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:
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).