Modified Van Der Pol Equation Research Articles

Λ -Fractional Forced Van der Pol Oscillator with Horizon

Extending the investigation of the modified Van der Pol equation with its chaotic behaviour into the context of Λ -fractional analysis, the solutions of the Λ-fractional Van der Pol oscillator are studied under the influence of a horizon restricting the region of non-locality. The various solutions to the equation depend upon the horizon, however, the chaotic behaviour of the oscillator mainly depends upon the fractional order. A comparison of the chaotic behaviour of the various Van der Pol oscillator systems is presented.

Feedback Stabilization Applied to Heart Rhythm Dynamics Using an Integro-Differential Method

In this paper, we applied a chaos control method based on integro-differential equations for stabilization of an unstable cardiac rhythm, which is described by a variation of the modified Van der Pol equation. Chaos control with this method may be useful for stabilization of irregular heartbeat using a small perturbation. This method differs from other stabilization strategies by the absence of adjustable parameters and the lack of rough approximations in determining control functions whose control parameters are fixed by the properties of the unstable system itself.

Central pattern generator network model for the alternating hind limb gait of rats based on the modified Van der Pol equation.

Herein, we employed a central pattern generator (CPG), a spinal cord neural network that regulates lower-limb gait during intra-spinal micro-stimulation (ISMS). Particularly, ISMS was used to determine the spatial distribution pattern of CPG sites in the spinal cord and the signal regulation pattern that induced the CPG network to produce coordinated actions. Based on the oscillation phenomenon of the single CPG neurons of Van der Pol (VDP) oscillators, a double-cell CPG neural network model was constructed to realise double lower limbs, six-joint modelling, the simulation of 12 neural circuits, the CPG loci characterising stimuli-inducing alternating movements and changes in polarity stimulation signals in rat hindlimbs, and leg-state change movements. The feasibility and effectiveness of the CPG neural network were verified by recording the electromyographic burst-release mode of the flexor and extensor muscles of the knee joints during CPG electrical stimulation. The results revealed that the output pattern of the CPG presented stable rhythm and coordination characteristics. The 12-neuron CPG model based on the improved VDP equation realised single-point control while significantly reducing the number of stimulation electrodes in the gait training of spinal cord injury patients. We believe that this study advances motor function recovery in rehabilitation medicine.

Stochastic dynamics of the FitzHugh-Nagumo neuron model through a modified Van der Pol equation with fractional-order term and Gaussian white noise excitation

The stochastic response of the FitzHugh-Nagumo model is addressed using a modified Van der Pol (VDP) equation with fractional-order derivative and Gaussian white noise excitation. Via the generalized harmonic balance method, the term related to fractional derivative is splitted into the equivalent quasi-linear dissipative force and quasi-linear restoring force, leading to an equivalent VDP equation without fractional derivative. The analytical solutions for the equivalent stochastic equation are then investigated through the stochastic averaging method. This is thereafter compared to numerical solutions, where the stationary probability density function (PDF) of amplitude and joint PDF of displacement and velocity are used to characterized the dynamical behaviors of the system. A satisfactory agreement is found between the two approaches, which confirms the accuracy of the used analytical method. It is also found that changing the fractional-order parameter and the intensity of the Gaussian white noise induces P-bifurcation.

Cardiac Conduction Model for Generating 12 Lead ECG Signals With Realistic Heart Rate Dynamics

We present an extended heterogeneous oscillator model of cardiac conduction system for generation of realistic 12 lead ECG waveforms. The model consists of main natural pacemakers represented by modified van der Pol equations, and atrial and ventricular muscles, in which the depolarization and repolarization processes are described by modified FitzHugh-Nagumo equations. We incorporate an artificial RR-tachogram with the specific statistics of a heart rate, the frequency-domain characteristics of heart rate variability produced by Mayer and respiratory sinus arrhythmia waves, normally distributed additive noise and a baseline wander that couple the respiratory frequency. The standard 12 lead ECG is calculated by means of a weighted linear combination of atria and ventricle signals and thus can be fitted to clinical ECG of real subject. The model is capable to simulate accurately realistic ECG characteristics including local pathological phenomena accounting for biophysical properties of the human heart. All these features provide significant advantages over existing nonlinear cardiac models. The proposed model constitutes a useful tool for medical education and for assessment and testing of ECG signal processing software and hardware systems.

