Van Der Pol Oscillator Research Articles

Integral resonant negative derivative feedback suppression control strategy for nonlinear dynamic vibration behavior model

One of the major problems in robotics research has been developing an actuator system for extremely dynamic-legged robots. High torque density and the capacity to control dynamic physical interactions are two design requirements for high-speed locomotion that are challenging for conventional actuators used in manufacturing applications to meet. To address this system and apply the desired control to reach the best stability position, the robot's foot was simulated with the Van der Pol equations, applied the required control, and studied that application. This work describes the actions of a new novel control mechanism known as the Integral Resonant Negative Derivative Feedback (IRNDF) controller, which reduces the vibration response of a double Van der Pol oscillator subjected to external excitations. This unique controller combines integral resonant control (IRC) and negative derivative feedback (NDF) controllers to provide a new controller effect for double Van der Pol oscillators. The multiple scale perturbation technique (MSPT) has been applied to solve the controlled system analytically. The MATLAB and MAPLE programs have been used to complete and clarify all of the numerical talks. The frequency response curves have been used to study the impact that altering the parameter values had on the amplitude. The controlled system vibration amplitude is governed by frequency-response equations (FREs), which have been constructed. In the vibration system, the IRC, NDF, and IRNDF controllers were compared to see which one was the best. Numerical results show that the unique IRNDF controller is the best at reducing oscillations and decreasing amplitude values. The effects of the effective parameters on the controlled system have been identified. The frequency-response equation that was derived has been used to plot the various response curves for the framework that show the stable and unstable zones when the controller is off and on. Lastly, excellent agreement between the derived numerical findings and the analytical ones was observed. Lastly, utilizing time histories and response curves to compare analytical and numerical solutions was fascinating and significant.

Enhanced numerical resolution of the Duffing and Van der Pol equations via the spectral homotopy analysis method employing chebyshev polynomials of the first kind

Enhanced numerical resolution of the Duffing and Van der Pol equations via the spectral homotopy analysis method employing chebyshev polynomials of the first kind

Data-driven predictor of control-affine nonlinear dynamics: Finite discrete-time bilinear approximation of koopman operator

This paper introduces a novel and efficient data-driven approach for approximating a finite discrete-time bilinear model of control affine nonlinear dynamical systems with a tunable parameter that balances model dimension and prediction accuracy. An approximation of the Koopman operator based on the evolutions of the nonlinear system measurements used to lift a control-affine nonlinear system to a higher dimensional model. However, higher dimensional spaces can result in a long learning time and the curse of dimensionality in control analysis. The proposed approach addresses these challenges by introducing a convex optimization which identifies informative observable functions. This technique allows for the adjustment of a parameter to strike a balance between model dimension and accuracy in prediction. The main contribution of this study is to introduce a reduced dimensional bilinear model for a nonlinear complex system. This achievement is made possible by implementing convex sparse optimization, enabling the exploration of informative estimated Koopman eigenfunctions while minimizing the number of system measurements required. The optimization problem is solved using the alternating direction method of multipliers. The effectiveness of the proposed method is evaluated on three different nonlinear systems: a numerical nonlinear system, a Van der Pol oscillator, and a Duffing oscillator. In the last simulation, an estimation of the Koopman linear model is considered as a special case, and the policy iteration algorithm is employed to evaluate optimal control designed for different reduced-dimensional models.

Optimal synchronization to a limit cycle.

In the absence of external forcing, all trajectories on the phase plane of the van der Pol oscillator tend to a closed, periodic trajectory-the limit cycle-after infinite time. Here, we drive the van der Pol oscillator with an external time-dependent force to reach the limit cycle in a given finite time. Specifically, we are interested in minimizing the non-conservative contribution to the work when driving the system from a given initial point on the phase plane to any final point belonging to the limit cycle. There appears a speed-limit inequality, which expresses a trade-off between the connection time and cost-in terms of the non-conservative work. We show how the above results can be generalized to the broader family of non-linear oscillators given by the Liénard equation. Finally, we also look into the problem of minimizing the total work done by the external force.

