Bifurcação de Hopf: uma análise de escala

• Autor(a) principal: Silva, Vinicius Barros da [UNESP]
• Data de Publicação: 2016
• Tipo de documento: Trabalho de conclusão de curso
• Idioma: por
• Título da fonte: Repositório Institucional da UNESP
• Texto Completo: http://hdl.handle.net/11449/155517; http://www.athena.biblioteca.unesp.br/exlibris/bd/capelo/2017-07-27/000885319.pdf
• topic: Teoria da bifurcação; Estabilidade; Leis de escala (Fisica estatistica)

The main goal of this work is to explore the evolution towards the steady state at and nearby the supercritical Hopf bifurcation with the ordinary differential equation (ODE) that describes the dynamics written in the normal form. In polar coordinates, the same system is then characterized by an angular and, a radial equation. We applied a scaling formalism similar the one used in statistical mechanics to describe phase transitions. By considering this formalism in the analysis of the radial equation at the bifurcation point, we used a phenomenological description characterized by scaling hypothesis to prove that a homogeneous function with three critical exponents describes the decay to the fixed point In turn, we discovered these exponents are related to each other by a scaling law of the type . In the other hand, the analysis of the angular equation at the bifurcation point shown that a homogeneous function with three new critical exponents related to each other by the same scaling law discussed previously also describes the decay to the steady state However, the analysis of the radial equation near the bifurcation point showed the convergence to the equilibrium point is described by an exponential decay with a relaxation time given by a power law of the type, where is a critical exponent. However, the analysis of the angular equation near the bifurcation point shown exactly the same critical exponents found at the bifurcation point. To sum up, we obtained by considering the convergence to steady state at and nearby the Hopf bifurcation the scaling properties and the critical exponents that characterizes the bifurcation in study. The knowledge of these exponents allows one to identify to what universality a bifurcation belongs to