Showing papers on "Van der Pol oscillator published in 2014"
TL;DR: In this article, the authors investigated the potential of using a piezoelectric energy harvester to concurrently harness energy from base excitations and vortex-induced vibrations, and the results showed that there is a significant improvement in the level of the harvested power which can attain 150% compared to using two separate harvesters.
Abstract: We investigate the potential of using a piezoelectric energy harvester to concurrently harness energy from base excitations and vortex-induced vibrations. The harvester consists of a multilayered piezoelectric cantilever beam with a circular cylinder tip mass attached to its free end which is placed in a uniform air flow and subjected to direct harmonic excitations. We model the fluctuating lift coefficient by a van der Pol wake oscillator. The Euler–Lagrange principle and the Galerkin procedure are used to derive a nonlinear distributed-parameter model for a harvester under a combination of vibratory base excitations and vortex-induced vibrations. Linear and nonlinear analyses are performed to investigate the effects of the electrical load resistance, wind speed, and base acceleration on the coupled frequency, electromechanical damping, and performance of the harvester. It is demonstrated that, when the wind speed is in the pre- or post-synchronization regions, its associated electromechanical damping is increased and hence a reduction in the harvested power is obtained. When the wind speed is in the lock-in or synchronization region, the results show that there is a significant improvement in the level of the harvested power which can attain 150 % compared to using two separate harvesters. The results also show that an increase of the base acceleration results in a reduction in the vortex-induced vibrations effects, an increase of the difference between the resonant excitation frequency and the pull-out frequency, and a significant effects associated with the quenching phenomenon.
182 citations
TL;DR: In this paper, the synchronization of dissipatively coupled van der Pol oscillators in the quantum limit was studied, where each oscillator is near its quantum ground state and the critical coupling increases with detuning.
Abstract: We study the synchronization of dissipatively coupled van der Pol oscillators in the quantum limit, when each oscillator is near its quantum ground state. Two quantum oscillators with different frequencies exhibit an entanglement tongue, which is the quantum analog of an Arnold tongue. It means that the oscillators are entangled in steady state when the coupling strength is greater than a critical value, and the critical coupling increases with detuning. An ensemble of many oscillators with random frequencies still exhibits a synchronization phase transition in the quantum limit, and we analytically calculate how the critical coupling depends on the frequency disorder. Our results can be experimentally observed with trapped ions or neutral atoms.
141 citations
TL;DR: Van der Pol's contributions during the period of 1926-1930 were investigated to show how, with Le Corbeiller's help, he popularized the "relaxation oscillations" using the previous experiments as examples and, turned them into a concept.
Abstract: Relaxation oscillations are commonly associated with the name of Balthazar van der Pol via his eponymous paper (Philosophical Magazine, 1926) in which he apparently introduced this terminology to describe the nonlinear oscillations produced by self-sustained oscillating systems such as a triode circuit. Our aim is to investigate how relaxation oscillations were actually discovered. Browsing the literature from the late 19th century, we identified four self-oscillating systems in which relaxation oscillations have been observed: i) the series dynamo machine conducted by Gerard-Lescuyer (1880), ii) the musical arc discovered by Duddell (1901) and investigated by Blondel (1905), iii) the triode invented by de Forest (1907) and, iv) the multivibrator elaborated by Abraham and Bloch (1917). The differential equation describing such a self-oscillating system was proposed by Poincare for the musical arc (1908), by Janet for the series dynamo machine (1919), and by Blondel for the triode (1919). Once Janet (1919) established that these three self-oscillating systems can be described by the same equation, van der Pol proposed (1926) a generic dimensionless equation which captures the relevant dynamical properties shared by these systems. Van der Pol's contributions during the period of 1926-1930 were investigated to show how, with Le Corbeiller's help, he popularized the "relaxation oscillations" using the previous experiments as examples and, turned them into a concept.
