Periodic and Quasiperiodic Solutions in the System of Three Hutchinson Equations with a Delayed Broadcast Connection

Abstract

The dynamics of an association of three coupled oscillators is studied. The link between the oscillators is a broadcast connection, that is, one element unilaterally effects the other two, which in turn interact with each other. An important property of the relation among the oscillators is the presence of a delay that obviously can often be found in applications. The studied system simulates the situation of population dynamics when populations are weakly connected, for example, are divided geographically. In this case one population can affect the other two, which in turn can influence each other but not the first one. Each individual oscillator is represented by the logistic equation with a delay (Hutchinson’s equation). Local asymptotic analysis of this system is done in the case of proximity of oscillator parameters to the values at which the Andronov-Hopf bifurcation occur, also the coupling coefficient in the system are assumed to be small. The method of normal forms is used. The study of the dynamics of the system in some neighborhood of a single equilibrium state is reduced to a system of ordinary differential equations on a stable integral manifold. For the construction of a normal form were found elementary modes obtained by using the symmetry of the problem, and the conditions for their stability. Taking into account the obtained asymptotic formulas, the phase reorganizations occurring in the system are numerically analyzed. It is shown that the delay in the communication circuits of the oscillators significantly affects the qualitative behaviour of the system solutions.