Lecture 7

18 Dezembro 2017, 15:30 • Maria Teresa Faria da Paz Pereira

Global stability for some selected examples: 1. A monotone scalar semi-dynamical system with one discrete delay and negative feedback. 2. A general autonomous scalar DDE with a weaker negative feedback condition; application to a logistic delay equation. 3. Wright's equation (statement with the 3/2-condition on a parameter a, and proof only for the case a<=1); some considerations about the general proof and about Wright's conjecture.
(This class took place on Wednesday Dec. 20, from 18h to 19h45.)

Lecture 6

11 Dezembro 2017, 15:30 • Maria Teresa Faria da Paz Pereira

Autonomous DDEs x'(t)=f(x_t) and semi-dynamical systems; (positive) orbits and omega-limit sets; if f is completely continuous, bounded positive orbits are pre-compact. Properties of the omega-limit sets; some consequences. Monotone semi-dynamical systems.
(This class took place on Wednesday Dec. 13, from 18h to 19h30.)

Lecture 5

4 Dezembro 2017, 15:30 • Maria Teresa Faria da Paz Pereira

Characteristic equations for linear autonomous DDEs (cont.) 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; the particular case of scalar equations (n=1) and sharp conditions for the asymptotic stability. 3. A predator-prey model with two delays and conditions for the absolute (i.e., independently of the size of the delays) local asymptotic stability of the positive equilibrium. 4. A chemostat model with one discrete delay and stability of its equilibria.(
(This class took place on Wednesday Dec. 6, from 18h to 19h45.)

Lecture 4

27 Novembro 2017, 15:30 • Maria Teresa Faria da Paz Pereira

Theorem of existence and uniqueness of solutions and consequences (cont): continuous dependence of solutions relative to initial conditions; continuation of solutions (for the future).

Definitions of stabilities (according to Lyapunov). Equilibria.

Linear autonomous DDEs; characteristic equation; properties of the roots of the characteristic equation; location of the roots and result on stability for linear DDEs (proof postponed). Linearisation about equilibria and principle of linearised stability for hyperbolic equilibria.

(This class took place on Wednesday Nov. 29, 2017, from 18h30 to 20h.)

Dependência suave dos valores próprios nos dados do problema

22 Novembro 2017, 11:00 • Maria Carlota da Rocha Xavier Rebelo Gonçalves

Apresentação feita pelo aluno Matthias Hofmann.