Method for determining the cyclicity of singular points for quadratic systems of differential equations

Abstract

In the theory of nonlinear dynamical systems described by differential equations with polynomial right-hand sides, the detection of hidden periodic modes (limit cycles) is still relevant. These periodic dynamic trajectories arise on the outskirts of the equilibrium point and determine the features of self-oscillating behavior that are characteristic only of purely nonlinear objects as a result of Andronov-Hopf bifurcations. The search for a rational method for determining the cyclicity of a singular point for a six-parameter system of differential equations with quadratic right-hand sides is relevant. This study is dedicated to solving this problem. Analysis of the mathematical model proposed in the work allowed us to calculate the first three Lyapunov quantities and, based on them, to formulate the conditions for the existence of limit cycles and propose a method for determining their multiplicity.