Most of the real-world problems in nature are multidimensional and when modeled, they arise as higher-order ordinary differential equations. We are interested in those models which are periodic and depend on time and are usually known as non-autonomous. This article contains several recent developments and advances in calculations of periodic solutions and their applications in various areas of the mathematical, physical and engineering sciences. We have investigated upper bounds for the non-autonomous ordinary differential equation (ODE) of the cubic degree [1–3]. The primary question striking in our minds is to investigate the maximum number of periodic solutions when the degree of non-autonomous (ODE) is increased from three to four; so, we started working for the quartic system. The analysis of periodic solutions is vital because they frequently arise as real-world problems from financial matters such as modelling economic processes to complex space robotics, from galaxies to weather forecasting models. Many physics problems are related to nonlinear analysis like relativity, elasticity, chaotic dynamics and Navier stokes equation in fluid dynamics. Bendixson’s negative criterion, based on the connection between compound matrices and (ODE’s), is applied to prove the nonexistence of periodic orbits and then nonexistence of Hopf bifurcation. Almost every bit of life needs this analysis, even inside the body as cardiac rhythms as blood flow, please see the example, [4–6]. The aesthetic appeal of periodic solutions may explain why so many people have become intrigued by these ideas. Alwash et al. [7,8] developed the theoretical foundations of periodic solutions of differential equations.

Alwash et al. [8] examine equation of the form:

(1) y˙=g(z)y3+d(z)y2+u(z)y,

to find the upper bound for the number of limit cycles for such system (1) in accordance with the second part of Hilbert’s sixteenth problem. The coefficients g; d; and u are real-valued continuous functions, but the independent variable is complex. We are principally concerned about the multiplicity of y=0 as a periodic solution, i.e., solutions satisfying

(2) y(0)=y(σ),

for σ ∈ R and look for information about the number of periodic solutions. We refer the reader to [8] for more extra pieces of information, however right here we do not forget that the multiplicity of a solution ϕ(t) of Eq. (1) satisfying Eq. (2) is the multiplicity of ϕ(0) as a root of the holomorphic function q:c→y(t,0,c)−c;y(t,t1,c) is the solution satisfies y(t,t1,c)=c . Here we take a multiplicity of the periodic solution as ‘k’ of the function ϕ(t) . Rouche’s theorem is used for the function q, it is assumed that we have at most k periodic solutions in the regions nearby ϕ (counting multiplicity): see, for example, ([2], Theorem 2.4) when sufficiently small perturbation of the equation is in an account for the following quartic equation:

(3) y˙=ϵy4+g(z)y3+d(z)y2+u(z)y,

For sufficiently small perturbation ϵ, the quartic equation preserves the same number of periodic solutions as Eq. (1). We used scaling y→ϵ1/3y to get the leading coefficient of Eq. (3) as one and this Eq. (3) becomes as:

(4) y˙=y4+ϵ−2/3(z)y3+ϵ−1/3(z)y2+u(z)y,

To make the above equation simple, which is helpful for lengthy calculations that arise in numerical integration, we rewrite Eq. (4) as:

(5) y˙=y4+g(z)y3+d(z)y2+u(z),

We are mainly focused on finding the multiplicity of periodic solutions of y = 0 greater than one for Eq. (1) as was done by researchers in [7], without loss of generality; we could take u(z) ≃ 0; as was taken in [1]. In this paper, we are considering the corresponding equation:

(6) y˙=y4+g(z)y3+d(z)y2,

There is another reason for our interest in Eq. (5). We say this system to be non-autonomous, nonlinear because all the ‘ y ’ on the right-hand side appear to a power greater than one. Because of the non-autonomous equation, equilibrium states are not usually associated with Eq. (6). The general form of the above Eq. (6) is as follows:

(7) y˙=yn+p1(z)yn−1+p2(z)yn−2+,…,+p0(z);here p0(z)=1.

