Sumários

Aula 7

17 Dezembro 2018, 15:30 • Maria Teresa Faria da Paz Pereira

Esta aula não se realizou neste dia, tendo sido compensada com tempo lectivo extra nas aulas 2,3,4,6.

Aula 6

10 Dezembro 2018, 15:30 • Maria Teresa Faria da Paz Pereira

Definition of 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.
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, whose proof was finally published in 2018.
Statement of the LaSalle invariance principle for DDEs. Considerations about the use of Liapunov functionals to prove global attractivity of equilibria.

Apresentação de alguns temas para trabalho final.

(Esta aula foi leccionada em português, no dia 14/12/18, e teve a duração de 2h.)

Aula 5

3 Dezembro 2018, 15:30 • Maria Teresa Faria da Paz Pereira

Study of characteristic equations for some selected examples (cont.): 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; Hopf bifurcation points. 4. A chemostat model with one discrete delay and stability of its equilibria. 5. Nicholson blowflies’s equation (homework).
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.

(Esta aula foi leccionada em português, no dia 07/12/18).

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).

Aula 3

19 Novembro 2018, 15:30 • 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 (according to Lyapunov); geometrical interpretation.

(Esta aula foi leccionada em português, no dia 23/11/18 e teve a duração de  cerca de 2 horas).