Analysis and stabilization of nonlinear chaotic systems

The qualitative analysis problem for chaotic dynamics of nonlinear control systems and stabilization of unstable periodic orbits (UPO stabilization) and stationary states were considered in the paper. Nonlinear control systems are analysed by the canonical transformation methods of affine systems and by the localization of periodic orbits method based on properties of Lie derivatives along smooth vector fields. It is proved that Lorenz and Rössler systems are not equivalent in the affine classification. For chaotic systems, a general approach to UPO stabilization and stabilization of stationary states based on canonical transformations was proposed. Estimates for chaotic regions in the phase space for Lorenz and Rössler systems were obtained. Relations between canonical transformations, localization, and UPO stabilization problems were studied.