scholarly journals Quasi-harmonic self-oscillations in discrete time: analysis and synthesis of dynamic systems

2022 ◽  
Vol 24 (4) ◽  
pp. 19-24
Author(s):  
Valery V. Zaitsev ◽  
Alexander V. Karlov
Keyword(s):  
Discrete Time ◽  
Oscillatory System ◽  
Discrete Models ◽  
Van Der Pol Oscillator ◽  
Time Analysis ◽  
Van Der Pol ◽  
Analog System ◽  
Analysis And Synthesis ◽  
The Difference ◽  
System Prototype

For sampling of time in a differential equation of movement of Thomson type oscillator (generator) it is offered to use a combination of the numerical method of finite differences and an asymptotic method of the slowl-changing amplitudes. The difference approximations of temporal derivatives are selected so that, first, to save conservatism and natural frequency of the linear circuit of self-oscillatory system in the discrete time. Secondly, coincidence of the difference shortened equation for the complex amplitude of self-oscillations in the discrete time with Eulers approximation of the shortened equation for amplitude of self-oscillations in analog system prototype is required. It is shown that realization of such approach allows to create discrete mapping of the van der Pol oscillator and a number of mappings of Thomson type oscillators. The adequacy of discrete models to analog prototypes is confirmed with also numerical experiment.

scholarly journals NON-LINEAR OSCILLATORS IN DISCRETE TIME: ANALYSIS AND SYNTHESIS OF DYNAMIC SYSTEMS

2021 ◽  
Author(s):  
Valeriy Zaycev ◽  
Alalvan Kasim
Keyword(s):  
Discrete Time ◽  
Dynamic Systems ◽  
Digital Filters ◽  
Time Analysis ◽  
Radio System ◽  
Oscillating Systems ◽  
Non Linear ◽  
Analysis And Synthesis

A physically justified method of synthesis of nonlinear oscillating systems oscillating in discrete time (DT) is proposed. Synthesized dynamic systems are used as nonlinear discrete (digital) filters and basic models of radio system elements.

Time analysis of forced variable-order fractional Van der Pol oscillator

2017 ◽  
Vol 226 (16-18) ◽  
pp. 3803-3810 ◽  
Author(s):  
Behrouz Parsa Moghaddam ◽  
José António Tenreiro Machado
Keyword(s):  
Van Der Pol Oscillator ◽  
Variable Order ◽  
Time Analysis ◽  
Van Der Pol

Analysis of the van der Pol System With Coulomb Friction Using the Method of Multiple Scales

2008 ◽  
Vol 130 (4) ◽  
Author(s):  
Hiroshi Yabuno ◽  
Yota Kunitho ◽  
Takuma Kashimura
Keyword(s):  
Steady State ◽  
Coulomb Friction ◽  
Multiple Scales ◽  
Time History ◽  
Steady States ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Trivial Equilibrium ◽  
The Difference ◽  
Unstable Focus

The effect of Coulomb friction on the nonlinear dynamics of a van der Pol oscillator is presented. A map from the magnitude of a peak to that of the succeeding valley in the time history is analytically described by considering both the exponential growth due to negative viscous damping and the switching condition due to Coulomb friction, which is a function of the sign of the velocity of the system. The steady states and their stability are clarified and the difference from those in the case without Coulomb friction is revealed. The addition of Coulomb friction makes the trivial equilibrium, which is an unstable focus in the system without friction, into a locally asymptotically stable equilibrium set. The branch of stable nontrivial steady states is not bifurcated from the trivial steady state by the effect of Coulomb friction and is different from the branch in the case without Coulomb friction, which is bifurcated from the trivial steady state through Hopf bifurcation. Furthermore, experiments are conducted and the theoretically predicted dynamics due to Coulomb friction is confirmed.

scholarly journals ABOUT NUMERICAL MODELLING OF THOMSON SELF-OSCILLATORY SYSTEMS

2017 ◽  
Vol 21 (6) ◽  
pp. 141-150
Author(s):  
V.V. Zaitsev ◽  
A.V. Karlov ◽  
Ar.V. Karlov
Keyword(s):  
Nonlinear Dynamics ◽  
Finite Difference ◽  
Numerical Integration ◽  
Discrete Time ◽  
Equations Of Motion ◽  
Integral Form ◽  
Van Der Pol Oscillator ◽  
Resonant System ◽  
Van Der Pol ◽  
Oscillatory Systems

The algorithm of numerical integration of a task of Cauchy for the equations of the movement of self-oscillatory systems of Thomson type is offered. The algorithm is based on the use of samples of impulse response of linear resonant system as discretization sequences at the transition to the discrete time in the integral form of the equations of motion. Estimates of an error of numerical decisions are given. Transformation of finite difference algorithm in object of nonlinear dynamics in discrete time is discussed. Version of discrete mapping of Van der Pol oscillator is proposed. The algorithm of numerical integration of a task of Cauchy for the equations of the movement of self-oscillatory systems of Thomson type is offered. The algorithm is based on the use of samples of impulse response of linear resonant system as discretization sequences at the transition to the discrete time in the integral form of the equations of motion. Estimates of an error of numerical decisions are given. Transformation of finite difference algorithm in object of nonlinear dynamics in discrete time is discussed. Version of discrete mapping of Van der Pol oscillator is proposed. The algorithm of numerical integration of a task of Cauchy for the equations of the movement of self-oscillatory systems of Thomson type is offered. The algorithm is based on the use of samples of impulse response of linear resonant system as discretization sequences at the transition to the discrete time in the integral form of the equations of motion. Estimates of an error of numerical decisions are given. Transformation of finite difference algorithm in object of nonlinear dynamics in discrete time is discussed. Version of discrete mapping of Van der Pol oscillator is proposed.

NUMERICAL SOLUTION FOR DUFFING-VAN DER POL OSCILLATOR VIA BLOCK METHOD

2020 ◽  
Vol 10 (1) ◽  
pp. 1857-8365
Author(s):  
A. F. Nurullah ◽  
M. Hassan ◽  
T. J. Wong ◽  
L. F. Koo
Keyword(s):  
Numerical Solution ◽  
Van Der Pol Oscillator ◽  
Block Method ◽  
Van Der Pol

scholarly journals Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yajie Li ◽  
Zhiqiang Wu ◽  
Guoqi Zhang ◽  
Feng Wang ◽  
Yuancen Wang
Keyword(s):  
Time Delay ◽  
White Noise ◽  
Gaussian White Noise ◽  
Singularity Theory ◽  
Van Der Pol Oscillator ◽  
Order System ◽  
Damping Force ◽  
Van Der Pol ◽  
Restoring Force ◽  
White Noise Excitation

Abstract The stochastic P-bifurcation behavior of a bistable Van der Pol system with fractional time-delay feedback under Gaussian white noise excitation is studied. Firstly, based on the minimal mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and the critical parametric conditions for stochastic P-bifurcation of system amplitude are determined according to the singularity theory. Finally, the types of stationary PDF curves of system amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

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