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

Олександр Юрійович Швець, Олександр Михайлович Макасєєв

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.

Keywords


Pendulum system; Systems with limited excitation; Maps of dynamical regimes; Regular and chaotic attractors; Delay factors

References


V.O. Kononenko, Vibrating System with a Limited Power Supply.London,Great Britain: Iliffe, 1969, 236 p.

Yu.A. Mitropolsky and A.Yu. Shvets, “About influence of delay on a stability of a pendulum with the vibrating suspension point”, in Analytical Methods of Non-Linear Oscillations. Kyiv, Ukraine: Institute of Mathematics, 1980, pp. 115–120 (in Russian).

Yu.A. Mitropolsky and A.Yu. Shvets, “About oscillations of a pendulum with a vibrating point of a suspension at presence delays”, in Analytical Methods of Non-Linear Oscillations, Kyiv, Ukraine: Institute of Mathematics, 1980, pp. 120–128 (in Russian).

T.S. Krasnopolskaya and A.Yu. Shvets, “Chaotic interactions in a pendulum-energy-source system”, Prikladnaya Mekhanika, 1990, vol. 26, no. 5, pp. 90–96 (in Russian).

T.S. Krasnopolskaya and A.Yu. Shvets, “Chaos in vibrating systems with limited power-supply”, Chaos, 1993, vol. 3, no. 3, pp. 387–395.

A.Yu. Shvets and A.M. Makaseyev, “Chaotic Oscillations of Nonideal Plane Pendulum Systems”, CMSIM J., 2012, no. 1, pp. 195–204.

T.S. Krasnopolskaya and A.Yu. Shvets, Regular and Chaotical Dynamics of Systems with Limited Excitation.Moscow,Izevsk,Russia: R&C Dynamics, 2008, 280 p. (in Russian).

N.A. Magnizkiy and S.V. Sidorov, New Methods of Chaotic Dynamics. Moscow,Russia: Editorial URSS, 2004, 320 p. (in Russian).

A.A. Samarskiy and A.V. Gulin, Numerical Methods. Мoscow,Russia: Nauka, 1989 (in Russian).

A.Yu. Shvets and A.M. Makaseyev, “The influence of delay factors on regular and chaotic oscillations of plane pendulum”, in Proc. Institute of Mathematics of NASU, 2012, vol. 9, no. 1, pp. 365–377 (in Russian).

A.Yu. Shvets and O.M. Makasyeyev, “Chaos in pendulum systems with limited excitation in the presence of delay”, CMSIM J., 2014, no. 3, pp. 233–241.


GOST Style Citations


  1. Kononenko V.O. Vibrating system with a limited powersupply. – London: Iliffe, 1969. – 236 p.

  2. Митропольский Ю.А., Швец А.Ю. О влиянии запаздывания на устойчивость маятника с вибрирующей точкой подвеса // Аналитические методы исследования нелинейных колебаний. – К.: Ин-т математики, 1980. – С. 115–120.

  3. Митропольский Ю.А., Швец А.Ю. О колебаниях маятника с вибрирующей точкой подвеса при наличии запаздывания // Аналитические методы исследования нелинейных колебаний. – К.: Ин-т математики. – 1980. – С. 120–128.

  4. Краcнопольская Т.С., Швец А.Ю. Хаотические режимы взаимодействия в системе “маятник–источник энергии” // Прикл. мех. – 1990. – 26, № 5. – С. 90–96.

  5. Krasnopolskaya T.S., Shvets A.Yu. Chaos in vibrating systems with limited power-supply // Chaos. – 1993. – 3, № 3. – P. 387–395.

  6. Shvets A.Yu., Makaseyev A.M. Chaotic Oscillations of Nonideal Plane Pendulum Systems // CMSIM J. – 2012. – № 1. –  P. 195–204.

  7. Краснопольская Т.С., Швец А.Ю. Регулярная и хаотическая динамика систем с ограниченным возбуждением. – М.; Ижевск: НИЦ “Регулярная и хаотическая динамика”, Ин-т компьютерных исследований, 2008. – 280 с.

  8. Магницкий Н.А., Сидоров С.В. Новые методы хаотической динамики. – М.: Едиториал УРСС, 2004. – 320 с.

  9. Самарский А.А., Гулин А.В. Численные методы. – М.: Наука, 1989. – 432 с.

  10. Швец А.Ю., Макасеев А.М. Влияние запаздывания на регулярные и хаотические колебания плоского маятника // Зб. праць Ін-ту математики НАН України. – 2012. – 9, № 1. – P. 365–377.

  11. Shvets A.Yu., Makasyeyev O.M. Chaos in pendulum systems with limited excitation in the presence of delay // CMSIM J. – 2014. – № 3. – P. 233–241.




DOI: https://doi.org/10.20535/1810-0546.2015.4.50580

Refbacks

  • There are currently no refbacks.


Copyright (c) 2017 NTUU KPI