Schauder fixed point theorem based existence of periodic solution for the response of Duffing’s oscillator

Abstract

An initial-boundary value problem that is Duffing's oscillator with time varying coefficients will be studied. Using Banach's fixed-point theorem, the existence of periodic solution of the equation will be predicted. The method applied in this paper is the Schauder second fixed point theorem, which includes the response of structures under vibratory force systems. As an example, the dynamics of nonlinear simply supported rectangular thin plate under influence of a relatively moving mass is studied. By expansion of the solution as a series of mode functions, the governing equations of motion are reduced to an ordinary differential equation for time development vibration amplitude, which is Duffing's oscillator. Finally, a parametric study is developed, after that some numerical examples are solved, and the validity of the present analysis is clearly shown.

Nontrivial Periodic Solutions of an -Dimensional Differential System and Its Application

“…From the several numerical results, it is noticed that the existent range of periodic solutions for the thin plate system becomes larger with the increase of the ratio of the modulus of nonlinear term's coefficient and parametric excitation term. By means of analytical arguments and numerical simulation runs, it is easy to find that the proposals given in this study are seldom obtained in the known literature, for example, [2,5,6].…”

“…Therefore, the nonlinear dynamic behavior of the thin plate received very considerable attention within many articles available in the technical and scientific literature. See [1][2][3][4][5][6], for example, and the references therein.…”

“…In the aforementioned works, based on the Schauder second fixed point theorem, Dizaji et al [2] predicted the existence of periodic solutions of the following governing equations of motion:…”

Robust Stable Limit Cycle Generation in Multi-Input Mechanical Systems

“…Repetitive patterns are demanded behaviors in many practical systems. [1][2][3][4] For example, in a walking robot, the foot motion follows a repetitive pattern. This behavior can be described by an attractive limit cycle in the phase trajectories of the dynamical system [5][6][7][8] The limit cycle is an isolated periodic orbit in the phase plane, which is a very rich dynamical behavior in the nonlinear dynamical systems.…”