Three-dimensional nonlinear vortex-induced vibrations of risers under internal gas-liquid and external shear flows

A novel three-dimensional (3D) dynamic model is proposed to explore the nonlinear vortex-induced vibration (VIV) characteristics of top-tension risers under the combined action of internal gas-liquid two-phase flow and external shear flow. The van der Pol equation is used to evaluate the interaction between the riser and the external shear flow. The nonlinear governing equations are established through Hamilton's principle, and they are discretized by the Galerkin method and solved via the Runge-Kutta method. The VIV responses calculated by the proposed nonlinear dynamic model are compared with the previously experimental and computational fluid dynamics (CFD) results, demonstrating the effectiveness and accuracy of the present theoretical method. The influences of the internal flow effect, the tension and the shear parameter of the external flow on the VIV responses of the riser are analyzed. The results show that when the internal flow effect is not considered, the transition of the external flow from uniform flow to shear flow leads to an increase in excited modes and a decrease in resonance responses of the riser in the cross-flow (CF) direction. Furthermore, with the increase of the external shear flow velocity, resonances at different positions of the riser are not synchronized, which is different from that of the riser in uniform flow. In the case of the riser conveying gas-liquid two-phase flow and being subjected to external uniform or shear flow, the increase in the velocity of the liquid phase leads to an increase in the maximum response amplitudes of the riser in the in-line (IL) and axial directions. When the internal flow changes from single-phase to gas-liquid two-phase, the maximum VIV responses of the riser in the IL and axial directions tend to decrease regardless of whether the riser is in uniform flow or shear flow.

Weakly nonlinear hyperbolic differential equation of the second order in Hilbert space

We consider nonlinear perturbations of the hyperbolic equation in the Hilbert space. Necessary and sufficient conditions for the existence of solutions of boundary-value problem for the corresponding equation and iterative procedures for their finding are obtained in the case when the operator in linear part of the problem hasn't inverse and can have nonclosed set of values. As an application we consider boundary-value problem for van der Pol equation in a separable Hilbert space.

Death transitions in attractive-repulsive coupled oscillators with higher-order interactions

The first-order phase transition from oscillation to steady state, known as explosive death (ED), is prevalent in various dynamical models. However, previous studies on ED have predominantly focused on interactions between pairs of elements. In this work, we investigate how death transitions occur when both pairwise and three-body interactions are present in an extended Van der Pol oscillator network with attractive-repulsive coupling. By examining both global and non-local interaction mechanisms, the impact of higher-order interactions on ED and the differences in the transition process are comprehensively analyzed. Firstly, we construct a diagram of the global dynamics in the context of higher-order and first-order coupling strengths, identifying that the higher-order interactions promote the onset of ED with a contribution comparable to that of first-order interactions. Specifically, for global coupling, the theoretical backward critical curves matching the numerical results are derived through linear stability analyses, showcasing a linear correlation with a slope of -1 between the higher-order and first-order coupling strengths. Under non-local coupling, fitting the numerically obtained backward critical curves likewise yields a consistent quantitative relationship. Additionally, during the transition process of ED, we discover intriguing coexisting states in the hysteresis area under non-local coupling, including the coexistence of chimera states with coherent or incoherent oscillation, and the coexistence of chimera states with oscillation death. This is attributed to symmetry breaking induced by non-local action. These findings enhance the understanding of higher-order interactions in complex systems and provide a fresh perspective for studying multi-stability behavior in biochemical and physical systems.

