Razaz Faroug Yousif Mustafa2017-10-312017-10-312015http://hdl.handle.net/123456789/6759The global behaviour of a dynamical system can be described by it’s Morse decomposition or its attractor and repller configuration. Conley index theory proves either existence or the non-existence of connection orbits between equilibria of dynamical systems. We provide some applications of Morse decomposition, and Conley index theory which include the use of graded module braids and chain complex braids in constructing connection matrices for each of the Morse sets. In particular, we give examples for the existence of connection matrices between the elements of a given Morse set and hence the existence of connecting orbits between the elements of the Morse set. This approach can help one in the study of global behaviour of a given dynamical system.dynamical systemsThesis