69 citations
Journal Article•
TL;DR: The synchronization of dissipatively coupled van der Pol oscillators in the quantum limit is studied, when each oscillator is near its quantum ground state and the critical coupling increases with detuning.
Abstract: We study the synchronization of dissipatively coupled van der Pol oscillators in the quantum limit, when each oscillator is near its quantum ground state Two quantum oscillators with different frequencies exhibit an entanglement tongue, which is the quantum analog of an Arnold tongue It means that the oscillators are entangled in steady state when the coupling strength is greater than a critical value, and the critical coupling increases with detuning An ensemble of many oscillators with random frequencies still exhibits a synchronization phase transition in the quantum limit, and we analytically calculate how the critical coupling depends on the frequency disorder Our results can be experimentally observed with trapped ions or neutral atoms
67 citations
TL;DR: In this paper, the primary resonance of van der Pol (VDP) oscillator with fractional-order derivative is studied analytically and numerically, and the results certify the correctness and satisfactory precision of the approximately analytical solution.
Abstract: In this paper the primary resonance of van der Pol (VDP) oscillator with fractional-order derivative is studied analytically and numerically. At first the approximately analytical solution is obtained by the averaging method, and it is found that the fractional-order derivative could affect the dynamical properties of VDP oscillator, which is characterized by the equivalent damping coefficient and the equivalent stiffness coefficient. Moreover, the amplitude–frequency equation for steady-state solution is established, and the corresponding stability condition is also presented based on Lyapunov theory. Then, the comparisons of several different amplitude–frequency curves by the approximately analytical solution and the numerical integration are fulfilled, and the results certify the correctness and satisfactory precision of the approximately analytical solution. At last, the effects of the two fractional parameters, i.e., the fractional coefficient and the fractional order, on the amplitude–frequency curves are investigated for some typical excitation amplitudes, which are different from the traditional integer-order VDP oscillator.
59 citations
TL;DR: In this article, a nonlinear dynamical response of a vertical riser concurrently subjected to hybrid excitations, namely, vortex-induced vibrations (VIVs) and base excitations are investigated.
57 citations
TL;DR: In this article, the approximately analytical solution of van der Pol (VDP) oscillator with two kinds of fractional-order derivatives is obtained based on averaging method, which could characterize the effects of the fractional parameters on the limit cycle in fractional order VDP oscillator.
Abstract: In this paper the approximately analytical solution of van der Pol (VDP) oscillator with two kinds of fractional-order derivatives is obtained based on averaging method. Two equivalent system parameters, i.e. equivalent damping coefficient and equivalent stiffness coefficient, are defined, which could characterize the effects of the fractional parameters on the limit cycle in fractional-order VDP oscillator. The same points and differences between the traditional integer-order and fractional-order VDP oscillator are analyzed and summarized in detail. The differences are focused on the convergence speed and frequency characteristic of the limit cycle in VDP oscillator. The comparison between the analytical and numerical solution verifies the correctness and satisfactory precision of the approximately analytical solution. At last, the effects of the fractional parameters on the convergence speed and frequency characteristic of the limit cycle in fractional-order VDP oscillator are illustrated based on some typical system parameters.
48 citations
TL;DR: It is found that the proposed method can easily be programmed and can predict accurate periodic approximations while the system parameters being unfolded.
40 citations
TL;DR: In this paper, an infinite-horizon optimal controller, based on a state-dependent Riccati equation approach, is proposed to solve the tracking problem for nonlinear systems, which results in a state feedback optimal control law plus a time varying term, which minimizes a quadratic performance index.
Abstract: This paper pressents an infinite-horizon optimal controller, based on a state-dependent Riccati equation approach to solve the tracking for nonlinear systems. The synthesized control law comes from solving the Hamilton-Jacobi-Bellman equation for state-dependent coefficient factorized nonlinear systems. The proposed controller results in a state feedback optimal control law plus a time varying term, which minimizes a quadratic performance index. In order to illustrate the tracking to a desired reference, the proposed optimal control law is applied to two systems with practical applications: the Van der Pol oscillator and a doubly fed induction generator. Simulation results illustrate the effectiveness of the control scheme.
