Third-order systems: Periodicity conditions

Recently a third-order existence theorem has been proven to establish the sufficient conditions of periodicity for the most general third-order ordinary differential equationx‴+f(t,x,x′,x″)=0x‴+f(t,x,x′,x″)=0In 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.x‴+g1(x′)x″+g2(x)x′+g(x,x′,t)=e(t)x‴+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.