Forced Van Der Pol Oscillator Research Articles

A multi-modal nonlinear dynamic model to investigate time-domain responses of a micro-cantilever in fluids

In this current work, a new nonlinear dynamic model based on the forced Van der Pol oscillator is introduced to demonstrate the time-domain sensitivities of the micro-cantilever to the varying properties of the surrounding fluids. Effects of diverse multi-frequency excitations on the hydrodynamically forced displacements are investigated for the Glycerol-water solutions with different concentrations. Driving forces at the eigenmode frequencies are applied simultaneously to actuate the micro-cantilever in multi-modal operations. The hydrodynamic force induces notable variations in the observables of high-frequency steady-state vibrations. To illustrate, the frequency of the displacements decreases with increasing dynamic viscosity and density of the fluids (among 55% and 85% Glycerol-water solutions) in bimodal- and trimodal-frequency excitations. Essentially, the observable responses are often used to distinguish the surrounding fluids in which the micro-cantilever operates. In addition, steady-state observables are achieved at only particular eigenmodes in single- and multi-frequency operations. It is highlighted that the periodic oscillations are obtained for the first and second eigenmodes with the highest value of forced Van der Pol parameter (μ = 1030). Clearly, higher eigenmodes require different values of the nonlinearity parameter to acquire periodic vibrations in multi-modal operations. In general, achieving steady-state observables is substantially critical in quantifying sensitivity to varying fluid properties. For instance, the vibration frequency of around 7.33 MHz and amplitude of around 0.03 pm are obtained at the first eigenmode for 75% Glycerol-water solution in tetra-modal operations. Note that femtometer amplitudes of deflections can be measured using quantum-enhanced AFM techniques. The frequency responses obtained in this work are compared with the measured ones in the literature and the results show satisfactory agreements. Therefore, a novel multi-modal nonlinear dynamic model enables to quantify observable sensitivity to micro-rheological properties at higher eigenmodes of the micro-cantilever.

Theoretical Study of Forced Van Der Pol Oscillator Equation Using Multiple Two-timing Regular Parameter Perturbation and Asymptotic Expansion Techniques

This paper presents the theoretical study of forced Van der Pol oscillator equation. Oscillatory systems are studied to know measures that can reduce the amplitude of oscillation of the oscillatory system. Here, multiple two-timing regular parameter perturbation is applied since it is a kind of perturbation among other perturbation techniques that enables the study of the behaviour of a system under certain conditions. Asymptotic expansion technique was also applied. Excel Microsoft was used to analyse the uniformly valid asymptotic solution of the Van der Pol oscillator equation obtained. The uniformly valid asymptotic solution in the independent variable obtained, showed that damping alters the amplitude of the oscillatory system thereby affecting its motion. Increase in damping decreases the amplitude of oscillation of the system. With damping incorporated in the system though very small damping, the amplitude of oscillation reduces with time.

Nested mixed-mode oscillations in the forced van der Pol oscillator

The forced van der Pol oscillator has played a fundamental role in the development of nonlinear science. It is notable that the van der Pol oscillator in the absence of an AC forcing term explains the underlying mechanism that induces limit cycles and relaxation oscillations; the forced van der Pol oscillator was the first electric oscillator that, via measurements of the emitted sound, was inferred to exhibit chaotic behavior (van der Pol and van der Mark, 1927). This oscillator thus had a significant influence on the development of chaos theory. It was subsequently demonstrated, via a nonstandard analysis undertaken in the 1980s, that a canard explosion was present in the dynamics exhibited by this oscillator. In previous works (Inaba and Kousaka, (2020); Inaba et al., (2023)), we established the existence of bifurcation structures that are referred to as nested mixed-mode oscillations (MMOs); these structures are generated by a forced Bonhoeffer–van der Pol (BVP) oscillator. Nested MMOs, despite their complexity, are robust, repeatable, and composed of higher dimensional oscillations. In this study, we show that nested MMOs can also be produced by the forced van der Pol oscillator. This paper is of extreme importance as it demonstrates that nested MMOs are not a phenomenon that can only be observed in specific BVP systems. Based on the fact that nested MMO bifurcations also occur in the forced van der Pol oscillator, this study indicates that nested MMOs could represent a phenomenon that is more widely observed than previously believed.

