In this thesis we have studied two decompositions of dynamic systems: POD and DMD. The focus was on understanding the theoretical background defining the algorithm and applying it to obtain the low-order model (for POD) and reconstruction and prediction (for DMD method). The POD decomposition is a method that seeks a projection of the system to an orthonormal basis to obtain a low-order model of the observed system, but such that the projection is optimal in the least-squares sense. Above mentioned orthonormal basis is called the POD basis. We have outlined the baseline POD calculation in several possible cases: in finite-dimension, when it’s as simple as calculating an SVD of a matrix, in more complex space with weighted scalar product, and by snapshot method. The low-order model was obtained by Galerkin’s projection to the space spanned by the POD basis. The total error of the reduced system is given as a function of the approximation error. The DMD method is based on finding coherent spatial structures that define the dynamics of the system. These are given by the eigenvectors of the mapping matrix \(A\). Matrix \(A\) is generally unknown, but two sets of snapshots are known. From the given sets of snaphots we can determine the operator \(\tilde{A}\), which is a linear approximation of \(A\). It can be shown that the eigenvalues and eigenvectors of \(\tilde{A}_k\) are aproximations of eigenvalues and eigenvectors of \(A\). We have provided three algorithms for computing DMD modes that differ in the constraints on the set of input data and the computation of eigenvectors of \(\tilde{A}\). The system’s time evolution is determined by eigenvalues of \(A\). Complete reconstruction of the system is obtained from DMD modes, associated eigenvalues and initial amplitudes. Finally, we applied the DMD method on dynamic system governed by Navier-Stokes equations, and POD on system governed by equations of convection diffusion reaction.