Calculation of focal values for first-order non-autonomous equation with algebraic and trigonometric coefficients

Saima Akram; Allah Nawaz; Thabet Abdeljawad; Abdul Ghaffar; Kottakkaran Sooppy Nisar

doi:10.1515/phys-2020-0105

Open Access Published by De Gruyter Open Access November 11, 2020

Calculation of focal values for first-order non-autonomous equation with algebraic and trigonometric coefficients

Saima Akram , Allah Nawaz , Thabet Abdeljawad , Abdul Ghaffar and Kottakkaran Sooppy Nisar

From the journal Open Physics

https://doi.org/10.1515/phys-2020-0105

Abstract

This article concerns with the development of the number of focal values. We analyzed periodic solutions for first-order cubic non-autonomous ordinary differential equations. Bifurcation analysis for periodic solutions from a fine focus z = 0 is also examined. In particular, we are interested to detect the maximum number of periodic solutions for various classes of higher order in which a given solution can bifurcate under perturbation of the coefficients. We calculate the maximum number of periodic solutions for different classes, namely, C 10 , 5 and C 12 , 6 with trigonometric coefficients, and they are found with nine and eight multiplicities at most. The classes C 8 , 3 and C 8 , 4 with algebraic coefficients have at most eight limit cycles. The new formula ϰ 10 is developed by which we succeeded to find highest known multiplicity ten for class C 9,3 with polynomial coefficient. Periodicity is calculated for both trigonometric and algebraic coefficients. Few examples are also considered to explain the applicability and stability of the methods presented.

Keywords: focal values; periodic coefficients; bifurcation; non-autonomous equation

1 Introduction

A periodic solution is a solution that periodically depends on the independent variable t. For a periodic solution x ( s ) , there is a number T ≠ 0 such that

x ( s + T ) = x ( s ) for S ∈ ℝ .

All possible such T are called periods of this periodic solution and are usually considered for a system of ordinary differential equations (ODEs). In the entire twentieth century, the investigation of periodic solutions of non-autonomous ODEs has been profoundly influential in the creation and development of fundamental parts of present mathematics, such as functional analysis (operator theory, iterative methods, etc.), algebraic topology (fixed point theorems, topological degree, Conley index, etc.), variational methods (dual action principle, minimax theorems, Morse theory, etc.), and symplectic techniques (Poincare–Birkhoff-type fixed point theorems, etc.). In real world, in general, most of the known phenomena have the tendency to repeat after some time. With the help of linear or non-linear differential equations (DEs), we are able to accurately define real or complex aspect that arises in our environment and impact the course of nature we evolve in. The concerned area of attention incorporates inter-disciplinary programs including natural sciences, mathematics and statistics, environmental protection and technology, electricity and energy, traveling waves, manufacturing and construction, and engineering and technology. It is found that the determination of limit cycles plays an important role in the smooth running of generators and many magneto-electric machines, which was observed by a French engineer Gerard Lescuyer in ref. [12]. In the nineteenth century, an electric arc was used for lighthouses and road lights, which has a significant downside; the noise delivered by electrical discharge was disturbing the inhabitants. W. D. B. Duddell (1872–1917) stopped the disturbing noise by embedding a high-frequency oscillating circuit by the utilization of limit cycle; henceforth, limit cycle was important for radioengineering.

The limit cycle is a common problem of both physics and mathematics. For the physical aspect, it commonly arises for nonlinear and linearized pendulum, pendulum clock. For example, consider a Van der Pol oscillator with its relation to physics is explained as follows: x ̈ − μ x ̇ + μ x 2 x ̇ + x = 0 .

The first and last term ( x ̈ + x = 0 ) constitute a harmonic oscillator in which energy is conserved and therefore amplitude of motion x is constant.
The second term ( − μ x ̇ ), which is velocity dependent, represents negative damping which adds energy to the system in each cycle continuously and therefore it tries to increase the amplitude unboundedly.
The third term which is also velocity dependent but it represents positive damping which removes energy from the system in each cycle and it tries to decrease the amplitude.

The damping coefficient ( μ x 2 ) is not constant but depends on position or displacement x and is called position-dependent damping. Its effect increases with absolute displacement.
When an oscillator starts with small displacement, initially amplitude increases due to negative damping and attains a limiting value due to control of negative damping by the increased positive damping (position dependent). Now positive damping becomes predominant and further reduces amplitude of motion to certain minimum value and again negative damping becomes predominant and continues to increase amplitude and so on.

Numerous phenomena in nature can be formed by systems of ODEs, which depend periodically on time. For instance, a linear or a non-linear oscillator can be forced by a periodic external force, and a significant question is to know whether the oscillator can exhibit a periodic response under this forcing. This question begins from problems in classical and celestial mechanics, before accepting significant applications in radio electricity and electronics. Nowadays, it also plays a great role in mathematical biology and population dynamics, just as in mathematical economics, where the considered systems are often subject to seasonal variations. Most of the previous studies on special DEs have been based on models from mechanics and electronics fields that are far from being totally understood but the recent applications to biology, demography, and economy will introduce new classes of DEs and systems with periodic time dependence, requiring the use of new analytical and topological tools. In this regard, the study of periodic solutions of non-autonomous DE is of increasing significance, see refs. [4,9,10,13,14,15,16,17,18,25,26,27].

The question for the maximum number of periodic solutions for (1) attracts more interest. Many results related to research on the development of such phenomena have been pointed out so far and other investigations are going on. Neto in ref. [19] stated that for equation (1), we cannot get highest upper bound for a number of periodic solutions until certain coefficients are restricted. Also, in this paper our basic focus is to acquire highest periodic solutions of any class of type (1). We consider first-order DE of the form as:

(1) z ̇ = γ ( s ) z 3 + δ ( s ) z 2 + υ ( s ) z ,

where independent variable s and coefficients γ , δ , υ are real functions, and z ∈ ℂ . Equation (1) is the part of the equation written by Lloyd [22] as follows:

(2) z ̇ = p 0 ( s ) z n + p 1 ( s ) z n − 1 + p 2 ( s ) z n − 2 + ⋯ + p n ( s ) ,

with the condition that p 0 ( s ) = 1 and p 1 ( s ) , … , p n ( s ) are real valued periodic continuous functions. This class of equation has received more attention in the literature. It was shown in refs. [6,23] that when p 0 ( s ) = 1 , there are three periodic solutions provided taking multiplicity into account. For n = 3 , equation (2) is known as Abel’s DE, that is important due to its connection with Hilbert’s sixteenth problem for DEs; see ref. [21] for details. When n ≥ 4 and p 0 ( s ) = 1 , the results of ref. [23] no longer hold; the examples given in Neto [19] can be used to show that there is no upper bound to the periodic solutions. Moreover, systems with constant angular velocities can be reduced to polynomial equations. For general cubic system, it is now known that as many as ten limit cycles can be bifurcated from single equilibrium point, see refs. [5,6,7]. General consequence about the number of periodic solutions can be used to obtain the number of limit cycles; see refs. [2,3,8].

Complexified form of equation (1) is used to find the maximum number of periodic solutions, see refs. [1,22,24] for details. Due to this reason, periodic solutions cannot be destroyed by small perturbation of the coefficients. Also consider that for equation (1) there exists β ∈ ℝ such that:

z ( β ) = z ( 0 ) .

