This dissertation studies quantum decoherence in anharmonic oscillator systems to monitor and understand the way the systems evolve. It also explores methods to control the systems' evolution, and the effects of decoherence when applicable. We primarily do this by finding the time evolution of the systems using their Lie algebraic structures. We solve for a generalized Caldirola-Kanai Hamiltonian, and propose a general way to produce a desired evolution of the system. We apply the analysis to the effects of Dirac delta fluctuations in mass and frequency, both separately and simultaneously. We also numerically demonstrate control of the generalized Caldirola-Kanai system for the case of timed Gaussian fluctuations in the mass term. This is done in a way that can be applied to any system that is made up of a Lie algebra. We also explore the evolution of an optomechanical coupled mirror-laser system while maintaining a second order coupling. This system creates anharmonic effects that can produce cat states which can be used for quantum computing. We find that the decoherence in this system causes a rotational smearing effect in the Husimi function which, with the second order term added, causes rotational smearing after a squeezing effect. Finally, we also address the dynamic evolution and decoherence of an anharmonic oscillator with infinite coupling using the Born-Markov master equation. This is done by using the Lie algebraic structure of the Born-Markov master equation's superoperators when applying a strategic mean field approximation to maintain dynamic flexibility. The system is compared to the Born-Markov master equation for the harmonic oscillator, the regular anharmonic oscillator, and the dynamic double anharmonic oscillator. Throughout, Husimi plots are provided to visualize the dynamic decoherence of these systems.