40 citations
TL;DR: Inclusion of cardiac muscle response allows to investigate interactions between pacemakers and resulting global heartbeat dynamics by means of clinically comparable realistic ECG signals.
39 citations
TL;DR: In this paper, the dynamics of a system of two coupled van der Pol oscillators is investigated, where the coupling between the two oscillators consists of adding to each oscillators amplitude a perturbation proportional to the other one.
Abstract: In this paper, the dynamics of a system of two coupled van der Pol oscillators is investigated. The coupling between the two oscillators consists of adding to each one’s amplitude a perturbation proportional to the other one. The coupling between two laser oscillators and the coupling between two vacuum tube oscillators are examples of physical/experimental systems related to the model considered in this paper. The stability of fixed points and the symmetries of the model equations are discussed. The bifurcations structures of the system are analyzed with particular attention on the effects of frequency detuning between the two oscillators. It is found that the system exhibits a variety of bifurcations including symmetry breaking, period doubling, and crises when monitoring the frequency detuning parameter in tiny steps. The multistability property of the system for special sets of its parameters is also analyzed. An experimental study of the coupled system is carried out in this work. An appropriate electronic simulator is proposed for the investigations of the dynamic behavior of the system. Correspondences are established between the coefficients of the system model and the components of the electronic circuit. A comparison of experimental and numerical results yields a very good agreement.
Journal Article•
TL;DR: This work investigates how quantum fluctuations affect phase locking of one or many van der Pol oscillators and finds that phase locking is much more robust in the quantum model than in the equivalent classical model.
Abstract: The van der Pol oscillator is the prototypical self-sustained oscillator and has been used to model nonlinear behavior in biological and other classical processes. We investigate how quantum fluctuations affect phase locking of one or many van der Pol oscillators. We find that phase locking is much more robust in the quantum model than in the equivalent classical model. Trapped-ion experiments are ideally suited to simulate van der Pol oscillators in the quantum regime via sideband heating and cooling of motional modes. We provide realistic experimental parameters for 171Yb+ achievable with current technology.
TL;DR: The obtained results show that hidden attractors exist around chaotic attractors in a general autonomous van der Pol–Duffing oscillator.
Abstract: In this paper, a general autonomous van der Pol–Duffing oscillator is studied. Several issues, such as periodic bifurcations and the dynamical structures of the system are investigated either analytically or numerically. Especially, a phenomenon of hidden attractors is noticed and an algorithm for the location of hidden attractors is given. The obtained results show that hidden attractors exist around chaotic attractors.
TL;DR: In this article, the effects of slowly varying parametric excitation on the dynamics of van der Pol system were investigated and an approximate calculation for the number of spikes in each cluster of repetitive spiking of mixed-mode oscillations was explored based on bifurcation delay behaviors.
Abstract: This paper investigates the effects of slowly varying parametric excitation on the dynamics of van der Pol system. Periodic bifurcation delay behaviors are exhibited when the parametric excitation slowly passes through Hopf bifurcation value of the controlled van der Pol system. The first bifurcation delay behavior relies on initial conditions, while the bifurcation delay behaviors that follow the first one are immune to initial conditions. These bifurcation delay behaviors result in a hysteresis loop between the spiking attractor and the rest state, which is responsible for the generation of mixed-mode oscillations. Then an approximate calculation for the number of spikes in each cluster of repetitive spiking of mixed-mode oscillations is explored based on bifurcation delay behaviors. Theoretical results agree well with numerical simulations.
01 Jan 2014
TL;DR: In-plane free vibration and stability analysis of high speed rotating disks and rings is discussed in this paper, where the authors present a tool for detecting and characterization of Levy flights in chaotic trajectories.
