Van Der Pol Type Oscillator Research Articles

An effective mathematical model of cardiac electrical activities for Bundle Branch Blocks

In this paper a mathematical model of cardiac induction system is developed. Sinoatrial node (SA), atrioventricular node (AV) and His Purkinje system (HP) are represented by modified Van der Pol-type oscillators (VDP) connected with time-delay coupling. Atrial and ventricular muscles are modeled using modified Aliev-Panfilov equations (AP), with stimulation current from the related pacemakers, to represent the P, QRS, Ta, and T waves. The main aim of this paper is to model bundle branch blocks (BBBs) for right (RBBB) and left (LBBB) branches. In fact, the right and left ventricles are directly affected by these blocks since the HP bundle is composed of right and left bundle branches that provide electrical signals, respectively, to the right and left ventricles. Four VDP oscillators are utilized, one for each of SA, AV, right HP, and left HP. The depolarization of the right and left ventricular muscles are represented by two systems of AP equations to produce right QRS and left QRS waves. The results of the simulation are compared with real normal and pathological electrocardiograms (ECGs) to show the effectiveness and the accuracy of the proposed model. The simulation proves that the model can reproduce the blocks in the HP bundle branches.

Towards the Resolution of a Quantized Chaotic Phase-Space: The Interplay of Dynamics with Noise.

We outline formal and physical similarities between the quantum dynamics of open systems and the mesoscopic description of classical systems affected by weak noise. The main tool of our interest is the dissipative Wigner equation, which, for suitable timescales, becomes analogous to the Fokker-Planck equation describing classical advection and diffusion. This correspondence allows, in principle, to surmise a finite resolution, other than the Planck scale, for the quantized state space of the open system, particularly meaningful when the latter underlies chaotic classical dynamics. We provide representative examples of the quantum-stochastic parallel with noisy Hopf cycles and Van der Pol-type oscillators.

Can Lévy noise induce coherence and stochastic resonances in a birhythmic van der Pol system?

The analysis of a birhythmic modified van der Pol type oscillator driven by periodic excitation and Lévy noise shows the possible occurrence of coherence resonance and stochastic resonance. The frequency of the harmonic excitation in the neighborhood of one of the limit cycles influences the coherence of the dynamics on the time scale of the oscillations. The autocorrelation function, the power spectral density and the signal-to-noise-ratio used in this analysis are shown to be maximized for an appropriate choice of the noise intensity. In particular, a proper adjustment of the Lévy noise intensity enhances the output power spectrum of the system, that is, promotes stochastic resonance. Thus, the resonance, as examined using standard measures, seems to occur also in the presence of nonstandard noise. The initial selection of the attractor seems to have an influence on the coherence, while the skewness parameter of the Lévy noise has not a notable impact on the resonant effect.

Dynamics of the forced negative conductance series LCR circuit

SummaryIn this paper, we make a revisit of a simple LCR circuit introduced earlier in 2005 by Thamilmaran et. al., and present the circuit simulation studies of its dynamics using the PSpice circuit simulator package. This is a simple sinusoidally forced series LCR circuit, employing an ordinary single PN junction diode in conjunction with a negative conductance as its nonlinearity. In circuit theoretic terms, this circuit is a modified form of the forced van der Pol type oscillator and is interesting because it exhibits a dual nature, namely, a forced van der Pol type circuit behaviour and an MLC circuit type behaviour for different parametric ranges. In addition, statistical studies have shown the circuit to possess strong chaoticity. Subsequent works on this circuit have shown the birth of strange nonchaos through four different routes, namely, the Heagy‐Hammel route, gradual fractalization route, intermittency route, and Bubbling route. In lieu of these, we look at the circuit afresh and study its dynamics in an exhaustive manner. These studies reveal many new bifurcations and routes to chaos. These results have been verified using hardware experiments in addition to numerical computations.

Piecewise Semi-Analytical Formulation for the Analysis of Coupled-Oscillator Systems

A new simulation technique to obtain the synchronized steady-state solutions existing in coupled oscillator systems is presented. The technique departs from a semi-analytical formulation presented in previous works. It extends the model of the admittance function describing each individual oscillator to a piecewise linear one. This provides a global formulation of the coupled system, considering the whole characteristic of each voltage-controlled oscillator (VCO) in the array. In comparison with the previous local formulation, the new formulation significantly improves the accuracy in the prediction of the system synchronization ranges. The technique has been tested by comparison with computationally demanding circuit-level harmonic balance simulations in an array of Van der Pol-type oscillators and then applied to a coupled system of FET-based oscillators at 5 GHz, with a very good agreement with measurements.

