Basins of Attraction: From unidimensional maps to complex networks

• Grant number: 24/06718-5
• Support Opportunities: Regular Research Grants
• Start date: October 01, 2024
• End date: September 30, 2026
• Field of knowledge: Physical Sciences and Mathematics - Physics
• Principal Investigator: Rene Orlando Medrano Torricos
• Grantee: Rene Orlando Medrano Torricos
• Host Institution: Instituto de Ciências Ambientais, Químicas e Farmacêuticas (ICAQF). Universidade Federal de São Paulo (UNIFESP). Campus Diadema. Diadema , SP, Brazil
• Associated researchers: Antonio Mihara

Abstract

Multistability is one of the most recurring and important phenomena in dynamic systems. It is characterized by the simultaneous presence of several attractors, accessed by appropriate initial conditions, for the same set of system control parameters. The set of initial conditions that converge to one of these states is called the basin of attraction of that attractor. Knowledge about the structure of basins of attraction is fundamental for the good performance from simple systems, such as one-dimensional maps, to complex networks, such as power grids, airline routes, social networks and neural systems.The phenomenon of multistability imposes a high degree of operational unpredictability. In low-dimensional systems, unpredictability manifests itself in basins composed of highly irregular boundaries, that is, with high fractal dimensions, as in the Wada basins. This type of basin is composed of at least three distinct basins simultaneously present on all boundaries between the basins. In this project, we propose to study the Wada basins in one-dimensional maps, in order to understand the dynamic mechanisms that promote the emergence of this peculiar phenomenon. In multidimensional systems, such as networks, this issue is critical in itself, considering that the coexistence of a large number of attractors is recurrent. In recent work with the Kuramoto model with N coupled oscillators, we associate the volume of the basins of attraction of the equilibrium states (q-twist) with their respective eigenvalues. We propose to extend this study to systems with delay and repulsive regimes of the Kuramoto model. This approach will open new perspectives on spatiotemporal dynamics in a wide range of applications. In particular, we are interested in the interpretation of results in terms of neural networks. (AU)