Van Der Pol Type Research Articles

Extended state observer-based control of heartbeat described by heterogeneous coupled oscillator model

In this paper, a novel framework to investigate the effect of external electrical signals on cardiac rhythm dynamics using the extended state observer-based control (ESOBC) is proposed. Heterogeneous coupled oscillator model (HCOM) is adopted for the generation of P, Ta, QRS, and T waves, and the realistic synthetic electrocardiogram (ECG) due to its capacity for simulating cardiac muscles response. Sinoatrial (SA) and atrioventricular (AV) nodes and His–Purkinje system are described by modified Van der Pol-type (MVDP) equations connected with time-delay velocity coupling. The electrical responses resulted from depolarization (De) and repolarization (Re) of cardiac muscles are separately presented by modified FitzHugh–Nagumo (MFHN) model. The SA and HP pacemakers are coupled with atrial (AT) and ventricles (VT) muscles, respectively, through the stimulation currents. To satisfy the integral chain form of ESO, and deal with the complex structure of the model, including the large number of parameters, and the coupling between the oscillators and interdependent states, three steps are considered. Firstly, a coordinate transformation is proposed for AT and VT responses described by MFHN model to handle mismatched condition. Following this, the nonlinearity of MFHN and the coupling between cardiac muscles and pacemakers are considered as lumped disturbances. Lastly, an extended state observer is designed to compensate the nonlinearity and estimate the unmeasured system states and handle the lumped disturbances. The stability of the closed-loop system is proved, and the effectiveness of the proposed framework for suppressing well-known rhythm disorder, i.e., sinus tachycardia is demonstrated. The framework presented in this work can be easily extended for a wide range of arrhythmia, and serve as an approach to investigate the effect of external stimulation to the heart muscles.

Chaotic Dynamical Behavior of Coupled One-Dimensional Wave Equations

In this paper, we consider the chaotic oscillation of coupled one-dimensional wave equations. The symmetric nonlinearities of van der Pol type are proposed at the two boundary endpoints, which can cause the energy of the system to rise and fall within certain bounds. At the interconnected point of the wave equations, the energy is injected into the system through an anti-damping velocity feedback. We prove the existence of the snapback repeller when the parameters enter a certain regime, which causes the system to be chaotic. Numerical simulations are presented to illustrate our theoretical results.

A simplified model to evaluate peak amplitude for vertical vortex-induced vibration of bridge decks

A simplified model is developed to conveniently evaluate the peak vortex-induced vibration (VIV) amplitudes of a bridge deck at various mass-damping conditions. As a simplified form of the describing function-based model previously developed by the authors, the key innovation of the new model is the introduction of an envelope curve of the aerodynamic describing functions (or amplitude-dependent flutter derivatives), which determines the maximum negative aerodynamic damping versus vibration amplitude in the lock-in range. Based on the envelope curve, the peak VIV amplitudes at different mass-damping conditions can be obtained directly without calculating all the VIV amplitudes in the entire lock-in range. The new model is more practical for engineers in the bridge engineering community since it only contains a single group of aerodynamic parameters which don't vary with the reduced wind speed. The envelope curve can be either identified based on the VIV decay-to-resonance and/or grow-to-resonance signals at a single mass-damping condition, or based on the VIV steady amplitudes at different mass-damping conditions. Numerical examples involving the VIV analyses of a rigid rectangular cylinder and a bridge deck sectional model are utilized to validate the simulation accuracy of the proposed model, and the model is applied to calculate the peak VIV amplitudes of two flexible bridge decks at various mechanical damping levels. The proposed model is capable of accurately and conveniently predicting the peak VIV amplitudes of a bridge deck sectional model at various mass-damping conditions. For a flexible bridge deck, the peak VIV amplitude calculated by the proposed model is slightly conservative due to the overestimated negative aerodynamic damping at some span-wise segments of the bridge deck. The superiority of the proposed model relative to the conventional van der Pol-type model is also demonstrated.