Moreover, the coefficients were considered to be periodic functions, please see the example, [8–11]. However, the questions related to Hilbert’s sixteenth problem are reduced to polynomial equations in which p0 has zeros. They showed that for n = 3, Eq. (7) has precisely three periodic solutions. Here, the results of [12] no longer hold; they have presented some examples in [13] which showed that there is no idea about the upper bound for the number of periodic solutions when n≥4 and p0(z) = 1; until some coefficients are restricted. Earlier, Shahshahani [14]; determined the multiplicity of Eq. (6) when g(z),d(z) and u(z) are polynomial functions of z. We also present in Section 3 that the conjecture made in [14] is false.

In Section 2, some results from the literature are employed and formulae for calculating the maximum number of periodic solutions, greater than eight are formulated and presented. In Section 3, we calculated the periodic multiplicity for various classes of coefficients. In the last section, we present conclusions and discussions.

In this section, we recall some important concepts necessary to understand the presented method; for instance, see [1]. For the sake of multiplicity of the zero solution of Eq. (5), we observe that in a neighborhood of y=0 , we can write

(8) y(z,0,c)=∑i=0∞ui(z)ci

for 0≤z≤σ ; the functions ui(z) are continuous and u1(0)=1 and ui(0)=0 for i>1 . The multiplicity (μ) is “ μ=k ” if

(9) u1(σ)=1,u2(σ)=u3(σ)=…=ui−1(σ)=0.}

However, ui(σ)≠0 . To calculate the functions ui(z) , we placed the expression (8) into (5) and achieved a set of linear differential equations which can be solved recursively. We will study from Eq. (9) that u1(z)=u1(z)υ(z) , where u1(z) is set as:

u_1(z)=e∫oσv(s)ds.

Hence, y=0 is a multiple solution iff ∫oσv(s)ds . Since we were right here interested in equations, wherein the multiplicity of the origin is greater than one, we recall that ∫oσv=0 . At that point, we apply the transformation y→ypn−1 , wherein p(z)=e∫ozv and obtain

(10) y=(p(z))3y4+(p(z))2y3+p(z)υ(z)y2.

Since p is periodic, the function q is unchanged through the transformation and therefore the periodic solution of Eq. (5) and Eq. (10) have identical preliminary points and multiplicities. Additionally, the transformation of the independent prompts to the Eq. (6). For Eq. (6), the functions ui(z) , for i>1 are calculated by the following relation:

(11) u˙i=(∑j+k+l+m=ij,k,l,m≥1)(ujukulum)+γ(z)(∑j+k+l=ij,k,l≥1)(ujukul)+δ(z)(∑j+k=ij,k≥1)(ujuk).

With u1(z)=1 . For i≤8 functions ui(z) and ϰi are provided in [1], for i=9 , we have calculated u9(z) and ϰ9 ; and present them below in Theorems 2.1 and 2.2. Some tremendously complicated computations are involved; hence, we omit some details and we write results.

In this article, we present a novel approach for finding periodic solutions. By using Theorem 2.2, we can find the highest periodic solutions as 9. We base this method on the construction of Akram et al. [1]. The approach can apply to other planar systems as well.

When we started working for quartic non-autonomous (ODE), for this investigation, formulae for finding upper bounds greater than 8 are not present in the literature up to now. We came up with many challenges while working with this problem of quartic type. The first task is to compute these new formulas, which are previously unavailable in the literature. By putting in many efforts, we succeeded in constructing the new formula u9 and ϰ9 , which are given in theorems 2.1 and 2.2. In the below theorem, we use the notation “ (.)¯ ” for indefinite integral such as δ¯=∫δ(s)ds .