Abstract: In-plane free vibration and stability analysis of high speed rotating disks and rings.- Patent licensing: Stackelberg versus Cournot models.- Privatization and government preferences in a mixed duopoly: Stackelberg versus Cournot.- Nonlinear self-adjointness for some generalized KdV equations.- Conservation Laws for a Family of Benjamin-Bona-Mahony-Burgers equations.- Energy dissipation through viscoelastic chain-based devices.- Simulation of Costas loop in phase-frequency space for general case of signals waveforms.- Simulation of drilling systems models and hidden oscillations.- Dynamical response of a Van der Pol system with an external harmonic excitation and fractional derivative.- Basins of attraction in a simple harvesting system with a stopper.- Milling of composites by recurrence plots and Hilbert-Huang transform.- Formations of Transitional Zones in Shock Wave with Saddle-Node Bifurcations.- Approaches for defining and measuring assembly supply chains complexity.- Tuning of Fractional order PI^lamda D^mu controller with Responce surface methodology.- Chaos in a piecewise linear system with periodic excitation.- 1D Cahn-Hilliard dynamics: coarsening and interrupted coarsening.- Tomographic analysis: a tool for detection and characterization of Levy flights in chaotic trajectories.- Analytical dynamics of nonlinear rotating blades under parametric excitation.- Dynamical synchronization of two gyroscope systems.- Analytical dynamics of a mass-damper-spring constrained system.- Fractional Maps as Maps with Memory.- New Trends in the Computational Methods in the Fractional Calculus.- Application of the local fractional Fourier series to fractal signals.- Analysis of the DNA Information.
TL;DR: In this paper, the dynamics of a memristor-based Van der Pol oscillator coupled to a linear circuit (VDPCL) is investigated. And the basic properties of the circuit are analyzed by means of bifurcation analysis.
Abstract: This paper investigates the dynamics of a memristor-based Van der Pol oscillator coupled to a linear circuit (VDPCL). This chaotic oscillator is a modification of the classical Van der Pol coupled to a linear circuit, and is obtained by replacing the classical cubic nonlinearity by the memristive one. The memristive VDPCL oscillator, in addition to having a very special stability property, exhibits interesting spectral characteristics, which makes it suitable for chaos-based secure communication applications. The memristor is realized by using off-the-shelf components. The basic properties of the circuit are analyzed by means of bifurcation analysis. Chaotic attractors from numerical and experimental analysis are presented, followed by a comparison of results obtained from the modified VDPCL oscillator and those from the classical VDPCL oscillator. An application to synchronization and chaos secure communication is also presented.
TL;DR: In this paper, the authors consider an extended three-dimensional Bonhoeffer-van der Pol oscillator with three distinct time-scales and derive asymptotic estimates for the corresponding parameter intervals.
Abstract: We consider an extended three-dimensional Bonhoeffer–van der Pol oscillator which generalises the planar FitzHugh–Nagumo model from mathematical neuroscience, and which was recently studied by Sekikawa et al. (Phys Lett A 374(36):3745–3751, 2010) and by Freire and Gallas (Phys Lett A 375:1097–1103, 2011). Focussing on a parameter regime which has hitherto been neglected, and in which the governing equations evolve on three distinct time-scales, we propose a reduction to a model problem that was formulated by Krupa et al. (J Appl Dyn Syst 7(2):361–420, 2008) as a canonical form for such systems. Based on results previously obtained in Krupa et al. (2008), we characterise completely the mixed-mode dynamics of the resulting three time-scale extended Bonhoeffer–van der Pol oscillator from the point of view of geometric singular perturbation theory, thus complementing the findings reported in Sekikawa et al. (2010). In particular, we specify in detail the mixed-mode patterns that are observed upon variation of a bifurcation parameter which is naturally obtained by combining two of the original parameters in the system, and we derive asymptotic estimates for the corresponding parameter intervals. We thereby also disprove a conjecture of Tu (SIAM J Appl Math 49(2): 331–343, 1989), where it was postulated that no stable periodic orbits of mixed-mode type can be observed in an equivalent extension of the Bonhoeffer–van der Pol equations.