Identification method for backbone curve of cantilever beam using van der Pol-type self-excited oscillation

This study presents an experimental method for identification of the backbone curves of cantilevers using the nonlinear dynamics of a van der Pol oscillator. The backbone curve characterizes the nonlinear stiffness and nonlinear inertia of the resonator, so it is important to identify this curve experimentally to realize high-sensitivity and high-accuracy sensing resonators. Unlike the conventional method based on the frequency response under external excitation, the proposed method based on self-excited oscillation enables direct backbone curve identification, because the effect of the viscous environment is eliminated under the linear velocity feedback condition. In this research, the method proposed for discrete systems is extended to give an identification method for continuum systems such as cantilever beams. The actuation is given with respect to both the linear and nonlinear feedbacks so that the system behaves as a van der Pol oscillator with a stable steady-state amplitude. By varying the nonlinear feedback gain, we can produce the self-excited oscillation experimentally with various steady-state amplitudes. Then, using the relationship between these steady-state amplitudes and the corresponding experimentally measured response frequencies, we can detect the backbone curve while varying the nonlinear feedback gain. The efficiency of the proposed method is determined by identifying the backbone curves of a macrocantilever with a tip mass and a macrocantilever subjected to atomic forces, which are representative sources of hardening and softening cubic nonlinearities, respectively.

System identification and early warning detection of thermoacoustic oscillations in a turbulent combustor using its noise-induced dynamics

Despite significant research, self-excited thermoacoustic oscillations continue to hinder the development of lean-premixed gas turbines, making the early detection of such oscillations a key priority. We perform output-only system identification of a turbulent lean-premixed combustor near a Hopf bifurcation using the noise-induced dynamics generated by inherent turbulence in the fixed-point regime, prior to the Hopf point itself. We model the pressure fluctuations in the combustor with a van der Pol-type equation and its corresponding Stuart–Landau equation. We extract the drift and diffusion terms of the equivalent Fokker–Planck equation via the transitional probability density function of the pressure amplitude. We then optimize the extracted terms with the adjoint Fokker–Planck equation. Through comparisons with experimental data, we show that this approach can enable prediction of (i) the location of the Hopf point and (ii) the limit-cycle amplitude after the Hopf point. This study shows that output-only system identification can be performed on a turbulent combustor using only pre-bifurcation data, opening up new pathways to the development of early warning indicators of thermoacoustic instability in practical combustion systems.

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.

Anti-collocated observer-based output feedback control of wave equation with cubic velocity nonlinear boundary and Dirichlet control

This study focuses on the observer and observer-based output feedback control design for a one-dimensional wave equation with van der Pol type nonlinear boundary condition and Dirichlet boundary control. The contribution of this work is divided into two parts. Firstly, the anti-collocated backstepping observer design for the nonlinear wave system is investigated, and the well-posedness and the asymptotical stability analysis of the observer error are studied. Secondly, an output feedback observer-based controller is set up by using boundary displacement measurement only to stabilise the nonlinear wave system; also the asymptotical stability of the closed-loop system is shown. The validity of the theoretical results is demonstrated by numerical simulations.

Output feedback stabilization for 1-D unstable wave equations with boundary control matched disturbance and van der Pol nonlinear boundary

In this paper we study the output feedback stabilization of a plant described by a one-dimensional wave equation with a van der Pol type nonlinear boundary condition, an unknown internal nonlinear uncertainty and an external disturbance acting at the control end (boundary). The simultaneous occurrence of nonlinear internal uncertainty, external disturbance, and van der Pol type nonlinear boundary term in the plant leads to the system is more complicated. First, we show that the open-loop system is well-posed and we design an infinite-dimensional estimator to estimate the disturbance. It is shown that the disturbance estimator can successfully estimate the total disturbance, in the sense that the estimation error signal is in L2(0,∞). Using the estimated disturbance, we propose an asymptotic state observer, and then we design an observer-based output feedback stabilizing controller. The closed-loop system is shown to be asymptotically stable. Finally, a numerical simulation is carried out to illustrate the theoretical results and effectiveness of the proposed control law.