These solutions are periodic, even if γ , δ , and υ are not periodic. If for equation (1) μ > 1 , as was described in ref. [1], we can substitute υ ( s ) = ˜ 0 and equation (1) could reduce to the form as:

(3) z ̇ = γ ( s ) z 3 + δ ( s ) z 2 ,

where γ and δ may be polynomials (i) in s and (ii) in cos ( s ) and sin ( s ) , and for more details see refs. [1,5,22,23]. The functions ξ i ( s ) , for i > 1 , are calculated by using the relation:

(4) ξ i = γ ∑ j + k + l = i j , k , l ≥ 1 ξ j ξ k ξ l + δ ∑ j + k = i j , k ≥ 1 ξ j ξ k ,

with ξ 1 ( s ) = 1 . On the other hand, as “i” increases, certain calculations become extremely complicated due to integration by parts utilized in it. Consider that ϰ i = ξ i ( β ) , at that point ϰ = i if ϰ 1 = 1 and ϰ k = 0 for 2 ≤ k ≤ i − 2 , but ϰ i ≠ 0 these ϰ i s are known as focal values. For i ≤ 8 functions ξ i ( s ) and ϰ i are present in ref. [1], and the maximum multiplicity 8 is calculated with the help of these formulas. To calculate the focal value more than 8, Yasmin in ref. [29] had calculated ξ 9 ( s ) and ϰ 9 , and now for i = 10 , we have calculated ξ 10 ( s ) and ϰ 10 .

In Section 2, essential formulae with which we calculate the maximum multiplicity are discussed. In Section 3, perturbation techniques and isolation conditions for center are given. Main results are presented in Section 4. Some examples are discussed in Section 5, and conclusions are discussed in Section 6.

2 Development of the formulas ξ 10 and ϰ 10

Alwash in ref. [1] had calculated functions ξ i ( s ) and ϰ i , for 2 ≤ i ≤ 8 . For i = 9 , 10 , these functions are written in the following theorems. These functions are very important because with the help of them we get the highest possible multiplicity. To develop algorithm for various classes, we use programming language Maple 18, which is used to calculate periodic solutions by using the following theorems.

Theorem 2.1

For equation (4) conclusive functions ξ 2 , ξ 3 , … , ξ 8 are given in ref. [1]. In ref. [29], ξ 9 is calculated and we also succeeded to calculate the conclusive function ξ 10 which is as follows:

ξ 9 = δ ¯ 8 + 7 δ ¯ 6 γ ¯ + δ ¯ 6 γ ¯ + 6 δ ¯ 5 δ ¯ γ ¯ + 2 δ ¯ 5 γ ¯ δ ¯ + 5 δ ¯ 4 δ ¯ 2 γ ¯ + 3 δ ¯ 4 γ ¯ γ ¯ + 3 δ ¯ 3 γ ¯ δ ¯ 2 + 5 δ ¯ 4 γ γ ¯ ¯ + 39 2 δ ¯ 4 γ ¯ 2 − 2 δ ¯ 3 δ γ ¯ 2 ¯ + 24 δ ¯ 3 δ ¯ γ ¯ γ ¯ + 6 δ ¯ 3 γ γ ¯ ¯ δ ¯ − 10 δ ¯ 3 γ δ ¯ γ ¯ ¯ + 12 δ ¯ γ ¯ δ ¯ 3 γ ¯ + 4 γ ¯ δ δ ¯ 3 γ ¯ ¯ + 4 δ ¯ 3 γ ¯ δ ¯ 3 + 43 6 δ ¯ 2 γ ¯ 3 + 4 γ ¯ 3 δ δ ¯ ¯ + 4 δ ¯ 2 γ ¯ γ δ ¯ 2 ¯ − 10 δ δ ¯ γ ¯ γ δ ¯ 2 ¯ ¯ + 15 2 γ ¯ 2 δ ¯ 2 γ ¯ + 2 δ ¯ 2 δ ¯ 2 γ ¯ − 2 δ ¯ 4 γ ¯ + 8 γ ¯ δ δ ¯ 3 ¯ + 2 δ ¯ δ ¯ 2 γ δ ¯ γ ¯ ¯ + 26 δ ¯ γ δ ¯ 2 γ ¯ δ ¯ + 6 δ ¯ 2 γ ¯ γ ¯ − 6 δ ¯ 2 γ γ ¯ ¯ + 12 δ ¯ 2 δ ¯ γ γ ¯ ¯ + 16 δ ¯ 2 γ ¯ δ δ ¯ γ ¯ − 16 δ ¯ 3 γ δ ¯ γ ¯ ¯ + 9 δ ¯ 2 ( δ ¯ γ ) 2 ¯ + 9 ( δ ¯ γ ) 2 ¯ γ ¯ − δ γ ¯ 3 ¯ δ ¯ + 35 8 γ ¯ 4 − 6 δ ¯ γ ¯ δ γ ¯ 2 ¯ + 6 γ ¯ δ γ ¯ 2 δ ¯ + 33 δ ¯ γ ¯ 2 δ ¯ γ ¯ − 24 δ γ ¯ 2 δ ¯ γ ¯ ¯ + 6 δ ¯ 2 γ ¯ γ ¯ γ ¯ − 4 δ γ ¯ δ γ ¯ 2 ¯ ¯ .

