ABOUT NUMERICAL MODELLING OF THOMSON SELF-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.

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Volterra-series approach to stochastic nonlinear dynamics: Linear response of the Van der Pol oscillator driven by white noise

Physical Review E ◽

10.1103/physreve.102.032209 ◽

2020 ◽

Vol 102 (3) ◽

Author(s):

Roman Belousov ◽

Florian Berger ◽

A. J. Hudspeth

Keyword(s):

Nonlinear Dynamics ◽

White Noise ◽

Linear Response ◽

Volterra Series ◽

Van Der Pol Oscillator ◽

Van Der Pol

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VAN DER POL AND RAYLEIGH OSCILLATORS IN DISCRETE TIME

Vestnik of Samara University Natural Science Series ◽

10.18287/2541-7525-2014-20-7-104-114 ◽

2017 ◽

Vol 20 (7) ◽

pp. 104-114

Author(s):

V.V. Zaitsev ◽

S.V. Lindt ◽

A.N. Shilin

Keyword(s):

Linear System ◽

Discrete Time ◽

Dynamic Systems ◽

Discrete Systems ◽

External Influence ◽

Van Der Pol ◽

Oscillatory Systems ◽

Combination Of Methods

New discrete displays of classical self-oscillatory systems - Van der Paul and Rayleigh’s oscillators are oﬀered. Displays with the kept temporary characteristics of response of linear system on external inﬂuence are received on the basis of combination of methods of parametrical synthesis and invariancy of pulse char- acteristics of dynamic systems. Examples of generation of regular and chaotic self-oscillations in discrete time are given. For the analysis of self-oscillations in the received discrete systems the method of slowly changing amplitudes is used. The eﬀect of substitution of frequencies in a range of self-oscillations with use of the improved ﬁrst approach is considered.

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THE MAPPINGS OF VAN DER POL — DYUFFING GENERATOR IN DISCRETE TIME

Vestnik of Samara University Natural Science Series ◽

10.18287/2541-7525-2017-23-2-51-59 ◽

2017 ◽

Vol 23 (2) ◽

pp. 51-59

Author(s):

V. V. Zaitsev ◽

A. N. Shilin

Keyword(s):

Nonlinear Dynamics ◽

Discrete Time ◽

Nonlinear Oscillations ◽

Dynamic Properties ◽

The Self ◽

Pulse Characteristic ◽

Van Der Pol ◽

Sampling Series ◽

Work Transition ◽

To Receive

In the work transition to discrete time in the equation of movement of van der Pol – Dyuffing generator is described. The transition purpose—to create mappings of the generator as subjects of the theory of nonlinear oscillations (nonlinear dynamics) in discrete time. The method of sampling is based on the use of counting of the pulse characteristic of an oscillatory contour as the sampling series for a signal in a self-oscillating ring ”active nonlinearity – the resonator – feedback”. The choice of the consecutive scheme of excitement of a contour allows to receive the iterated displays in the form of recurrent formulas. Two equivalent forms of discrete displays of the generator of van der Pol – Dyuffing—complex and valid are presented. In approximation of method of slow-changing amplitudes it is confirmed that the created discrete mappings have dynamic properties of an analog prototype. Also within the numerical experiment it is shown that in case of the high power of generation the effect of changing of frequencies of harmonicas of the generated discrete signal significantly influence dynamics of the self-oscillators. In particular, in the discrete generator of van der Pol – Dyuffing the chaotic self-oscillations are observed.

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

Physics of Wave Processes and Radio Systems ◽

10.18469/1810-3189.2021.24.4.19-24 ◽

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.

