Aula 4

26 Novembro 2018, 15:30 • Maria Teresa Faria da Paz Pereira

Equilibria; linearization about equilibria.
Linear autonomous DDEs x'(t)=L(x_t); characteristic equation;  properties of the roots of the characteristic equation.
Elements of the theory of Hille-Phillips for C_0 semigroups of linear operators in Banach spaces; exponential behaviour and connection with the sprectrum of the infinitesimal generator.
 The solution operator for x'(t)=L(x_t) is a C_0 semigroup; infinitesimal generator.
Location of the roots and result on stability for linear DDEs via the Hille-Phillip theory (basic ideas for proof). Linearisation about equilibria and principle of linearised stability for hyperbolic equilibria.
Characteristic equations for linear autonomous DDEs and applications: 1. The logistic equation (or Wright's equation) with one discrete delay and the equation x'(t)=-ax(t-r) revisited. 2. The linear DDE x′(t) = Ax(t) + Bx(t − r) (r > 0), where A,B are nxn matrices: small delays are harmless. The particular case of scalar equations (n=1) and sharp conditions for the asymptotic stability.


(Esta aula foi leccionada em português, no dia 30/11/18 e teve a duração de  cerca de 2 horas).