Review of: "Periodic second-order systems and coupled forced Van der Pol oscillators"

Review of: "Periodic second-order systems and coupled forced Van der Pol oscillators"

Effect of Oscillatory Heat Load on Pressure Drop Oscillation

Microchannel and minichannel evaporator cooling systems can help remove large heat fluxes with high heat transfer coefficients. However, flow instabilities during microchannel boiling, like pressure drop oscillation (PDO), can deteriorate the cooling performance. In this study, we numerically analyze the effect of an oscillatory evaporator heat load on PDO characteristics occurring in a pumped liquid cycle (PLC). Our model indicates that a specific range of heat load oscillation amplitude and frequency can suppress the PDO amplitude significantly. For the PLC considered in this study, the PDO amplitude can be reduced by 60% when the baseline heat load of 300 W is superposed with an oscillating component of amplitude 100 W and frequency 6 Hz. The study finds that a smaller heat load oscillation amplitude would show a weaker effect on PDO suppression. For an oscillation amplitude of 60 W, the PDO amplitude reduction was only 40% for a wide range of heat load frequency. When the combined effect of heat load oscillation amplitude and frequency was investigated, results showed that the PDO would reach a state of invariance for all heating amplitudes when the frequency was large enough. We believe this could be utilized as an active control strategy to suppress flow oscillation and optimize the overall cooling performance. Furthermore, the PDO characteristics under oscillatory evaporator heat load are compared with the forced Van der Pol oscillator. The similarities between the two systems noted in this study are initial steps towards understanding the effect of various oscillatory system parameters on PDO occurring in a two-phase system.

Analytical and Numerical Study to a Forced Van der Pol Oscillator

In this paper we show the way to apply some and analytical and numerical techniques in order to solve the forced Van der Pol oscillator. We illustrate the obtained results with examples. A comparison with Runge–Kutta numerical method is made in order to see the accuracy of the approximated analytical solution.

Finding the closed-form solutions of dissipative oscillatory systems

This paper shows how to use the approximate Hamiltonian approach for the non-conservative system not capable of possessing Hamiltonian. Using the approximate Hamiltonian method for a non-conservative system is not possible in general. We propose a way to obtain the closed-form solutions for such systems. We use the approximate dual Hamiltonian method to construct the first integrals and closed-form solutions of the Van der Pol equation. First the solutions of the initial value VdP equation is obtained using approximate dual Hamiltonian method. Then a good agreement is observed in the comparison between the numerical results and the results through approximate dual Hamiltonian method. Finally, we use the approximate dual Hamiltonian method to find the dual Hamiltonian and first integrals of the forced Van der Pol oscillator and Liénard system. These significant results can be applied to any Van der Pol equation.

Viability, viscosity, and storage functions in model-predictive control with terminal constraints

Under terminal constraints, the set of states such that a model-predictive control strategy is feasible, and thus stable, is equivalent to the viable-reachable set of the controlled system. We prove that viscosity and storage function approaches to viability theory are equivalent to each other. Based on these results we propose inner and outer bounds of the feasible set by storage functions, which for polynomial systems can be computed using sum-of-squares programmes. Numerical, discrete-time and continuous-time examples are given for lateral car dynamics and the forced Van der Pol oscillator, respectively.

Nonlinear dynamics of a RLC series circuit modeled by a generalized Van der Pol oscillator

Abstract In this paper, nonlinear dynamics study of a RLC series circuit modeled by a generalized Van der Pol oscillator is investigated. After establishing a new general class of nonlinear ordinary differential equation, a forced Van der Pol oscillator subjected to an inertial nonlinearity is derived. According to the external excitation strength, harmonic, subharmonic and superharmonic oscillatory states are obtained using the multiple time scales method. Bifurcation diagrams displayed by the model for each system parameter are performed numerically through the fourth-order Runge–Kutta algorithm.

