Sumários

Lecture 7

27 Maio 2020, 14:30 • Maria Teresa Faria da Paz Pereira

This lecture did not take place; the material for this lecture  was covered in Lecture 6.
(Remark: Lectures 1,2,4,5,6 all had extra time.)

Lecture 6

20 Maio 2020, 14:30 • Maria Teresa Faria da Paz Pereira

Global attractivity: study of global attractivity and stability for some selected examples:

1. A monotone scalar semi-dynamical system with one discrete delay and negative feedback: presentation by a student.
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 historical notes about Wright's conjecture, whose proof was finally published in 2018.


This lecture took place on Tuesday, May 26, from 17h to 19h.

Lecture 5

13 Maio 2020, 14:30 • Maria Teresa Faria da Paz Pereira

Presentation by a student: ``small delays are harmless"; study of the characteristic equations for some selected examples: a predator-prey model with two delays; a chemostat model with one discrete delay and stability of its equilibria; Nicholson blowflies’s equation.

Addenda to the predator prey model: conditions for the absolute (i.e., independently of the size of the delays) local asymptotic stability of the positive equilibrium.

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.


This lecture took place on May 19, from 17h to 19h.

Lecture  4 (video-conf)

6 Maio 2020, 14:30 • 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.
Location of the roots and result on stability for linear DDEs. Linearisation about equilibria and principle of linearised stability for hyperbolic equilibria.

Characteristic equations for linear autonomous DDEs and applications to some concrete examples.

Theory of Hille-Phillips for C_0 semigroups of linear operators in Banach spaces applied to linear DDEs: the solution operator for x'(t)=L(x_t) is a C_0 semigroup; infinitesimal generator. Exponential behaviour and connection with the sprectrum of the infinitesimal generator via the Hille-Phillip theory (basic ideas for proof).

This lecture took place on May 12, from 17h to 18h45.

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Lecture 3 (video-conference)

29 Abril 2020, 14:30 • Maria Teresa Faria da Paz Pereira

Statements of the general results about existence and existence and uniqueness of local solutions for an abstract DDE x'(t)=f(t,x_t); proof of the existence and uniqueness theorem only in the case of f(t,\phi) continuous and Lipschitzian relative to \phi in each bounded set of $\R\times C([-r,0],\R^n), by using the contraction principle (rather than Schauder's fixed point theorem).
Consequences: continuous dependence of solutions relative to initial conditions; continuation of solutions (for the future).

Definitions of stabilities.

(Tutorial.)

This lecture took place on May 5, at 17h.