With the developments of biotechnology and other disciplines such as computational science, synthetic biology has made great progresses in theoretical analysis, functional design, and experimental implementation, which is attracted extensively in the interdisciplinary fields such as computational biology and artificial intelligence. From the perspective of mathematical science, theories of designing various synthetic biological elements with specific functions have been emerging, such as gene switches, gene oscillators, and biological logic gates. From the perspective of technological innovation, great progresses have been made in biosynthesis and functionalization strategies such as genetic engineering and chemical modifications on proteins (enzymes) for self-assembly. The rapid developments of these related aspects have also greatly promoted the development of synthetic biology. This review specifically focuses on theoretical basis and analysis methods behind various synthetic biological networks with specific functions from the perspective of biomolecular network dynamics, including functional biological devices such as switches and oscillators, as well as factors related to mathematics and network theory, including correlations between positive and negative feedback loops and nonlinear dynamics, nonlinear factors and the causes of time delays, stability and bifurcation-related theories, and theoretical basis and analysis methods related to dynamics, such as the robustness and period tunability of periodic oscillators are also addressed, which provides theoretical analysis methods that can be used as reference for further design of more complex or easily synthesized biological devices. Therefore, synthetic biology based on dynamics can start with mathematical modeling and dynamical system theory to construct synthetic gene regulatory networks with specific functions. By applying gene editing technology and adopting reasonable assembly strategies for experimental manipulations, we can verify the theoretical designs. By analyzing gene expression profiles, the feasibility and performance of the theoretical design can be explored. Further analysis of the topology and function of synthetic gene regulatory networks, as well as relationship between dynamics and parameters can help us better understand adjustable design strategies and key factors for redesign.