Coherence and stochastic resonance in a birhythmic van der Pol system

We consider the response to uncorrelated noise and harmonic excitation of a birhythmic van der Pol-type oscillator. This system, as opposed to the standard van der Pol oscillator, is characterized by two stable orbits. The noisy oscillator can be analytically mapped, with the technique of stochastic averaging, onto an ordinary bistable system with a bistable (quasi)potential. The birhythmic oscillator can also be numerically characterized through the diagnostics of coherent resonance and the signal-to-noise-ratio. The analysis shows the presence of noise-induced coherent states, influenced by the different time scales of the oscillator.

2DOF CFD calibrated wake oscillator model to investigate vortex-induced vibrations

In this study a new two degrees-of-freedom wake oscillator model is proposed to describe vortex-induced vibrations of elastically supported cylinders capable of moving in cross-flow and in-line directions. The total hydrodynamic force acting on the cylinder is obtained here as a sum of lift and drag forces, which are defined as being proportional to the square of the magnitude of the relative flow velocity around the cylinder. The two van der Pol type oscillators are then used to model fluctuating drag and lift coefficients. As the relative velocity around the cylinder depends both on the fluid flow velocity and the velocity of the cylinder, the equations of motions of the cylinder in cross-flow and in-line directions become coupled through the fluid forces. It is shown that such approximation of the fluid forces allows to obtain the well known low dimensional models in the limit case, and the model proposed by Facchinetti et al. [1] to describe the cross-flow vibrations is used as an example. Existing experimental data and Computational Fluid Dynamics (CFD) results are used to calibrate the proposed model and to verify the obtained predictions of complex fluid-structure interactions for different mass ratios. The "super upper" branch phenomenon, exclusive for a two degrees-of-freedom motion at low mass ratios, has been observed. The influence of the empirical parameters of the wake oscillators and fluid forces coefficients on the dynamic responses is also discussed.

VIV fatigue reliability analysis of marine risers with uncertainties in the wake oscillator model

Uncertainties are rife in the fatigue life prediction of marine risers subjected to vortex-induced vibration (VIV). Industry deals with this issue by imposing large factors of safety that may not be properly justified, resulting in over-conservative riser designs in general. One important source of uncertainty arises from the VIV prediction models. This paper focusses on identifying the uncertainties of a wake oscillator model which approximates the fluctuating hydrodynamic force coupled with the riser equation of motion for nonlinear fluid–structure interaction analysis. This van der Pol-type oscillator relies on two wake coefficients which are described deterministically by empirical equations obtained via curve-fitting. However, the underlying data exhibit wide scatter; thus, it is proposed to model the two key coefficients as random variables. Based on experimental data, the joint probability density function of the variables is approximated. A new fast reliability approach is proposed for the VIV fatigue reliability analysis, while Monte Carlo simulations are performed for comparisons. Case studies of a vertical riser in a uniform flow show that the proposed method compares favorably with Monte Carlo in terms of predicting the failure probability as well as safety factors conforming to prescribed reliability levels. Moreover, this study reveals that the randomness of wake coefficients leads to large variability in the riser fatigue damage. The correlation between the coefficients should be properly incorporated as it affects the fatigue reliability of risers experiencing VIV.

Stochastic bifurcations induced by correlated noise in a birhythmic van der Pol system

We investigate the effects of exponentially correlated noise on birhythmic van der Pol type oscillators. The analytical results are obtained applying the quasi-harmonic assumption to the Langevin equation to derive an approximated Fokker–Planck equation. This approach allows to analytically derive the probability distributions as well as the activation energies associated to switching between coexisting attractors. The stationary probability density function of the van der Pol oscillator reveals the influence of the correlation time on the dynamics. Stochastic bifurcations are discussed through a qualitative change of the stationary probability distribution, which indicates that noise intensity and correlation time can be treated as bifurcation parameters. Comparing the analytical and numerical results, we find good agreement both when the frequencies of the attractors are about equal or when they are markedly different.

