6 = 0, then either b = 0 or

because l ≠ 0. If b = 0 then d = c = 0, by using these values, γ(t) and δ(t) takes the following form:

$γ(t)=e(t4-15),$
$δ(t)=l(t4-15).$

Let σ(t) = t5t, then $\stackrel{·}{\sigma }\left(t\right)=5{t}^{4}-1$. Also, σ(0) = σ(1). Therefore, we can write $\gamma \left(t\right)=\frac{1}{5}e\stackrel{·}{\sigma }$ and $\delta \left(t\right)=\frac{1}{5}l\stackrel{·}{\sigma }.$ From theorem 3.1, The origin is a center having f(σ)= $\frac{1}{5}e$ and Thus, suppose that b ≠ 0. By using (63), we have η7 as follows:

$η7=-3011201l4(4904530106070545l2+18047929302961536e)1193751333049572276388321296384000000.$

Recalling that l ≠ 0 (considered above), if η7 = 0 then

If (64) holds, then we find

$η8=3739528414309673192972599618926786014498207710495466937428606344400899932160000000l7.$

which is constant multiple of l7, and l ≠ 0. Thus we conclude that (59) have eight periodic solutions.

Example: Consider the differential equation:

With γ(t) =equation of circle = ax2 + by2 + cx + dy + f, δ(t) =quadratic equation= gx2h, we calculate the periodic solutions.

SOLUTION: By using theorem 2.2, we calculate:

$η2=g3-h,$
$η3=13a+13b+12c+12d+f.$

Thus, the multiplicity of z = 0 is μ = 2 if η2 ≠ 0, and multiplicity μ = 3 if η2 = 0 but η3 ≠ 0. If η2 = η3 = 0, then we put $h=\frac{g}{3},$ and $f=-\frac{1}{3}a-\frac{1}{3}b-\frac{1}{2}c-\frac{1}{2}d.$ By using these values, we get

$η4=-1360cg.$

If η4 = 0 then c = g = 0. If g = 0, then δ(t) = 0 and η3 = 0 shows the mean value of γ(t) is zero. So by corollary 3.2, the origin is a center. Suppose that g ≠ 0. Now by using the value of c = 0, we calculate η5 = 0. Thus we conclude that (65) have four periodic solutions.

## 6. Conclusion and Discussion

In this article, periodic solutions are calculated. The solutions satisfying 𝔷(β) = 𝔷(0), are called periodic orbits of the Equation (1). The periodic orbits that are isolated in the set of all periodic orbits are usually called the limit cycle. Periodic solutions are found for algebraic coefficients for various classes by using bifurcation analysis. We examined classes C3,8, C4,3, C7,3, C7,5, C7,6,C9,1. We could only get a maximum multiplicity of 10 by using the classical formulas existing in the literature. We succeeded in developing the formula η10 by which classes C7,3, and C9,1 have maximum multiplicity 10, which is the highest known until this time. We also improved some already calculated results of Yasmin and Ashraf [20] for class C4,3, where μmax is improved from 5 to 8. A systematic procedure has been established in defining coefficients of higher-order polynomial functions. A perturbation method has been properly established for making the maximal number of limit cycles in section 3, which was used numerically to calculate all the classes mentioned in the article. Some examples are also presented to show the applicability of the method. Since the journey toward solving Hilbert's 16th problem is still far at an end, searching for more limit cycles and raising the general lower bound form could be an effective choice for approaching the problem.

## Data Availability Statement

All datasets generated for this study are included in the article/supplementary material.

## Author Contributions

SA and AN: conceptualization. SA, AN, and NY: writing original draft. DB and KN: methodology, review, and editing. AG and DB: formal analysis. SA, AG, and KN: software. All authors contributed to the article and approved the submitted version.

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## References

1. Hilbert D. Mathematical problems. Bull Am Math Soc. (1902) 8:437–79. doi: 10.1090/S0002-9904-1902-00923-3

2. Poincare H. Memoire sur les courbes definies par une equation differentielle. I, II. J Math Pures Appl. (1881) 7:375–422.

3. Poincare H. Memoire sur les courbes definies par une equation differentielle. I, II. J Math Pures Appl. (1882) 8:251–296.

4. Poincare H. Sur les courbes definies par les equations differentielles. III, IV. J Math Pures Appl. (1885) 1:167–224.

5. Poincare H. Sur les courbes definies par les equations differentielles. III, IV. J Math Pures Appl. Paris. (1886) 2:155–217.

6. Guckenheimer J, Holmes P. Nonlinear Oscillations Dynamical Systems and Bifurcations of Vector Fields. 2nd ed. New York, NY: Springer-Verlag (1992).

7. Van der pol B. On relaxation-oscillations. Phil. Mag. (1926) 2:978–92. doi: 10.1080/14786442608564127

8. Lynch S. Dynamical Systems With Applications Using Matlab, Dynamical Systems With Applications Using Maple. Boston, MA: Birkhauser (2004).

