INFLUENCE OF TURNING POINTS ON DECISION DELAY NEAR AN UNSTABLE EQUILIBRIUM POSITION

This paper considers a system of equations consisting of three first-order equations of fast variables and one equation of a slow variable. The first approximation matrix of the system of fast variables has different eigenvalues, with one pair of complex conjugates. The eigenvalues have zeros and they are turning points. The system has an equilibrium position, and the stability of the equilibrium position is lost at a certain value of the slow variable. The problem is set to study the delay in solving a system of fast variables near an unstable equilibrium position and the influence of turning points on the delay. The delay of the solution near an unstable position is proven and the influence of turning points is clarified. The proof used geometric constructions using the level line of harmonic functions, methods of successive approximations, Laplace, stationary phase, integration by parts and comparison of infinitesimals.