Analytical and approximate solutions to autonomous, nonlinear, third-order ordinary differential equations

A Volterra integral formulation for determining the periodic solutions of some autonomous, nonlinear, third-order ordinary differential equations

“…Furthermore, by requiring that these solutions be free from secular terms and using Eqs. (16) and (17) with ¼ 1, it is an easy exercise to show that this formulation is identical to the Volterra integral one presented in this paper, by means of the fundamental theorem of calculus [24][25][26].…”

“…(16) and (17) into Eq. (15) yields a power series expansion in and, by setting the coefficients of this series that multiply the monomials n ; n ¼ 0; 1; .…”

“…This behavior has been analyzed in the past by means of harmonic balance methods [6,7], linearized harmonic balance procedures [8], asymptotic perturbation techniques which combine the harmonic balance procedure and the method of multiple time scales [9], the Linstedt-Poincaré procedure [10], the method of averaging of Krylov-Bogoliubov-Mitropolskii [10][11][12], parameter-perturbation Linstedt-Poincaré techniques [13,14] which employ an artificial or book-keeping parameter and expand both the solution and some constants that appear (or are introduced) in the differential equation in terms of this parameter, artificial parameter-Linstedt-Poincaré techniques [15,16] based on the introduction of a linear term proportional to the unknown frequency of oscillation and a new independent variable and the use of either the third-order equation or a system of a first-order and a second-order ordinary differential equations, etc. Parameter-perturbation methods are extensions of the homotopy perturbation technique which introduces an artificial parameter and expands both the solution and the unknown frequency of oscillation in terms of this parameter and has been applied to a variety of problems [17].…”

“…Such an expansion can be avoided entirely by reducing the third-order nonlinear Eqs. (1)-(3) with p = 1 to systems consisting of a first-order linear and a second-order nonlinear ordinary differential equations and expanding both the solution and the frequency of oscillation in power series of the artificial parameter [16].…”

Study of differential control method for solving chaotic solutions of nonlinear dynamic system

“…Although scholars have conducted exhaustive research regarding this subject, how to obtain the analytical solutions of nonlinear differential equations still is a problem that needs to be studied [1][2][3][4][5]. Especially the expressions of the influence of a strongly nonlinear subsystem upon the dynamic behaviors of such dynamic system cannot be directly obtained by numerical methods.…”