Limit cycles as stationary states of an extended harmonic balance ansatz

A limit cycle is a self-sustained, periodic, isolated motion appearing in autonomous differential equations. As the period of a limit cycle is unknown, finding it as a stationary state of a rotating ansatz is challenging. Correspondingly, its study commonly relies on numerical methodologies (e.g., brute-force time evolution, and variational shooting methods) or circumstantial evidence such as instabilities of fixed points. Alas, such approaches are (i) , as they rely on specific initial conditions, and (ii) do not provide analytical intuition about the physical origin of the limit cycles. Here, we (I) develop a multifrequency rotating ansatz with which we (II) find limit cycles as stationary-state solutions via a semianalytical homotopy continuation. We demonstrate our approach and its performance on the Van der Pol oscillator. Moving beyond this simple example, we show that our method captures all coexisting fixed-point attractors and limit cycles in a modified nonlinear Van der Pol oscillator. Our results facilitate the systematic mapping of out-of-equilibrium phase diagrams, with implications across multiple fields of the natural sciences. Published by the American Physical Society 2024

Effect of colored noise on precursors of thermoacoustic instability in model gas turbine combustors

In this work, we numerically investigate the dynamics of a prototypical thermoacoustic system, the generalized Van der Pol oscillator, in the presence of additive noise of varying color (correlation time) and intensity while the system undergoes supercritical and subcritical Hopf bifurcation. We specifically investigate the influence of noise color on trends in the coherence factor and the Hurst exponent in the subthreshold region to assess their reliability as instability precursors. The Hurst exponent is found reliable only for correlation times much larger than the time scale of the instability while the coherence factor is found to be reliable for the entire range of noise color investigated. These inferences are found to hold for both supercritical and subcritical bifurcation cases.

Approximations in Mean Square Analysis of Stochastically Forced Equilibria for Nonlinear Dynamical Systems

Motivated by important applications to the analysis of complex noise-induced phenomena, we consider a problem of the constructive description of randomly forced equilibria for nonlinear systems with multiplicative noise. Using the apparatus of the first approximation systems, we construct an approximation of mean square deviations that explicitly takes into account the presence of multiplicative noises, depending on the current system state. A spectral criterion of existence and exponential stability of the stationary second moments for the solution of the first approximation system is presented. For mean square deviation, we derive an expansion in powers of the small parameter of noise intensity. Based on this theory, we derive a new, more accurate approximation of mean square deviations in a general nonlinear system with multiplicative noises. This approximation is compared with the widely used approximation based on the stochastic sensitivity technique. The general mathematical results are illustrated with examples of the model of climate dynamics and the van der Pol oscillator with hard excitement.

Reliability of early warning indicators of critical transition in stochastic Van der Pol oscillators with additive correlated noise

Reliability of early warning indicators of critical transition in stochastic Van der Pol oscillators with additive correlated noise

Crises of a Fractional Birhythmic Van der Pol Oscillator Using the Improved Cell Mapping Method

Birhythmicity oscillators have been extensively applied in fields such as biology, physics, and engineering. Studying their global dynamics is crucial for gaining a comprehensive understanding of the intrinsic mechanisms that govern oscillator behavior. This paper focuses on investigating the influence of memory effects on the global dynamics of the fractional birhythmic van der Pol (BVDP) oscillator. To determine the system’s global properties, we employ an improved cell mapping method. Specifically, the system’s evolution is computed by introducing additional auxiliary variables, creating a space for storing historical information. This method allows us to examine the system’s global properties without memory loss. Through a comparison of the global dynamic behaviors of BVDP oscillators with memory to those without memory, we observe that the presence of memory effects results in the emergence of chaotic attractors in the system. This, in turns, results in system instability and heightened sensitivity to initial conditions. Furthermore, our findings suggest that changes in the fractional-order can induce various crises in the oscillator and may have the potential to suppress chaotic oscillations.

Study of aerodynamic damping of a crosswind-excited circular cylinder and its application to full-scale flexible structures

The accurate assessment of aerodynamic damping is essential when estimating the impact of aero-elastic effects on crosswind-excited line-like structures. This study aims to determine the crosswind aerodynamic damping of a circular cylinder by curve-fitting to the response probability density function. The generalized Van der Pol oscillator model is employed to depict the nonlinearity associated with amplitude-dependent aerodynamic damping. The initial section of this study presents the theoretical foundation of the identified approach used in this research. Subsequently, the efficacy of the identification methodology is validated by means of a stochastic simulation of the crosswind response of a circular cylinder. Moreover, identification of nonlinear aerodynamic damping is further accomplished by the utilization of an aero-elastic model wind tunnel test. The identified nonlinear aerodynamic damping is then utilized to determine the response statistical quantities for a total of 16 chimneys. The determined aerodynamic damping provides a satisfactory assessment of the peak response, which is crucial for design considerations.

