Equivalence problem for 3-dimensional dynamical systems

In this thesis, local equivalence problem for an autonomous dynamical system on a three dimensional manifold is considered by means of Cartan's method of equivalence. The solution of equivalence problem for an autonomous dynamical systems separates into two branches determined by integrability and nonintegrability of differential 1-form 1 which is a 1-form along the integral curve of the dynamical system. For integrable case, it is proved that all prolonged analytic coframes, and therefore all prolonged analytic dynamical systems, can be mapped to each other by a class of diffeomorphisms determined by a single function of a one variable. For the latter case, it is also proved that all prolonged analytic coframes, and therefore all prolonged analytic dynamical systems, can be mapped to each other by a class of diffeomorphisms determined by a single function of a one variable. For non-integrable case, problem is reduced to the base manifold and it is proved that number of the fundamental structures invariants and their dependence on a coordinate along the integral curve are determined according to whether the vector field is divergence free or not. For a divergence free vector field a flat connection, whose components are corresponding to the functions of Hamiltonians, is obtained. Accordingly, it is shown that fundamental invariants of the problem and their coframe derivatives are invariant along the flow of the dynamical system, in other words, they can be interpreted as a functions of Hamiltonians and therefore, it is seen that necessary and sufficient conditions for equivalence of dynamical system are determined only by the Hamiltonian functions. Also for a vector field with non-zero divergence a flat connection is obtained on a base manifold and structure equations of coframe, representing a dynamical system, are figured out. In the case of vanishing both of the structures functions, it is seen that Lie algebra determined by structure equations is isomorphic to Heisenberg algebra and thereby underlying manifold is locally Heisenberg group.