TL;DR: Using the model of a generalized Van der Pol oscillator in the regime of subcritical Hopf bifurcation, it is shown that for appropriate choices of time delay, either suppression or enhancement of coherence resonance can be achieved.
Abstract: Using the model of a generalized Van der Pol oscillator in the regime of subcritical Hopf bifurcation we investigate the influence of time delay on noise-induced oscillations. It is shown that for appropriate choices of time delay either suppression or enhancement of coherence resonance can de achieved. Analytical calculations are combined with numerical simulations and experiments on an electronic circuit.
TL;DR: A couple of single dynamic linear/non-linear systems under additive and multiplicative random excitations are discussed using FEM as a solution tool of the FP equation.
TL;DR: In this paper, a phenomenological model and analytical-numerical approach to systematically characterize variable hydrodynamic coefficients and maximum achievable responses in two-dimensional vortex-induced vibrations with dual two-to-one resonances are presented.
Abstract: A phenomenological model and analytical-numerical approach to systematically characterize variable hydrodynamic coefficients and maximum achievable responses in two-dimensional vortex-induced vibrations with dual two-to-one resonances are presented. The model is based on double Duffing and van der Pol oscillators which simulate a flexibly-mounted circular cylinder subjected to uniform flow and oscillating in simultaneous cross-flow/in-line directions. Depending on system quadratic and cubic nonlinearities, amplitudes, oscillation frequencies and phase relationships, analytical closed-form expressions are derived to parametrically evaluate key hydrodynamic coefficients governing the fluid excitation, inertia and added mass force components, as well as maximum dual-resonant responses. The amplification of the mean drag is ascertained. Qualitative validations of numerical predictions with experimental comparisons are discussed. Parametric investigations are performed to highlight the important effects of system nonlinearities, mass, damping and natural frequency ratios.
TL;DR: In this paper, the authors presented a numerical scheme using uniform Haar wavelet approximation and quasilinearization process for solving some nonlinear oscillator equations, which is applied on three types of oscillators namely, Duffing, Van der Pol, and Duffing-van der Pol.
TL;DR: In this article, the authors considered a deterministic dynamical system with the global attractor 𝒜 which supports a unique ergodic probability measure P. The measure P can be considered as the uniform long-term mean of the trajectories staying in a bounded domain D containing &#d 49c; and solved the first exit time and location problem from D in the limit of ϵ↘0.
Abstract: We consider a finite-dimensional deterministic dynamical system with the global attractor 𝒜 which supports a unique ergodic probability measure P. The measure P can be considered as the uniform long-term mean of the trajectories staying in a bounded domain D containing 𝒜. We perturb the dynamical system by a multiplicative heavy tailed Levy noise of small intensity ϵ > 0 and solve the asymptotic first exit time and location problem from D in the limit of ϵ↘0. In contrast to the case of Gaussian perturbations, the exit time has an algebraic exit rate as a function of ϵ, just as in the case when 𝒜 is a stable fixed point studied earlier in [9, 14, 19, 26]. As an example, we study the first exit problem from a neighborhood of the stable limit cycle for the Van der Pol oscillator perturbed by multiplicative α-stable Levy noise.
TL;DR: A structure of the oscillation frequencies parameter space for three and four dissipatively coupled van der Pol oscillators is discussed and an organization of quasi-periodic areas of different dimensions is considered.
TL;DR: In this article, the authors proposed a technique to obtain limit cycles and quasi-periodic solutions of forced nonlinear oscillators and applied it to the forced Van der Pol oscillator and the forced van der Pol Duffing oscillator.