Two-stroke relaxation oscillators

Two-stroke relaxation oscillations consist of two distinct phases per cycle—one slow and one fast—which distinguishes them from the well-known van der Pol-type ‘four-stroke’ relaxation oscillations. This type of oscillation can be found in singular perturbation problems in non-standard form, where the slow-fast timescale splitting is not necessarily reflected in a slow-fast variable splitting. The existing literature on such non-standard problems has developed primarily through applications—we complement this by illustrating the suitability of a more general framework for geometric singular perturbation theory to prove existence and uniqueness for a general class of two-stroke relaxation oscillators. While this result can be derived from a more general result in de Maesschalck et al (2011 Indagationes Math. 22 165–206), our methods emphasise the scope, simplicity and applicability of this non-standard approach. We apply this non-standard geometric singular perturbation toolbox to a collection of examples arising in the dynamics of nonlinear transistors and models for mechanical oscillators with friction.

Study of a piezoelectric plate based self-sustained electric and electromechanical oscillator

This work deals with the electrical characterization of a piezoelectric plate and its use to generate self-sustained electrical and mechanical oscillations. An experimental frequency-impedance curve of the piezoelectric plate is obtained leading to the electric equivalent of the piezoelectric plate constituted of a capacitive branch in parallel with many RLC resonant branches. The values of the electric components for each branch are determined using the resonant and anti-resonant frequencies. It is demonstrated theoretically that these branches can be understood in terms of vibration modes of the plate; each of which has its own resonant frequency. It is then demonstrated theoretically and experimentally that putting the piezoelectric plate in series with a resistor presenting a nonlinear I − V characteristics having a negative slope, autonomous oscillations of the Van der Pol type can be observed. A theoretical development has led to the appearance of self-sustained periodic and mixed modes mechanical oscillations of the plate deflection.

A study on multi-frequency patterns in nonlinear network oscillators using incremental harmonic balance method

This paper presents a study on multi-frequency pattern (MFP) vibration of nonlinear network systems based on the incremental harmonic balance (IHB) method. Instead of solving the considered highly-dimensional systems straightforwardly, the IHB method is proposed to solve a single-degree-of-freedom system with a time delay. The time delay is also incorporated into the iteration scheme, by introducing its relations with the phase lag and the angular frequency of limit cycle solutions. As the delay is adjusted to be in preset certain relations with the phase lag and the angular frequency, limit cycle solutions of the time-delayed system can be utilized to generate solutions of the original system. Two network systems are investigated to verify the proposed approach. The presented method can provide both stable and unstable limit cycles, and hence multiple solutions can be tracked successfully. Surprisingly, the coexistence of stable MFP is revealed in the van der Pol-type network oscillators. The attained results are useful for in-depth understanding of the mechanism of MFP. With such high efficiency and accuracy, moreover, the presented analysis method could be applicable in more nonlinear network systems.

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.

Input-output system identification of a thermoacoustic oscillator near a Hopf bifurcation using only fixed-point data.

We present a framework for performing input-output system identification near a Hopf bifurcation using data from only the fixed-point branch, prior to the Hopf point itself. The framework models the system with a van der Pol-type equation perturbed by additive noise, and identifies the system parameters via the corresponding Fokker-Planck equation. We demonstrate the framework on a prototypical thermoacoustic oscillator (a flame-driven Rijke tube) undergoing a supercritical Hopf bifurcation. We find that the framework can accurately predict the properties of the Hopf bifurcation and the limit cycle beyond it. This study constitutes an experimental demonstration of system identification on a reacting flow using only prebifurcation data, opening up pathways to the development of early warning indicators for nonlinear dynamical systems near a Hopf bifurcation.

Chaotic oscillations of one-dimensional coupled wave equations with mixed energy transports

In this paper, we consider the chaotic behavior of one-dimensional coupled wave equations with mixed partial derivative linear energy transport terms. The van der Pol-type symmetric nonlinearities are proposed at two boundary endpoints, which cause the energy of the coupled system to rise and fall within certain ranges. At the interconnected point of the two wave equations, the energy is injected into the system through a middle-point velocity feedback. We prove that when the parameters satisfy some conditions, the coupled wave equations have snapback repellers which can make the whole system chaos in the sense of Li-Yorke. Numerical simulations are presented to verify the theoretical results.

