Data-driven analysis has seen explosive growth with widespread availability of data and unprecedented computational power. Time-series data is of particular interest from a systems theory perspective which seeks to predict behaviour of involved entities and uncover unforeseen events. While system identification has been vastly successful in constructing models for linear systems, the Koopman operator framework has gained popularity due to its relations to nonlinear systems. The Koopman representation's linearity in functions of state and beyond local validity has been exploited by the dynamic mode decomposition (DMD) class of algorithms for their ease of identifying system models from data.

We study the effect of some attributes of data -- number of time-points, power-spectrum, periodicity, sampling, and number of trajectories -- on the results of Extended-DMD (EDMD). After a brief overview of the Koopman framework, we derive EDMD as a projection onto a basis without assumptions on desired functions lying in the span of that basis. We apply persistence of excitation (PE) to the Koopman framework and prove necessary and sufficient condition on the data for accuracy of EDMD approximation of the Koopman operator for nonlinear systems - using the basis at the data-points. Findings from this motivated us to pursue an order reduction algorithm for EDMD that has been known to suffer from the curse of dimensionality. We prove properties of matrices introduced for EDMD to list conditions and give solutions compiled into Reduced Order EDMD (RODMD), an algorithm that modifies the EDMD basis such that it minimizes the error in approximating desired functions while also keeping the dimension at its lowest. Alongside, we use results on convergence of EDMD estimation to the Koopman operator in literature to compare sampling characteristics of data used. For a certain number of data-points, we prove quantitative convergence guarantee for transient behaviour in terms of number of trajectories that the data is spread across. We use these derivations and proofs to show the rather basic relationship between EDMD and system identification, and use it to show the PE condition on data for uncontrolled systems as necessary for estimation of the Koopman operator.

We also present an application of dynamical systems theory to artificial neural networks to give an example of the interdisciplinary applicability of the Koopman framework. Using a dynamical system perspective of the optimization of neural networks, we prove the existence of explicit representation of some parameters of the neural network in terms of the other parameters and data, thereby paving way for one-shot learning through DMD algorithms.