Showing papers on "Van der Pol oscillator published in 2020"
TL;DR: The aim of this paper is to propose the Atangana–Baleanu fractional methodology for fathoming the Van der Pol damping model by using the reproducing kernel algorithm.
Abstract: The aim of this paper is to propose the Atangana–Baleanu fractional methodology for fathoming the Van der Pol damping model by using the reproducing kernel algorithm. To this end, we discuss the ma...
85 citations
TL;DR: In this paper, a new wake oscillator model is proposed to describe the coupled cross-flow and in-line vortex-induced vibrations of an elastically supported rigid cylinder, which can predict the appearance of the super-upper branch at small mass ratios without changing the tuning parameters.
40 citations
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TL;DR: A hierarchy of deep neural network time-steppers is developed to approximate the flow map of the dynamical system over a disparate range of time-scales, enabling numerical integration and forecasting that is both accurate and highly efficient.
Abstract: Nonlinear differential equations rarely admit closed-form solutions, thus requiring numerical time-stepping algorithms to approximate solutions. Further, many systems characterized by multiscale physics exhibit dynamics over a vast range of timescales, making numerical integration computationally expensive due to numerical stiffness. In this work, we develop a hierarchy of deep neural network time-steppers to approximate the flow map of the dynamical system over a disparate range of time-scales. The resulting model is purely data-driven and leverages features of the multiscale dynamics, enabling numerical integration and forecasting that is both accurate and highly efficient. Moreover, similar ideas can be used to couple neural network-based models with classical numerical time-steppers. Our multiscale hierarchical time-stepping scheme provides important advantages over current time-stepping algorithms, including (i) circumventing numerical stiffness due to disparate time-scales, (ii) improved accuracy in comparison with leading neural-network architectures, (iii) efficiency in long-time simulation/forecasting due to explicit training of slow time-scale dynamics, and (iv) a flexible framework that is parallelizable and may be integrated with standard numerical time-stepping algorithms. The method is demonstrated on a wide range of nonlinear dynamical systems, including the Van der Pol oscillator, the Lorenz system, the Kuramoto-Sivashinsky equation, and fluid flow pass a cylinder; audio and video signals are also explored. On the sequence generation examples, we benchmark our algorithm against state-of-the-art methods, such as LSTM, reservoir computing, and clockwork RNN. Despite the structural simplicity of our method, it outperforms competing methods on numerical integration.
36 citations
TL;DR: In this article, a new wake oscillator model with nonlinear coupling is proposed for the modelling of vortex-induced vibration, which is capable of reproducing both free and forced vibration experiments.
33 citations
TL;DR: In this paper, the frequency-locking phenomenon and transition to quasi-periodic oscillations via Hopf bifurcation of the second kind are determined analytically by the multiple time scales method up to the second-order perturbation.
Abstract: The regular and chaotic vibrations of a nonlinear structure subjected to self-, parametric, and external excitations acting simultaneously are analysed in this study. Moreover, a time delay input is added to the model to control the system response. The frequency-locking phenomenon and transition to quasi-periodic oscillations via Hopf bifurcation of the second kind (Neimark–Sacker bifurcation) are determined analytically by the multiple time scales method up to the second-order perturbation. Approximate solutions of the quasi-periodic motion are determined by a second application of the multiple time scales method for the slow flow, and then, slow–slow motion is obtained. The similarities and differences between the van der Pol and Rayleigh models are demonstrated for regular, periodic, and quasi-periodic oscillations, as well as for chaotic oscillations. The control of the structural response, and modifications of the resonance curves and bifurcation points by the time delay signal are presented for selected cases.
31 citations
29 Apr 2020
TL;DR: In this article, the authors studied synchronization along single quantum trajectories of two quantum van der Pol oscillators under homodyne detection, characterizing the phenomenon and finding a new link between the statistics of synchronization and entanglement.
Abstract: This work studies synchronization along single quantum trajectories of two quantum van der Pol oscillators under homodyne detection, characterizing the phenomenon and finding a new link between the statistics of synchronization and entanglement.
28 citations
TL;DR: In this article, a numerical inverse Laplace transform method using Bernoulli polynomials operational matrix of integration (BOMI) is proposed. But the solution approach is to adopt Laplace Adomian decomposition method for solving nonlinear differential equations.
Abstract: A numerical inverse Laplace transform method is established using Bernoulli polynomials operational matrix of integration. The efficiency of the method is demonstrated through some standard nonlinear differential equations: Duffing equation, Van der Pol equation, Blasius equation and jerk equation. The solution approach is to adopt Laplace Adomian decomposition method for solving nonlinear differential equations and then at each step employ the numerical inverse Laplace transform using the developed method based on Bernoulli polynomials operational matrix of integration. The numerical results exemplify that the estimated solutions are in good agreement with exact or numerical methods available in literature wherever the exact solutions are not known.