Numerical simulation of a wind excited quasi-periodic regime of a stochastic van der Pol-type equation

The contribution regards a mathematical single-degree-of-freedom model of a slender structure vibrating in an air flow. Based on an experimental investigation, movement of such structures can be expressed by van der PolDuffing-type equations. Several particular configuration parameter settings for a white and non-white Gaussian random excitation together with deterministic harmonic forcing are considered and numerically analysed. The results support recently published analytic formulas.

Vortex-induced vibration of bridge decks: Describing function-based model

A describing function (DF)-based model is introduced for the simulation of vortex-induced vibration (VIV) of bridge decks. Similar to the linear frequency response function, the DF is the complex ratio of the first-order component of the nonlinear output to the harmonic input. The DF can be either identified using the forced vibration technique or based on the VIV nonlinear response time history. An iterative procedure is accordingly developed to predict the VIV response with the DF-based model, and an equivalent-damping-ratio-based simplified method is further proposed to efficiently obtain the limit cycle oscillation (LCO) amplitude of VIV. It is demonstrated that the conventional van der Pol-type model is equivalent to a special case of the DF-based model for VIV. Three case studies involving various cross-sections are utilized to validate the simulation accuracy and efficiency of the proposed DF-based model for typical features at VIV lock-in such as LCO and hysteresis phenomena. The vertical VIVs can be well captured by the DF-based model, while its capability of simulating the torsional VIVs requires further improvement. Furthermore, the predictive capability of DF-based model for vertical VIVs of bridge decks within a wide range of mass-damping conditions is highlighted.

Synchronization of Spatiotemporal Irregular Wave Propagation Via Boundary Coupling

Abstract Wave dynamics reflect a broad spectrum of natural phenomena and are often characterized by wave equation such as in the development of meta-devices used to steer wave propagation. Modeling synchronization of wave dynamics is critical in various applications such as in communications and neuroscience. In this paper, we study the synchronization problem for oscillations governed by wave equation with nonlinear (van der Pol type) boundary conditions through a single boundary coupling. The dynamics of the master system is self-excited and presents sensitive and rapid oscillations. With the only signal received at one end of the boundary, by constructing a mathematical model, we show the existence of a slave system that can be synchronized with the master system via the study of wave reflections on the boundary to recover the actual wave dynamics. The coupling gain, which represents the strength of the connection between the master system and the slave system, has been identified. The obtained result can be also viewed as an observer construction when the measurable output is only on the boundary. Numerical simulations are provided to demonstrate the effectiveness of the theoretical outcomes.

Role of spontaneous emission in the formation of the steady-state optical spectrum of a diode laser

Based on Maxwell’s equations, we derive a van der Pol-type equation for self-oscillations in the cavity of a diode laser. Its solution is self-oscillations (lasing) even in the absence of spontaneous emission. This solution differs fundamentally from solutions obtained using rate equations containing an additional term in the form of spontaneous emission intensity. Taking into account spontaneous emission in the van der Pol model makes calculation results more realistic and allows one to find parameters characterising diode laser output coherence, including the spectral densities of laser light amplitude and phase fluctuations.

Stabilization of One-Dimensional Wave Equation With Nonlinear Boundary Condition Subject to Boundary Control Matched Disturbance

In this paper, we consider the stabilization of a one-dimensional wave equation with nonlinear van der Pol type boundary condition that covers the antistable boundary, and subject to boundary control matched disturbance on the other side. Due to the nonlinear boundary condition and disturbance, the uncontrolled system may present spatiotemporal chaotic, period-doubling bifurcation, and some other dynamical behaviors. We will deal with this disturbance, which is supposed to be bounded only, by the integral sliding mode control. The well-posedness of the system for the closed-loop system is proved and the “reaching condition” is obtained. Finally, we provide some numerical simulations to illustrate the theoretical outcomes.

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.