and

ξ 10 = δ ¯ 9 − 23 2 δ ¯ 7 γ ¯ − 1235 6 δ ¯ 5 γ ¯ γ ¯ + 3 δ ¯ 5 γ ¯ γ ¯ + 111 γ ¯ δ ¯ 4 δ ¯ γ ¯ − 444 γ ¯ δ δ ¯ 3 δ ¯ γ ¯ ¯ + 20 γ ¯ δ ¯ δ ¯ 4 γ ¯ − 12 γ ¯ δ δ ¯ 4 γ ¯ ¯ + 214 3 γ ¯ δ ¯ 3 δ ¯ 2 γ ¯ − 160 γ ¯ δ δ ¯ 2 δ ¯ 2 γ ¯ ¯ + 15 2 γ ¯ 2 δ ¯ 3 γ ¯ − 970 3 γ γ ¯ 2 δ ¯ 3 ¯ + 30 γ ¯ δ ¯ 2 δ ¯ 3 γ ¯ − 68 γ ¯ δ δ ¯ δ ¯ 3 γ ¯ ¯ + 9 γ ¯ δ ¯ 3 γ γ ¯ ¯ + 1015 9 γ ¯ 3 δ ¯ 3 − 237 δ δ ¯ 2 γ ¯ 3 ¯ − 11 2 γ ¯ δ ¯ 2 δ γ ¯ 2 ¯ + 26 γ ¯ δ δ ¯ δ γ ¯ 2 ¯ ¯ + 319 2 γ ¯ 2 δ ¯ 2 δ ¯ γ ¯ − 174 δ δ ¯ γ ¯ 2 δ ¯ γ ¯ ¯ − 90 γ ¯ γ δ ¯ δ ¯ 2 γ ¯ ¯ + 24 γ ¯ δ ¯ 2 γ δ γ ¯ ¯ + 40 γ ¯ δ ¯ γ γ ¯ δ ¯ 2 ¯ − 24 γ ¯ δ γ γ ¯ δ ¯ 2 ¯ + 3 γ ¯ δ ¯ 2 γ δ ¯ γ ¯ ¯ − 154 γ ¯ γ δ ¯ 2 δ ¯ γ ¯ ¯ − 24 γ γ ¯ 2 δ ¯ 2 δ ¯ + 70 γ ¯ δ ¯ ( δ ¯ γ ¯ ) 2 + 42 γ ¯ δ ( δ ¯ γ ¯ ) 2 ¯ − 70 γ ¯ δ ¯ 3 γ 2 ¯ − 3 2 γ ¯ δ γ ¯ 3 ¯ − 21 δ γ ¯ 4 ¯ − 15 4 γ ¯ 2 δ γ ¯ 2 ¯ + 169 4 γ ¯ 4 δ ¯ + 24 γ γ ¯ 2 δ ¯ δ ¯ 2 γ ¯ − 24 γ ¯ 2 δ δ ¯ 2 γ ¯ ¯ + 10 γ ¯ 3 δ ¯ γ ¯ + 9 2 δ ¯ 4 δ ¯ 3 γ ¯ + 8 γ ¯ δ ¯ 7 − 74 γ γ ¯ 3 δ ¯ ¯ + 8 δ ¯ δ δ ¯ γ ¯ 3 ¯ + 34 3 γ ¯ δ ¯ 3 δ ¯ 2 γ ¯ + 2 δ ¯ δ ¯ 6 γ ¯ + 7 δ ¯ 6 δ ¯ γ ¯ + 3 γ δ ¯ 7 − 5 γ δ ¯ 6 ¯ + 6 δ ¯ 5 δ ¯ 2 γ ¯ − 6 γ δ ¯ δ ¯ 4 γ ¯ ¯ + 2 δ ¯ 3 δ ¯ 3 γ ¯ + 10 δ ¯ γ γ ¯ δ ¯ 4 ¯ + 26 γ ¯ 2 δ ¯ 5 − 5 2 δ ¯ 4 δ γ ¯ 2 ¯ + 5 2 δ δ ¯ 4 γ ¯ 2 ¯ + 73 2 γ ¯ δ ¯ 4 δ ¯ γ ¯ − 127 2 δ ¯ 4 γ δ ¯ γ ¯ ¯ + 9 δ ¯ 2 γ γ ¯ δ ¯ 3 ¯ − 20 δ ¯ δ ¯ 3 γ δ ¯ γ ¯ ¯ + 19 γ γ ¯ 2 δ ¯ 3 γ ¯ − 21 δ ¯ 2 γ δ ¯ 3 γ ¯ ¯ + 8 δ ¯ γ ¯ δ δ ¯ 3 γ ¯ ¯ − 160 3 γ δ ¯ 3 δ ¯ 2 γ ¯ ¯ + 32 δ ¯ 4 γ δ γ ¯ ¯ ¯ − 20 δ ¯ γ ¯ δ ¯ δ δ ¯ 2 γ ¯ ¯ + 24 δ ¯ γ ¯ 2 δ ¯ 2 γ ¯ + 4 3 δ ¯ 3 δ ¯ 2 γ ¯ − 31 30 γ δ ¯ 5 ¯ − 4 5 γ ¯ δ ¯ 5 + 16 δ ¯ δ ¯ 3 γ ¯ δ ¯ − 16 γ ¯ δ δ ¯ 4 ¯ + 3 δ ¯ 2 δ ¯ 2 γ δ ¯ γ ¯ ¯ + 42 δ ¯ 2 δ ¯ 2 γ ¯ δ ¯ γ ¯ + 12 γ ¯ δ ¯ δ ¯ 2 γ ¯ − 12 γ δ ¯ δ ¯ 2 γ ¯ ¯ − 12 δ ¯ δ ¯ 2 γ ¯ γ ¯ + 12 δ ¯ 3 δ ¯ 2 γ ¯ γ ¯ + 32 δ 2 γ ¯ δ ¯ δ ¯ 2 γ ¯ ¯ − 32 δ ¯ δ ¯ 3 γ δ γ ¯ ¯ ¯ + 14 δ ¯ 3 ( δ ¯ γ ¯ ) 2 − 15 δ ¯ 5 γ 2 ¯ − 28 γ δ ¯ ( δ ¯ γ ¯ ) 2 − 3 2 δ ¯ 2 δ γ ¯ 3 ¯ + 13 2 δ ¯ 2 γ δ γ 2 ¯ ¯ + 12 δ ¯ δ γ ¯ 2 ¯ δ γ ¯ ¯ − 36 δ δ ¯ γ ¯ 2 δ γ ¯ ¯ ¯ − 48 δ ¯ γ ¯ 2 δ δ ¯ γ ¯ ¯ − 16 γ δ ¯ γ γ ¯ δ ¯ 2 ¯ ¯ − 8 δ ¯ δ γ ¯ δ γ ¯ 2 ¯ ¯ + 2 δ ¯ 3 δ ¯ 4 γ ¯ + 8 δ ¯ 2 δ ¯ 2 γ δ γ ¯ ¯ − 8 δ δ ¯ 4 γ γ ¯ ¯ − 2 γ δ ¯ δ ¯ 4 γ ¯ ¯ + 1 2 δ ¯ 4 δ ¯ γ ¯ + 2 δ ¯ δ ¯ γ ¯ δ ¯ 3 γ ¯ + δ δ ¯ γ ¯ δ γ ¯ 2 ¯ ¯ + δ ¯ ( δ ¯ 2 γ ¯ ) 2 .

Proof

As this theorem contains the symbolic representations, the proof is extremely difficult, time consuming, and is very long to write. So, it is omitted.□

Now, by taking integration by parts of the functions ξ i ( s ) for 2 ≤ i ≤ 10 , of Theorem 2.1, we get Theorem 2.2. It is helpful in finding maximum multiplicity. Here integrals are like ∫ γ ( s ) δ ( s ) ¯ d s , and bar “−” shows that the integral δ ( s ) ¯ = ∫ 0 s δ ( s ) d s , is definite.

Theorem 2.2

The solution z = 0 of (3) has a multiplicity k , wherever 2 ≤ k ≤ 10 iff ϰ n = 0 for 2 ≤ n ≤ k − 1 and ϰ n ≠ 0 where:

ϰ 2 = ∫ 0 β δ

ϰ 3 = ∫ 0 β γ

ϰ 4 = ∫ 0 β γ δ ¯

ϰ 5 = ∫ 0 β γ δ ¯ 2

ϰ 6 = ∫ 0 β γ δ ¯ 3 − 1 2 γ ¯ 2 δ

ϰ 7 = ∫ 0 β γ δ ¯ 4 + 2 γ δ ¯ 2 γ ¯

ϰ 8 = ∫ 0 β γ δ ¯ 5 + 3 γ δ ¯ 3 γ ¯ + γ δ ¯ 2 δ ¯ γ ¯ − 1 2 γ ¯ 3 δ

ϰ 9 = ∫ 0 β γ δ ¯ 6 − 5 γ δ ¯ 4 γ ¯ − 2 δ ¯ 3 δ γ ¯ ¯ + 20 δ γ ¯ 2 ¯ + 2 δ γ ¯ ¯ δ γ ¯ 2

and

ϰ 10 = ∫ 0 β γ δ ¯ 7 − 1235 6 γ γ ¯ δ ¯ 5 − 970 3 γ γ ¯ 2 δ ¯ 3 − 237 δ δ ¯ 2 γ ¯ 3 − 24 γ γ ¯ 2 δ δ ¯ 2 − 70 γ ¯ δ ¯ 3 γ 2 − 21 γ ¯ 4 δ − 74 γ γ ¯ 3 δ ¯ + 5 2 γ ¯ 2 δ δ ¯ 4 + 32 δ ¯ 4 γ δ γ ¯ ¯ − 16 δ δ ¯ 4 γ ¯ − 15 δ ¯ 5 γ 2 − 36 δ δ ¯ γ ¯ 2 δ γ ¯ ¯ − 8 δ δ ¯ 4 γ γ ¯ .

Note: here Theorem 2.2 is the only basic theorem with which we calculate the periodic solutions. As our aim is to find the maximum number of periodic solutions, for this some useful conditions for the origin to be a center are described in Section 3.

3 Perturbation techniques and isolation conditions for center

This section describes briefly about conditions for center. Maximum focal values for different polynomial classes of non-autonomous equation of type (3) can be determined with the help of Theorem 2.2. In the investigation of the number of periodic solutions of equation (3), the local question is of bifurcation, that is, to understand the way in which solutions are created and destroyed. The study of local bifurcation of limit cycles is related to the form of the Poincare return map defined about the origin of equation (3). This in turn is determined by the vanishing of certain number of polynomials in the coefficients of γ and δ, which we previously called focal values. For calculating these focal values, we developed a program using Maple 18. We named maximum multiplicity as ϰ k and for stopping criteria we consider some conditions which tell us the need to proceed further ϰ k or not. These conditions which are sufficient to determine z = 0 as a center are described in the form of theorems and corollaries as follows.

Corollary 3.1

If any δ or γ is identically 0 and other has mean value zero, then the origin is a center.

