Modified Van Der Pol Oscillator Research Articles

Design of intelligent hybrid NAR-GRNN paradigm for fractional order VDP chaotic system in cardiac pacemaker with relaxation oscillator

Managing cardiac disease and abnormal heart rate variability is a challenging problem with its psychological impact on lifesaving intervention. The research presents a novel machine learning approach to paradigm dynamic pacemaker design based on fractional order modified Van der Pol oscillator system implemented to generate rich non-sinusoidal signals for cardiac intervention and treatment. The physical framework of the novel contribution comprises Sino-Atrial (SA) and Atrio-Ventricular (AV) nodes, designed to utilize fractional order excitation signals derived from the VDP dynamic system. Hybrid paradigm is designed by integrating the dynamical nonlinear autoregressive (NAR) neural networks and generalized regression neural networks (GRNNs) to analyze the parametric fractal impulse model for cardiac muscle. The chaotic pattern of the proposed nonlinear system is explored in terms of phase portraits and Lyapunov exponents. The exceptional performance of the cutting-edge fractional order multimodal computing paradigm is validated by achieving an RMSE of 0.1E-14. The dynamic chaotic and fractal trajectory of relaxation oscillator based on its computed control parameters can provide valuable insight to design robust and efficient electronic implantable cardioverter defibrillators (ICDs) as pacemakers to stabilize cardiac impulse variability and alleviate healthcare burden as compared to traditional cardiac rehabilitation procedures.

Stochastic bifurcations induced by Lévy noise in a fractional trirhythmic van der Pol system

Trirhythmical nature has aroused considerable interest in describing dynamic behaviors of a fractional self-sustained system. In this paper, the fractional trirhythmic system is considered and a bifurcation analysis in a stochastic fractional trirhythmic self sustained system subjected to Lévy noise perturbation is presented. The fractional electronic circuit has been used to model the system and the oscillations are described by a nonlinear fractional differential equation to show a new bifurcation parameter. Then, based on the minimum mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of the damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Using the predictor–corrector simulation method, it is shown a good agreement between the fractional and the equivalent integer order system. The detailed parameter space study reveals that the multi-rhythmic properties of the oscillation of this system can be efficiently controlled by fractional order parameter and Lévy noise parameters. Finally, the analytical solutions are validated by numerical results of the Monte Carlo simulation of the original fractional modified van der Pol oscillator. The trirhythmic region can be detected in the parameter space (α,β,γ,δ) by choosing an approximate fractional order. The stability index and the noise intensity can in any case govern the dynamics of the self sustained system, but regulating the skewness parameter cannot bring about transitions between unimodal, bimodal and trimodal in this multi-rhythmic system. More bifurcations appear by adjusting the fractional order in this system. These results may be conducive to further exploring bifurcations in the real-world applications. This study sheds new insight lights on the understanding nontrivial effects of fractional derivative on a trirhythmic self sustained system.

Efficacy of vibro-impact energy harvesting absorbers on controlling dynamical systems under vortex-induced vibrations and base excitation

This study investigates improving the efficacy of an energy harvesting absorber's ability to control a structure under vortex-induced vibrations, base excitation, and a combination of the two by including mechanical amplitude stoppers. The nonlinear reduced-order model is developed through modifying trilinear spring models to represent the impact forces, a modified van der Pol oscillator to represent the forcing due to the vortex-induced vibrations and using the Euler-Lagrange principle to express the equations of motion. It is seen that a soft stopper stiffness and a 5mm gap performs the most effectively of increasing the power generated from the absorber while still greatly reducing the primary structure's amplitude. By changing the stopper's location towards the middle of the energy harvesting absorber, the large effects of the impact forces are reduced and improves the efficacy of medium and hard stopper stiffnesses to generate near the amount of power the soft stopper does, while greatly improving the control of the primary structure. When the system is under combined loadings, the large oscillations of the synchronization region cause the effective configuration to be that of a 27.5mm gap with soft stiffnesses. The results shows that medium stiffness stoppers with small gaps generate large aperiodic regions due to the high impact force. When the oscillations are close to the stoppers, the beating phenomenon is observed and is not overpowered by the vibro-impact force.

On Certain Noise Filtering Techniques in Fixed Point Iteration-based Adaptive Control

In control applications the use of noise-burdened sensor signals cannot be evaded. Also, certain signals can be lost. This problem traditionally is tackled by the use of Kalman filters that provide some “optimal solution” to these problems based on reasonable assumptions that are not always well underpinned in the practice. These are assumptions are made with regard to the system model and the statistical distribution of the noise signals. The Fixed Point Iteration-based adaptive controller is applicable for various strongly nonlinear models. Because feeding back the order of time-derivative of the system’s variable that immediately can be varied by the control signal, its noise sensitivity can be considerable. In this paper the operation of an unscented Kalman filter-based technique is compared with that of a simple moving window with affine signal approximation, and the use of a third order low pass filter in the control of a modified van der Pol oscillator. In this model a quadratic drag term is added to the original model to describe the motion of the system in turbulent fluid environment. According to the numerical simulations it can be stated that the simpler methods can replace the more complicated Kalman filter.