Theorem 2.1 For Eq. (6); the functions u₂, u₃, … , u₉ in the expansion (11) are as follows:

u2=δ¯,

u3=δ¯2+γ¯,

u4=δ¯3+2δ¯γ¯+δ¯γ¯+z,

u5=δ¯4+3δ¯2γ¯+δ¯2γ¯+3δ¯δ¯γ¯+2δ?+32γ¯2+2zδ¯,

u6=δ¯5+4δ¯3γ¯+δ¯3γ¯+3δ¯2δ¯γ¯+3δ¯2¯+4δ¯δ?+2δ¯δ¯2γ¯−12δγ¯2¯+92δ¯γ¯2+3δ¯γ¯γ¯+γ?+3z(δ¯2+γ¯),

u7=δ¯6+5δ¯4γ¯+δ¯4γ¯+4δ¯3δ¯γ¯+2δ¯δ¯3γ¯+4δ¯3¯+3δ¯2δ¯2γ¯+6δ¯δ¯2¯+6δ¯2δ?+272δ¯2γ¯2+3δ¯2γ¯γ¯−δγ¯2¯δ¯−2δδ¯γ¯2¯+6(δ¯γ¯)2+2δ¯γ?+4δ¯γ¯¯−4(δγ¯¯)2+6δ?⁡γ¯+52γ¯3+4z(δ¯3+3δ¯γ¯+2δγ¯¯)+2z,

u8=δ¯7+6δ¯5γ¯+δ¯5γ¯+2δ¯δ¯4γ¯+5δ¯4δ¯γ¯+3δ¯2δ¯3γ¯+4δ¯3δ¯2γ¯+8δ¯δ¯3¯+9δ¯2δ¯2¯−8δ¯3δ?+5δ¯4¯+752δ¯3γ¯2+3δ¯3γ¯γ¯+12δ¯2γ¯δ¯γ¯−δ¯2δγ¯¯γ¯−6δ¯2δγ¯2¯−25δ¯2δγ¯¯γ¯−32δ¯2δγ¯2¯−4δ¯2γ¯δγ¯¯+5δ¯(δγ¯¯)2−4δ¯δδ¯γ¯2¯+9δ¯2γ¯¯−δ¯2γ?−2δ?⁡δ¯γ¯+9δ¯2¯γ¯+3δ¯2γ¯+16δ¯δ?⁡γ¯+8δ¯δ¯γ¯¯+10δ?⁡δ¯γ¯−2δ¯?−12δγ¯3¯−32δγ¯2¯γ¯+10δ¯γ¯3+152δ¯γ¯γ¯2+32γ¯2¯+3γ¯γ?+z(5δ¯$4$+15δ¯2γ¯+5δ¯2γ¯+10δ¯δ¯γ¯+10γ?+152γ¯2)+5z2δ¯,

