Gene expression is one of the core areas in comprehending and assessing how biological cells work. Gene regulatory networks (GRNs), representing the intri-cate mechanism between genes and their regulatory modules, are instrumental in controlling gene expression and cell functions. These models shed light on how transcription factors interact with their regulatory modules within a cell. Despite the multitude of studies focusing on the analysis and enhancement of GRNs, there is still room for contributions. This thesis investigates a novel framework inspired by the gene networks constructed using synthetic biology, and presents stability analyses of the nonlinear infinite dimensional dynamical system models arising in this framework. In the first part of the thesis, we extend a previously studied benchmark GRN model including time delay, and present an analysis of the extended frame-work. We utilize unmodeled dynamics and possibly ignored interactions, including higher-order dynamics, in our system design. The stability of the extended system is analyzed by considering various nonlinearity functions and design pa-rameters, and the results are compared with those of the benchmark original model. In the second part, we employ an extension of a gene network model using a multiplicative perturbation of the dynamical system. Each cascaded subsystem in this extended framework has an additional block, including a multiplicative term with a high-pass filter, and the effect of additional parameters on the robustness and delay margin of the system is investigated. Experiments with various design parameters yield that the stability characteristics of GRNs can be improved using the model pertaining to the extension under specific perturbations. Finally, the third part covers the analysis of nonlinear dynamics and chaos in GRNs, particularly focusing on the two-gene original and extended gene net-works. Chaotic dynamics depend strongly on the inclusion of time delays, but the circuit motifs that show chaos differ when both original and extended models are considered. Our results suggest that for a particular higher-order extension of the gene network, it is possible to observe the chaotic dynamics in a two-gene system without adding any self-inhibition. This finding can be explained as a result of the modification of the original benchmark model induced by unmodeled dynamics. We argue that regulatory gene circuit models with additional parameters demonstrate non-periodic dynamics much more easily.