A heterogeneous coupled oscillator model for simulation of ECG signals

We present a novel model of cardiac conduction system including main pacemakers and heart muscles. Sinoatrial node, atrioventricular node and His–Purkinje system are represented by modified van der Pol-type oscillators connected with time-delay velocity coupling. For description of atrial and ventricular muscles, where depolarization and repolarization processes are considered as separate waves, we use modified FitzHugh–Nagumo model. In this work, we obtained synthetic ECG as a combined signal of atrial and ventricular muscles and reproduced several normal and pathological rhythms. Inclusion of cardiac muscle response allows to investigate interactions between pacemakers and resulting global heartbeat dynamics by means of clinically comparable realistic ECG signals. This feature distinguishes our model from existing cardiac oscillator models. To solve the system of differential equations describing the proposed heterogeneous coupled oscillator model we developed a software in MATLAB environment utilizing special DDE23 function.

Exploring a noisy van der Pol type oscillator with a stochastic approach

Based on conventional Ito or Stratonovich interpretation, zero-mean multiplicative noise can induce shifts of attractors or even changes of topology to a deterministic dynamics. Such phenomena usually introduce additional complications in analysis of these systems. We employ in this paper a new stochastic interpretation leading to a straightforward consequence: The steady state distribution is Boltzmann-Gibbs type with a potential function severing as a Lyapunov function for the deterministic dynamics. It implies that an attractor corresponds to the local extremum of the distribution function and the probability is equally distributed right on an attractor. We consider a prototype of nonequilibrium processes, noisy limit cycle dynamics. Exact results are obtained for a class of limit cycles, including a van der Pol type oscillator. These results provide a new angle for understanding processes without detailed balance and can be verified by experiments.

One-dimensional Fermi accelerator model with moving wall described by a nonlinear van der Pol oscillator

A modification of the one-dimensional Fermi accelerator model is considered in this work. The dynamics of a classical particle of mass m, confined to bounce elastically between two rigid walls where one is described by a nonlinear van der Pol type oscillator while the other one is fixed, working as a reinjection mechanism of the particle for a next collision, is carefully made by the use of a two-dimensional nonlinear mapping. Two cases are considered: (i) the situation where the particle has mass negligible as compared to the mass of the moving wall and does not affect the motion of it; and (ii) the case where collisions of the particle do affect the movement of the moving wall. For case (i) the phase space is of mixed type leading us to observe a scaling of the average velocity as a function of the parameter (χ) controlling the nonlinearity of the moving wall. For large χ, a diffusion on the velocity is observed leading to the conclusion that Fermi acceleration is taking place. On the other hand, for case (ii), the motion of the moving wall is affected by collisions with the particle. However, due to the properties of the van der Pol oscillator, the moving wall relaxes again to a limit cycle. Such kind of motion absorbs part of the energy of the particle leading to a suppression of the unlimited energy gain as observed in case (i). The phase space shows a set of attractors of different periods whose basin of attraction has a complicated organization.

A generalized van der Pol type oscillator: Investigation of the properties of its limit cycle

A generalized van der Pol type oscillator is considered, with power-form nonlinearities in the restoring and damping force. The amplitude of the limit cycle is obtained by applying the generalized Krylov–Bogoliubov method for purely nonlinear oscillators. The influence of the values of the powers of the damping and the restoring force on this amplitude is investigated both analytically and numerically. The limiting values of this amplitude are found and the time for the amplitude to ‘nearly’ reach the limit cycle is estimated.

Preliminary analytical and numerical investigations of a van der Pol type oscillator having discontinuous dependence on the velocity

Preliminary analytical and numerical investigations of a van der Pol type oscillator having discontinuous dependence on the velocity

A simpler model of the human circadian pacemaker.

Numerous studies have used the classic van der Pol oscillator, which contains a cubic nonlinearity, to model the effect of light on the human circadian pacemaker. Jewett and Kronauer demonstrated that Aschoff's rule could be incorporated into van der Pol type models and used a van der Pol type oscillator with higher order nonlinearities. Kronauer, Forger, and Jewett have proposed a model for light preprocessing, Process L, representing a biochemical process that converts a light signal into an effective drive on the circadian pacemaker. In the paper presented here, the authors use the classic van der Pol oscillator with Process L and Jewett and Kronauer's model of Aschoff's rule to model the human circadian pacemaker. This simpler cubic model predicts the results of a three-pulse human phase response curve experiment and a two-pulse amplitude reduction study with as much, or more, accuracy as the models of Jewett and Kronauer and Kronauer, Forger, and Jewett, which both employ a nonlinearity of degree 7. This suggests that this simpler cubic model should be considered as a potential alternative to other models of the human circadian system currently available.

