The main contribution of this thesis can be formulated in terms of a synchronization problem describing interactions between a system of harmonic oscillators (HO) and a given dynamical system, called HO synchronization. This thesis investigates the applications of HO synchronization in the area of controlling chaos. It proposes an innovative feedback stabilization technique, called HO feedback control, that utilizes an output of the synchronized system to stabilize periodic orbits of dynamical systems. A particularly attractive application of this stabilization technique is found in controlling chaotic systems, where the aim is to stabilize unstable periodic orbits embedded in the chaotic attractors. This thesis utilizes the same concept of synchronization to develop a novel method of detecting unstable periodic motions in chaotic time series, called HO time series analysis. To do so, the proposed method does not require any information about the underlying dynamics beyond a recorded time history. Therefore, it is appealing in experimental situations. The information obtained from the HO time series analysis can be used in various methods of controlling chaos, including the HO feedback control. In a different, but related application, this thesis proposes a novel transformation of a time delay system to a system of ordinary differential equations featuring the same concept of synchronization, called HO transformation. This transformation yields an efficient finite dimensional approximation to the original time delay system. It is practically important as it allows implementation of classical theories and conventional tools, developed for finite dimensional systems, to analyze time delay systems. This thesis utilizes the HO transformation to reveal the relation between the delayed feedback control and the HO feedback control.