Synchronization of chaotic systems: a generalized Hamiltonian systems approach

Two techniques for the control of nonlinear systems with periodic coefficients are presented. The control methods are based on ether delaying the bifurcations that lead to chaos by using a linear control or changing nonlinear features of limit cycles by employing a nonlinear controller. In the linear control strategy the system is driven to a desired periodic orbit by means of a state-feedback controller. The state transition matrix of the closed loop system is computed symbolically, containing the unknown control gains. The gains are obtained by placing the eigenvalues of the state transition matrix inside the unit circle of the complex plane. The nonlinear control strategy employs a local bifurcation analysis of time-periodic systems. Properties such as stability or rate of growth of a bifurcated limit set can be adjusted by a nonlinear state-feedback control. First the Lyapunov-Floquet transformation is applied such that the linear part of the equation becomes time-invariant. Then through an application of time-periodic center manifold reduction and time-dependent normal form theory a completely time-invariant form is obtained for codimension one bifurcations. This normal form is found suitable for the application of control strategies developed for autonomous systems. By controlling the appropriate bifurcation on the route to chaos, the chaotic behavior can be delayed. The control strategies are illustrated through an example of a parametrically excited simple pendulum undergoing a symmetry breaking bifurcation. Comparison of the two methods is also made