Van Der Pol Oscillator Research Articles

Primary and super-harmonic simultaneous resonance of van der Pol oscillator

In this paper the stability and the dynamic behavior of primary and super-harmonic simultaneous resonance of van der Pol (VDP) oscillator are researched. At first, the first-order approximate analytical solution is obtained by the method of multiple scales, and the correctness and satisfactory precision of the approximate solution are verified by comparing with the numerical solution. Moreover, the amplitude–frequency and phase-frequency equations of steady-state solution is derived from the approximate analytical solution, and the stability condition is discussed based on Lyapunov theory. Then the stability region and the types of equilibrium points are qualitatively analyzed and the result shows that the amplitude of the super-harmonic excitation is the major contributor to the stability and the types of equilibrium points. Finally, the periodic motion and chaos of VDP oscillator are analyzed numerically. It is found the VDP oscillator with small primary excitation amplitude and large super-harmonic excitation amplitude will generate complex dynamic phenomena including periodic and chaotic motions, which are certified by bifurcation diagram, largest Lyapunov exponent and power spectrum. These results are helpful to the analysis and design of the similar systems.

Time-energy suboptimal control of nonlinear input-affine systems

In this paper, we derive the necessary conditions for time-energy optimal control of input-affine nonlinear systems using Hamilton–Jacobi–Bellman equation. The objective function is chosen as a weighted combination of control effort and time-horizon minimisation. The necessary conditions of optimality are transformed such that the resultant conditions are structurally similar to a differential matrix Riccati equation. A closed-form approximate solution to these necessary conditions is derived such that the resulting closed-loop dynamics is locally exponentially stable around the origin. Furthermore, it is also shown that the states converge within a ball of an arbitrarily small radius around the origin within a finite time. Finally, the performance of the proposed controller is tested using numerical simulations for Vander Pol's oscillator as well as spacecraft detumbling problem. Furthermore, the performance of the proposed controller is also tested in comparison to the infinite-horizon state-dependent Riccati equation controller.

Three-dimensional nonlinear vortex-induced vibrations of top-tension risers considering platform motion

A novel three-dimensional (3-D) dynamic model considering nonlinearity of tension is proposed to explore in-line (IL), cross-flow (CF) and axial (AX) vortex-induced vibration (VIV) responses of riser under platform motion. The interaction between riser and external flow is simulated by the van der Pol equation, and the nonlinear equations of motion are established by Hamilton's principle and the Galerkin method, and solved by Runge-Kutta method. The influences of heave and swing motions of platform on VIV responses of the riser are investigated. It is found that for large velocity of external flow, the axial displacement of riser results in an increase in IL excitation mode and also an increase in IL oscillating displacement. When the riser is excited by swing motion of platform, with the increase of excitation amplitude, the maximum response amplitudes will also increase in all the three directions, and the amplitudes in high frequency domain increase more obviously. Combined effect of heave and swing motions of the platform results in the increase of VIV amplitudes in IL and AX directions, and combined action of forced and parametric excitations has little effect on response amplitude of riser in CF direction.

Benchmarking Nonlinear Oscillators for Grid-Forming Inverter Control

Virtual oscillator control (VOC) is a time-domain strategy that leverages the dynamics of nonlinear oscillator circuits for synchronizing and regulating grid-forming inverters. In this article, we examine a class of second-order circuits composed of a harmonic oscillator and nonlinear state-dependent damping that has found extensive interest in the context of VOC. We center our analysis on the Van der Pol, Dead-zone, and Andronov-Hopf oscillators; these are characterized by several distinguishing attributes but they all share the common structure noted above. Analytical methods based on averaging and perturbation theory are outlined to derive several performance metrics related to harmonic and dynamic properties of these oscillators under a unified framework. Our results reveal that the Andronov-Hopf oscillator is well suited for grid-forming inverter applications since it can yield harmonics-free waveforms without compromising dynamic performance. Analytical results are validated with numerical simulations and experiments, and a multiinverter hardware setup is used to illustrate a practical use case.