Corollary 3.2

Consider for continuously differentiable functions γ and δ of period ω and there exist θ ∈ [0, ω] such that γ(θ − s) = −γ(s), δ(θ − s) = −δ(s). The origin is a center.

Theorem 3.1

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

f ( ζ ( s ) ) ζ ̇ = γ ( s ) ,

g ( ζ ( s ) ) ζ ̇ = δ ( s ) ,

then the origin is a center for (3).

Corollary 3.1

For odd continuously differentiable functions γ and δ of period ω , the origin is a center.

First we determine the maximum multiplicity μ max , then a series of perturbations of the coefficients is to be made, by doing so we get one periodic solution to bifurcate out from origin, at each step. For more details see refs. [1,22].

Consider equation (3), having multiplicity μ = j (suppose). Let U be in the region nearby 0 in the complex plane containing no periodic solution except z = 0 . Using Theorem 2.4 from ref. [1], the initial point which remains unchanged regarding the highest number of periodic solutions, our goal is to get ϰ 2 = ϰ 3 = … = ϰ j − 2 = 0 but ϰ j − 1 ≠ 0 by perturbing coefficients of γ and δ , if possible. As a result, the most effective solution in U and ψ is zero solution while we get periodic solution ψ ( s ) where ψ(0) ∈ U as non-trivial solution. By considering the underlying fact that the complex solutions always occur in conjugate pairs, we can say that ψ is real. Let now U ₁ and V ₁ be neighborhood of zero and ψ , respectively, such that V 1 ∪ U 1 ⊂ U and V 1 ∩ U 1 = ϕ . The periodic solutions around V 1 and U 1 are preserved when we take small perturbation in presented coefficients. By applying the same procedure, our choice is to perturb the coefficients such that ϰ k = 0 for k = 2 , 3 , … , j − 3 but ϰ j ≠ 0 . So we get μ = j − 2 . In this way, we get two non-trivial real periodic solutions with zero solution having multiplicity j − 2 . Continuously, in the same way we end up for equation (3) having μ = 2 , and j − 2 distinct non-trivial real periodic solutions.

4 Main results

In this section, we have presented a variety of classes with algebraic and trigonometric coefficients with polynomial of different degree. During the long history of research in Hilbert’s 16th problem, people usually focused on lower degree systems, which have less critical points, less coefficients and are easier in controlling the behaviors and computing the results. However, for the uniform bound research, studying higher degree polynomial systems was inevitable. But we have tried various polynomials by considering higher order polynomials. We find the maximum number of periodic solutions for these polynomials. We end up with the fact that upper bound for periodic solutions cannot be judged or depends upon the degree of polynomial and it is always unpredictable. This concept is in-line with the Hilbert’s 16th problem that there is no upper bound for the maximum number of periodic solutions.

4.1 Polynomial coefficients for classes C 9,3 , C 8,4 , and C 8,3

To compute the higher order periodic solutions, we use programming language Maple 18. Moreover, method for calculating periodic solution involves integration by parts, which is very time consuming as the multiplicity increases. For polynomial “s,” suppose C p , q shows the classes of equation of the form (3) with degree p and q for γ and δ correspondingly see refs. [1,28], the aforementioned classes are presented below in the form of theorems. We refer the readers to see refs. [5,6,7,20], for more classes having greatest multiplicity as ten.

Theorem 4.1

Let C _9,3 be class of equations of the form (3), with

γ ( s ) = a + b s + d s 3 + e s 4 + h s 7 + j s 9 .

δ ( s ) = m + p s 3 .

Then we conclude μ max ( C 9 , 3 ) ≥ 10 .

Proof

Using Theorem 2.2, we take:

ϰ 2 = m + 1 4 p ,

ϰ 3 = a + 1 2 b + 1 4 d + 1 5 e + 1 8 h + 1 10 j .

Thus, multiplicity of z = 0 is μ = 2 if ϰ 2 ≠ 0 . And multiplicity μ = 3 if ϰ 2 = 0 but ϰ 3 ≠ 0 . If ϰ 2 = ϰ 3 = 0 , then γ ( s ) and δ ( s ) can be written as follows:

(5) γ ( s ) = b s − 1 2 + d s 3 − 1 4 + e s 4 − 1 5 + h s 7 − 1 8 + j s 9 − 1 10 ,

(6) δ ( s ) = m s 3 − 1 4 .

And we compute ϰ 4 as follows:

ϰ 4 = − p ( − 2916 j − 2695 h − 1232 e + 4620 b ) 1108800 .

If ϰ 4 = 0 , then either p = 0 or:

(7) j = − 2695 2916 h − 1232 2916 e + 4620 2916 b .

If m = 0 , then δ ( s ) = 0 and ϰ 3 = 0 give that the mean value of γ ( s ) is zero. So, origin is a center from Corollary 3.1. Thus, m ≠ 0 , consider that (7) holds then:

ϰ 5 = − p 2 ( 2401 h + 2264 e + 1716 b ) 382112640 .

If ϰ 5 = 0 , then as we already consider p ≠ 0 implies:

(8) h = − 2264 2401 e − 1716 2401 b .

And by using (8) we take ϰ 6 as:

ϰ 6 = − p ( e + 5 b ) ( 350372 b + 164997 p 2 − 17924 e ) 20866904298240 .

If ϰ 6 = 0 , then, as we already consider p ≠ 0 either e = − 5 b or

(9) b = − 164997 350372 p 2 + 17924 350372 e .

If e = − 5 b , then f = 3 b , j = 0 from (8) and (7). By using these values, equations (5) and (6) become:

γ ( s ) = c s 2 − 1 3 + b 3 s 3 − 3 s s 2 − 1 3 ,

δ ( s ) = m s 2 − 1 3 .

If ζ ( s ) = s 3 − s , then we can easily see that ζ ̇ ( s ) = 3 s 2 − 1 and ζ ( 0 ) = ζ ( 1 ) . So, the aforementioned equations are written as:

γ ( s ) = 1 3 [ c + b ( 3 s 3 − 3 s ) ] ζ ̇ ,

δ ( s ) = m 3 ζ ̇ .

This shows that origin is the center, from Theorem 3.1, having:

f ( ζ ( s ) ) = 1 3 [ c + b ( 3 s 3 − 3 s ) ] ,

and g ( ζ ( s ) ) = m 3 . So, we take d ≠ − 4 b . If (9) holds, then we compute ϰ 7 as:

ϰ 7 = − 873 p 2 ( − 15 p 2 + 8 e ) ( 2004714660761864 e − 2504896436724845 p 2 + 3257612032613400 d ) 1718428405731478011194624000 .

If ϰ 7 = 0 recalling that p ≠ 0 , then either e = 15 8 p 2 or:

(10) e = 2504896436724845 2004714660761864 p 2 − 3257612032613400 2004714660761864 d .

If e = 15 8 p 2 , then,

γ ( s ) = 1 3 [ c + m 2 ( − 2 s 3 + 2 s ) ] ζ ̇ ,

δ ( s ) = m 3 ζ ̇ .

Using Theorem 3.1, origin is the center with f ( ζ ( s ) ) = 1 3 [ c + m 2 ( − 2 s 3 + 2 s ) ] and g ( ζ ( s ) ) = m 3 . So consider e ≠ 15 8 p 2 . By using (10), we calculate ϰ 8 as:

ϰ 8 = 97 p ( 2168532 d + 834727 p 2 ) ϖ 2154476782025955696318164067500193721621069511080614440960 ,

where

ϖ = 136588632749185622319477352487034900 d 2 − 1498419783235042611352047121429860375 d p 2 − 1454638394978310393540107904927553045372 p 4 .

Now if ϰ 8 = 0 , then either ϖ = 0 or

(11) d = − 834727 2168532 p 2 .

Because p ≠ 0 . If (11) holds but ϖ ≠ 0, p ≠ 0 , then we compute ϰ 9 as:

ϰ 9 = − p 5 ( 468140924873533 p + 46325757047490400 ) 360399749176193126400 .