Periodic second-order systems and coupled forced Van der Pol oscillators

We present an existence and localization result for periodic solutions of second-order non-linear coupled planar systems, without requiring periodicity for the non-linearities. The arguments for the existence tool are based on a variation of the Nagumo condition and the Topological Degree Theory. The localization tool is based on a technique of orderless upper and lower solutions, that involves functions with translations. We apply our result to a system of two coupled Van der Pol oscillators with a forcing component.

Multi-type synchronization for coupled van der Pol oscillator systems with multiple coupling modes.

In this paper, we investigate synchronous solutions of coupled van der Pol oscillator systems with multiple coupling modes using the theory of rotating periodic solutions. Multiple coupling modes refer to two or three types of coupling modes in van der Pol oscillator networks, namely, position, velocity, and acceleration. Rotating periodic solutions can represent various types of synchronous solutions corresponding to different phase differences of coupled oscillators. When matrices representing the topology of different coupling modes have symmetry, the overall symmetry of the oscillator system depends on the intersection of the symmetries of the different topologies, determining the type of synchronous solutions for the coupled oscillator network. When matrices representing the topology of different coupling modes lack symmetry, if the adjacency matrices representing different coupling modes can be simplified into structurally identical quotient graphs (where weights can be proportional) through the same external equitable partition, the symmetry of the quotient graph determines the synchronization type of the original system. All these results are consistent with multi-layer networks where connections between different layers are one-to-one.

Influence of the magnetic flux on the dynamics of a self-sustaining system: analytical, numerical and analogical investigations

This paper investigates the nonlinear dynamics of a ferroelectric enzyme-substrate reaction modeled by the birhythmic van der Pol oscillator coupled to the magnetic flux. We derive the equilibrium points and study their stability. We analyze some bifurcation structures and the variation of the Lyapunov exponents. The phenomena of symmetric attractors and the anti-monotonicity are observed. By increasing the magnetic flux, we find that the equilibrium points are stable, tends to control chaotic regimes, and affects regular and quasi-regular ones. As the magnetic flux increases, the amplitude of the oscillations around the equilibrium points decreases and the amplitude of the limit cycles at the Hopf bifurcation tends to disappear. Further increasing the magnetic flux gives rise to chaotic dynamics. The electrical circuit and analogical simulations are derived using the PSpice software. The agreement between analogical and numerical results is acceptable.

Characterization of cracking-healing cycle of asphalt mixture based on air-coupled second harmonic measurement using Duffing-van der pol oscillator

Second harmonic generation (SHG) technique using Rayleigh surface wave has been successfully attempted for the non-destruction evaluation of asphalt mixture during the self-healing process. The air-coupled transducer placement is desirable for the convenient implementation of SHG for the large-scale detection. Nevertheless, the pronounced attenuation of ultrasonic wave by air-coupled propagation will cause the second harmonic induced by microcracks to be barely detectable, particularly under the influence of inevitable noise. In this study, a novel approach by combining Duffing oscillator and van der pol equation is developed to identify and quantify the second harmonic in noise-contained signals based on the phase trajectory and periodic trajectory area exponent. The multi-physics finite element model is first established to simulate the air-asphalt propagation of Rayleigh waves, and proof-of-concept numerical simulations are carried out to estimate the second harmonic amplitude of noise-contaminated signals. The air-coupled SHG experiments are conducted, and the approach of the Duffing-van der pol oscillator is applied to extract the second harmonic in experimental data, and the self-healing behavior of asphalt mixture is analyzed through the variation of the nonlinear parameter of SHG measurements. The results indicate that the contactless SHG method combined with the Duffing-van der pol oscillator can effectively realize the convenient and reliable estimation of the nonlinear ultrasonic signal during the cycle of asphalt deterioration and healing, which is important for the property characterization of asphalt mixture considering its high attenuation to the ultrasonic waves.