u9=δ¯8−δ¯3γ¯+4δ¯3γ+58δ¯γ¯δ¯2¯+12δ¯2+20(δ)2+10δ¯2γ¯δ¯¯−2δγ¯2+2δ¯γ¯2¯+24δ¯γ¯γ¯−2γ+38zδ¯2¯−38δ¯2+24zγ+35δ¯6γ¯−1612δ¯4γ¯2+3δ¯4γ¯γ¯+30δ¯γ¯δ¯3γ¯+90δδ¯2δ¯γ¯¯+4δ¯3γ¯γ¯δ¯−4δ¯3γ¯γ¯δ¯+12δ¯3¯γ¯+11γ¯δ¯2δ¯2γ¯+363γ¯δ¯δ¯2¯+34γ¯δ¯2δ+392δ¯2γ¯3+152γ¯2δ¯2γ¯+21γ¯(δ¯γ¯)2−2γ¯δ¯δγ¯2¯+4γ¯γ⁡δ¯+358γ¯4+12γ¯δ¯γ¯¯+15γ¯2δ+583γ¯δ¯3z−58γ¯δδ¯2z¯+24γ¯δγ¯¯−24γ¯2δ¯+9γ¯z2−36γγ¯δ¯δ¯γ¯¯−66γ¯δδ¯2z¯+42γ¯δ¯γ¯z−42γ¯δ¯γ¯¯−42γ¯2δ¯z+42γ¯2δz¯+9γ¯δ¯3δ¯2γ¯−9γ¯δ¯5γ¯+27γ¯δδ¯2δ¯2γ¯¯−12γ¯(δγ¯¯)2+24γ¯2δδγ¯¯¯−6γδδ¯γ¯2¯¯+2δ¯δ¯5γ¯+7γ¯δ¯6+3δ¯2δ¯4γ¯+325δ¯5δ¯γ¯+4δ¯3δ¯3γ¯+5δ¯4δ¯2γ¯+12δ¯3δ¯2¯+12δ¯2δ¯3¯+10δ¯4δ+10δ¯δ¯4¯+1013δ¯4γ¯+8γ¯δ¯δ¯3γ¯+19γ¯δ¯2δ¯2γ¯−2δ¯δ¯2δγ¯¯γ¯+823γδ¯3δγ¯¯¯−12δ¯γ¯2δδ¯2¯−523γ¯δ¯3δγ¯¯−2δ¯3δγ¯2¯−8δδ¯2γ¯δγ¯¯¯+δ¯2(δγ¯¯)2−6δ¯2δδ¯γ¯2¯+18δ¯δ¯2γ¯¯−2δ¯δ¯2γ−4δ¯δ⁡δ¯γ¯+12γ¯δ¯δ¯2¯+4γ⁡δ¯3−12γ¯δ⁡δ¯2+12δ¯2δ¯γ¯+10δ¯2δ¯γ¯−10δ¯3γ¯−4δ¯δ−δγ¯3¯δ¯+2γ¯δ¯δγ¯2¯+492γ¯3δ¯2+24γ¯2δ¯δ¯γ¯+3γ¯2¯δ¯+8γ¯δγ¯¯−20γ⁡δ¯¯+10δ¯3γ¯δ¯γ¯+4δ⁡δ¯δ¯γ¯−4δ⁡δ¯2γ¯+2δ¯γ¯δ¯δ¯2γ¯−2δ¯2γδ¯2γ¯¯+δ¯5z+563γ¯δ¯3z24γ¯2δ¯z−9δ¯2δγ¯¯z+9δ¯2δγ¯¯¯z+9δ¯2(δ¯γ¯)2z+4δ⁡δ¯z+9δ¯2z2+12δδ¯2γ¯z+10δ¯2δ¯γ¯z+20γ⁡δ¯z−20γ¯δ¯¯z+24γ¯2δ¯z−9δ¯2δγ¯¯z+9δ¯2δγ¯¯¯z+9δ¯2(δ¯γ¯)2z+4δ⁡δ¯z+9δ¯2z2

By using the above mentioned functions of Theorem 2.1, we can obtain Theorem 2.2, under some suitable conditions.

Theorem 2.2 The solution y=0 of (6) has a multiplicity k, wherever 2≤k≤9 if and only if ϰ2=ϰ3=,…,ϰk−1=0 and ϰk≠0 where