If ϰ 9 = 0 , then, as p ≠ 0 considered above, we take value of p as:

(12) p = − 46325757047490400 468140924873533 .

If equation (11) ≠ 0, l ≠ 0, but ϖ = 0 holds, then c = v i m 2 3 for i = 1 , 2 . With v 1 = 45.009923140 , v 2 = − 17.20305420 . If (12) holds, then we calculate ϰ 10 as:

ϰ 10 = − 246170968089040020843132555199309318758721656531805611422973490949770117415 433037004469929324894110408796222133520824777771678887690189430000000000 672721066620747881588679008241701869018568763020767458314634780816136018016 89019208490477278758468072653806776549667214338780680840263904483442061 ,

which is the non-zero constant number. Thus, we finish up with the multiplicity of class C 9,3 as 10, i.e. μ max ( C 9 , 3 ) ≥ 10 .

As the maximum multiplicity is 10, even and also negative, so from remark 5.1, the origin is stable.□

Theorem 4.2

For the following equation:

(13) z ̇ = γ ( s ) z 3 + δ ( s ) z 2

with

(14) γ ( s ) = 834727 8674128 − 46325757490400 468140924873533 + ε 1 2 + 9242545245767 1002357380932 ε 2 − 4792770 30044399 ε 3 − 1525 2401 ε 4 − 95 2916 ε 5 − 1 10 ε 6 + ε 7 + − 3 8 − 463257547490400 468140924873533 + ε 1 2 − 20831230118475 2505932595233 ε 2 + 4481 87593 ε 3 + ε 4 s + − 46325757490400 4681409273533 + ε 1 2 + ε 2 s 3 + − 46325757047490400 468140924873533 + ε 1 2 − 40720076675 250589395233 ε 2 + ε 3 s 4 + − 3 2 + ε 1 2 + 3988548175300 2505893395233 ε 2 − 2675324 2731309 ε 3 − 1716 2401 ε 4 + ε 5 s 7 + − 229589151075900 250589332595233 ε 2 + 31428 55741 ε 3 + 110 ε ε 4 − 2695 2916 ε 5 + ε 6 s 9 ,

(15) δ ( s ) = 11581439261872600 468140924873533 − 1 4 ε 1 + ε 8 + − 46325757047490400 468140924873533 + ε 1 s 3 .

Choose ε r for 1 ≤ r ≤ 8 to be non-zero and small as compared to ε r − 1 . Then, (13) has eight distinct non-trivial real periodic solutions.

Proof

If we substitute ε r = 0 , ∀ r = 1 , 2 , … , 8 and coefficients are as given in equations (14) and (15). So, multiplicity of the origin ϰ is 10 . Choose ε 1 ≠ 0 and ε r = 0 for 2 ≤ r ≤ 8 , then it can be easily seen that ϰ 9 is a constant multiple of ε 1 but ϰ 2 = ϰ 3 = … = ϰ 7 = ϰ 8 = 0 . So, the multiplicity reduces by one and ϰ = 9 . For that reason, one periodic solution bifurcates out of the origin. Now set ε 1 ≠ 0, ε 2 ≠ 0 , and ε r = 0 for 3 ≤ r ≤ 8 , then we have ϰ r = 0 for r = 2 , 3 , … , 7 but ϰ 8 results in the form of ε 2 with some constant multiple. So, ϰ = 8 . Now set ε 1 ≠ 0, ε 2 ≠ 0, ε 3 ≠ 0 , and ε r = 0 for 4 ≤ r ≤ 8 , then we have ϰ r = 0 for r = 2 , 3 , … , 6 but ϰ 7 results in the form of ε 3 with some constant multiple. If ε 2 is sufficiently small, then there are two non-trivial real periodic solutions. Further moving on the present way, we own eight real periodic non-trivial solutions.□

Corollary 4.1

For the following equation,

(16) z ̇ = γ ( s ) z 3 + δ ( s ) z 2 + τ + υ ,

if τ , υ are small enough, and γ ( s ) , δ ( s ) are as used in Theorem 4.2, equation (16) has ten real periodic solitons.

Proof

Given that τ = 0 , υ = 0 , and μ = 2 , then (16) has eight real periodic solutions. If τ ≠ 0 but small enough, then μ = 1 , also using the same arguments as above, we have nine periodic solutions. That are distinct and non-trivial, z = 0 is another solution, thus we take ten real periodic solutions.□

In the following theorem, two different classes are discussed and we are using the variable 2s instead of s.

Theorem 4.3

Suppose classes C 8, j for equation (3), for j = 3 , 4 . We cease with the result μ max ( C 8 , j ) ≥ 8 , for all j = 3 , 4 .

Proof

First, we are considering the class C 8,4 . The class C 8,3 , where degree of γ ( s ) is 3, is the special case. Let we have:

(17) γ ( s ) = a + b ( 2 s ) + d ( 2 s ) 3 + e ( 2 s ) 4 + i ( 2 s ) 8 , δ ( s ) = j + m ( 2 s ) 3 + n ( 2 s ) 4 .

Suppose that m = 0 in (17). By utilizing Theorem 2.2, we calculate:
ϰ 2 = j + 16 5 n ,

ϰ 3 = a + b + 2 d + 16 5 e + 256 9 i .

Thus, multiplicity of z = 0 is μ = 2 if ϰ 2 ≠ 0 . And multiplicity μ = 3 if ϰ 2 = 0 but ϰ 3 ≠ 0 . If ϰ 2 = ϰ 3 = 0 , then
(18) γ ( s ) = b ( 2 s − 1 ) + d ( ( 2 s ) 3 − 2 ) + e ( 2 s ) 4 − 16 5 + i ( 2 s ) 8 − 256 9 ,

(19) δ ( s ) = n ( 2 s ) 4 − 16 5 .

And ϰ 4 is calculated as:
ϰ 4 = − n 16 ( − 2048 i + 42 d + 45 b ) 4725 .

If ϰ 4 = 0 , then, either n = 0 or:
(20) i = 42 2048 d + 45 2048 b .

If n = 0 , then ϰ 2 = 0 gives j = 0 , hence, δ ( s ) = 0 . If ϰ 2 = 0 , it means that the mean value of γ ( s ) is zero. By Corollary 3.1, origin is a center. So, n ≠ 0 . Substituting (20), we get:
ϰ 5 = − 128 n 2 ( − 64 d + 135 b ) 1645875 .

If ϰ 5 = 0 , then:
(21) d = 135 64 b .

Because we have already supposed n ≠ 0 . If (21) holds, then:
ϰ 6 = b n ( 17612809728 n 2 + 912321575 b ) 10708457760000 .

If ϰ 6 = 0 , then either b = 0 or:
(22) b = − 17612809728 912321575 n 2 ,
because n ≠ 0 . If b = 0 , then equations (18) and (19) reduce to the following form as:
γ ( s ) = e ( 2 s ) 4 − 16 5 ,

δ ( s ) = n ( 2 s ) 4 − 16 5 .

Let ζ ( s ) = 16 5 s 5 − 16 5 s then ζ ̇ ( s ) = 16 s 4 − 16 5 . Also, ζ ( 0 ) = ζ ( 1 ) . So, we can write it as:
γ ( s ) = e ζ ̇ ,

δ ( s ) = n ζ ̇ .

Applying Theorem 3.1, origin is the center having f ( ζ ( s ) ) = e and g ( ζ ( s ) ) = n . So, we take b ≠ 0 . If (22) holds, then ϰ 7 is as follows:
ϰ 7 = − 5870936576 n 4 ( − 8428089836600759 n 2 + 222152861854264 e ) 30806201439395903630484375 .

Now if ϰ 7 = 0 , recalling that n ≠ 0 , then:
(23) e = 8428089836600759 222152861854264 n 2 .

With holding (23) we have:
ϰ 8 = 28867635868851347067252018453282286 489029822575050718567583980424296875 n 7 .