26 citations
TL;DR: This paper exploits the nonlocality of fractional calculus, aiming to enhance the identification accuracy of the VdPDO-based nonlinear systems, and proposes the proposed combined FEM-LMS (CFEM- LMS) algorithm, which is based on the FLANN structure.
23 citations
16 Sep 2020
TL;DR: In this article, the authors investigate the quantum van der Pol oscillator near the ground state with additional dissipation and show that it leads to an increase in synchronization, which is not the case with the ground-state oscillator.
Abstract: The authors investigate the quantum van der Pol oscillator near the ground state with additional dissipation and show that it leads to an increase in synchronization.
23 citations
22 citations
TL;DR: In this article, the authors describe spatiotemporal patterns in a network of identical van der Pol oscillators coupled in a two-dimensional geometry, and show that the system under study demonstrates a plethora of different spatio-temporal structures including chimera states when the coupling parameters are varied.
Abstract: We describe spatiotemporal patterns in a network of identical van der Pol oscillators coupled in a two-dimensional geometry. In this study, we show that the system under study demonstrates a plethora of different spatiotemporal structures including chimera states when the coupling parameters are varied. Spiral wave chimeras are formed in the network when the coupling strength is rather large and the coupling range is short enough. Another type of chimeras is a target wave chimera. It is shown that solitary states play a crucial role in forming an incoherence cluster of this chimera state. They can also spread within the coherence cluster. Furthermore, when the coupling range increases, the target wave chimera evolves to the regime of solitary states which are randomly distributed in space. Growing the coupling strength leads to the attraction of solitary states to a certain spatial region, while the synchronous regime is set in the other part of the system. This spatiotemporal pattern represents a solitary state chimera, which is firstly found in the system of continuous-time oscillators. We offer the explanation of these phenomena and describe the evolution of the regimes in detail.
TL;DR: It is shown that the optimized waveform for the entrainment stability yields fasterEntrainment to the driving signal than the case with a simple sinusoidal waveform, while that for the phase coherence yields little improvement from thesinusoidal case.
Abstract: Optimal entrainment of a quantum nonlinear oscillator to a periodically modulated weak harmonic drive is studied in the semiclassical regime. By using the semiclassical phase-reduction theory recently developed for quantum nonlinear oscillators [Y. Kato, N. Yamamoto, and H. Nakao, Phys. Rev. Res. 1, 033012 (2019)10.1103/PhysRevResearch.1.033012], two types of optimization problems, one for the stability and the other for the phase coherence of the entrained state, are considered. The optimal waveforms of the periodic amplitude modulation can be derived by applying the classical optimization methods to the semiclassical phase equation that approximately describes the quantum limit-cycle dynamics. Using a quantum van der Pol oscillator with squeezing and Kerr effects as an example, the performance of optimization is numerically analyzed. It is shown that the optimized waveform for the entrainment stability yields faster entrainment to the driving signal than the case with a simple sinusoidal waveform, while that for the phase coherence yields little improvement from the sinusoidal case. These results are explained from the properties of the phase sensitivity function.
TL;DR: In this article, a two-dimensional wake oscillator model for VIV of a horizontal flexible structure with pinned-pinned ends in uniform flow is presented. And the model is calibrated with the published experimental data by Sanaati and Kato (2012) for the middle cross-section.
TL;DR: In this paper, a nonlinear time-domain simulation model for predicting two-dimensional vortex-induced vibration (VIV) of a flexibly mounted circular cylinder in planar and oscillatory flow is presented.
TL;DR: The distributed-order hyperchaotic unforced and forced complex van der Pol oscillators with complex parameter are introduced and investigated and a scheme to achieve the complete synchronization between the two oscillators is state.
Abstract: The distributed-order hyperchaotic unforced and forced complex van der Pol oscillators with complex parameter are introduced and investigated in this paper. The basic dynamical properties including equilibrium point and its stability and chaotic behavior of the unforced oscillator are studied. The intervals of the parameters values at which this oscillator has periodic, chaotic, and hyperchaotic behaviors are calculated using Lyapunov exponents. These intervals of chaotic and hyperchaotic behaviors can be used in many applications such as secure communication and electronic circuits. Using the linear feedback control, the control of solutions of our oscillator(unforced) converge to a fixed point are studied. We state a scheme to achieve the complete synchronization between two distributed-order hyperchaotic unforced complex van der Pol oscillators. The analytical formula of the controller is derived and used to achieve synchronization. Secure communications via hyperchaotic masking for a text which contains alphabets, numbers, space, and symbols are investigated using the proposed scheme of this work. The dynamics of the distributed-order hyperchaotic forced complex van der Pol oscillator with complex parameter is investigated. Synchronization and secure communications can be similarly studied for the forced oscillator.