Dynamics and energy harvesting from parametrically coupled self-excited electromechanical oscillator

The investigated parametrically coupled electromechanical structure is composed of a mechanical Duffing oscillator whose mass sits on a moving belt surface. The driving electrical network is a van der Pol oscillator whose aim is to actuate the attached DC motor to provide some rotatry unbalances and parametric coupling in the vibrating structure. The coupled oscillator is applied to energy harvesting and overcomes the limitation of low energy generation associated with a single oscillator of this kind. The system was solved analytically and validated by numerical methods. The global dynamics of the structure were investigated, and nonlinear phenomena such as Neimark–Sacker bifurcation, discontinuity-induced bifurcation, grazing–sliding, and bifurcation to multiple tori were identified. These nonlinear behaviors affect the harvested energy at bifurcation points, resulting in jumps from one energy level to another. In addition to harnessing the highest energy under hard parametric coupling, the coupling ensures that higher and more useful energy is harvested over a wider range of belt speeds. Finally, the qualitative validation of the numerical concept by experimental setup verifies the workings of the model.

The masking technique for forced nonlinear oscillator stability behavior analysis using the non-perturbative approach

The study utilized the masking technique to explore the stability behavior of a forced nonlinear oscillator through the non-perturbative approach, with a particular focus on a Van der Pol oscillator subjected to external force, characterized by both cubic and quadratic nonlinearities. The application of the non-perturbative method (NPM) in conjunction with the masking technique was a pivotal aspect of this research, transforming the inherently non-homogeneous, nonlinear system into a homogeneous linear system. This transformation was crucial as it simplified the complex dynamics of the system, rendering it more amenable to analysis. Through this method, the research successfully established the system’s overall frequency, meticulously accounting for the impact of the periodic external force. The study also identified a distinct type of resonance response, where the system’s frequency incorporates the excited frequency in a nonlinear relationship. The masking technique proved to be an invaluable tool for examining the stability behavior of forced vibrations in oscillators via the NPM, providing profound insights into stability under external forces and enhancing the understanding and control of oscillatory behaviors in nonlinear dynamical systems. A critical confirmation of the current methodology is provided by the remarkable agreement found between the numerical solution and the provided analytical solution. This agreement shows that the analytical method produces trustworthy predictions and appropriately describes the system’s behavior. The plotted stability diagrams, which demonstrate that the model’s simulation of stability behavior is consistent with observed events, particularly resonance phenomena, offer further validity for the findings. In the resonance case, the effects of the damping coefficient and the external force’s magnitude are significant. The results of the analysis show that an increase in the damping coefficient has a destabilizing effect that causes unstable zones to expand. In contrast, in the resonance state, the quadratic and cubic nonlinearity factors both contribute to stabilization. Understanding how various system factors impact stability dynamics particularly in relation to resonance phenomena is made easier with the help of this insight.

Model-Free Optimal Vibration Control of a Nonlinear System Based on Deep Reinforcement Learning

Optimal control of nonlinear vibration requires precise knowledge of the system and the solution to Hamilton–Jacobi–Bellman (HJB) equation. However, in practical engineering applications, acquiring precise system parameters poses challenges, and the analytical solutions for the HJB equation are difficult to obtain. In this paper, two reinforcement learning algorithms, Twin Delayed Deep Deterministic Policy Gradient (TD3) algorithm and Soft Actor-Critic (SAC) algorithm, are employed to train neural network-based optimal controllers for the van der Pol vibration system in the presence of unknown system parameters. To validate their performance, the controllers undergo testing in a series of experiments, including assessments of free vibration, frequency sweep excitation, and Gaussian noise excitation. The results indicate that both the TD3-trained and SAC-trained neural network-based controller are capable of proficiently suppress the vibration of the van der Pol oscillator. Additionally, these two model-free controllers can approximate the optimal control law which solved based on the dynamic model of the nonlinear system.