Bifurcating Solutions and Asymptotics in Logistic Equation with State-Dependent Delays

Abstract

The article considers the question of local dynamics of the logistic equation with delays and the delay coefficients being the nonlinear functions. It is supposed that one of the parameters, characterizing the delay value, is sufficiently large. The stability criterion is formulated and critical cases in the problem of balance state stability are defined. The main content of the paper is focused on studying the local dynamics in cases close to critical. In nature, these critical cases are infinite dimensional. Special nonlinear boundary-value problems of the parabolic type are constructed, which play the role of normal forms. Their nonlocal dynamics defines the behavior of the solutions from the small neighborhood of the balance state of the initial equation. The conclusions on the role of nonlinearity in the delay coefficients, in the behavior of the dynamic properties of the solutions are summarized.