Multiple Time-Scales Dynamics of a Cardiac Pacemaker Model with Application to Heart Rhythm Modeling: Theoretical Study and FPGA Implementation

The aim of this work is to investigate the nonlinear dynamics of a forced cardiac pacemaker model represented by a forced modified van der Pol oscillator by using the method of harmonic balance and the method of multiple time scales. The amplitudes of the forced harmonic, primary resonant state, super-harmonic and sub-harmonic resonant state oscillations are obtained and the effects of some system parameters on these amplitudes are investigated; hysteresis and jump phenomena are found. Three similar oscillators are then coupled together via time-delay coupling to model the heart rhythm and the effect of the coupling factor and time delay is also investigated. The overall model is then discretized using the fourth-order Runge–Kutta algorithm. A corresponding very high-speed integrated circuit hardware description language code is then written, synthesised and implemented using the Vivado-2017.4 software for the Artix7-xc7a35tftg256-1 field programmable gate array chip. The chip statistics, time series electrocardiograms and phase portraits of the model are derived; a good correlation between theoretical results (obtained with MATLAB program and Vivado software) and practical results is observed.

A Modified Van Der Pol Oscillator Model for the Unsteady Lift Produced by a Flapping Flat Plate for Different Positions of the Rotation Axis

To understand the nonlinear interaction between unsteady aerodynamic forces and the kinematics of structures, we theoretically and numerically investigated the characteristics of lift coefficients produced by a flapping thin flat plate controlled by the rotation axis position. The flat plate was placed in a 2-D incompressible flow at a very low Reynolds number (Re = 300). We showed that the behavior of the unsteady aerodynamic forces suggests the existence of a limit cycle. In this context, we developed a Reduced Order Model (ROM) by resolving the modified van der Pol oscillator using the Taylor development method and computational fluid dynamics (CFD) solutions. A numerical solution was obtained by integrating the differential equation of the modified van der Pol oscillator using the fourth-order Runge–Kutta method (RK4). The model was validated by comparing this solution with the reformulated equation of the added mass lift coefficient. Using CFD and ROM solutions, we analyzed the dependency of the unsteady lift coefficient generation on the kinematics of the flapping flat plate. We showed that the evolution of the lift coefficient is influenced by the importance of the rotation motion of the Leading Edge (LE) or Trailing Edge (TE), according to the position of the rotation axis. Indeed, when the rotation axis is moved towards the LE, the maximum and the minimum values of the lift coefficient are proportional to the downward and upward motions respectively of the TE and the rotation axis. However, when the rotation axis is moved towards the TE, the maximum and the minimum values of the lift coefficient are proportional to the downward and upward motions respectively of the LE and the rotation axis.

Improvement of the Cardiac Oscillator Based Model for the Simulation of Bundle Branch Blocks

In this paper, we propose an improvement of the cardiac conduction system based on three modified Van der Pol oscillators. Each oscillator represents one of the components of the heart conduction system: Sino-Atrial node (SA), Atrio-Ventricular node (AV) and His–Purkinje system (HP). However, while SA and AV nodes can be modelled through a single oscillator, the modelling of HP by using a single oscillator is a rough simplification of the cardiac behaviour. In fact, the HP bundle is composed of Right (RB) and Left Bundle (LB) branches that serve, respectively, the right and left ventricles. In order to describe the behaviour of each bundle branch, we build a phenomenological model based on four oscillators: SA, AV, RB and LB. For the characterization of the atrial and ventricular muscles, we used the modified FitzHugh–Nagumo (FHN) equations. The numerical simulation of the model has been implemented in Simulink. The simulation results show that the new model is able to reproduce the heart dynamics generating, besides the physiological signal, also the pathological rhythm in case of Right Bundle Branch Block (RBBB) and Left Bundle Branch Block (LBBB). In particular, our model is able to describe the communication interruption of the conduction system, when one of the HP bundle branches is damaged.

