Applications of Fast-Slow Dynamics to Coevolutionary Networks

Complex networks are everywhere, from social interactions and ecosystems to power grids and neural systems. Traditionally, these networks have been studied by either fixing their structure while observing dynamic interactions (dynamics on networks) or allowing the structure itself to evolve (dynamics of networks). However, real-world systems often involve a reciprocal interplay between the two, where the dynamics of nodes and the network's structure influence each other. This feedback-driven framework, known as coevolutionary networks, enables the study of adaptive systems that are more resilient, efficient, and realistic.

This thesis delves into the intricate behavior of coevolutionary networks through three interconnected themes: decision-making, synchronization patterns, and the adaptive control of neuromorphic networks. First, we explore resource management systems using fast-slow dynamics to identify critical transitions and design strategies for sustainable decisions. Next, we analyze networks of Kuramoto oscillators, revealing conditions for stabilizing chimera states, fascinating patterns combining synchronized and desynchronized behaviors relevant to neural and power grid stability. Finally, we apply coevolutionary principles to neuromorphic networks, demonstrating how adaptive external control can stabilize complex rhythmic activity, offering insights for robotics and real-time AI applications.

By combining coevolutionary dynamics with mathematical tools like Geometric Singular Perturbation Theory, this work provides a deeper understanding of adaptive systems across multiple domains. The findings highlight the importance of flexibility, feedback, and multiscale processes in designing robust networks capable of thriving in dynamic environments.