Influence of the delay on the occurrence of deterministic chaos in some non-ideal pendulum systems

Abstract

Background. The influence of the delay of interaction between pendulum and electric motor and the delay of the medium on the dynamics of non-ideal pendulum systems of the type “pendulum-electric motor” is considered. Mathematical model of this system is a system of ordinary differential equations with delay.

Objective. The influence of delay factors on steady-state oscillations of non-ideal pendulum systems of the type “pendulum–electric motor” is studied.

Methods. The approaches that reduce the mathematical model of the system to a system of three or fifteen differential equations without delay are suggested. For general analysis of nonlinear dynamics the maps of dynamical regimes are constructed. These maps allow conducting a qualitative identification of the type of steady-state regime. The construction of dynamical regimes maps is based on analysis and processing of spectrum of Lyapunov characteristic exponents. Phase portraits of regular and chaotic attractors are constructed.

Results. The use of three-dimensional mathematical model to study the dynamics of “pendulum–electric motor” systems is sufficient only at small values of the delay. For relatively high values of the delay multi-dimensional system of fifteen equations should be used.

Conclusions. The essential influence of the delay on qualitative change in the dynamic characteristics in “pendulum–electric motor” systems is shown. In some cases the delay is the controlling factor in the process of chaotization of pendulum systems.