Hydrodynamic response of an annular swirling liquid sheet surrounding a forced gas jet

Gas–liquid coaxial injectors invariably involve a shear layer between a fast-moving gas stream and a slow-moving liquid stream. This shear layer within the confinement of a recess has a region of absolute instability beyond a critical momentum flux ratio. This causes the spray to exhibit self-pulsation at discrete natural frequencies. We apply sinusoidal acoustic forcing to the gas jet over a range of frequencies and amplitudes to explore the frequency response of the spray. The fluctuating spray width near the injector orifice is measured as a time-resolved tracer characteristic, and its power spectral density is used to determine the spectral response of the spray. The non-pulsating spray that involves a primarily convectively unstable shear layer responds unconditionally to all the forcing frequencies. However, for a self-pulsating spray, when the forcing frequency is far from the natural frequency and both are incommensurate, the spectral response involves both the frequencies, their linear combinations and harmonics representing a state of quasi-periodicity. When the forcing frequency is close enough or the amplitude is high enough, 1 : 1 lock-in is observed where the natural mode is suppressed completely and the spray behaves just like the unforced flow with the peak shifted to the forcing frequency. Combining the experimental observations and application of the van der Pol oscillator model, we could demonstrate the analogous behaviour of this multiphase system with other hydrodynamically self-excited systems with external forcing. The results presented here can prove significant in understanding the dynamics of the atomization process in thermoacoustic coupling and possible control of it.

Spatiotemporal pattern of periodic rhythms in delayed Van der Pol oscillators for the CPG-based locomotion of snake-like robot

This is the further research on the delayed half-center oscillator (DHCO) neural system presented in our previous paper (Song and Xu in Nonlinear Dyn 108:2595–2609, 2022. https://doi.org/10.1007/s11071-022-07222-y). The DHCO is used to construct a CPG (central pattern generator) neural system to control locomotion of a snake-like robot with pitch-yaw connecting configuration. To this end, we firstly give an improved model of the VDP (Van der Pol) oscillator. Employing mutually coupled delay, a pair of VDP oscillator is connected to produce an half-center oscillator (HCO) module with time delay that is called as a DHCO (delayed HCO) model. Based on the analysis of the Hopf bifurcation, periodic rhythm and their spatiotemporal patterns of the DHCO are illustrated in the different regions of parameters. The DHCO presents periodic rhythms with synchronous and anti-synchronous patterns, which is to control joint actuators combined in snake-like robot with pitch-yaw connecting configuration. To realize a backward propulsive wave to promote snake-like robotic locomotion, based on the DHCOs, we construct a chain type of the CPG neural system combined with a new unidirectional delay in which phase difference can be regulated. Numerical simulations are illustrated that the CPG neural system can control snake-like robot to move with serpentine, rectilinear, and side-winding patterns in the forward and backward directions. The results show that the snake-like robot can be controlled in expected locomotion patterns for a region but not a fixed value of the controlling parameters. Further, the corresponding regions of the parameters are obtained by using theoretical dynamical analysis but not a trial-and-error method. The snake-like robot gets smooth and stable gait transition with parameter changing.

Solving Perturbed Dynamic Systems Using Schur Decomposition

This work introduces a methodology for the generation of an approximate analytical solution to perturbed ordinary differential equations using Schur decomposition. This methodology is based on the use of operator theory to find a linear approximation to the ordinary differential equation in an expanded space of configuration. Once this linearization is performed, the Schur decomposition is used to transform the resultant differential equation into an upper triangular system that can be solved sequentially following the upper triangular structure. Based on these results, a perturbation technique is also proposed to study these problems. Finally, this work provides a set of algorithms to automate all the methodologies presented, with a special focus on polynomial differential equations and the use of Legendre polynomials to represent the dynamical system. Several examples of application are provided, including the Duffing oscillator, the Van der Pol oscillator, and the zonal harmonics problem around the moon.