That is constant multiple of n 7 and n is also non-zero, n ≠ 0 taken above. Thus, μ max ( C 8 , 4 ) ≥ 8 .

As the maximum multiplicity is 8 (even) and is also positive, so from remark 5.1, the origin is unstable.
For the class C 8,3 , we put n = 0 in system (17). So,

γ ( s ) = a + b ( 2 s ) + d ( 2 s ) 3 + e ( 2 s ) 4 + i ( 2 s ) 8 ,

δ ( s ) = j + m ( 2 s ) 3 .

Utilizing Theorem 2.2, we calculate:

ϰ 2 = j + 2 m ,

ϰ 3 = a + b + 2 d + 16 5 e + 256 9 i .

Thus, multiplicity of z = 0 is μ = 2 if ϰ 2 ≠ 0 . And multiplicity μ = 3 if ϰ 2 = 0 but ϰ 3 ≠ 0 . If ϰ 2 = ϰ 3 = 0 , then:

(24) a = − b − 2 d − 16 5 e − 256 9 i ,

(25) j = − 2 m .

By using equations (24) and (25), we have:

(26) γ ( s ) = b ( 2 s − 1 ) + d ( ( 2 s ) 3 − 2 ) + e ( 2 s ) 4 − 16 5 + i ( 2 s ) 8 − 256 9 ,

(27) δ ( s ) = m ( ( 2 s ) 3 − 2 ) .

Also, we calculate ϰ 4 as:

ϰ 4 = − m ( − 15360 i − 416 e + 195 b ) 2925 .

If ϰ 4 = 0 , either m = 0 or:

(28) i = − 416 15360 e + 195 15360 b .

For m = 0 , (25) gives j = 0 , so, δ ( s ) = 0 . For ϰ 2 = 0 , it means that γ ( s ) has mean value zero. By Corollary 3.1, origin is the center. So, suppose m ≠ 0 . If (28) holds, then ϰ 5 is:

ϰ 5 = − 2 m 2 ( 47360 e + 13143 b ) 20675655 .

If ϰ 5 = 0 , then either m = 0 or:

(29) e = − 13143 47360 b ,

as already supposed that m ≠ 0 . Thus, by using (29), we have:

ϰ 6 = − b m ( 6632594100 m 2 + 495544543 b ) 265577512200000 .

Now if we take ϰ 6 = 0 , then as m ≠ 0 either b = 0 or:

(30) b = − 6632594100 495544543 m 2 .

If b = 0 , then equations (26) and (27) take the form as follows:

γ ( s ) = d ( ( 2 s ) 3 − 2 )

and

δ ( s ) = m ( ( 2 s ) 3 − 2 ) .

If ζ ( s ) = 2 ( s ) 4 − 2 s , we can see that ζ ̇ ( s ) = ( 2 s ) 3 − 2 , ζ ( 0 ) = ζ ( 1 ) . So, the above equations reduce as:

γ ( s ) = d ζ ̇ ,

δ ( s ) = m ζ ̇ .

By Theorem 3.1, having f ( ζ ) = d and g ( ζ ) = m , the origin is a center. So, suppose b ≠ 0 . By using (30), we have ϰ 7 as:

ϰ 7 = 953 m 4 ( − 20383512043888891 m 2 + 14791762980133920 d ) 31628693959703755751200 .

If ϰ 7 = 0 , recalling that m ≠ 0 (considered above), then:

(31) d = 20383512043888891 14791762980133920 m 2 .

With holding (31) we find:

ϰ 8 = − 4859156546838859822405999250792431561 25723503136148881839056844556697054208000 m 7 .

As m ≠ 0 considered above, we cannot go ahead. So, μ max ( C 8 , 3 ) ≥ 8 .

As the maximum multiplicity is even and negative, so from remark 5.1, the origin is stable.□

Now, we are using the perturbation method for the classes of Theorem 4.3.

Theorem 4.4

The class C 8 , 3 for equation (3),with:

γ ( s ) = 10521471646179553459 1634489809304798160 m 2 − 20331 29600 ε 2 − 328 328 ε 3 − 2 ε 1 − 256 9 ε 4 + ε 5 + 2 − 6632594100 495544543 m 2 + ε 2 s + 8 20383512043888891 14791762980133920 m 2 + ε 1 s 3 + 16 9061557615 2439603904 m 2 − 13143 47360 ε 2 + ε 3 s 4 + 256 − 1242266949 4592195584 m 2 + 153153 7577600 ε 2 − 13 480 ε 3 + ε 4 s 8 .

and

δ ( s ) = − 2 m + ε 6 + 8 m s 3 .

Choose ε r for 1 ≤ r ≤ 6 to be non-zero and small as compared to ε r − 1 . Then (3) has six distinct non-trivial real periodic solutions.

Proof

If we substitute ε r for 1 ≤ r ≤ 6, then proof is similar to that in Theorem 4.2. So it is omitted.□

Theorem 4.5

For equation (3), suppose that

γ ( s ) = σ + b s + d s 3 + e s 4 + i s 8 ,

δ ( s ) = − 16 5 n + ε 6 + 16 n s 4

with

σ = 200081843882254612 13190326172596925 n 2 − 1811 256 ε 2 − 16 5 ε 1 − 31 12 ε 3 − 256 9 ε 4 + ε 5 ,

b = − 17612809728 912321575 n 2 + ε 2 ,

d = − 7430404104 182464315 n 2 + 135 64 ε 2 + ε 3 ,

e = 8428089836600759 222152861854264 n 2 + ε 1 ,

i = − 309600171 245846656 n 2 + 4275 65536 ε 2 + 21 1024 ε 3 + ε 4 ,

and

j = − 16 5 n + ε 6 .

If ε l ( 1 ≤ l ≤ 6 ) are taken to be non-zero and also

| ε 6 | ≪ | ε 5 | ≪ ⋯ ≪ | ε 1 | ,

then (3) has six non-trivial real periodic solutions for the class C 8,4 .

Proof

The proof is similar to that in Theorem 4.2. So it is omitted.□

In Section 4.2, we have considered the non-homogeneous trigonometric coefficients for different classes and then periodic solutions are calculated.

4.2 Trigonometric coefficients for classes C 10,5 and C 12,6

In this section, we consider equation (3), having polynomials γ ( s ) and δ ( s ) in sin s and cos s; for this we suppose that ω = 2 π . This brings us closer to Hilbert’s 16th problem that is original motivation. Equations of the form (1) are of Hilbert’s type which comes from (3) with υ = 0 , γ , and δ of degree 2 ( n + 1 ) and ( n + 1 ) , respectively, are non-homogeneous polynomials. Under these conditions, there are restrictions on the possible values of multiplicity “ ϰ ” that did not apply to the equation considered in Section 4.1; this makes problem very complicated. Consider that γ and δ have only terms of even and odd degree accordingly; the same situation is in Hilbert’s type with n is even in (1), for degree of γ and δ . Suppose γ ( π + s ) = γ ( s ) , δ ( π + s ) = − δ ( s ) ; therefore, − ϕ ( s + π ) is a periodic solution if ϕ ( s ) is such a solution. As complex solutions always occur in conjugate pairs due to which the total number of non-trivial periodic solutions is even; see complex analysis for more details. Furthermore, the multiplicity of the origin is odd; for ϰ > 1, we can show ( ϰ − 1 ) is even.

In accordance with Section 4.1, we now go off to various classes of equation (3). By using programming language Maple18, the calculations in this module are verified. The expressions related to Theorem 2.2 are of such size that direct use of Maple18, integration operator either fails or becomes very slow due to insufficient pile space. This difficulty was overcome by individually computing a list of definite integrals of the form ∫ 0 2 π s j cos s sin n s d s using Maple18, first storing this in a file and then entering it into an array during a Maple execution. Then integrals of polynomials in s , sin s , and cos s are calculated. In the following theorems, focal values are calculated for non-homogenous trigonometric coefficients.

