Synchronization of Multiple Chaotic Gyroscopes

Abstract

In the course of our examination into the relationships found within gyrodynamics, we present a method to control the motions of multiple ‘slave’ nonlinear mechanical systems synchronized with the motions of a ‘master’ mechanical system, even in the situation where multiple slave systems are unique from one another. Recent development in analytical dynamics has led us to establish a set of control forces allowing the synchronization between highly nonlinear dynamical systems. Some of these qualities include the absence of linearization or nonlinear cancellation and the establishment of synchronizing control forces in a closed form. We apply this methodology in our study to distinct, chaotic gyroscopes. These slave gyroscopes will show synchronization with the motion of a master gyro exhibiting chaotic or regular motion. By providing two sample examples, the first system utilizing three separate gyros and the second utilizing five, we will demonstrate the simplicity and effectiveness of our approach. 1. Introduction Studying gyroscopes has offered us an integral understanding of the sciences as gyrodynamics retains unique properties that allow tremendous functionality in real world applications such as vehicles, aircraft and other complex engineering systems. One of these properties is that if harmonic vertical base excitations are applied to symmetric gyros, the system will display dynamic behavior exhibiting an array of regular to chaotic motion [1−6]. Some of these types of behavior include fixed points, periodic behavior, period doubling, quasi-periodic and chaotic motions. The work of Pecorra and Carrol [7, 8] have spurred tremendous interest in the synchronization of two chaotic systems. The process by which a master gyro controls the synchronized activity of slave gyros in identical fashion presents unique opportunities for study. A common method of attaining synchronized motion between chaotic systems can be described as a type of generalized feedback control. This control signal is applied to the slave system which is often some linear or nonlinear function of the difference in the motion between the master and the slave. As applied by Chen [4], the slave gyro reaches synchronization when the feed-back gain goes over a measured value. Numeric experimentation gives us the feedback value above the synchronization point. Additionally, we can utilize nonlinear control theory to examine gyro synchronization issues. In this model, we envision an independent set of first order nonlinear differential equations with the master to slave differential represented as an error signal. We commence by applying a control to the slave gyro to zero the error signal, oftentimes utilizing feedback linearization. This allows for the use of standard linear feedback control theory as nonlinear terms involving the error signal are gone [10]. However, this method is limited when there are several different slave gyros that need to be synchronized to the motion of the master.