ϰ2=∫oσδ,

ϰ3=∫oσγ,

ϰ4=∫oσ(γδ¯+1),

ϰ5=∫oσ(γδ¯2+2δ¯),

ϰ6=∫oσ(2γδ¯3+6δ¯2−δγ¯2+2γ¯),

ϰ7=∫oσ(γδ¯4+4δ¯3−2δδ¯γ¯2+4δ¯γ¯),

ϰ8=∫oσ(γδ¯5+5δ¯4−6δ¯2−δ¯2δγ¯¯γ−δ¯2γ¯+9δ¯2γ¯−2γδ¯δ?−12γ¯3δ−2δ?+32γ¯2),

and

ϰ9=∫oσ(γδ¯6+10δ¯5+959γ¯δ¯3+2903δ¯γ¯δ¯2+20δ¯2¯+503δ¯δ¯2γ¯−103δ¯γ¯2+103δ¯γ¯2+40γ¯δ¯γ¯−1003γ?+150δδ¯2δ¯γ¯−203γ¯δδ¯3γ¯−60γγ¯δ¯δ¯γ¯−70γ¯δ¯γ¯−15δ¯5γ¯γ+40δγ¯δγ¯¯+45δδ¯2γ¯δ¯2γ¯−10γδδ¯γ¯2¯+4109γδ¯3δγ¯¯−403δδ¯2γ¯δγ¯¯−1003δ¯γ?+15δ¯2δγ¯¯−203δ¯δ?⁡γ−1003γδ¯2δ¯2γ¯+70δγ¯2z−6203γ¯δ¯2δz−1003γ¯δ¯z).

Assume that ϰi=ui(σ) , at that point μ=i if ϰ1=1 and ϰk=0 for 2≤k≤i−2 but ϰi≠0 . These ϰi ’s are known as focal values.

Proof By definition, the multiplicity of the zero solution is k if an(σ)=0 for 2≤n≤k−1 and uk(σ)≠0 . Write ξk=uk(σ) and let ϰk be the value of ξk when ξn=0 for n<k .

Since u2(z)=∫oσδ, we have ξ2=∫oσδ . With ξ2=0 , we have ξ3=∫oσγ and ξ4=∫oσ(γδ¯+1) ; hence ϰ3 and ϰ4 as stated. Next, suppose that ξ2,ξ3=0 and substitute the relations ∫oσδ=∫oσγ=0 into the expressions for u5(σ) and u6(σ) given in theorem 2.1; we obtain ϰ4=∫oσ(γδ¯+1),ϰ5=∫oσ(γδ¯2+2δ¯) .

Finally, we suppose that ξ4=0 (as well as ξ2=ξ3=0 ). We substitute the relations ∫oσδ=∫oσγ=∫oσ(γδ¯+1)=0 into the expressions for u7(σ) and u8(σ) ; then ϰ7=∫oσ(γδ¯4+4δ¯3−2δδ¯γ¯2+4δ¯γ¯) ,

and ϰ8=∫oσ(γδ¯5+5δ¯4−6δ¯2−δ¯2δγ¯¯γ−δ¯2γ¯+9δ¯2γ¯−2γδ¯δ−−12γ¯3δ−2δ−+32γ¯2) .

Continuing in the same way we get: ϰ9=∫oσ(γδ¯6+10δ¯5+959γ¯δ¯3+2903δ¯γ¯δ¯2+20δ¯2¯+503δ¯δ¯2γ¯−103δ¯γ¯2+103δ¯γ¯2+40γ¯δ¯γ¯−1003γ?+150δδ¯2δ¯γ¯−203γ¯δδ¯3γ¯−60γγ¯δ¯δ¯γ¯−70γ¯δ¯γ¯−15δ¯$5$γ¯γ+40δγ¯δγ¯¯+45δδ¯2γ¯δ¯2γ¯−10γδδ¯γ¯2¯+4109γδ¯3δγ¯¯−403δδ¯2γ¯δγ¯¯−1003δ¯γ?+15δ¯2δγ¯¯−203δ¯δ?⁡γ−1003γδ¯2δ¯2γ¯+70δγ¯2z−6203γ¯δ¯2δz−1003γ¯δ¯z ).

Now we are going to define the center and some necessary conditions for the center.

Definition An equilibrium point surrounded in its immediate neighborhood (not necessarily over the whole plane) by a closed path is called a center.

Conditions for Center

Corollary 2.1 If any g(z) or d(z) is identically zero and the other has a mean value of zero, then the origin is a center.

Corollary 2.2 For continuously odd differentiable functions g(z) and d(z) of period σ: The origin is a center.

We consider various classes in which two types of coefficients, namely (z or cos (z) and sin (z)) for g(z) and d(z) in Eq. (6), are considered for the calculations of periodic solutions.