Identification of van der Pol oscillator parameters

2021 ◽  
Vol 34 ◽  
pp. 11-18
Author(s):  
Iryna Baraniukova ◽  
Volodymyr Shcherbak
Keyword(s):  
Stability Property ◽  
Original System ◽  
Oscillatory Behavior ◽  
Parameters Estimation ◽  
Van Der Pol Oscillator ◽  
Extended System ◽  
Van Der Pol ◽  
Simulation Results ◽  
The Stability

An invariant relations method for parameters estimation is used for the wellknown van der Pol oscillator. The approach is based on dynamical extension of original system and synthesis of appropriate invariant relations, from which the unknowns can be expressed as a functions of the known quantities on the trajectories of extended system. The stability property is formally checked taking into account the oscillatory behavior of the system. The simulation results confirm efficiency of the proposed scheme of nonlinear identificator design.

Dynamics of Two Van Der Pol Oscillators Coupled via a Bath

2003 ◽  
Author(s):  
Erika Camacho ◽  
Richard Rand ◽  
Howard Howland
Keyword(s):  
Software Package ◽  
Bifurcation Theory ◽  
Floquet Theory ◽  
Van Der Pol Oscillator ◽  
Direct Connection ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Existence And Stability ◽  
Van Der Pol Oscillators ◽  
The Stability

In this work we study a system of two van der Pol oscillators, x and y, coupled via a “bath” z: x¨−ε(1−x2)x˙+x=k(z−x)y¨−ε(1−y2)y˙+y=k(z−y)z˙=k(x−z)+k(y−z) We investigate the existence and stability of the in-phase and out-of-phase modes for parameters ε > 0 and k > 0. To this end we use Floquet theory and numerical integration. Surprisingly, our results show that the out-of-phase mode exists and is stable for a wider range of parameters than is the in-phase mode. This behavior is compared to that of two directly coupled van der Pol oscillators, and it is shown that the effect of the bath is to reduce the stability of the in-phase mode. We also investigate the occurrence of other periodic motions by using bifurcation theory and the AUTO bifurcation and continuation software package. Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We present a simplified model of a circadian oscillator which shows that it can be modeled as a van der Pol oscillator. Although there is no direct connection between the two eyes, they can influence each other by affecting the concentration of melatonin in the bloodstream, which is represented by the bath in our model.

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

2014 ◽  
Vol 706 ◽  
pp. 149-158 ◽  
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.

Period-m Motions in a Periodically Forced van der Pol Oscillator

2013 ◽  
Author(s):  
Albert C. J. Luo ◽  
Arash Baghaei Lakeh
Keyword(s):  
Fourier Series ◽  
Periodic Motion ◽  
Approximate Solutions ◽  
Van Der Pol Oscillator ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Stable Period ◽  
Stability And Bifurcation Analysis ◽  
Series Expression ◽  
The Stability

Period-m motions in a periodically forced, van der Pol oscillator are investigated through the Fourier series expression, and the stability and bifurcation analysis of such periodic motions are carried out. To verify the approximate solutions of period-m motions, numerical illustrations are given. Period-m motions are separated by quasi-periodic motion or chaos, and the stable period-m motions are in independent periodic motion windows.

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

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
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.

scholarly journals Nonlinear Dynamics of a Periodically Driven Duffing Resonator Coupled to a Van der Pol Oscillator

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
X. Wei ◽  
M. F. Randrianandrasana ◽  
M. Ward ◽  
D. Lowe
Keyword(s):  
Strong Coupling ◽  
Internal Resonance ◽  
Weak Coupling ◽  
Response Pattern ◽  
Coupled System ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Periodically Driven ◽  
Coupling Case ◽  
The Stability

We explore the dynamics of a periodically driven Duffing resonator coupled elastically to a van der Pol oscillator in the case of 1 : 1 internal resonance in the cases of weak and strong coupling. Whilst strong coupling leads to dominating synchronization, the weak coupling case leads to a multitude of complex behaviours. A two-time scales method is used to obtain the frequency-amplitude modulation. The internal resonance leads to an antiresonance response of the Duffing resonator and a stagnant response (a small shoulder in the curve) of the van der Pol oscillator. The stability of the dynamic motions is also analyzed. The coupled system shows a hysteretic response pattern and symmetry-breaking facets. Chaotic behaviour of the coupled system is also observed and the dependence of the system dynamics on the parameters are also studied using bifurcation analysis.

Analytical Solutions of Period-1 Motions in a Periodically Forced van der Pol Oscillator

2012 ◽  
Author(s):  
Albert C. J. Luo ◽  
Arash Baghaei Lakeh
Keyword(s):  
Bifurcation Analysis ◽  
Analytical Solutions ◽  
Eigenvalue Analysis ◽  
Van Der Pol Oscillator ◽  
Comprehensive Understanding ◽  
Van Der Pol ◽  
Stability And Bifurcation ◽  
Approximate Analytical Solutions ◽  
Stability And Bifurcation Analysis ◽  
The Stability

In this paper, the approximate analytical solutions of period-1 motion in the periodically forced van der Pol oscillator are obtained by the generalized harmonic balanced method. The stability and bifurcation analysis of the period-1 solutions is completed through the eigenvalue analysis, and numerical illustrations of periodic-1 solutions are given to verify the approximate motion. This investigation provides more accurate solutions of period-1 motions in the van der pol oscillator for a better and comprehensive understanding of motions in such an oscillator.