Theorem 4.6

Let C _10,5 be the class for equation (3), if the coefficients are as follows:

δ ( s ) = ( a cos 3 s + b sin 3 s ) ( cos 2 s + sin 2 s ) ,

γ ( s ) = ( e cos s sin 3 s + c cos 3 s sin s ) ( cos 2 s + sin 2 s ) 3 ,

then we calculate ϰ max ( C 10 , 5 ) ≥ 9 .

Proof

From Theorem 2.2, we now calculate ϰ 2 = ϰ 3 = ϰ 4 = 0 and by proceeding further we calculate ϰ 5 as:

ϰ 5 = − 11 a b π ( c + e ) 64 .

If ϰ 5 = 0 , then either a = b = 0 or:

(32) c + e = 0 .

If a = b = 0 , then δ ( s ) = 0 , and for ϰ 2 = 0 the mean value of γ ( s ) is zero. As a result, origin is a center from Corollary 3.1. From (32), we substitute c = − e and calculate ϰ 6 = 0 and ϰ 7 as:

ϰ 7 = − 75 e a b ( a + b ) ( a − b ) 2048 .

If ϰ 7 = 0 as a b ≠ 0, also for e = a = b = 0 , γ ( s ) is zero, and for ϰ 3 = 0 δ ( s ) has mean value zero. As a result, origin is a center from Corollary 3.1. So, we take e a b ( a 2 − b 2 ) ≠ 0 . Furthermore, by using (32) we calculate ϰ 8 = 0 and moving in the same manner we calculate ϰ 9 as:

ϰ 9 = − 151267 e b 4 π 1290240 .

If ϰ 9 = 0 , then the only possibility is e = b = 0 , but they are non-zero, as shown above. Hence, ϰ max ( C 10,5 ) ≥ 9 .□

Theorem 4.7

Let C _12,6 be the class for equation (3), if the coefficients are as follows:

γ ( s ) = ( c cos s sin s 3 + e cos s 3 sin s ) ( cos 2 s + sin 2 s ) 4 ,

δ ( s ) = ( a cos 6 s + b sin 4 s cos 2 s ) ,

then ϰ max ( C 12,6 ) ≥ 8 .

Proof

With the use of Theorem 2.2, it can easily be calculated that:

ϰ 2 = π ( b + 5 a ) 8 .

If ϰ 2 = 0 , then we substitute value of “b” and calculate ϰ 3 = 0 , further we calculate the value of ϰ 4 as:

ϰ 4 = a π ( 2 c + 3 e ) 32 .

For ϰ 2 = ϰ 4 = 0 , by substituting values of “b” and “e” we get ϰ 5 = 0 and ϰ 6 as:

ϰ 6 = − a c π ( − 16 c + 61 a 2 ) 49152 .

If ϰ 6 = 0 , then for c = 61 16 a 2 and we get ϰ 7 = 0 . The value for ϰ 8 is given as follows:

ϰ 8 = 70656605 a 7 π 43486543872 .

Now suppose that ϰ i = 0 for i = 2 , 4 , 6 , 8 . There are two possibilities as follows:

If a = 0 , then b = 0 . For both values as zero, δ ( s ) = 0 . From Corollary 3.1, origin is a center.
If a c = 0 , possibility for a = 0 is discussed, then c = 0 results e = 0 gives γ ( s ) = 0 . From Corollary 3.1, origin is a center. Hence, we conclude that ϰ max ( C 12 , 6 ) ≥ 8 .□

5 Examples

For perturbation methods an essential remark is listed below. With the help of this, we can make decision about the periodic solutions that they are stable or unstable.

Remark 5.1

In the perturbation techniques defined in Section 3, we conclude ( j − 2 ) (real periodic solutions) at the end. But the full complement of ( j − 2 ) real periodic solutions fails to yield, if it happens to such an extent that there is j < k thus ϰ j = 0 whenever ϰ j − 1 = 0 . This happens when the multiplicity is definitely odd. However, for the number of real periodic solutions, we can say from “exchange of stability” reasoning that, If multiplicity μ is even, the origin is stable ϰ μ < 0 and unstable if ϰ μ > 0 . If μ is odd, then the origin is stable on the right and unstable on the left if ϰ μ < 0 , while it is stable on the left and unstable on the right if ϰ μ > 0 .

Now we will give some examples that show the feasibility of our results, for more examples see refs. [5,6,7,11]. By using the above mentioned remark, the stability behavior is also discussed here.

Example 5.1

Consider the following equation:

(33) d x d t = a × γ ( t ) x 3 + b × δ ( t ) x 2 .

Here γ ( t ) = e ι π t = cos ( π t ) + ι sin ( π t ) , δ ( t ) = e ι 2 π t = cos ( 2 π t ) + ι sin ( 2 π t ) .

Solution: By using Theorem 2.2, the calculations of periodic solutions are as follows:

ϰ 2 = 0.6035009250 b ( ι + 0.8105543097 ) π

ϰ 3 = 0.3703182747 a ( ι + 2.097798766 ) π .

Thus, multiplicity of x = 0 is μ = 2 if ϰ 2 ≠ 0 . And multiplicity μ = 3 if ϰ 2 = 0 but ϰ 3 ≠ 0 . If ϰ 2 = ϰ 3 = 0 , then as 0.6035009250 ( ι + 0.8105543097 ) ≠ 0 , 0.3703182747 ( ι + 2.097798766 ) ≠ 0 , so by using the only possibility that a = b = 0 , we calculate ϰ 4 = 0 . Now for ϰ 4 = 0 , we are unable to proceed further. Hence, for equation (33), we end up with the four periodic solutions. As the multiplicity is 3 ( odd ) and is positive, it is stable on the left and unstable on the right.

Example 5.2

For the following equation:

(34) d x d t = γ ( t ) x 3 + δ ( t ) x 2 .

Here γ ( t ) = ( c 0 + c 1 cos t + c 2 sin t ) and δ ( t ) = ( b 0 + b 1 cos t + b 2 sin t ) .

Solution: By using Theorem 2.2, the calculations of periodic solutions are as follows:

ϰ 2 = 2 ( b 0 π ) ,

ϰ 3 = 2 ( c 0 π ) .

Thus, multiplicity of x = 0 is μ = 2 if ϰ 2 ≠ 0 . And multiplicity μ = 3 if ϰ 2 = 0 but ϰ 3 ≠ 0 . If ϰ 2 = ϰ 3 = 0, then as 2 π ≠ 0, so by using the only possibility c 0 = b 0 = 0 , we calculate ϰ 4 as:

ϰ 4 = − ( c 1 b 2 − c 2 b 1 ) π

Now for ϰ 4 = 0 , we are unable to proceed further. Hence for equation (34), we end up with the four periodic solutions. The multiplicity is 4 ( even ) and negative by the above equation, so the origin is stable.

6 Conclusion

In this paper, we have obtained some existence results for first-order cubic non-autonomous DEs by using the method of perturbation. The solutions satisfying z ( β ) = z ( 0 ) are called periodic orbits of equation (3). The periodic orbits that are isolated in the set of all periodic orbits are usually called the limit cycle. Focal values are found for two types of coefficients called algebraic and trigonometric coefficients for various classes of higher order. We examined classes C 10,5 and C 12,6 with trigonometric coefficients while C 8,3 and C 8,4 with polynomial coefficients. The maximum periodic solution for these classes is 9. By evaluating equation (3), we succeeded to get new formulas ξ 10 and ϰ 10 . Now, using our newly developed formulas we obtained the highest known multiplicity 10 for class C 9,3 with polynomial coefficient. To check the implementation of the methods we have taken few examples, calculated their limit cycles, and discussed about the stability and found that our method is applicable. As a future work, one can calculate the maximum multiplicity greater than ten by first generalizing Theorem 2. This can be calculated by substituting the value of i > 10 in equation (4).

n.sooppy@psau.edu.sa

Acknowledgements

Thabet Abdeljawad would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

Availability of data and material: None.
Conflict of interest: None.
Funding: None.
Author contributions: All authors contributed towards writing draft, software and reviewed final version of the manuscript.