Relationship between Mathematical Parameters of Modified Van der Pol Oscillator Model and ECG Morphological Features

The mathematical model describes the electrical and mechanical activity of the cardiac conduction system thought set of differential equations. By changing the value of parameters included in these equations, it is possible to change the amplitude and the period of ECG waves. Although this model is a powerful tool for modeling the electrical activity of the heart, its use is often limited to those familiar with the differential equations that describe the system. The purpose of this work is to provide a system that allows generating an ECG signal using Ryzhii model without knowing the details of differential equations. First, we provide the relationships between the ECG wave features and the model parameters; then we generalize them through fitting neural networks. Finally, putting in series fitting neural network and heart model, we provide a system that allows generating a synthetic signal by setting as input only the morphological ECG feature. We computed numerical simulation in Simulink environment and implemented the fitting neural networks in Matlab. Results show that non-linear trends characterize the correlation functions between ECG morphological features and model parameters and that the fitting neural networks can generalized this trend by providing the model parameters given in input the respectively ECG feature.

Parameter Identification and Adaptive Control of Uncertain Goodwin Oscillator Networks with Disturbances

This paper investigates the dynamic properties of a differential equation model of mammals’ circadian rhythms, including parameter identification, adaptive control, and outer synchronization. The circadian oscillator network is described by a Goodwin oscillator network, the couplings of which are from vasoactive intestinal polypeptides described by modified Van der Pol oscillators. We build up a drive‐response system consisting of two networks with unknown parameters and disturbances. Then, we propose effective parameter updating laws to identify the unknown parameters and design adaptive control strategies to achieve outer synchronization in the drive‐response system. As special cases, two succinct corollaries are presented for different instances. All the theoretical results are proved through strict mathematical deduction based on Lyapunov stability theory, and a numerical example is also carried out to illustrate the effectiveness.

Impact of viscoelastic coupling on the synchronization of symmetric and asymmetric self-sustained oscillators

We investigate a system of two viscoelastically coupled, modified Van der Pol oscillators to compare their synchronization properties due to elastic and viscoelastic coupling. We show that viscoelastic coupling leads to in-phase synchronization while elastic coupling favours anti-phase synchronization. To study the impact of symmetry and nonlinearity, the restoring forces in the Van der Pol oscillators are extended to include nonlinear and asymmetric components. If the asymmetry, or rather the nonlinearity, of the restoring forces exceeds a certain threshold, only in-phase synchronized motion is found to be stable. Another important finding is that chaotic solutions can only be found if the restoring forces are asymmetric and the coupling incorporates viscosity.

Nonlinear Reduced-Order Models for Aerodynamic Lift of Oscillating Airfoils2

For the present study, setting Strouhal number as the control parameter, we perform numerical simulations for the flow over oscillating NACA-0012 airfoil at a Reynolds number of 1000. This study reveals that aerodynamic forces produced by oscillating airfoils are independent of the initial kinematic conditions suggesting the existence of limit cycle. Frequencies present in the oscillating lift force are composed of the fundamental harmonics and its odd harmonics. Using these numerical simulations, we analyze the shedding frequencies close to the excitation frequencies. Hence, considering it as a primary resonance case, we model the unsteady lift force with a modified van der Pol oscillator. Using the method of multiple scales and spectral analysis of the steady-state computational fluid dynamics (CFD) solutions, we estimate the frequencies and the damping terms in the reduced-order model (ROM). We show the applicability of this model to all planar motions of the airfoil; heaving, pitching, and flapping. With increasing the Strouhal number, the nonlinear damping terms for all types of motion approach similar magnitudes. Another important aspect in one of the proposed model is its ability to capture the time-averaged value of the aerodynamic lift force. We also notice that increase in the magnitude of the lift force is due to the effect of destabilizing linear damping parameter.

Chaos control applied to cardiac rhythms represented by ECG signals

The control of irregular or chaotic heartbeats is a key issue in cardiology. In this regard, chaos control techniques represent a good alternative since they suggest treatments different from those traditionally used. This paper deals with the application of the extended time-delayed feedback control method to stabilize pathological chaotic heart rhythms. Electrocardiogram (ECG) signals are employed to represent the cardiovascular behavior. A mathematical model is employed to generate ECG signals using three modified Van der Pol oscillators connected with time delay couplings. This model provides results that qualitatively capture the general behavior of the heart. Controlled ECG signals show the ability of the strategy either to control or to suppress the chaotic heart dynamics generating less-critical behaviors.

Nonlinear analysis in a modified van der Pol oscillator

In this paper we study the nonlinear dynamics of a modified van der Pol oscillator. More precisely, we study the local codimension one, two and three bifurcations which occur in the four parameter family of differential equations that models an extension of the classical van der Pol circuit with cubic nonlinearity. Aiming to contribute to the understand of the complex dynamics of this system we present analytical and numerical studies of its local bifurcations and give the corresponding bifurcation diagrams. A complete description of the regions in the parameter space for which multiple small periodic solutions arise through the Hopf bifurcations at the equilibria is given.