Memristor Based van der Pol Oscillation Circuit

2014 ◽  
Vol 24 (12) ◽  
pp. 1450154 ◽  
Author(s):  
Yimin Lu ◽  
Xianfeng Huang ◽  
Shaobin He ◽  
Dongdong Wang ◽  
Bo Zhang
Keyword(s):  
Analog Circuit ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Circuit Element ◽  
Nonlinear Properties ◽  
Oscillation Circuit ◽  
Excited Oscillation ◽  
Exciting Source ◽  
The Stability ◽  
Passive Circuit

The memristor is referred to as the fourth fundamental passive circuit element of which inherent nonlinear properties offer to construct the chaos circuits. In this paper, a flux-controlled memristor circuit is developed, and then a van der Pol oscillator is implemented based on this new memristor circuit. The stability of the circuit, the occurring conditions of Hopf bifurcation and limit circle of the self-excited oscillation are analyzed; meanwhile, under the condition of the circuit with an external exciting source, the circuit exhibits a complicated nonlinear dynamic behavior, and chaos occurs within a certain parameter set. The memristor based van der Pol oscillator, furthermore, has been created by an analog circuit utilizing active elements, and there is a good agreement between the circuit responses and numerical simulations of the van der Pol equation. In the consequence, a new approach has been proposed to generate chaos within a nonautonomous circuit system.

Stochastic Stability of Some Hamiltonian Systems

2002 ◽  
Author(s):  
F. Jedrzejewski
Keyword(s):  
Hamiltonian Systems ◽  
Stochastic Stability ◽  
Stochastic Modelling ◽  
Original System ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Van Der Pol Oscillator ◽  
Stochastic Fluctuations ◽  
The Stability ◽  
Averaged Hamiltonian

Stochastic stability plays an important role in modern theories of nonlinear structural dynamics. Recently, more realistic models based on stochastic modelling and Itoˆ calculus, like flow induced vibrations and seismic excitations have been proposed. In this paper, the almost-sure asymptotic stability of some hamiltonian systems subjected to stochastic fluctuations is investigated. Dynamical systems are reduced to Itoˆ stochastic differential equations for the averaged hamiltonian by using a new stochastic averaging method. The stability of the original system is determined approximately by examining the behavior of the averaged hamiltonian. Analytical expressions for the stochastic stability exponents are obtained. The proposed procedure is illustrated on the Rayleigh Van der Pol Oscillator.

A New Fractional Order Observer Design for Fractional Order Nonlinear Systems

2011 ◽  
Author(s):  
Sara Dadras ◽  
Hamid Reza Momeni
Keyword(s):  
Fractional Order ◽  
Original System ◽  
Observer Design ◽  
Fractional Order Systems ◽  
State Variables ◽  
Lyapunov Approach ◽  
Error System ◽  
Simulation Results ◽  
System States ◽  
The Stability

In this paper, a class of fractional order systems is considered and simple fractional order observers have been proposed to estimate the system’s state variables. By introducing a fractional calculus into the observer design, the developed fractional order observers guarantee the estimated states reach the original system states. Using the fractional order Lyapunov approach, the stability (zero convergence) of the error system is investigated. Finally, the simulation results demonstrate validity and effectiveness of the proposed scheme.

scholarly journals Stability Switches and Double Hopf Bifurcation Analysis on Two-Degree-of-Freedom Coupled Delay van der Pol Oscillator

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2444
Author(s):  
Yani Chen ◽  
Youhua Qian
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form ◽  
Degree Of Freedom ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Double Hopf Bifurcation ◽  
Central Manifold ◽  
The Stability ◽  
Two Degree Of Freedom

In this paper, the normal form and central manifold theories are used to discuss the influence of two-degree-of-freedom coupled van der Pol oscillators with time delay feedback. Compared with the single-degree-of-freedom time delay van der Pol oscillator, the system studied in this paper has richer dynamical behavior. The results obtained include: the change of time delay causing the stability switching of the system, and the greater the time delay, the more complicated the stability switching. Near the double Hopf bifurcation point, the system is simplified by using the normal form and central manifold theories. The system is divided into six regions with different dynamical properties. With the above results, for practical engineering problems, we can perform time delay feedback adjustment to make the system show amplitude death, limit loop, and so on. It is worth noting that because of the existence of unstable limit cycles in the system, the limit cycle cannot be obtained by numerical solution. Therefore, we derive the approximate analytical solution of the system and simulate the time history of the interaction between two frequencies in Region IV.

Sign in / Sign up

Export Citation Format

Share Document