In this article, the maximum possible numbers of periodic solutions for the quartic differential equation are calculated. In this regard, for the first time in the literature, we developed new formulae to determine the maximum number of periodic solutions greater than eight for the quartic equation. To obtain the maximum number of periodic solutions, we used a systematic procedure of bifurcation analysis. We used computer algebra Maple 18 to solve lengthy calculations that appeared in the formulae of focal values as integrations. The newly developed formulae were applied to a variety of polynomials with algebraic and homogeneous trigonometric coefficients of various degrees. We were able to validate our newly developed formulae by obtaining maximum multiplicity nine in the class C_{4,1} using algebraic coefficients. Whereas the maximum number of periodic solutions for the classes C_{4,4}; C_{5,1}; C_{5,5}; C_{6,1}; C_{6:6}; C_{7,1} is eight. Additionally, the stability of limit cycles belonging to the aforementioned classes with algebraic coefficients is briefly discussed. Hence, we conclude from the above-stated facts that our new results are a credible, authentic and pleasant addition to the literature.

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 [

Alwash et al. [

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

for

For sufficiently small perturbation

To make the above equation simple, which is helpful for lengthy calculations that arise in numerical integration, we rewrite

We are mainly focused on finding the multiplicity of periodic solutions of y = 0 greater than one for

There is another reason for our interest in

Moreover, the coefficients were considered to be periodic functions, please see the example, [

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 [

for

However,

Hence,

Since

With

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

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

_{2}, u_{3}, … , u_{9} in the expansion

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

and

Assume that

Since

Finally, we suppose that

and

Continuing in the same way we get:

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

We consider various classes in which two types of coefficients, namely

We have to consider

The above lemma is from Alwash et al. [

Which are illustrated in the following theorems, throughout this paper, we use the symbol “C” to represent the class and we show the maximum multiplicity for classes with μ_{max}.

_{4,4}_{5,5} _{6,6}

_{4,4} for

Then we calculate

From

If

If

with _{7} as:

Hence our conclusion is

Then

_{2} = _{3} = 0 and:

If

If

Now,

where

Hence,

_{6,6} for

Then

_{5} as:

If

Further, for

Here it is a nonzero constant number. So,

Now, for the polynomial z; we will explain the calculations of the maximum possible periodic solution for many classes of the

Let

Thus multiplicity of

If

we calculate

Now for _{6}

Using _{7} as:

If

For

As the outcome of obtained multiplicity is even having the negative sign. So, by using remark 1, it can be concluded that the origin is stable.

and

If suppose

If

we calculate

Now for

If

Using _{7} as:

If

Now, we cannot proceed further with more calculations. So, concluded that multiplicity is 8, i.e.,

Then by using theorem 2.2, we calculate

If

Substituting these

Also we calculate

If

and calculate

If

For

Now if we take

And we calculate

If

As _{8} is even and its value has negative behavior. By using remark 1, it can be concluded that the origin is stable.

and

Utilizing theorem 2.2, we get

Thus multiplicity of

If

By using

If

If

Moreover, _{7} takes form as:

Now, if

Withholding

If _{9} as follows:

This ϰ_{9} is a non-zero constant number. It is concluded that the class

The multiplicity of this class is the highest one up to date. The outcome of maximum multiplicity for

After calculating the periodic solutions, we will make a series of perturbations of the coefficients resulting from the periodic solution to bifurcate out of origin. In Hopf bifurcation, the creation of limit cycles near a fixed point is described. As the bifurcation parameter approaches some critical value, the limit cycle approaches the fixed point and the amplitude of the limit cycle approaches zero. The presented below method is followed by Alwash et al. [

where

and

Then seven real non-trivial periodic solutions exists, if

and

If

If

Finally, the concluding remarks of the article are described below.

In this article, we calculate the possible maximum number of periodic solutions of the quartic differential equation. Earlier in the literature, there are no formulae available to calculate maximum multiplicity greater than 8. Therefore, we calculated the formulae _{4,4}; C_{5,5}; C_{6,6}; C_{7,1}; C_{6,1}; C_{5,1}; where we consider homogenous trigonometric and algebraic coefficients for the sake of variety regarding different polynomials. The most challenging task is to get the multiplicity of any class greater than eight, which assures us that the newly developed formula _{4,1}. In this way, we develop new formulae and validate them through the use of previous literature. As future work, one can develop formula ϰ_{10} for calculating multiplicity greater than nine concerning quartic equation.

Methodology, A. N. and S. A.; writing–original draft preparation, S. A., A. N., M. B. R. and M. R.; formal analysis, S. A., A. N., M. R. and M. B. R.; writing-review and editing, S. A., M. R., M. B. R. and A. N.

^{n}+ p

_{1}(t) z

^{n−1}+ p

_{2}(t) z

^{n−2}+, … , + p

_{0}(z)

_{j = 0}

^{n}a

_{j}(z) z, 0 ≤ z ≤ 1 for which x(0) = x(1)