The Duffing Oscillator And Linearization Techniques For Its Motion Constants
Analyzing the characteristics of higher order nonlinear dynamic systems is really difficult. This can involve giving solutions with respect to time. Motion constants are another way of studying the behavior of the dynamic system. If the motion constant is known, solving the system is no longer needed to analyze the characteristics of the system. Motion constants are time independent integrals that are hard to find for nonlinear dynamic systems. We chose the Duffing Oscillator as a higher order nonlinear dynamic system to have its motion constants investigated. The Duffing Oscillator was chosen because studying it gives a better view of how rigid bodies act. It forms a clear dynamic analog of the general torque-free motion of an arbitrary rigid body, meaning it covers most of the arbitrary rigid body dynamics. Investigating the motion constants for a finite dimensional nonlinear system, such as the Duffing Oscillator (can be quite difficult) but finding the motion constants for a linear autonomous system, regardless of its dimension, is easier and has recently been found. In this study we propose finding the motion constants of the Duffing Oscillator through the motion constant of a linear representation. A linear representation is found through Carleman Linearization. This is a technique used to linearize a finite dimensional nonlinear system of differential equations to an infinite dimensional, linear, autonomous system of differential equations. Using Carleman Linearization, the Duffing equation is linearized; the motion constant was found, and compared to the true known value of the real system.
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