Self-oscillating reactions have been widely observed and studied since the last century because they exhibit unique behaviors different from the traditional chemical reactions. Self-oscillating systems, such as the Belousov-Zhabotinsky (BZ) reaction, oxidation reaction of CO on single crystal Pt, and calcium waves in the heart tissue, are of great interest in a variety of scientific areas. This thesis contributes to the understanding of wave transition in BZ reaction, and to possible applications of non-equilibrium behaviors of polymer systems. In BZ reaction, two types of wave patterns, target and spiral, are frequently observed. The transition from one to another is not fully understood. Hence, a systematic investigation has been performed here to investigate the mechanism by which heterogeneity affects the formation of wave patterns. A BZ reaction catalyst was immobilized in ion exchange polystyrene beads to form active beads. Then active and inactive beads with no catalyst loading were mixed together with various ratios to achieve various levels of heterogeneity. In the same reaction environment, different wave patterns were displayed for the bead mixtures. We observed a transition from target patterns to spiral patterns as the percentage of the active beads in the beads mixture decreased. The increase of the heterogeneity led to wave pattern transition. Heterogeneity hindered the propagation of target waves and broke them into wavelets that generated spiral waves. In an effort to develop practical applications based on non-equilibrium phenomena, we have established a novel drug delivery system. A proton generator Zirconium Phosphate (ZrP) was imbedded inside a pH sensitive polymer matrix, poly acrylic acid (PAA). Through the ion exchange with sodium cation (Na+), ZrP generates protons to control the swelling/shrinking behaviors of PAA. The drug encapsulated in the matrix can be released in a controlled manner by adjusting the supply of Na+. This system might be developed into vehicles to deliver drugs to specific targets and release at a proper time. This new delivery technique will be convenient and significantly increase the efficiency of medicines.