Dynamical analysis of the FitzHugh–Nagumo oscillations through a modified Van der Pol equation with fractional-order derivative term

The nonlinear dynamics of action potentials in the FitzHugh–Nagumo model is addressed using a modified Van der Pol equation with fractional-order derivative and periodic parametric excitation. Through the averaging method, the approximately analytical and the steady-state solutions are obtained, and their existence condition and stability are investigated. Analytical calculations are confirmed numerically and one insists on the coupled effects of the parametric excitation, system parameters and fractional-order parameter to discuss the various dynamical behaviors of the studied system. Mainly, the fractional-order derivative modifies the features of the amplitude–frequency curves. This might be an efficient tool to control the dynamics of the action potentials, with important biological implications that are discussed.

A higher-order nonlinear oscillator model for coupled cross-flow and in-line VIV of a circular cylinder

ABSTRACTIn this paper, a higher-order nonlinear oscillator model has been proposed to predict the characteristics of the vortex-induced vibration (VIV) of a cylinder with two degrees of freedom. First, the vibration equations of the cylinder fixed by four symmetric springs were deduced based on the Lagrangian second kind of kinetic equation and the Taylor expansion formula of the binary function. Then, the fluctuating lift and drag were derived by using the discrete point vortex theory. Finally, the coupling model of the VIV was proposed by introducing the modified Van der Pol equation. On this basis, the VIV prediction of the cylinders with different mass ratios and damping ratios was carried out, and the results were compared with the experimental data to verify correctness and universality of the model. Besides, the sensitivity of the parameters in the model was analysed to investigate the predicting characteristics of the model systematically.

Reduced-Order Modeling of Torque on a Vertical-Axis Wind Turbine at Varying Tip Speed Ratios

Vertical-axis wind turbine (VAWT) has received significant attention due to its application in urban environment. Torque produced by VAWT determines its efficiency and power output. In this paper, we develop a reduced-order model of torque VAWT at different tip speed ratios (TSR). We numerically simulate both 2D and 3D flows past a three-bladed Darrieus H-type VAWT and compute overall torque acting on the turbine. We then perform higher-order spectral analysis to identify dominant frequencies and nonlinear couplings. We propose a reduced-order model of torque in the form of modified van der Pol equation with additional quadratic term to allow for even harmonics in addition to odd harmonics present in the system. Using, a perturbation approach of method of multiple scales, we solve the proposed model and compute the coefficients at different TSR. The model not only predicts torque accurately in time domain but also in spectral domain. These reduced-order models provide an accurate and computationally efficient means to predict overall performance and output of the turbine with varying free-stream conditions even in predictive setting.

A modified van der Pol equation with delay in a description of the heart action

Abstract In this paper, a modified van der Pol equation is considered as a description of the heart action. This model has a number of interesting properties allowing reconstruction of phenomena observed in physiological experiments as well as in Holter electrocardiographic recordings. Our aim is to study periodic solutions of the modified van der Pol equation and take into consideration the influence of feedback and delay which occur in the normal heart action mode as well as in pathological modes. Usage of certain values for feedback and delay parameters allows simulating the heart action when an accessory conducting pathway is present (Wolff-Parkinson-White syndrome).

Numerical Analysis of the Coupled Modified van der Pol Equations in a Model of Heart Action

In this paper, a modified van der Pol equations are considered as a description of the heart action. Wide ranges of the model parameters yield interesting qualitative results, e.g. Hopf bifurcation, Bogdanov-Takens bifurcation, transcritical and pitchfork bifurcations but also some stable solutions can be found. The physiological model works in the narrowest range of parameters which allows to obtain a stable behaviour what is important in biological problem. When some kinds of pathologies appear in the heart, it is possible to obtain chaotic behaviour. My aim is to compare the influence of these two types of coupling (unidirectional and bidirectional) on the behaviour of the van der Pol system. The coupling takes place in a system with healthy conductivity, between two nodes: SA and AV, but in some circumstances, a pathological coupling may occur in the heart. The van der Pol oscillator is a type of relaxation oscillator which can be synchronized. In this paper, synchronization properties of such a system are studied as well. For the purpose of a numerical analysis of the system in question, a numerical model was created.

