Investigation on Dynamics of the Extended Duffing-Van der Pol System

Abstract The chaotic motion in periodic self-excited oscillators has been extensively investigated through experiments and computer simulations. However, with the advent of the study of chaotic motion by means of strange attractors, Poincar´e map, fractal dimension, it has become necessary to seek for a better understanding of nonlinear system with higher order nonlinear terms. In this paper we consider an extended Duffing-Van der Pol oscillator by introducing a nonlinear quintic term. The dynamical behaviour of the system is investigated by using Melnikov analysis and numerical simulation. The results can help one to understand the essence of given nonlinear system.

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Computer and Hardware Modeling of Periodically Forcedϕ6-Van der Pol Oscillator

Active and Passive Electronic Components ◽

10.1155/2016/3426713 ◽

2016 ◽

Vol 2016 ◽

pp. 1-7 ◽

Cited By ~ 1

Author(s):

A. O. Adelakun ◽

A. N. Njah ◽

O. I. Olusola ◽

S. T. Wara

Keyword(s):

Numerical Simulation ◽

Chaotic Dynamics ◽

Random Number ◽

Dynamical Behaviour ◽

Van Der Pol Oscillator ◽

Secure Communications ◽

Multiple Time ◽

Van Der Pol ◽

Simulation Results ◽

Good Agreement

Numerical simulation results for the dynamics ofϕ6-systems abound in the literature but their experimental results are yet to be known. This paper presents the chaotic dynamics ofϕ6-Van der Pol oscillator via electronic design, simulation, and hardware implementation. The results obtained are found to be in good agreement with numerical simulation results. The condition for stability of the fixed points is also computed and the method of multiple time scale is used to investigate the dynamical behaviour of the system. Therefore, theϕ6-circuits which have rich dynamics and may have important applications in secure communications, random number generations, cryptography, and so forth have been practically implemented.

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BIFURCATION SCENARIO OF A THREE-DIMENSIONAL VAN DER POL OSCILLATOR

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127493000283 ◽

1993 ◽

Vol 03 (02) ◽

pp. 399-404 ◽

Cited By ~ 5

Author(s):

T. SÜNNER ◽

H. SAUERMANN

Keyword(s):

Bifurcation Diagram ◽

Bifurcation Analysis ◽

Chaotic Behavior ◽

Three Dimensional ◽

Global Bifurcation ◽

Dynamical Behaviour ◽

Van Der Pol Oscillator ◽

Two Dimensional ◽

Van Der Pol ◽

Dimensional Models

Nonlinear self-excited oscillations are usually investigated for two-dimensional models. We extend the simplest and best known of these models, the van der Pol oscillator, to a three-dimensional one and study its dynamical behaviour by methods of bifurcation analysis. We find cusps and other local codimension 2 bifurcations. A homoclinic (i.e. global) bifurcation plays an important role in the bifurcation diagram. Finally it is demonstrated that chaos sets in. Thus the system belongs to the few three-dimensional autonomous ones modelling physical situations which lead to chaotic behavior.

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NEIMARK BIFURCATIONS OF A GENERALIZED DUFFING–VAN DER POL OSCILLATOR WITH NONLINEAR FRACTIONAL ORDER DAMPING

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127413501770 ◽

2013 ◽

Vol 23 (11) ◽

pp. 1350177 ◽

Cited By ~ 1

Author(s):

A. Y. T. LEUNG ◽

H. X. YANG ◽

P. ZHU

Keyword(s):

Fractional Order ◽

Harmonic Balance ◽

Harmonic Balance Method ◽

Viscoelastic Behavior ◽

Higher Order ◽

Balance Method ◽

Van Der Pol Oscillator ◽

Homotopy Continuation ◽

Van Der Pol ◽

Steady State Responses

A generalized Duffing–van der Pol oscillator with nonlinear fractional order damping is introduced and investigated by the residue harmonic homotopy. The cubic displacement involved in fractional operator is used to describe the higher-order viscoelastic behavior of materials and of aerodynamic damping. The residue harmonic balance method is employed to analytically generate higher-order approximations for the steady state responses of an autonomous system. Nonlinear dynamic behaviors of the harmonically forced oscillator are further explored by the harmonic balance method along with the polynomial homotopy continuation technique. A parametric investigation is carried out to analyze the effects of fractional order of damping and the effect of the magnitude of imposed excitation on the system using amplitude-frequency curves. Jump avoidance conditions are addressed. Neimark bifurcations are captured to delineate regions of instability. The existence of even harmonics in the Fourier expansions implies symmetry-breaking bifurcation in certain combinations of system parameters. Numerical simulations are given by comparing with analytical solutions for validation purpose. We find that all Neimark bifurcation points in the response diagram always exist along a straight line.

