In this thesis we have studied the interaction between ellipticity and dissipation in the equations proposed by Hsieh, and established the relation between this interaction and chaos. This study is the first systematical exploration ever made on the basis of Hsieh's system of partial differential equations. Theoretical results are obtained including existence, uniqueness, well-posedness, asymptotic behavior and steady states. Extensive numerical simulations with Hsieh's equations are made, and different routes to chaos are found. The numerical studies have revealed the chaotic nature of the solution. Moreover, the modified form, i.e. the conservative form of Hsieh's system, is found to admit only steady state solutions represented by a set of discrete peaks. The number of peaks increases as the dissipation decreases.