Numerical Analysis of the Coupled Modified van der Pol Equations in a Model of Heart Action

In this paper, a modified van der Pol equations are considered as a description of the heart action. Wide ranges of the model parameters yield interesting qualitative results, e.g. Hopf bifurcation, Bogdanov-Takens bifurcation, transcritical and pitchfork bifurcations but also some stable solutions can be found. The physiological model works in the narrowest range of parameters which allows to obtain a stable behaviour what is important in biological problem. When some kinds of pathologies appear in the heart, it is possible to obtain chaotic behaviour. My aim is to compare the influence of these two types of coupling (unidirectional and bidirectional) on the behaviour of the van der Pol system. The coupling takes place in a system with healthy conductivity, between two nodes: SA and AV, but in some circumstances, a pathological coupling may occur in the heart. The van der Pol oscillator is a type of relaxation oscillator which can be synchronized. In this paper, synchronization properties of such a system are studied as well. For the purpose of a numerical analysis of the system in question, a numerical model was created.

ECG Generation Based on Action Potential Using Modified Van Der Pol Equation

Aims: The aim of this paper is to propose a mathematical model of the cardiac electrical activity in accordance with heart physiology and utilizing this model to generate ECG (electrocardiogram) signal. Study Design: Modeling the cardiac electrical activity based on physiology of the heart. Place and Duration of Study: The paper reports the result of a year research in Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran. Methodology: In order to model the cardiac electrical activity, cardiac cells are divided into six parts whose action potentials are modeled by modified Van der Pol equation. Moreover, the parts are coupled unidirectionally and also the propagation delays are included. Results: The ECG is computed by adding a trial and ventricular potential differences. Conclusion: The resultant action potentials and also ECG signal are compared with real data. The results verify the proposed approach.

Chaos control applied to heart rhythm dynamics

The dynamics of cardiovascular rhythms have been widely studied due to the key aspects of the heart in the physiology of living beings. Cardiac rhythms can be either periodic or chaotic, being respectively related to normal and pathological physiological functioning. In this regard, chaos control methods may be useful to promote the stabilization of unstable periodic orbits using small perturbations. In this article, the extended time-delayed feedback control method is applied to a natural cardiac pacemaker described by a mathematical model. The model consists of a modified Van der Pol equation that reproduces the behavior of this pacemaker. Results show the ability of the chaos control strategy to control the system response performing either the stabilization of unstable periodic orbits or the suppression of chaotic response, avoiding behaviors associated with critical cardiac pathologies.

Synthesis and experimental verification of sinusoidal oscillator based on the modified Van der Pol equation

The article presents a method for synthesis of sinusoidal oscillators based on the modified Van der Pol equation. The synthesis applies the Melnikov theory, which allows obtaining a differential equation with amplitude and frequency assigned in advance. Formulas for determination of basic parameters of the oscillations are presented. The modified Van der Pol equation is practically modelled by a Field Programmable Analogue Array AN221E04 of Anadigm Inc. The obtained experimental results demonstrate the basic theoretical relations.

Autoparametric oscillatory system with chaotic dynamics

A mathematical model of the new chaotic system based on a modified van der Pol equation featuring delayed parametric self-action is considered. The chaotization of oscillations in this “autoparametric” system is analyzed by numerical methods.

Synchronization of diffusively coupled oscillators near the homoclinic bifurcation.

It has been known that a diffusive coupling between two limit cycle oscillations typically leads to the in-phase synchronization and also that it is the only stable state in the weak-coupling limit. Recently, however, it has been shown that the coupling of the same nature can result in the distinctive dephased synchronization when the limit cycles are close to the homoclinic bifurcation, which often occurs especially for the neuronal oscillators. In this paper we propose a simple physical model using the modified van der Pol equation, which unfolds the generic synchronization behaviors of the latter kind and in which one may readily observe changes in the sychronization behaviors between the distinctive regimes as well. The dephasing mechanism is analyzed both qualitatively and quantitatively in the weak-coupling limit. A general form of coupling is introduced and the synchronization behaviors over a wide range of the coupling parameters are explored to construct the phase diagram using the bifurcation analysis.

