Dynamic properties of one impulse Cauchy problem

A model of a fully connected association of neurons with a synaptic electrical connection which is a system of $m$ differential equations with delay is considered. By a special substitution, this system is reduced to a system of impulsive ordinary differential equations. For the corresponding dynamical system in the case $m=3$, we study the existence, stability, and asymptotic representation of periodic solutions on the basis of a bifurcation analysis of a two-dimensional mapping, a shift operator along trajectories of a solution to a special system of two differential equations. Particular attention is paid to the number of coexisting stable regimes. We study the problem of finding parameters for which the number of such modes is maximum. In order to search the fixed points of the resulting two-dimensional mapping, a numerical study is used based on the following iterative procedure. Selected the starting point, the Runge-Kutta method with a given step calculates the solution values on the segment $[0,T].$ At the end point $T$ of this segment, the solution value is compared with the initial one and if the deviation exceeds the specified value, then the value at the end point is taken as the initial one and the calculation cycle by the Runge-Kutta method is repeated. The calculations terminate when the required small deviation is reached, i.e., a fixed point of the shift operator is found, and so is the corresponding stable periodic mode, or when the number of iterations reaches a given large number, and this indicates the absence of a fixed point. The paper presents the results of a numerical study that made it possible to demonstrate the main rearrangements occurring in the phase space of a two-dimensional mapping. The obtained fixed points allow us to find asymptotic stable solutions of the original problem.