This thesis investigates signal propagation and stability in spider web-like networks, focusing on how velocity differences, structural geometry, and complexity influence network behavior. Spider webs, known for their resilience, flexibility, and efficient vibration transmission, offer valuable insights into designing robust artificial networks. By employing mathematical and physical modeling, this study explores force distribution, signal propagation dynamics, and collision phenomena within these networks.

The study introduces distinct propagation approaches, ranging from simple discrete collision analysis to advanced continuous simulations incorporating energy dissipation, adaptive weighting, and refined collision detection algorithms. Key methodologies include simulations of force distribution using recurrence relations, random walk models, and wavefront propagation models to examine how signals traverse complex network topologies. These simulations reveal that network topology significantly impacts signal efficiency, propagation speed, collision frequency, and signal loss, with central nodes emerging as critical hubs of activity and congestion. Additionally, structural defects such as inactive nodes, altered masses, and weakened edges are systematically introduced to evaluate their influence on the overall stability and signal propagation efficiency. These imperfections profoundly affect network performance, demonstrating the necessity for structural adaptability and redundancy to maintain integrity under stress.

This thesis aims to identify the parameter combinations necessary for the Brusselator and Oregonator models to exhibit oscillating behavior. Initially, the Brusselator model is reviewed, reproducing the results from [1]. Using Bendixson’s Criterion and the Poincaré-Bendixson Theorem, the existence of stable limit cycles is proven, and the Hopf-Bifurcation locations are derived analytically. The Brusselator model is then extended from a batch reactor to a Continuous Stirred-Tank Reactor to include the inflow of two components previously considered constant. An extensive eigenvalue investigation in Matlab is conducted to determine the parameter combinations that induce oscillating behavior, with Hopf-Bifurcation locations presented in a three-dimensional plot. Lastly, the Oregonator model is introduced and analysed. The existence of limit cycles is proven using similar methods, and an eigenvalue analysis yields the analytic expressions for the Hopf-Bifurcation locations. Periodic solutions are found in all three models, and the necessary parameter combinations are identified.