Hidden Symmetry in Networked Systems

• Grant number: 25/26595-8
• Support Opportunities: Scholarships in Brazil - Doctorate (Direct)
• Start date: May 01, 2026
• End date: April 30, 2030
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Eeltje Cornelis Nijholt
• Grantee: Ana Cláudia Rodrigues e Silva
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 24/00930-2 - Hidden symmetry in networked systems, AP.JP

Abstract

This grant addresses challenges posed by real-world complex systems described as networks of interconnected dynamical elements. These systems feature in diverse fields such as ecology, biology and physics. Changes in the interaction structure have far-reaching e¿ects. Indeed, disorders like Parkin- son's disease, schizophrenia and epilepsy are believed to be linked to abnormal interaction patterns among neurons. Predicting disturbances and anticipating their consequences is crucial for averting disasters.We will develop comprehensive mathematical tools for classifying bifurcations and other dynamical behav- ior in network systems. My approach cuts right to the heart of many problems in the field, by realizing the very network structure itself as an intrinsic geometrical property of the system. Consequently, a foothold is gained for a wide variety of interdisciplinary techniques, which I will use to systematically analyze the impact of interaction structures.The main question from the field of network dynamical systems asks: what is the influence of a network structure on a dynamical system? The importance of this problem follows from countless applications, ranging from ecology to power grids and from neuroscience to disease control. Over the years, researchers have obtained many powerful results by approaching this question from an equally impressive number of directions. In spite of this, some of the most elementary techniques from dynamical systems theory remain unavailable within the network setting. Most notably, bifurcation theory, the study of qualitative changes in dynamical behavior, is still poorly understood for such systems. Not only are local bifurcations essential in countless results and applications -even considered the 'bread and butter' of the dynamical systems community-, they are also observed to be surprisingly rich for network systems. At the same time, established bifurcation techniques are simply not capable of dealing with network structure. Considering the historical success of bifurcation theory, this is a major shortcoming that is significantly holding back progress in the field. In recent years, I have developed a unifying framework for solving this longstanding open problem, based on intrinsic geometrical descriptions of network structure. (AU)