Third-order systems: Periodicity condition

Recently a third-order existence theorem has been proven to establish the sufficient conditions of periodicity for the most general third-order ordinary differential equation, x'+f(t,x,x',x')=0. In this paper we prove a new theorem, and establish a new sufficient condition for periodicity of a more restricted and better classified third-order system obeying the following third-order ordinary differential equation: xt'+g1(x')x'+g2(x)x'+g(x,x',t)=e(t). In order to obtain conditions that guarantee the existence of periodic solutions and stable responses, the Schauder's fixed-point theorem has been implemented to prove the third-order periodic theorem for the differential equation. We show the applicability of the new third-order existence theorem by analyzing an independent suspension for conventional vehicles has been modeled as a non-linear vibration absorber with a non-linear third-order ordinary differential equation. Furthermore a numerical method has been developed for rapid convergence, and applied for a sample model. The correctness of sufficient conditions and solution algorithm has been shown with appropriate figures.