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DETECTING COUPLING DIRECTIONS IN MULTIVARIATE OSCILLATORY SYSTEMS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127407019664 ◽

2007 ◽

Vol 17 (10) ◽

pp. 3735-3739 ◽

Cited By ~ 8

Author(s):

MATTHIAS WINTERHALDER ◽

BJÖRN SCHELTER ◽

JENS TIMMER

Keyword(s):

Nonlinear Dynamics ◽

Nonlinear Systems ◽

Stochastic Processes ◽

Stochastic Systems ◽

Partial Directed Coherence ◽

Van Der Pol ◽

Multivariate Processes ◽

Oscillatory Systems ◽

Multivariate Systems

Determination of synchronization phenomena between pairs of coupled multivariate processes is of particular interest in Nonlinear Dynamics. Besides synchronization phenomena, coupling directions between the processes are investigated. We present an approach to analyze coupling directions in multivariate oscillatory stochastic systems. We propose usage of partial directed coherence developed in the framework of linear stochastic processes. We show that partial directed coherence is also applicable to detect coupling directions in nonlinear systems such as coupled stochastic van der Pol and stochastic Rössler systems. Furthermore, a differentiation between direct and indirect couplings in multivariate systems is possible when applying partial directed coherence.

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A New Approach for the Numerical Integration of Nonlinear Dynamic Systems

10.1115/imece2000-2173 ◽

2000 ◽

Author(s):

Zhao Chang Zheng ◽

Zhi Xiao Su ◽

Ray P. S. Han ◽

Zefang Xu

Keyword(s):

Exact Solution ◽

Numerical Integration ◽

Forced Vibration ◽

Initial Conditions ◽

Duffing Oscillator ◽

Van Der Pol Oscillator ◽

Essential Difference ◽

Van Der Pol ◽

New Approach ◽

System Equations

Abstract In this paper, the exact solution of a linearlized equation is employed as initial conditions for the next timestep in the numerical integration of the end displacement and velocity. This exact solution is calculated by means of the Duhamel integration. The system equations are satisfied continuously and not discretely as done traditionally. The essential difference of the present method from other works is that the performance of dynamics systems can be traced continuously. Comparisons between the proposed method with traditional techniques are presented. Examples investigated include the large amplitude nonlinear vibration of a simple pendulum of a conservation system, the period of vibration and chaos in the forced vibration of the Duffing oscillator and forced vibration of the van der Pol oscillator of a non-conservation system. The results obtained indicate that the accuracy of the proposed method supersede that of the traditional techniques.

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Design of evolutionary optimized finite difference based numerical computing for dust density model of nonlinear Van-der Pol Mathieu’s oscillatory systems

Mathematics and Computers in Simulation ◽

10.1016/j.matcom.2020.10.004 ◽

2021 ◽

Vol 181 ◽

pp. 444-470

Author(s):

Ihtesham Jadoon ◽

Muhammad Asif Zahoor Raja ◽

Muhammad Junaid ◽

Ashfaq Ahmed ◽

Ata ur Rehman ◽

...

Keyword(s):

Finite Difference ◽

Density Model ◽

Van Der Pol ◽

Dust Density ◽

Oscillatory Systems ◽

Numerical Computing

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Nonlinear dynamics of a RLC series circuit modeled by a generalized Van der Pol oscillator

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2019-0031 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Yélomè Judicaël Fernando Kpomahou ◽

Clément Hodévèwan Miwadinou ◽

Richard Gilles Agbokpanzo ◽

Laurent Amoussou Hinvi

Keyword(s):

Nonlinear Dynamics ◽

System Parameter ◽

Nonlinear Ordinary Differential Equation ◽

Van Der Pol Oscillator ◽

Multiple Time Scales ◽

Multiple Time ◽

Bifurcation Diagrams ◽

External Excitation ◽

Van Der Pol ◽

Multiple Time Scales Method

Abstract In this paper, nonlinear dynamics study of a RLC series circuit modeled by a generalized Van der Pol oscillator is investigated. After establishing a new general class of nonlinear ordinary differential equation, a forced Van der Pol oscillator subjected to an inertial nonlinearity is derived. According to the external excitation strength, harmonic, subharmonic and superharmonic oscillatory states are obtained using the multiple time scales method. Bifurcation diagrams displayed by the model for each system parameter are performed numerically through the fourth-order Runge–Kutta algorithm.

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