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On the Delayed van der Pol Oscillator with Time-Varying Feedback Gain

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.706.149 ◽

2014 ◽

Vol 706 ◽

pp. 149-158 ◽

Cited By ~ 2

Author(s):

Mustapha Hamdi ◽

Mohamed Belhaq

Keyword(s):

Numerical Simulation ◽

Type System ◽

Stationary Solutions ◽

Van Der Pol Oscillator ◽

Feedback Gain ◽

Time Varying ◽

Van Der Pol ◽

Existence Region ◽

The Stability ◽

Time Delayed Feedback

This work studies the effect of time delayed feedback on stationary solutions in a van derPol type system. We consider the case where the feedback gain is harmonically modulated with a resonantfrequency. Perturbation analysis is conducted to obtain the modulation equations near primaryresonance, the stability analysis for stationary solutions is performed and bifurcation diagram is determined.It is shown that the modulated feedback gain position can influence significantly the steadystates behavior of the delayed van der Pol oscillator. In particular, for appropriate values of the modulateddelay parameters, the existence region of the limit cycle (LC) can be increased or quenched.Moreover, new regions of quasiperiodic vibration may emerge for certain values of the modulatedgain. Numerical simulation was conducted to validate the analytical predictions.

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On using block pulse transform to perform equivalent linearization for a nonlinear Van der Pol oscillator under stochastic excitation

Nonlinear Engineering ◽

10.1515/nleng-2015-0022 ◽

2016 ◽

Vol 0 (0) ◽

Author(s):

Amir Younespour ◽

Hosein Ghaffarzadeh

Keyword(s):

Nonlinear System ◽

Present Method ◽

Gaussian White Noise ◽

Van Der Pol Oscillator ◽

Equivalent Linearization ◽

Van Der Pol ◽

White Noise Excitation ◽

Linearization Methods ◽

The Mean ◽

Methods Numerical

AbstractThis paper applied the idea of block pulse (BP) transform in the equivalent linearization of a nonlinear system. The BP transform gives effective tools to approximate complex problems. The main goal of this work is on using BP transform properties in process of linearization. The accuracy of the proposed method compared with the other equivalent linearization including the stochastic equivalent linearization and the regulation linearization methods. Numerical simulations are applied to the nonlinear Van der Pol oscillator system under Gaussian white noise excitation to demonstrate the feasibility of the present method. Different values of nonlinearity are considered to show the effectiveness of the present method. Besides, by comparing the mean-square responses for divers values of nonlinearity and excitation intensity depicted the present method is able to approximate the behavior of nonlinear system and is in agreement with other methods.

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Algorithms for Controlling chaos: Application to BVP Oscillator

International Astronomical Union Colloquium ◽

10.1017/s025292110006591x ◽

1993 ◽

Vol 132 ◽

pp. 45-45

Author(s):

S. Rajeskar

Keyword(s):

Computer Simulations ◽

Limit Cycles ◽

Digital Computer ◽

Chaotic Dynamics ◽

Van Der Pol Oscillator ◽

Time Dependent ◽

Van Der Pol ◽

Small Magnitude ◽

Controlling Chaos ◽

Regular Motion

AbstractWe discuss how chaotic dynamics can be converted into regular motion in Bonhoeffer-van der Pol oscillator. Using a control signal proportional to the actual and desired outputs we study the control of fixed points and limit cycles by making time-dependent perturbations of amplitude of external force. We show the round-off induced periodicity in the digital computer simulations of orbits on chaotic attractor. We illustrate the stabilization of unstable periodic or bits by adding periodic pulses of small magnitude.

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DYNAMICS OF SELF-OSCILLATIONS IN A TWO-STAGE VAN DER POL OSCILLATOR

Vestnik of Samara University Natural Science Series ◽

10.18287/2541-7525-2013-19-6-141-146 ◽

2017 ◽

Vol 19 (6) ◽

pp. 141-146

Author(s):

V.V. Zaitsev ◽

S.V. Lindt ◽

I.V. Stulov

Keyword(s):

Numerical Simulation ◽

Spatial Distribution ◽

Van Der Pol Oscillator ◽

Ring Oscillator ◽

Van Der Pol ◽

Two Stage ◽

Generation Threshold

The results of numerical simulation of self-oscillations in a two-stage ring oscillator with active cells of Van der Pol are presented. It is shown that for large exceedances of generation threshold in a system with identical cells, non-uniform spatial distribution of amplitudes of self-oscillations is observed.

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