The concept of Dynamics exists everywhere in our lives. As an integral component of daily reality, engineers and scientists research, analyze, and observe the behaviors of dynamics and dynamical systems and their application in our real world activities. As a general rule, dynamical systems are characterized by their nonlinear quality. By their very nature, such systems have presented scientists with enormous difficulty in finding unequivocal solutions to systems, and even when a solution is possible, the path to finding one is extremely complicated. Oftentimes, it has proven virtually impossible to find these solutions. Given the inherent challenges in coming up with solutions to systems, the research that we've focused on has involved four separate but related areas.