Vortex-induced vibration of flexible riser transporting high-speed spiral flow in deep-sea mining

According to Hamilton's principle and a mechanical decomposition model proposed for high-speed spiral flow, vibration governing equations of flexible riser transporting high-speed spiral flow are developed for vortex-induced vibration (VIV) analysis for lifting pipe system in deep-sea mining. The classic van der Pol oscillator is used to simulate the dynamic characteristics of VIV. The governing equations are then discretized and solved by the finite element method and the Runge-Kuta method. Furthermore, dynamic characteristics of VIV of the flexible riser transporting straight flow and spiral flow are investigated and the effects of the spiral flow incidence velocity and angle controlled by the jet device on VIV responses are evaluated. Also, the VIV dynamics considering different damping ratios are examined. The results show that the rotational angular velocity of the spiral flow would amplify the mean deflection of the in-line (IL) direction, and the incidence velocity has a significant nonlinear effect on the dominant frequency of the riser, which would induce the high-order dominant vibration modes when the incidence velocity is large sufficiently. However, the incidence angle has relatively less effects on the VIV responses of flexible riser. Moreover, the locked interval of VIV is less influenced by damping ratio below 0.1.

Multi-Mode Motion Dynamics of a Top Tensioned Riser Excited by Vortices and Varying Tension

A coupled model of a top tensioned riser (TTR) under combined excitations from vortices and time-varying tension is proposed. And the van der Pol oscillator is introduced to simulate the time-varying characteristics of vortex shedding. The first three mode approximations as a six degree-of-freedom (DOF) system of vibrations of the fluid-structure interaction system are derived. Two cases of resonances are analyzed by using the numerical method. One case is on the vortex induced vibration (VIV) of the six DOF system, where the 1-to-1 internal resonance (also named “lock-in” phenomenon) between frequencies of the first modes of structure and the wake oscillator are investigated. Bifurcation curves between dynamic responses in terms of the shedding frequency are derived. Comparing to the vibrations of the first order mode, the “lock-in” phenomenon for vibrations of the higher order modes can occur in the wider parameter range with large values. The second case is on the combined resonance between the 1-to-1 internal and 1-to-2 parametric resonance. Response curves versus the shedding frequency under different amplitude ratios are calculated by the Runge–Kutta method. Effects of system parameters including the amplitude ratio of the parametric excitation, shedding frequency on the response curves are investigated. Comparing to the case when only considering the 1-to-1 internal resonance in the VIV, results of the combined resonance show those large-amplitude responses of the structure and wake oscillator can be induced. Responses will increase rapidly as the amplitude of the varying tension increases. Unstable responses can occur within a large frequency band of nondimensional shedding frequency. Furthermore, the dynamic characteristics of the responses of the six DOF system including the bifurcation diagrams, time histories, phase-plane projection and Poincaré sections are computed for different amplitude ratios. Results indicate that complex dynamics such as the quasi-periodic and chaotic solutions can occur for different amplitude ratios. The analysis can be helpful for the parameter design of deep-water risers.

Oscillations in mixed-feedback systems

A new method is presented for the analysis of limit cycle oscillations in mixed-feedback systems. The calculation of the limit cycle is reformulated as the zero finding of a mixed-monotone relation, that is, of the difference of two maximally monotone relations. The problem can then be solved efficiently by borrowing existing algorithms that minimize the difference of two convex functions. The potential of the method is illustrated on the classical Van der Pol oscillator.

Anti-phase synchronization of waves in a multiplex network of van der Pol oscillators

We study numerically the spatiotemporal dynamics and anti-phase synchronization of complex structures in a two-layer multiplex network of bidirectionally and repulsively coupled two-dimensional lattices. Each layer consists of locally and attractively coupled identical van der Pol oscillators. Depending on randomly distributed initial conditions the isolated layers can exhibit either a spiral or a target wave regime. The latter is found for first time for the locally coupled van der Pol lattice considered in this work. We vary the inter-layer coupling strength for either fixed or changing intra-layer coupling strengths in the interacting layers. The numerical results are summarized in the two-dimensional diagrams plotted in the “intra-layer coupling strength – inter-layer coupling strength” parameter plane. Anti-phase synchronization is identified by evaluating the local and global cross-correlation coefficients which are equal to −1. We also explore the impact of random de-multiplexing on the robustness of the observed spatiotemporal structures and anti-phase synchronized regimes in the two-layer network.