Bifurcation threshold of the delayed van der Pol oscillator under stochastic modulation

We obtain the location of the Hopf bifurcation threshold for a modified van der Pol oscillator, parametrically driven by a stochastic source and including delayed feedback in both position and velocity. We introduce a multiple scale expansion near threshold, and we solve the resulting Fokker-Planck equation associated with the evolution at the slowest time scale. The analytical results are compared with a direct numerical integration of the model equation. Delay modifies the Hopf frequency at threshold and leads to a stochastic bifurcation that is shifted relative to the deterministic limit by an amount that depends on the delay time, the amplitude of the feedback terms, and the intensity of the noise. Interestingly, stochasticity generally increases the region of stability of the limit cycle.

Model Comparison for $1/f$ Noise in Oscillators With and Without AM to PM Noise Conversion

A new generalized 1/ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> Kurokawa noise analysis applicable to both low- and high- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> oscillators is proposed for 1/ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> phase modulation (PM) and amplitude modulation (AM) noise. A theoretical correspondence between the new generalized Kurokawa theory and the impulse sensitivity function and the perturbation projection vector (PPV) is also derived in the absence and presence of AM to PM noise conversion. The importance of AM to PM noise conversion effect is investigated next using a modified Van der Pol oscillator with fourth-order nonlinear capacitances and a harmonic short. The same analytic solution for the phase noise is derived for both the new generalized 1/ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> Kurokawa and the PPV analysis and is found to match the results from the conversion matrix method in two commercial harmonic-balance circuit simulators.

Chaotic Control in a Fractional-Order Modified Van Der Pol Oscillator

The dynamics of fractional-order systems have attracted increasing attention in recent years. In this paper, we study the chaotic behaviors in a fractional-order modified van der Pol oscillator. We find that chaos exists in the fractional-order modified van der Pol oscillator with order less than 3. In addition, the lowest order we find for chaos to exist in such system is 2.4. Finally, a simple, but effective, linear feedback controller is also designed to stabilize the fractional order chaotic van der Pol oscillator.

Nonlinear oscillations of the FitzHugh–Nagumo equations under combined external and two-frequency parametric excitations

The continuous FitzHugh–Nagumo (FHN for short) model is transformed into modified van der Pol oscillator with asymmetry under external and two-frequency parametric excitations. At the first, the dependence of the solutions on a combined external and two-frequency parametric stimulus forcing is investigated. By using the multiple scale method, ranges of applied current and/or parametric forcing in which nonlinear oscillations are observed are described. Second, when the multiple scale method cannot be used, we numerically prove that in the modified van der Pol oscillator with asymmetry under external and two-frequency parametric excitations, chaos and periodic solution depending on the combination between different frequencies of the model should appear. We also show that the amplitude of the oscillations can be reduced or increased. To do this, we perform the study of the FHN model by choosing a range of parameters exhibiting Hopf bifurcation and two qualitative different regimes in phase portrait.

Chaos in a Fractional-Order Modified Van Der Pol Oscillator

The dynamics of fractional-order systems have attracted increasing attention in recent years. In this paper, we study the chaotic behaviors in a fractional-order modified van der Pol oscillator . We find that chaos exists in the fractional-order modified van der Pol oscillator with order less than 3. In addition, the lowest order we find for chaos to exist in such system is 2.4.

Construction of approximate periodic solutions to a modified van der Pol oscillator

A new iteration scheme is proposed and applied for the modified van der Pol oscillator. A simple and effective iteration procedure to search for the periodic solutions of the equation is given. This procedure is a powerful tool for the determination of the approximate frequencies and periodic solutions of the nonlinear differential equations. The solutions obtained using the present iteration procedure are in good agreement with the numerical integration obtained by a fourth order Runge–Kutta method, which shows the applicability of the procedure.

Stability of the synchronization manifold in nearest neighbor nonidentical van der Pol-like oscillators

We investigate the stability of the synchronization manifold in a ring and an open-ended chain of nearest neighbors coupled self-sustained systems, each self-sustained system consisting of multi-limit cycles van der Pol oscillators. Such model represents, for instance, coherent oscillations in biological systems through the case of an enzymatic-substrate reaction with ferroelectric behavior in brain waves model. The ring and open-ended chain of identical and non-identical oscillators are considered separately. By using the Master Stability Function approach (for the identical case) and the complex Kuramoto order parameter (for the non-identical case), we derive the stability boundaries of the synchronized manifold. We have found that synchronization occurs in a system of many coupled modified van der Pol oscillators and it is stable even in presence of a spread of parameters.