Resumen A first order method is considered for the discretization in time of an integro-differential equation, which can be written as $D^{\alpha} u(t) = A u(t) + f(t)$, $1 < \alpha < 2$, where $A : D(A) \subset X \to X$ is a sectorial operator in a Banach space X. Qualitative properties of the numerical solution, such as contractivity and positivity, are studied. A numerical illustration is provided.
Resumen
RESUMEN
This paper explores analytically and numerically, in the context of the FitzHugh-Nagumo model of nerve membrane excitability, an interesting phenomenon that has been described as a delay or memory effect. It can occur when a parameter passes slowly through a Hopf bifurcation point and the system's response changes from a slowly varying steady state to slowly varying oscillations. On quantitative observation it is found that the transition is realized when the parameter is considerably beyond the value predicted from a straightforward bifurcation analysis which neglects; the dynamic aspect of the parameter variation. This delay and its dependence on the speed of the parameter variation are described. The model involves several parameters and particular singular limits are investigated. One in particular is the slow passage through a low frequency Hopf bifurcation where the system's response changes from a slowly varying steady state to slowly varying relaxation oscillations. We find in this case the onset of oscillations exhibits an advance rather than a delay. This paper shows that in general delays in. the onset of oscillations may be expected but that small amplitude noise and periodic environmental perturbations of near resonant frequency may decrease the delay and destroy the memory effect. This paper suggests that both deterministic and stochastic approaches will be important for comparing theoretical and experimental results in systems where slow passage through a Hopf bifurcation is the underlying mechanism for the onset of 'oscillations.
