Research On Anti-control Of Chaos Of First-order Partial Difference Equations

In the 1970s,chaos rised,many scientists dedicated to researching on chaos.They defined different chaos from different perspectives,studied the mechanism of generating of chaos and its application,and established some criteria of chaos in different spaces.How to make an initially nonchaotic dynamical system become chaotic,or enhance a chaotic system to present stronger or different type of chaos?Starting from this problem,people try to establish some anti-control of chaos(chaotification)schemes,and prove that the controlled systems are chaotic by applying the criteria of chaos.This thesis is concerned with anti-control of chaos of first-order partial difference equations.By applying the coupled-expansion theory of general discrete dynamical systems,we established some different anti-control of chaos schemes for first-order partial difference equations.All the controlled systems are proved to be chaotic in the sense of Li-Yorke or of both Li-Yorke and Devaney.Firstly,the following first-order partial difference equations with polynomial maps:is studied,where f:D(?)R2→R is a map,n≥0 is the time step,m is the lattice point,ε is a constant,and The system is proved to be chaotic in the sense of Li-Yorke or of both Li-Yorke and Devaney by applying the coupled-expansion theory of general discrete dynamical systems.Then,an example is given and computer simulation results show that the controlled system has complex dynamic behaviors.Secondly,four anti-control of chaos schemes for first-order partial difference equations with tangent and cotangent functions:are studied.Under periodic and no-periodic boundary conditions,the systems are proved to be chaotic in the sense of Li-Yorke or of both Li-Yorke and Devaney.In previous studies,sine functions,sawtooth functions and mod operations were selected as controllers.Now,we choose tangent and cotangent functions as controllers,which widens the selection range of controllers.Finally,two criteria of chaos for first-order partial difference equations with general controllers:are established.All the control systems are proved to be chaotic in the sense of Li-Yorke or of both Li-Yorke and Devaney by applying the coupled-expansion theory of general discrete dynamical systems.When the systems are chaotic,the equations and controllers need to satisfy very weak conditions,which makes anti-control of chaos of first-order partial difference equations more easy.Then,examples and computer simulation results are given to illustrate the correctness of the above theorems.