Van der Pol behavior of virtual anode oscillations in the sheath around a grid in a double plasma device

Experiments are reported on oscillations that arise in a double plasma device when plasma production is restricted to the source chamber and the separating grid between the two chambers is biased negatively. The free oscillating system shows periodic pulling which is a typical behavior of driven van der Pol type oscillators. The second interacting frequency is identified to be half the ion plasma frequency at the sheath edge on the source side. With the help of particle in cell simulations the concept of virtual anode oscillations (VAO’s) as the underlying oscillation mechanism is investigated and the van der Pol character of these is revealed. When applied to the experimental conditions, the VAO-model predicts correct oscillation frequencies. It gives a new interpretation of the scaling of these with plasma density and grid bias, and is compatible with earlier findings.

Power Spectral Density of Nonlinear System Response: The Recursion Method

Spectral densities of the response of nonlinear systems to white noise excitation are considered. By using a formal solution of the associated Fokker-Planck-Kolmogorov equation, response spectral densities are represented by formal power series expansion for large frequencies. The coefficients of the series, known as the spectral moments, are determined in terms of first-order response statistics. Alternatively, a J-fraction representation of spectral densities can be achieved by using a generalization of the Lanczos algorithm for matrix tridiagonalization, known as the “recursion method.” Sequences of rational approximations of increasing order are obtained. They are used for numerical calculations regarding the single-well and double-well Duffing oscillators, and Van der Pol type oscillators. Digital simulations demonstrate that the proposed approach can be quite reliable over large variations of the system parameters. Further, it is quite versatile as it can be used for the determination of the spectrum of the response of a broad class of randomly excited nonlinear oscillators, with the sole prerequisite being the availability, in exact or approximate form, of the stationary probability density of the response.

Synchronization of infinitely many coupled limit-cycle type oscillators

Phenomena of synchronization are studied for systems of infinitely many van der Pol type oscillators which have symmetrically distributed native frequencies and are coupled with each other via mean-field interactions of linear form. Totally synchronized (TS), quenched (Q), partially synchronized (PS), and totally dissynchronized (NS) states are shown to appear depending on the values of the coupling strength and the width of the native frequency distribution of the systems. The phase boundary between the TS and Q phases for the systems with strong couplings is rigorously determined on the basis of mean-field theoretical analysis involving the concept of order parameter.

Nonlinear parametric oscillations in certain stochastic systems: A random van der Pol oscillator

Astochastic model for some class ofnonlinear oscillators, which includes a van der Pol-type oscillator with random parameters, is analyzed in thediffusion limit. That is, small random fluctuations and long time are considered, while the nonlinearity is also assumed to be small. We show that there existstationary distributions, independent of the phase of the oscillator, a result proved earlier by R. L. Stratonovich assuming the random perturbations of the frequency to be delta correlated. The time behavior of the moments of the displacement of the oscillator from its rest position is also investigated and the results are compared with the corresponding ones for the linear random oscillator. A numerical study is also performed for the first two moments and plots are given.

Multimode oscillations in mutually coupled van der Pol type oscillators with fifth-power nonlinear characteristics

Coupled van der Pol oscillators containing a fifth-order conductance characteristic are analyzed using a method due to Endo and Mori. This structure has been proposed for modeling the myoelectrical activity of the human large intestine since it produces stable conditions of zero, two single-mode frequencies, and also a double-mode condition. The paper first considers the case of oscilators having equal frequencies and it is shown that for a certain range of parameters double modes are possible for two oscillators, which is in contrast to the case of conventional van der Pol dynamics. The theoretical results have been experimentally verified using both analog simulations and electronic circuitry. In digestive tract modeling, the coupled oscillators must often have unequal intrinsic frequencies and this case is also considered and shown to give similar but more complex parameter conditions for the establishment of double modes.