Abstract: In this paper we propose a technique to obtain limit cycles and quasi-periodic solutions of forced nonlinear oscillators. We apply this technique to the forced Van der Pol oscillator and the forced Van der Pol Duffing oscillator and obtain for the first time their limit cycles (periodic) and quasi-periodic solutions analytically. We introduce a modification of the homotopy analysis method to obtain these solutions. We minimize the square residual error to obtain accurate approximations to these solutions. The obtained analytical solutions are convergent and agree well with numerical solutions even at large times. Time trajectories of the solution, its first derivative and phase plots are presented to confirm the validity of the proposed approach. We also provide rough criteria for the determination of parameter regimes which lead to limit cycle or quasi-periodic behaviour.
TL;DR: In this paper, the response of the Van der Pol oscillator subjected to combined harmonic and random excitations is investigated by a technique combining two excellent methods, namely the stochastic averaging method and the equivalent linearization method.
TL;DR: In this article, two lead zirconate titanate (PZT) beams which had high power density were installed on a circular cylinder undergoing vortex-induced vibration (VIV) and a theoretical model has been presented to describe the electromechanical coupling of the open-circuit voltage output and the vibration amplitudes based on second-order nonlinear Van der pol equation and Gauss law.
Abstract: A concept of generating power from a circular cylinder undergoing vortex-induced vibration (VIV) was investigated. Two lead zirconate titanate (PZT) beams which had high power density were installed on the cylinder. A theoretical model has been presented to describe the electromechanical coupling of the open-circuit voltage output and the vibration amplitudes based on a second-order nonlinear Van der pol equation and Gauss law. A numerical computation was applied to measure the capacity of the power generating system. The lift and drag coefficient and the vortex shedding frequency were obtained to verify how the nondimensional parameter reduced velocity affects the fluid field. Meanwhile, a single-degree of freedom system has been added to describe the VIV, presynchronization, and synchronization together with postsynchronization regimes of oscillating frequencies. And the amplitudes of the vibration have been obtained. Finally, the vibrational amplitudes and the voltage output could go up to a high level in the synchronization region. The maximum value of the voltage output and the corresponding reduced velocity were 8.42 V and 5.6, respectively.
TL;DR: A quantitative asymptotic behavior of coupled Kuramoto oscillators with frustrations is presented and some sufficient conditions for the parameters and initial condition leading to phase or frequency synchronization are given.
Abstract: We present a quantitative asymptotic behavior of coupled Kuramoto oscillators with frustrations and give some sufficient conditions for the parameters and initial condition leading to phase or frequency synchronization.
We consider three Kuramoto-type models with frustrations.
First, we study a general case with nonidentical oscillators; i.e., the natural frequencies are distributed.
Second, as a special case, we study an ensemble of two groups of identical oscillators. For these mixture of two identical Kuramoto oscillator groups, we study the relaxation dynamics from the mixed stage to the phase-locked states via the segregation stage. Finally, we consider a Kuramoto-type model that was recently derived from the Van der Pol equations for two coupled oscillator systems in the work of Luck and Pikovsky [27]. In this case, we provide a framework in which the phase synchronization of each group is attained.
Moreover, the constant frustration causes the two groups to segregate from each other, although they have the same natural frequency. We also provide several numerical simulations to confirm our analytical results.
TL;DR: In this article, the synchronization of two dissipatively coupled Van der Pol oscillators in the quantum regime was studied and a crossover from weak to strong frequency entrainment was proposed.
Abstract: We study synchronization of two dissipatively coupled Van der Pol oscillators in the quantum regime. Due to quantum noise strict frequency locking is absent and is replaced by a crossover from weak to strong frequency entrainment. We discuss the differences to the behavior of one quantum Van der Pol oscillator subject to an external drive. Moreover, we describe a possible experimental realization of two coupled quantum van der Pol oscillators in an optomechanical setting.
TL;DR: In this paper, the dynamics of van der Pol oscillators are considered and the Lyapunov chart is presented in the parameter plane and the bifurcations of tori in the system at large frequency detuning of the oscillators.
TL;DR: A flux-controlled memristor circuit is developed, and then a van der Pol oscillator is implemented based on this new mem Bristor circuit, creating a new approach to generate chaos within a nonautonomous circuit system.
Abstract: 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.