Control and Chaos

Abstract

Control and chaos in the same phrase would seem contradictory, but the reader knows by now that chaos is both predictable and structured in phase space. That phase space structure contains within it many simpler topological features which are unstable [DN79] and through which the system may have passed as we altered the driving on the system by varying parameters. It may not come as a surprise then that careful adjustment of these same parameters using the known stable and unstable directions of the vector field might allow one to stabilize what had become unstable. In this chapter we examine a very interesting set of methods for control which slightly change the vector field as given, by adding dynamical degrees of freedom to the system. These make formerly constant parameters time dependent. After doing this phase space structures “nearby” the unstable structures of the original system can now become stable structures of the augmented system. In particular unstable periodic orbits of the original system may be slightly altered but acquire stability as the vector field develops a carefully selected set of time dependencies. Another kind of control scheme we will examine traces its lineage to standard control theory [Owe81, Bro85] even though that may often cast in a linear context. In this approach an external force acting on the dynamical variables is added to the vector field while the parameters of the original dynamics are unaltered.