The appearance of topological singularities, namely exceptional points (EPs) is an intriguing feature of parameter-dependent open quantum or wave systems. EPs are the special type of non-Hermitian degeneracies where two (or more) eigenstates of the underlying system coalesce. In this paper, we present a three level non-Hermitian Hamiltonian which hosts three interacting eigenstates. The matrix elements are optimized in such a way that the intermediate eigenstate interacts with both the other states and the underlying system hosts at least two different second order EPs. This coupling scheme has proven to be highly effective to encounter higher order EPs. The impact of quasi-static parameter variation along a cyclic contour around the embedded EPs on the dynamics of interacting eigenvalues has been investigated comprehensively in the context of cascaded state conversion. Such dynamics of the eigenvalues show a clear signature of the third order EP. Moreover, we examine the accumulation of phases around the identified EPs and study the hallmark of phase exchange during cascaded state conversions accompanied by the parametric encirclement of the third order EP. A device level implementation of the scheme is possible exploiting multimode anisotropic waveguides or coupled waveguide systems. © 2019 IOP Publishing Ltd.