Local bifurcation analysis and control of nonlinear dynamical systems with time-periodic coefficients

The subject of this study is the analysis and control of nonlinear dynamic systems with periodic coefficients. Such systems arise in the modeling of structures with periodic loads, helicopter rotor blade vibration in forward flight, rotor-bearing systems and fluid flows in micro-gravity environment, to name a few.; In this dissertation, an analytical technique is suggested for the construction of versal deformations of normal forms for periodic nonlinear systems. This is accomplished via a series of transformations. The first step is the application of the Lyapunov-Floquet transformation which converts the linear part of the periodic equation into a dynamically equivalent time-invariant matrix preserving all the stability and bifurcation characteristics. This time-invariant linear part allows the use of nonlinear simplification techniques such as the time-periodic center manifold reduction and the time-dependent normal form theory. The normal form of the equation is obtained at a given bifurcation point. In order to describe the dynamics in the neighborhood of this point versal deformation of the normal form is constructed. This is achieved by finding an approximate relationship between the parameters of the original system and the eigenvalues of the normal form. The versal deformation equations are suitable to study the dynamics quantitatively the neighborhood of a bifurcation point.; These normal forms also make it possible to control bifurcations of time-periodic systems. Purely nonlinear state feedback controllers are designed for each of the codimension one bifurcations to stabilize post-bifurcation limit sets such as limit cycles and quasi-periodic attractors and to modify the size and rate of growth of these limit sets. The idea is applicable to chaotic systems as well. Since chaos occurs through specific routes of bifurcations, controlling bifurcations can delay or even eliminate chaos.; The method of analysis and the controller design are illustrated through the examples of a commutative system, a parametrically excited simple pendulum and a double inverted pendulum. Chaos control of the simple pendulum is also included. The method of analysis is compared to small parameter methods such as averaging. The control technique is also compared with other existing control ideas. The results are verified by numerical integration.