Periodic Solutions for Different Classes of Abel’s Type Equation

Saima Akram, Allah Nawaz


The maximum number of periodic solutions for the ordinary differential equation is computed in this article. Periodic orbits for first order non-autonomous differential equation of Able's type from a fine focus z=0 are investigated. The research goal of the article is to achieve the maximum possible periodic solutions for two distinct polynomials; algebraic and trigonometric coefficients. The scientific novelty of the article lies in the fact that we have used our newly developed formula ϰ_10. With this formula, we determined the highest known multiplicity 10 for the classes C10,1, C10,2 with algebraic and C8,8, C20,10 with trigonometric coefficients, which is never done before in literature. We used the computer algebra program Maple 18 to handle the complicated and time-consuming integral computations required for calculating the periodic solutions. We used the systematic bifurcation method, which occurred when the coefficients are perturbed. The stability of limit cycles belonging to the classes, as mentioned earlier, is also briefly discussed. For the implementation of theoretical concepts, numerous examples are presented. Consequently, the results presented here are original, authentic, and a valuable contribution to the literature.

Keywords: Abel’s equation, periodic solutions, perturbation method, homogeneous and non-homogeneous trigonometric coefficients.

Full Text:



ALWASH M. Periodic solutions of polynomial non-autonomous differential equations. Electronic Journal of Differential Equations, 2005, 2005(84):1-8.

ALWASH A, LLYOD N. Nonautonomous equation related to the two-dimensional polynomial system. Proceedings of Royal Society Edinburgh, 1987, 105(1): 129-152.

AKRAM S, NAWAZ A, KALSOOM H. Existence of multiple periodic solutions for a cubic non-autonomous differential equation. Mathematical problems in engineering, 2020, 2020: 1-14.

AKRAM S, NAWAZ A, ABDELJAWAD T. Calculation of focal values for first-order non-autonomous equation with algebraic and trigonometric coefficients. Open Physics, 2020, 18(1): 738-750.

AKRAM S, NAWAZ A, YASMIN N. Periodic solutions of some classes of one-dimensional non-autonomous system. Frontiers in Physics, 2020, 8(1): 1-14.

AKRAM S, NAWAZ A, YASMIN N. Periodic solutions for a first-order cubic non-autonomous differential equation with bifurcation analysis. Journal of Taibah university for sciences, 2020, 14 (1): 1208-1217.

HILBERT D. Mathematical problems. Bull. Amer math society, 1902, 8: 437-479.

LLOYD N. The number of periodic solutions of the equation z ̇ =z^n+ p_1 (t) z^(n-1)+ p_2 (t) z^(n-2)+,…,+ p_0 (z). Proceedings of London Mathematical Society, 1973, 27(4): 667-700.

LLOYD N. Limit cycles of certain polynomial systems. Nonlinear functional analysis and its applications, 1986, 173: 317-326.

KUMAR S, KUMAR R, AGARWAL R. A study of fractional Lotka‐Volterra population model using Haar wavelet and Adams‐Bashforth‐Moulton methods. Mathematical Methods in the Applied Sciences, 2020, 43(8): 5564-5578.

KUMAR S, KUMAR R, OSMAN M. A wavelet-based numerical scheme for fractional-order SEIR epidemic of measles by using Genocchi polynomials. Numerical Methods for Partial Differential Equations, 2021, 37(2): 1250-1268.

NETO L. On the number of solutions of the equations ∑_(j=0)^n a_j (z)z 0≤t≤1 for which x(0)=x(1). Invent Math, 1980, 59(6): 67-76.

YASMIN N. Closed orbits of certain two-dimensional cubic systems. U. C. W Aberystwith Uk, 1989, 1-149.

NALLAPPAN G, SRINIVASAN S, ZHAI G. Dynamical analysis and sampled-data stabilization of memristor-based Chua's circuits. IEEE Access, 2021, 9: 25648-25658.

UNYONG B, GOVINDAN V, Bowmiya S. Generalized linear differential equation using Hyers-Ulam stability approach. AIMS Mathematics, 2021 6(1): 1607-1623.


  • There are currently no refbacks.