Relaxation Oscillations in the Logistic Equation with Delay and Modified Nonlinearity

We consider the dynamics of a logistic equation with delays and modified nonlinearity, the role of which is to bound the values of solutions from above. First, the local dynamics in the neighborhood of the equilibrium state are studied using standard bifurcation methods. Most of the paper is devoted to the study of nonlocal dynamics for sufficiently large values of the 'Malthusian' coefficient. In this case, the initial equation is singularly perturbed. The research technique is based on the selection of special sets in the phase space and further study of the asymptotics of all solutions from these sets. We demonstrate that, for sufficiently large values of the Malthusian coefficient, a 'stepping' of periodic solutions is observed, and their asymptotics are constructed. In the case of two delays, it is established that there is attractor in the phase space of the initial equation, whose dynamics are described by special nonlinear finite-dimensional mapping.