Route to, and control of chaos in multidimensional systems

Abstract

The goal of this thesis is to analyze the dynamic behavior of a given nonlinear system and then control the chaotic behavior of the system within some ranges of the parameters. In the first part of this thesis, we study the effect of a harmonic forcing function and the strength of the nonlinearity on a specific two-degree of freedom system namely, an elastic pendulum, with internal resonance. The forcing function amplitude is used here as the control parameter and the system's dynamics are studied through the variation of this parameter. It was found that the route to chaos always begins, in the pendulum system, with a secondary Hopf Bifurcation followed by consecutive torus doubling bifurcations, ending with torus breaking. A comparison is also made between the use of a linear spring, a weakly nonlinear spring and a strongly nonlinear spring. We observed that the nonlinearity could aid in the stabilization of the periodic orbit beyond the previously seen threshold of instability. Yet, if the strength of the nonlinearity is sufficiently large, it can lead to chaos in frequency regimes where chaos previously was not observed. The second part of this thesis is meant as both a comparison and description of the various methods currently used to control chaos, with application to a rotor supported on nonlinear ball bearings. There are two main types of control schemes. The first is the OGY method and the second is the periodic perturbation method. Both methods depend on the fact that when dealing with chaos small perturbations applied to a parameter of the system can affect the dynamics drastically. The periodic perturbations method is an open loop non-feedback method while the OGY is a closed loop method that depends largely on feedback to achieve the desired effect. We attempted in this thesis to distinguish the conditions for the success and failure of each method. It was found that the OGY method consumes the least energy in its application, and is most suitable when there is only one unstable manifold found for the saddle orbit. It was also found that from an application point of view, the periodic perturbation method is much simpler and can be applied on-line (in real time), no matter how many unstable manifolds the system has.