Convergence to the Steady State in Two-Dimensional Maps

• Grant number: 25/09985-7
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: August 01, 2025
• End date: July 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Physics - General Physics
• Principal Investigator: Edson Denis Leonel
• Grantee: João Pedro Cruz Ferreira
• Host Institution: Instituto de Geociências e Ciências Exatas (IGCE). Universidade Estadual Paulista (UNESP). Campus de Rio Claro. Rio Claro , SP, Brazil
• Associated scholarship(s): 25/20012-0 - Convergence to steady state in two-dimensional mappings, BE.EP.IC

Abstract

In this project, we aim to understand and describe how convergence to the steady state occurs in dissipative two-dimensional discrete maps. The system dynamics is governed by a discrete-time map in two variables, where the determinant of the Jacobian matrix is less than one. This condition implies phase-space area contraction, leading to convergence towards an attractor. It is known from one-dimensional maps that, at bifurcations, the dynamics converges to the steady state following a generalized homogeneous function characterized by three critical exponents. Near the bifurcation point, convergence occurs through exponential decay, where the relaxation time obeys a power-law behavior that defines a fourth critical exponent. The knowledge of these four exponents, three at the bifurcation and one near it, allows for the identification and classification of the local bifurcation involved. The central question of this investigation is: in two-dimensional maps, do local bifurcations exhibit the same universal behavior with respect to critical exponents? If so, what is the underlying explanation? If not, what causes the deviation? This is the main focus of the research.