Energy harvesting from a rotational nonlinear energy sink in vortex-induced vibrations

The concept of nonlinear energy sinks (NES) has been studied recently for passive fluid-induced vibration mitigation. A NES presents a fast and irreversible capacity of efficiently dissipating energy for various vibration problems. Besides, energy harvesting (EH) applications are suitable for generating electricity in vibrating mechanical structures. This work investigates the combination of a rotational NES with EH for the vortex-induced vibrations (VIV) of a cylinder exposed to airflow. The study considers an energy harvesting device comprising an electric generator ideally attached to the rotational NES mass pendulum shaft, resulting in a cylinder-NES-EH assembly. The model consists of an elastically mounted cylinder subjected to transverse airflow. The NES-EH is attached to the cylinder as a rigid pendulum arm with a mass at its tip that can freely oscillate, simultaneously absorbing the cylinder vibration and generating electricity. The fluid modeling is based on a wake model represented by the Van der Pol oscillator. The study is carried out by computational simulations of the differential equations of motion, and the rotational NES parameters optimization is done with the particle swarm method. Results comprise the analyses of the cylinder-NES-EH nonlinear dynamics characterization, identification of response regimes of the NES-EH system through a parametric study, and comparison with results from the literature. An optimal configuration was presented to maximize the generation of electric energy. Combining an EH with a rotational NES has shown an adequate potential for the design of energy extraction devices.

Passive suppression of vortex-induced vibrations using a nonlinear energy sink—Numerical and analytical perspective

This study investigates the suppression mechanism of instabilities induced by fluid–structure interactions (FSI) using passive vibration absorption devices, such as nonlinear energy sink (NES). The present FSI framework comprises a low-order phenomenological model, wherein the wake effect is modeled using the classical Van der Pol oscillator. The structure is represented as a cylindrical bluff body with degree-of-freedom along the cross-flow direction. The response of the NES-augmented structure exhibits specific relaxation type oscillations, referred to as strongly modulated response (SMR), passively suppressing the high amplitude vortex-induced vibrations (VIV). The underlying mechanism of SMR is studied using an analytical approach based on the Complexification-Averaging (CXA) technique. Using the CXA technique, the slow flow for the coupled FSI system with the NES attachment is modeled effectively, revealing nonlinear beating regimes and initiation of the targeted energy transfer (TET) mechanism. Subsequently, a transient resonance capture, implying a significant energy transfer from the structure to the NES, results in effective VIV suppression. The occurrence of SMRs is explained by analyzing the global dynamics using the slow invariant manifold (SIM) derived from the slow flow of the coupled system. The resultant SIM topology reveals a jump phenomenon between the stable branches, wherein the flow jumps from a lower stable branch to a higher stable branch through SMR. The novelty of this study lies in identifying the occurrence of SMRs as the mechanism of energy transfer and uses the CXA technique to explain the global dynamics by modeling the SIM for the 3-DOF coupled FSI framework with the NES attachment. Furthermore, the optimal operational parameter ranges for an efficient NES design are identified through a parametric study using the slow-flow model.

Quantum synchronization in quadratically coupled quantum van der Pol oscillators

We implement nonlinear anharmonic interaction in the coupled van der Pol oscillators to investigate the quantum synchronization behavior of the systems. We study the quantum synchronization in two oscillator models, coupled quantum van der Pol oscillators and anharmonic self-oscillators. We demonstrate that the considered systems exhibit a high-order synchronization through coupling in both classical and quantum domains. We show that, due to the anharmonicity of the nonlinear interaction between the oscillators, the system exhibits phonon blockade in the phase-locking regime, which is a pure nonclassical effect and has not been observed in the classical domain. We also demonstrate that, for coupled anharmonic oscillators, the system shows a multiple resonance phase-locking behavior due to nonlinear interaction. We point out that the synchronization blockade arises due to strong anticorrelation between the oscillators, which leads to phonon antibunching in the same parametric regime. In the anharmonic oscillator case we illustrate the simultaneous occurrence of bunching and antibunching effects as a consequence of simultaneous negative and positive correlation between the anharmonic oscillators. We examine the aforementioned characteristic features in the frequency entrainment of the oscillators using a power spectrum where one can observe normal mode splitting and the Mollow triplet in the strong coupling regime. Finally, we propose a possible experimental realization for the considered system in the trapped ion and optomechanical setting.

