A note on application and analysis of the non-linear and non-ideal dynamic behavior of the Mathieu equation

Abstract

Cargo transport to the space station and the transport of satellites to orbit the Earth are necessary for not only scientific analysis, but also for the telecommunications sector; therefore, the use of rockets with propellant fuel has been growing. Therefore, our manuscript proposes a mathematical model to analyze nonlinear dynamic behavior. Such a model is based on Mathieu's solution. However, we consider that the application of force for rocket propulsion is not ideal. The formulation of the applied force is based on Bessel's equations and has a dependence on parameters ($a_0$ and $b_0$) that enable interesting characteristics, such as $a_0 = 0$ which makes it possible to turn the non-ideal force into a force ideal, that is, a simple oscillatory force. And as a result, the parametric set was established to diagnose the chaotic and periodic behavior that can be observed in mathematical modeling. Establishing these analyzes allows the application of future work in the development of control projects to reduce vibrations that can cause anomalies in the trajectories for the system's entry into orbit. Another application of nonlinear dynamic analysis is to establish regions for future suppression of the chaotic regime that were observed in our numerical analyses.