TL;DR: In this paper, a model based on the vector form intrinsic finite element (VFIFE) method is proposed to study vortex-induced vibration (VIV) problems of top-tensioned risers, which is applicable to the fully-coupled cross-flow and in-line riser VIV scenarios.
06 Oct 2020
TL;DR: In this paper, the phase reduction in networks of coupled oscillators in the higher orders of the coupling parameter is studied. But the phase is not explicitly expressed in the phase equation.
Abstract: We explore the phase reduction in networks of coupled oscillators in the higher orders of the coupling parameter. For coupled Stuart-Landau oscillators, where the phase can be introduced explicitly, we develop an analytic perturbation procedure to allow for the obtaining of the higher-order approximation explicitly. We demonstrate this by deriving the second-order phase equations for a network of three Stuart-Landau oscillators. For systems where explicit expressions of the phase are not available, we present a numerical procedure that constructs the phase dynamics equations for a small network of coupled units. We apply this approach to a network of three van der Pol oscillators and reveal components in the coupling with different scaling in the interaction strength.
TL;DR: In this article, the authors studied the limit cycles of some cubic family of differential equations, containing the well-known Van der Pol-Duffing and Rayleigh Duffing oscillators, and provided their global phase portraits in the Poincare disk.
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TL;DR: In this article, the phase reduction in networks of coupled oscillators in the higher orders of the coupling parameter is studied. But the phase is not explicitly expressed in the phase equation.
Abstract: We explore the phase reduction in networks of coupled oscillators in the higher orders of the coupling parameter. For coupled Stuart-Landau oscillators, where the phase can be introduced explicitly, we develop an analytic perturbation procedure to allow for the obtaining of the higher-order approximation explicitly. We demonstrate this by deriving the second-order phase equations for a network of three Stuart-Landau oscillators. For systems where explicit expressions of the phase are not available, we present a numerical procedure that constructs the phase dynamics equations for a small network of coupled units. We apply this approach to a network of three van der Pol oscillators and reveal components in the coupling with different scaling in the interaction strength.
TL;DR: In this article, the authors investigated the canard explosion in a van der Pol electronic oscillator, a fast transition from a small amplitude periodic orbit to a relaxation oscillation, and developed an effective procedure based on the nonlinear time transformation method, that uses elementary trigonometric functions.
TL;DR: This model can be used to perform inexpensive numerical control experiments to suppress synchronization and thereby to mitigate unwanted oscillations in physical systems and report the existence of symmetry breaking phenomena during this transition from desynchronized chaos to synchronized periodicity.
Abstract: Some physical systems with interacting chaotic subunits, when synchronized, exhibit a dynamical transition from chaos to limit cycle oscillations via intermittency such as during the onset of oscillatory instabilities that occur due to feedback between various subsystems in turbulent flows. We depict such a transition from chaos to limit cycle oscillations via intermittency when a grid of chaotic oscillators is coupled diffusively with a dissimilar chaotic oscillator. Toward this purpose, we demonstrate the occurrence of such a transition to limit cycle oscillations in a grid of locally coupled non-identical Rossler oscillators bidirectionally coupled with a chaotic Van der Pol oscillator. Further, we report the existence of symmetry breaking phenomena such as chimera states and solitary states during this transition from desynchronized chaos to synchronized periodicity. We also identify the temporal route for such a synchronization transition from desynchronized chaos to generalized synchronization via intermittent phase synchronization followed by chaotic synchronization and phase synchronization. Further, we report the loss of multifractality and loss of scale-free behavior in the time series of the chaotic Van der Pol oscillator and the mean field time series of the Rossler system. Such behavior has been observed during the onset of oscillatory instabilities in thermoacoustic, aeroelastic, and aeroacoustic systems. This model can be used to perform inexpensive numerical control experiments to suppress synchronization and thereby to mitigate unwanted oscillations in physical systems.
TL;DR: In this article, the van der Pol oscillator in the deep quantum regime was studied and the results showed that squeezed driving loses its effect, noise boosts synchronization, bounded synchronization is bounded, and the limit-cycle is insensitive to strong driving.