9. Alwash MAM. Polynomial differential equations with small coefficients. Discrete Contin Dyn Syst. (2005) 25:1129–41. doi: 10.3934/dcds.2009.25.1129

10. Akram S. Periodic solutions of non-autonomous equations (M.Phil. thesis) (2007). p. 1–41.

11. Gasull A, Llibre J, Sotomayer J. Limit cycles of vector field of the form X(v) = Av+f(v)Bv. J Differs Equat (1987) 67:90–110. doi: 10.1016/0022-0396(87)90140-9

12. Hua N. The fixed point theory and the existence of the periodic solution on a nonlinear differential equation. Js Appl Math. (2018) 2018:6725989. doi: 10.1155/2018/6725989

13. Bohner M, Gasull A, Valls C. Periodic solution of linear, Riccati and Abel dynamic equations. J Math Anals Appl. (2019) 470:733–49. doi: 10.1016/j.jmaa.2018.10.018

14. Huang J, Liang H. A uniqueness criterion of limit cycles for planar polynomial system with homogenous nonlinearities. J Maths Anal Appl. (2018) 457:498–521. doi: 10.1016/j.jmaa.2017.08.008

15. Huang J, Liang H, Libre J. Non-existence and uniqueness of limit cycles for planar polynomial differential system with homogenouss nonlinearities. J Differ Equat. (2018) 265:3888–913. doi: 10.1016/j.jde.2018.05.019

16. Alvarez MJ, Bravo JL, Fernandez M, Prohens R. Alien limit cycles in Abels equation. J Math Anal Appl. (2020) 482:123525. doi: 10.1016/j.jmaa.2019.123525

17. Alwash MAM. Periodic solutions of polynomial non-autonomous differential equations. Electron J Differ Equat. (2005) 84:1–8.

18. Llyod NG. Limit cycles of certain polynomial systems. In: Singh SP, editor. Non-linear Functional Analysis and Its Applications. NATO ASI Series (C: Mathematical and Physical Sciences), Vol. 173. Dordrecht: Springer (1986). p. 317–26. doi: 10.1007/978-94-009-4632-3_25

CrossRef Full Text

19. Yasmin N. Bifurcating periodic solutions of polynomial system. Punjab Univ J Math. (2001) XXXiV:43–6.

20. Yasmin N, Ashraf M. Bifurcating periodic solutions of class C2,4 and C4,3 of research SCI. BZU. (2003) 14.

21. Neto L. On the number of solutions of the equations $\frac{dx}{dt}={\sum }_{j=0}^{n}{a}_{j}\left(t\right){t}^{\prime }$, 0 ≤ t ≤ 1 for which x(0) = x(1). Invent Math. (1980) 59:67–76.

22. Alwash MAM, Llyod NG. Non-autonomous equation related to polynomial two-dimensional system. Proc R Soc Edinb. (1987) 105:129–152. doi: 10.1017/S0308210500021971

23. Lloyd NG. Small amplitude limit cycles of polynomial differential equations. In: Everitt WN, Lewis RT, editors. Ordinary Differential Equations and Operators. Lecture Notes in Mathematics, Vol. 1032 (Berlin; Heidelberg: Springer) (1982). p. 346–57. doi: 10.1007/BFb0076806

24. Llyod NG. The number of periodic solutions of the equation $\stackrel{˙}{Z}={z}^{n}+{p}_{1}\left(t\right){z}^{n-1}+{P}_{2}\left(t\right){z}^{n-2}+...+{P}_{n}\left(t\right)$.+Pn(t). Proc Lond Math Soc. (1973) 27:667–700. doi: 10.1112/plms/s3-27.4.667

25. Dosary A, Khalil IT. Limit cycles for planar differential systems with quasi-homogenous nonlinearities. J Math Comp Sci. (2019) 9:365–71.

26. Zhou Z, Yan Y. On the composition center for a class of rigid system. Bull Braz Math Soc. (2020) 51:139–55. doi: 10.1007/s00574-019-00147-y

27. Yasmin N. Closed orbits of certain two dimensional cubic systems (Ph.D. thesis), University College of Wales Aberystwyth, United Kingdom (1989). p. 1–169.

28. Nawaz A. Bifurcation of periodic solutions of certain classes of cubic non-autonomous differential equations (M.Phil. thesis) (2018). p. 1–104.

Keywords: multiplicity, periodic solution, non-autonomous equation, bifurcation method, trigonometric coefficients

Citation: Akram S, Nawaz A, Yasmin N, Ghaffar A, Baleanu D and Nisar KS (2020) Periodic Solutions of Some Classes of One Dimensional Non-autonomous Equation. Front. Phys. 8:264. doi: 10.3389/fphy.2020.00264

Received: 06 March 2020; Accepted: 11 June 2020;
Published: 15 September 2020.

Edited by:

Grienggrai Rajchakit, Maejo University, Thailand

Reviewed by:

Gunasekaran Nallappan, Shibaura Institute of Technology, Japan
Chanikan Emharuethai, Maejo University, Thailand

Copyright © 2020 Akram, Nawaz, Yasmin, Ghaffar, Baleanu and Nisar. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Kottakkaran S. Nisar, n.sooppy@psau.edu.sa; ksnisar1@gmail.com