Hopf bifurcation for a class of partial differential equation with delay

Author(s)

Katia Andreia Gonçalves de Azevedo

• Document type: Doctoral Thesis
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2002-04-18

Examining board members

Luiz Augusto da Costa Ladeira; Sergio Muniz Oliva Filho; Gerson Petronilho; Jair Silverio dos Santos; Plácido Zoega Táboas

• Advisor: Luiz Augusto da Costa Ladeira

Abstract

In this work we study the retarded reaction-diffusion equation {∂U/∂t (t,x) = ∂2U/∂x2(t, x) + kU(t,x) + k/δ ∫-r + δ-r g(U(t,x), U(t + s, x)ds, U(t, 0) = U(t, π) = 0, t≥0 U(t,x) = ψ(t, x), (t, x) ∈ [-r, 0] X [0, π]. We show the existence of a sequence of values {Tkn}n= 0,1,2... of the parameter T such that a Hopf bifurcation occurs when the delay passes through each value {Tkn}. The main techniques used here are some results on nonlinear eigenvalue problems, the analysis of the characteristic equation of the linearized problem, the Liapunov-Schmidt method and the Implicit Function Theorem. (AU)