Simulation of physical systems with few degrees of freedom

In physics, the models that describe our universe from macro to micro scales involve a set of equations which denotes the relationships between variables. These systems are generally represented by differential equations and are classified as dynamical systems. Dynamical systems are studied in many science fields. With the use of chaos theory within dynamical systems, many hitherto unknown properties of the systems have been discovered. Most real systems are formalized under Hamiltonian mechanism and Hamiltonian systems both display regular and chaotic behavior. In this study, we investigate a Hamiltonian system, namely Matinyan-Yang-Mills-Higgs system, and we try to investigate the regions where the system makes transition from regular to chaotic regions. In the second part of the thesis, we study the power of artificial neural networks in modeling dynamical systems and estimating nonlinear time series. In the third part, we use dynamical system as random number generators and we use these random number generators in image encryption.