References

[1] Alwash MAM, Llyod NG. Non-autonomous equation related to polynomial two-dimensional system. Proc R Soc Edinb. 1987;105:129–52.10.1017/S0308210500021971Search in Google Scholar

[2] Alwash MAM. Periodic solutions of polynomial non-autonomous differential equations. Electron J Differential Equ. 2005;84:1–8.Search in Google Scholar

[3] Alwash MAM. Polynomial differential equations with small coefficients. Discret continuos dynamical Syst. 2009;25:1129–41.10.3934/dcds.2009.25.1129Search in Google Scholar

[4] Alshabanat A, Jleli M, Kumar S, Samet B. Generalization of Caputo-Fabrizio fractional derivative and applications to electrical circuits. Front Phys. 2020;1–10.10.3389/fphy.2020.00064Search in Google Scholar

[5] Akram S, Nawaz A, Kalsoom H, Idrees M, Chu YM. Existence of multiple periodic solutions for cubic nonautonomous differential equation. Math Probl Eng. 2020;2020:14.10.1155/2020/7618097Search in Google Scholar

[6] Akram S, Nawaz A, Yasmin N, Ghaffar A, Baleanu D, Nisar KS. Periodic solutions of some classes of one dimensional non-autonomous system. Front Phys. 2020;8. 10.3389/fphy.2020.00264.Search in Google Scholar

[7] Akram S, Nawaz A, Yasmin N, Kalsoom H, Chu YM. Periodic solutions for first order cubic non-autonomous differential equation with bifurcation analysis. J Taibah Univ Sci. 2020;14(1):1208–17.10.1080/16583655.2020.1810429Search in Google Scholar

[8] Bukaty YN, Bibikov VR. Bifurcation of an oscillatory mode under a periodic perturbation of a special oscillator. Differ Equ. 2019;55(6):753–9.10.1134/S001226611906003XSearch in Google Scholar

[9] Goufo EFD, Kumar S, Mugisha SB. Similarities in a fifth-order evolution equation with and with no singular kernel. Chaos Solitons Fractals. 2020;130:109467.10.1016/j.chaos.2019.109467Search in Google Scholar

[10] Ghanbari B, Kumar S, Kumar R. A study of behavior for immune and tumor cells in immunogenetic tumor model with non-singular fractional derivative. Chaos Solitons Fractals. April 2020;133:109619.Search in Google Scholar

[11] Hua N. The fixed point theory and the existence of the periodic solution on a nonlinear differential equation. J Appl Math. 2018;11:6725989.10.1155/2018/6725989Search in Google Scholar

[12] Gerard-Lescuyer JMA. Philos Mag. 1880;10:215.10.1080/14786448008626922Search in Google Scholar

[13] Jleli M, Kumar S, Kumar R, Samet B. An analysis for heat equations arises in diffusion process using new Yang-Abdel-Aty Cattani fractional operator. Math Methods Appl Sci. 2020;43:6062–80.10.1002/mma.6347Search in Google Scholar

[14] Kumar S, Ahmadian A, Kumar R, Kumar D, Singh J. An efficient numerical method for fractional SIR epidemic model of infectious disease by using Bernstein wavelets. MDPI. 2020;8(4):558.10.3390/math8040558Search in Google Scholar

[15] Kumar S, Kumar A, Baleanu D. Two analytic methods for time fractional nonlinear coupled Boussinesq-Burger’s equation arise in propagation of shallow water waves. Nonlinear Dyn. 2017;85(2):699–715.10.1007/s11071-016-2716-2Search in Google Scholar

[16] Kumar S, Kumar A, Abbas S, Al Qurash M. A modified analytical approach with existence and uniqueness for fractional Cauchy reaction-diffusion equations. Adv Differ Equ. 2020;2020:1–18.10.1186/s13662-019-2488-3Search in Google Scholar

[17] Kumar S, Kumar R, Cattani C, Samet B. Chaotic behavior of fractional predator-prey dynamical system. 2020;135:109811.10.1016/j.chaos.2020.109811Search in Google Scholar

[18] Kumar S, Kumar R, Agarwal RP, Samet B. A study of fractional Lotka-Volterra Population model using Haar wavelet and Adam’s-Bashforth-Moulton methods. Math method Appl Sci. 2020;43:5564–5578.10.1002/mma.6297Search in Google Scholar

[19] Neto L. On the number of solutions of the equations dxds=∑j=0naj(s)s′, 0≤s≤1 for which x(0)=x(1). Invent Math. 1980;59:67–76.10.1007/BF01390315Search in Google Scholar

[20] Nawaz A. Bifurcation of periodic solutions of certain classes of cubic non-autonomous differential equations [MPhil Thesis]. 2018. p. 1–104.Search in Google Scholar

[21] Hilbert D. Mathematical problems. Bull Amer math Soc. 1902;8:437–79.10.1201/9781351074315-35Search in Google Scholar

[22] Llyod NG. The number of periodic solutions of the equation ż=p0(s)zn+p1(s)zn−1+P2(s)zn−2+⋯+Pn(s). Proc Lond Math Soc. 1973;27(3):667–700.Search in Google Scholar

[23] Llyod NG. Limit cycles of certain polynomial systems, non-linear Functional analysis and its applications. SP Singh NATO ASI Ser C. 1986;173:317–26.Search in Google Scholar

[24] Lloyd NG. Small amplitude limit cycles of polynomial differential equations, ordinary differential equations and operators. In: Everitt WN, Lewis RT, editors. Lecture Notes in Mathematics. Springer-Verlag; 1982. p. 346–57.10.1007/BFb0076806Search in Google Scholar

[25] Sharma B, Kumar S, Cattani C, Baleanu D. Nonlinear dynamics of Cattaneo-Christov heat flux for third grade power law fluid. J Computational Nonlinear Dyn. 2020;15(1):011009.10.1115/1.4045406Search in Google Scholar

[26] Veeresha P, Prakasha DG, Kumar S. A fractional model for propagation of classical optical solitons by using non-singular derivative. Math methods Appl Sci. 2020. 10.1002/mma.6335.Search in Google Scholar

[27] Wilezynski P. Planar autonomous polynomial equations, the Riccati equation. J Differential Equ. 2008;244(6):1304–28.10.1016/j.jde.2007.12.008Search in Google Scholar

[28] Yasmin N. Bifurcating periodic solutions of polynomial system. Punjab Univ J Mathematics. 2001;xxxiv:43–64.Search in Google Scholar

[29] Yasmin N. Closed orbits of certain two dimensional cubic systems, Ph.D Thesis. UK: U.C.W. Aberystwith; 1989. p. 1–190.Search in Google Scholar

Received: 2020-03-29

Revised: 2020-08-27

Accepted: 2020-09-23

Published Online: 2020-11-11

This work is licensed under the Creative Commons Attribution 4.0 International License.

Calculation of focal values for first-order non-autonomous equation with algebraic and trigonometric coefficients

Abstract

1 Introduction

2 Development of the formulas ξ 10 and ϰ 10

Theorem 2.1

Proof

Theorem 2.2

3 Perturbation techniques and isolation conditions for center

Corollary 3.1

Corollary 3.2

Theorem 3.1

Corollary 3.1

4 Main results

4.1 Polynomial coefficients for classes C 9,3 , C 8,4 , and C 8,3

Theorem 4.1

Proof

Theorem 4.2

Proof

Corollary 4.1

Proof

Theorem 4.3

Proof

Theorem 4.4

Proof

Theorem 4.5

Proof

4.2 Trigonometric coefficients for classes C 10,5 and C 12,6

Theorem 4.6

Proof

Theorem 4.7

Proof

5 Examples

Remark 5.1

Example 5.1

Example 5.2

6 Conclusion

Acknowledgements

References

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