The Chaos Exploration And Control Of A Class Of Variable Hydraulic Stiffness Third-order Nonlinear Systems

At present, chaos has become a hot point for research, not only many chaossystems have been discovered and it has also been very common to see theirapplications in actual engineering. Meanwhile, as the research results increases, muchhas been acquired in the control methods related to chaos.The thesis has made chaos exploration about a class of variable hydraulic stiffnessthird-order nonlinear mathematical model and discovered several chaos systems;different control methods have been taken to meet different goals and the related chaossystems have successfully been taken under control.This article includes the following three parts:1. The chaos exploration has been made for the variable hydraulic stiffness modelwith three nonlinear items that are yy″′, yy″and yy′. It is proved by Lyapunov index thatthe third-order nonlinear system is a chaos system. It uses single-variable linearfeedback control method, and by changing the feedback gains to eliminate the chaos ofthe three-order model.2. The chaos exploration has been made for the variable hydraulic stiffness modelabout yy″′, yy″and yy′with two nonlinear items, and in the process another chaos systemwith two nonlinear items have been discovered. Under Lyapunov index distributionmethod, the chaos system has been stabilized to an expectation state; and it isdiscovered by simulation that the size of the distributed Lyapunov index can affect thespeed to the expectation state.3. The chaos exploration has been made for the variable hydraulic stiffness modelabout yy″′, yy″and yy′with only one nonlinear item, and in the process a chaos system isonce again discovered, in which there is only one non-linear item, neither square itemnor cubic; adaptive control process has been adopted to successfully suppress the chaosof the unknown system. By use of Lyapunov direct method, the error between thenewly discovered chaos system and Genesio-Tesi chaos system has been stabilized andthe control law to get the two systems synchronous has also been elicited, and in thisprocess, it has been seen that the size of error gains can affect the speed ofsynchronization of the two chaos systems.