Abstract
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 C4,1 using algebraic coefficients. Whereas the maximum number of periodic solutions for the classes C4,4; C5,1; C5,5; C6,1; C6:6; C7,1 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.
Highlights
Most of the real-world problems in nature are multidimensional and when modeled, they arise as higherorder ordinary differential equations
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
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
Summary
Most of the real-world problems in nature are multidimensional and when modeled, they arise as higherorder ordinary differential equations. We are interested in those models which are periodic and depend on time and are usually known as non-autonomous. We have investigated upper bounds for the non-autonomous ordinary differential equation (ODE) of the cubic degree [1,2,3]. 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
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