Two timescale harmonic balance. I. Application to autonomous one-dimensional nonlinear oscillators

Two timescale harmonic balance is a semi-analytical/numerical method for deriving periodic solutions and their stability to a class of nonlinear autonomous and forced oscillator equations of the form ẍ + x = f(x,ẋ,λ) and ẍ + x = f(x,ẋ,λ,t) , where λ is a control parameter. The method incorporates salient features from both the method of harmonic balance and multiple scales, and yet does not require an explicit small parameter. Essentially periodic solutions are formally derived on the basis of a single assumption: ‘that an N harmonic, truncated, Fourier series and its first two derivatives can represent x(t) , ẋ(t) and ẍ(t) respectively’. By seeking x(t) as a series of superharmonics, subharmonics, and ultrasubharmonics it is found that the method works over a wide range of parameter space provided the above assumption holds which, in practice, imposes some ‘problem dependent’ restriction on the magnitude of the nonlinearities. Two timescales, associated with the amplitude and phase variations respectively, are introduced by means of an implicit parameter Є . These timescales permit the construction of a set of amplitude evolution equations together with a corresponding stability criterion. In Part I the method is formulated and applied to three autonomous equations, the van der Pol equation, the modified van der Pol equation, and the van der Pol equation with escape. In this case an expansion in superharmonics is sufficient to reveal Hopf, saddle node and homoclinic bifurcations which are compared with results obtained by numerical integration of the equations. In Part II the method is applied to forced nonlinear oscillators in which the solution for x(t) includes superharmonics, subharmonics, and ultrasubharmonics. The features of period doubling, symmetry breaking, phase locking and the Feigenbaum transition to chaos are examined.

Active control of flow-induced vibrations of a flexible cylinder using direct velocity feedback

The use of a direct velocity feedback controller to attenuate actively vortex-induced vibrations of a flexible circular cylinder is described. The controller relies on its operation on an electromagnetic actuator, placed at a critical location inside the cylinder, to generate the control action necessary to resist the flow-induced excitations. The actuator is powered by a signal proportional and opposite to the velocity of the cylinder in order to enhance the overall viscous damping characteristics of the cylinder-actuator system. A prototype of the system was built, with a 3·175 cm diameter cylinder 26 cm long which has a first resonant frequency of 57·5 Hz. The system was tested in a wind tunnel, over a range of Reynolds number between 17 160 and 26 555, to demonstrate the feasibility of the controller in damping out the vortex-induced vibrations. The effect of varying the controller gain and the flow velocity on the system performance was determined in the frequency domain. The results obtained indicate that attenuations in excess of 80% are attained at resonant conditions. The experimental results are compared with the theoretical predictions obtained from a wake oscillator model the lift coefficients of which satisfy a modified van der Pol equation. Adequate agreement is found between theory and experiments.

Active vibration control of a flexible cylinder subjected to periodically shedding vortices

A mathematical model is developed to simulate the dynamics of a flexible cylinder subjected to vortex-induced excitation. The finite element method is used to describe the dynamic characteristics of the cylinder. The fluctuating lift forces induced on the cylinder are assumed to satisfy a modified Van der Pol equation and are determined by the wake-oscillator approach. The dynamic equations of the cylinder are coupled with the modified independent modal space control (MIMSC) method, which actively suppresses the dominant mode of the vibration. The performance of the active controller of the vortex-induced vibrations is obtained at various flow speeds and control weighting parameter. To verify the mathematical model, the computed results are compared with experimental data obtained in a low-speed wind tunnel over a range of Reynolds numbers between 5500 and 7500. Good agreement is observed between the theory and experiments. It was found that the controller is effective in attenuating the transverse vibrations of the cylinder by approximately 40% for all the flow speeds considered.

Wind‐Induced Vibration of Two Flexible Cylindrical Structures

This paper presents a simple mathematical model that can be used to predict the vortex‐excited resonant response of two flexible cylindrical structures in line with the wind direction. A modified van der Pol equation is employed to account for the effect of fluctuating lift forces on the cylinders, and normal mode technique is used for treating the flexibility of the cylinders. The governing equations are nonlinear in nature, and the solutions of these equations are assumed to be in the form of sinusoidal functions. Results obtained are in good agreement with the experimental data.