Nested mixed-mode oscillations in a canard-generating driven Bonhoeffer–van der Pol oscillator

In previous works (Inaba and Kousaka, 2020; Inaba and Tsubone, 2020), the existence of nested mixed-mode oscillation (MMO) bifurcation structures, generated by a driven slow–fast Bonhoeffer–van der Pol (BVP) oscillator, was confirmed. This oscillator has a stable focus and a stable relaxation oscillation caused by a subcritical Hopf bifurcation in the absence of perturbation. In this study, we assert that nested MMOs in this circuit can occur regardless of whether the oscillation is caused by a supercritical or subcritical Hopf bifurcation when no perturbation is applied, and that therefore, this phenomenon could be widespread. We show that MMOs that occur under weak periodic perturbations, generated in a classical BVP oscillator, where the oscillation is caused by a supercritical Hopf bifurcation, can be at least doubly nested. To show that nested MMOs can be at least doubly nested, we utilize first return plots, which are approximated one-dimensional maps and well explain mixed-mode oscillation-incrementing bifurcations.

Geometrical modelling of neuronal clustering and development

The dynamic geometry of neuronal development is an essential concept in theoretical neuroscience. We aimed to design a mathematical model which outlines stepwise in an innovative form and designed to model neuronal development geometrically and modelling spatially the neuronal-electrical field interaction. We demonstrated flexibility in forming the cell and its nucleus to show neuronal growth from inside to outside that uses a fractal cylinder to generate neurons (pyramidal/sphere) in form of mathematically called ‘surface of revolution’. Furthermore, we verified the effect of the adjacent neurons on a free branch from one-side, by modelling a ‘normal vector surface’ that represented a group of neurons. Our model also indicated how the geometrical shapes and clustering of the neurons can be transformed mathematically in the form of vector field that is equivalent to the neuronal electromagnetic activity/electric flux. We further simulated neuronal-electrical field interaction that was implemented spatially using Van der Pol oscillator and taking Laplacian vector field as it reflects biophysical mechanism of neuronal activity and geometrical change. In brief, our study would be considered a proper platform and inspiring modelling for next more complicated geometrical and electrical constructions.

Coupled van der Pol and Duffing oscillators: emergence of antimonotonicity and coexisting multiple self-excited and hidden oscillations

Coupled van der Pol and Duffing oscillators: emergence of antimonotonicity and coexisting multiple self-excited and hidden oscillations

Hierarchical deep learning of multiscale differential equation time-steppers.

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 expensive. In this work, we develop a hierarchy of deep neural network time-steppers to approximate the dynamical system flow map over a range of time-scales. The model is purely data-driven, enabling accurate and efficient numerical integration and forecasting. Similar ideas can be used to couple neural network-based models with classical numerical time-steppers. Our hierarchical time-stepping scheme provides advantages over current time-stepping algorithms, including (i) capturing a range of timescales, (ii) improved accuracy in comparison with leading neural network architectures, (iii) efficiency in long-time 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 numerous 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.This article is part of the theme issue ‘Data-driven prediction in dynamical systems’.

Symplectic Dynamics and Simultaneous Resonance Analysis of Memristor Circuit Based on Its van der Pol Oscillator

A non-autonomous memristor circuit based on van der Pol oscillator with double periodically forcing term is presented and discussed. Firstly, the differences of the van der Pol oscillation of memristor model between Euler method and symplectic Euler method, four-order Runge–Kutta method (RK4) and four-order symplectic Runge–Kutta–Nyström method (SRKN4), symplectic Euler method and RK4 method, and symplectic Euler method and SRKN4 method in preserving structure are compared from theoretical and numerical simulations, the symmetry and structure preserving and numerical stability of symplectic scheme are demonstrated. Moreover, the analytic solution of the primary and subharmonic simultaneous resonance of this system is obtained by using the multi-scale method. Finally, based on the resonance relation of the system, the chaotic dynamics behaviors with different parameters are studied.

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.