Abstract: Synchronization occurs ubiquitously in nature. The van der Pol oscillator has been a favorite model to investigate synchronization. Here we study the oscillator in the deep quantum regime, where nonclassical effects dominate the dynamics. Our results show: (i) squeezed driving loses its effect, (ii) noise boosts synchronization, (iii) synchronization is bounded, and (iv) the limit-cycle is insensitive to strong driving. We propose a synchronization measure and analytically calculate it. These results reflect intrinsic differences between synchronization in the quantum and deep quantum regimes.
TL;DR: In this paper, a van der Pol Oscillator (VDP) was used to represent the oscillatory nature of wake dynamics caused by the vortex shedding and the damping term in the VDP oscillator was assumed to be nonlinear.
Abstract: This article focuses on the dynamics of a modified van der Pol–Duffing circuit (MVDPD hereafter) (Fotsin and Woafo in Chaos Solitons and Fractals 24(5):1363–1371, 2005) whose symmetry is explicitly broken with the presence an offset term. When ignoring offset terms, the system displays an exact symmetry which is reflected in the location of the equilibrium points, the attractor topologies and the attraction basins shapes as well. In this mode of operation, the system displays typical behaviors such as period doubling sequences; spontaneous symmetry breaking, symmetry recovering,
and multistability involving several pairs of mutually symmetric attractors. In the presence of offset terms, the MVDPD circuit is non-symmetric and more complex nonlinear phenomena arise such as parallel bifurcation branches, coexisting multiple (i.e. two, three, four or five) asymmetric attractors, and crises. It should be noted that for each case of multistability discussed in this work, a hidden attractor (period-1 limit cycle) coexists with self-excited others. To the best of our knowledge, the coexistence of five attractors (symmetrical or asymmetrical), one of which is hidden has not yet been reported in the MVDPD circuit and thus deserves dissemination. PSpice simulation investigations based on the implementation of the MVDPD confirm the theoretical predictions.
TL;DR: This work analyzes the statistics of solutions and proposes a method to estimate parameter values from the oscillator's time series and uses experimental data of active oscillations in a biophysical system to demonstrate how the method applies to real observations and can be generalized for more complex models.
Abstract: The Van der Pol equation is a paradigmatic model of relaxation oscillations. This remarkable nonlinear phenomenon of self-sustained oscillatory motion underlies important rhythmic processes in nature and electrical engineering. Relaxation oscillations in a real system are usually coupled to environmental noise, which further enriches their dynamics, but makes theoretical analysis of such systems and determination of the equation parameter values a difficult task. In a companion paper we have proposed an analytic approach to a similar problem for another classical nonlinear model---the bistable Duffing oscillator. Here we extend our techniques to the case of the Van der Pol equation driven by white noise. We analyze the statistics of solutions and propose a method to estimate parameter values from the oscillator's time series. We use experimental data of active oscillations in a biophysical system to demonstrate how our method applies to real observations and can be generalized for more complex models.
TL;DR: This work studies explicit numerical integrators for “conditionally linear” systems of ordinary differential equations in neuronal dynamics using the Hodgkin--Huxley model of neuronal dynamics.
Abstract: Motivated by the Hodgkin--Huxley model of neuronal dynamics, we study explicit numerical integrators for “conditionally linear” systems of ordinary differential equations. We show that splitting an...
TL;DR: In this paper, the center-of-mass motion of a nanoparticle is cooled by a purely quadratic coupling between its motion and the optical field of a high finesse cavity.
Abstract: We report on cooling the center-of-mass motion of a nanoparticle due to a purely quadratic coupling between its motion and the optical field of a high finesse cavity The resulting interaction gives rise to a Van der Pol nonlinear damping, which is analogous to conventional parametric feedback where the cavity provides passive feedback without measurement We show experimentally that like feedback cooling the resulting energy distribution is strongly nonthermal and can be controlled by the nonlinear damping of the cavity As quadratic coupling has a prominent role in proposed protocols to generate deeply nonclassical states, our work represents a first step for producing such states in a levitated system
TL;DR: An independent bifurcation tree of period-2 motions to chaos coexisting with period-1 motions in a periodically driven van der Pol–Duffing oscillator is presented semi-analytically and nonlinear systems can be applied in nonlinear circuit design and fluid-induced oscillations.
Abstract: In this paper, an independent bifurcation tree of period-2 motions to chaos coexisting with period-1 motions in a periodically driven van der Pol–Duffing oscillator is presented semi-analytically. ...
TL;DR: In this paper, a model-based, output-only parameter identification method for self-sustained oscillators forced by dynamic noise is presented, which is based on analyzing stationary time series of a single key observable: the acoustic pressure fluctuations inside the bottle.