Осциллятор ФитцХью-Нагумо с переменной наследственностью и внешним воздействием

The paper studies oscillograms, phase trajectories and bifurcation diagrams of the nonlinear FitzHugh-Nagumo oscillator with variable heredity (fractional FitzHugh-Nagumo oscillator). The model equation for the fractional FitzHugh-Nagumo oscillator contains derivatives of fractional variables of the Gerasimov-Caputo type, as well as a function of external influence, which depend on time. Local initial conditions are valid for the model equation of the fractional oscillator. Due to the nonlinearity of the fractional FitzHugh-Nagumo oscillator, a numerical algorithm based on a nonlocal finite-difference scheme of the first order of accuracy was used. Then, using the Runge rule and computer experiments, an estimate of the computational accuracy is given. The numerical algorithm was implemented in the Maple 2021 environment for constructing oscillograms and phase trajectories and in Python for constructing bifurcation diagrams. It is shown that with an increase in the nodes of the computational grid, the computational accuracy tends to theoretical estimates. Further in the article the dynamic modes of the fractional FitzHugh-Nagumo oscillator are studied using bifurcation diagrams. It is shown that regular modes can be characterized not only by relaxation oscillations, but also by damped ones. The mode type depends on the values of the external action parameters, as well as on the values of the initial conditions. It is also shown that limit cycles may not always be stable. Bifurcation diagrams were constructed, which confirmed the dynamics obtained using oscillograms and phase trajectories. The question of rigorous justification of the existence of a unique stable limit cycle remains open. For a more rigorous definition of the conditions for the existence and uniqueness of a limit cycle, as well as its stability, analogs of the theorems of Liénard and Bendixson are needed. However, the results obtained in the article allow the fractional FitzHugh-Nagumo oscillator to be used to describe self-oscillatory processes. Further development of research can be associated with the construction of maps of dynamic modes, which is a rather labor-intensive task in computational terms. Such problems can be solved on a computing server using parallel programming methods.