STUDY POINTS OF REST HEREDITARITY DYNAMIC SYSTEMS VAN DER POL-DUFFING

Using numerical modeling, oscillograms and phase trajectories were constructed to study the limit cycles of a van der Pol Duffing nonlinear oscillatory system with a power memory. The simulation results showed that in the absence of a power memory (α = 2, β = 1) or the classical van der Pol Duffing dynamical system, there is a single stable limit cycle, i.e. Lienar theorem holds. In the case of viscous friction (α = 2, 0 < β < 1), there is a family of stable limit cycles of various shapes. In other cases, the limit cycle is destroyed in two scenarios: a Hopf bifurcation (limit cycle-limit point) or (limit cycle-aperiodic process). Further continuation of the research may be related to the construction of the spectrum of Lyapunov maximal exponents in order to identify chaotic oscillatory regimes for the considered hereditary dynamic system (HDS). В работе с помощью численного моделирования построены осциллограммы и фазовые траектории с целью исследования предельных циклов нелинейной колебательной системы Ван-дер-Поля Дуффинга со степенной памятью. Результаты моделирования показали, что в случае отсутствия степенной памяти (α = 2, β = 1) или классической динамической системы Ван-дер-Поля Дуффинга, существует единственный устойчивый предельный цикл, т.е. выполняется теорема Льенара. В случае вязкого трения (α = 2, 0 < β < 1), существует семейство устойчивых предельных циклов различной формы. В остальных случаях происходит разрушение предельного цикла по двум сценариям: бифуркация Хопфа (предельный цикл-предельная точка) или (предельный циклапериодический процесс). Дальнейшее продолжение исследований может быть связано с построением спектра максимальных показателей Ляпунова с целью идентификации хаотических колебательных режимов для рассматриваемой эредитарной динамической системы (ЭДС).

Periodic solutions of a nonlinear oscillatory system with two degrees of freedom

We study a nonlinear oscillatory system with two degrees of freedom. By using the continuation theorem of coincidence degree theory, some sufficient conditions are obtained to establish the existence of periodic solutions of the system.

A Study on the Characteristics of a Nonlinear Oscillatory System With Dry Friction

Nonlinear oscillatory system involved with friction is very common in nonlinear dynamics of engineering fields. This paper is to investigate the motions a nonlinear oscillatory system with involvement of dry friction. The cases of weakly and highly nonlinearity of the system are considered. Approximate and numerical solutions for the system are developed via the author’s newly developed P-T method. As demonstrated in the present work, the properties of the weakly and highly nonlinear systems exhibit great differences, though the governing equations of the two systems employ identical system parameters. The approximate solutions developed for the system are continuous everywhere on the time range considered. Under the conditions of weakly nonlinearity, the approximate solutions developed can therefore be conveniently implemented for the purpose of an analytical studying the properties of the system with numerous system parameters and various initial conditions. Taking this advantage, the behavior of motion of the weakly nonlinear system is analyzed and compared with the corresponding solutions developed with Van der Pol’s method. It is found in the present work, the system may undergo a self-excited oscillation under certain conditions. The highly nonlinear system is a physically much involved one. Its behavior is thus much complex in comparing with that of the weakly nonlinear system. Based on the approximate solutions developed for the highly nonlinear system, recurrence relations are generated for numerical calculations. For the sake of comparison with the oscillation of the weakly nonlinear system, numerical simulations for the highly nonlinear system are performed under the same initial conditions and identical system parameters. The conditions of convergence and divergence of the weakly nonlinear system are also established for application. Behavior of the oscillatory motion of the highly nonlinear system is investigated on the basis of the corresponding numerical solutions developed.