Retrieving highly structured models starting from black-box nonlinear state-space models using polynomial decoupling

Nonlinear state-space modelling is a very powerful black-box modelling approach. However powerful, the resulting models tend to be complex, described by a large number of parameters. In many cases interpretability is preferred over complexity, making too complex models unfit or undesired. In this work, the complexity of such models is reduced by retrieving a more structured, parsimonious model from the data, without exploiting physical knowledge. Essential to the method is a translation of all multivariate nonlinear functions, typically found in nonlinear state-space models, into sets of univariate nonlinear functions. The latter is computed from a tensor decomposition. It is shown that typically an excess of degrees of freedom are used in the description of the nonlinear system whereas reduced representations can be found. The method yields highly structured state-space models where the nonlinearity is contained in as little as a single univariate function, with limited loss of performance. Results are illustrated on simulations and experiments for: the forced Duffing oscillator, the forced Van der Pol oscillator, a Bouc-Wen hysteretic system, and a Li-Ion battery model.

Dynamics of distributed-order hyperchaotic complex van der Pol oscillators and their synchronization and control

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.

Coherence resonance in low-density jets

Coherence resonance (CR) is a phenomenon in which the response of a stable nonlinear system to external noise exhibits a peak in coherence at an intermediate noise amplitude. We report the first experimental evidence of CR in a hydrodynamic system, a low-density jet capable of undergoing both supercritical and subcritical Hopf bifurcations. By applying noise to the jet in its unconditionally stable regime, we find that, for both types of bifurcation, the coherence factor peaks at an intermediate noise amplitude and increases as the stability boundary is approached. We also find that the autocorrelation function decays differently between the two types of bifurcation, indicating that CR can reveal information about the nonlinearity of a system even before it bifurcates to a limit cycle. We then model the CR dynamics with a stochastically forced van der Pol oscillator calibrated in two different ways: (i) via the conventional method of measuring the amplitude evolution in transient experiments and (ii) via the system-identification method of Lee et al. (J. Fluid Mech., vol. 862, 2019, pp. 200–215) based on the Fokker–Planck equation. We find better experimental agreement with the latter method, demonstrating the deficiency of the former method in identifying the correct form of system nonlinearity. The fact that CR occurs in the unconditionally stable regime, prior to both the Hopf and saddle-node points, implies that it can be used to forecast the onset of global instability. Although demonstrated here on a low-density jet, CR is expected to arise in almost all nonlinear dynamical systems near a Hopf bifurcation, opening up new possibilities for the development of global-instability precursors in a variety of hydrodynamic systems.

Frequency-amplitude characteristics of periodic motions in a periodically forced van der Pol oscillator

In this paper, nonlinear frequency-amplitude characteristics of periodic motions in a periodically forced van der Pol oscillator are studied systematically. The periodic motions of the van der Pol oscillator are determined by the semi-analytical method, and the corresponding stability and bifurcation analysis is completed through the eigenvalue analysis. From the finite Fourier series analysis, the nonlinear frequency-amplitude characteristics of periodic motions are analyzed. From the frequency-amplitude analysis, the limit cycle of the van der Pol oscillator can be obtained analytically as excitation amplitude approach to zero, rather than numerically. For the van der Pol oscillator, most of periodic motions in the van der pol oscillator are symmetric. However, an asymmetric period-1 motion in the van der Pol oscillator is discovered. Thus a bifurcation tree of period-1 motion to chaos can be found.

Data-driven reconstruction of nonlinear dynamics from sparse observation

We present a data-driven model to reconstruct nonlinear dynamics from a very sparse times series data, which relies on the strength of the echo state network (ESN) in learning nonlinear representation of data. With an assumption of the universal function approximation capability of ESN, it is shown that the reconstruction problem can be formulated as a fixed-point problem, in which the trajectory of the dynamical system is a fixed point of the ESN. An under-relaxed fixed-point iteration is proposed to reconstruct the nonlinear dynamics from a sparse observation. The proposed fixed-point ESN is tested against both univariate and multivariate chaotic dynamical systems by randomly removing up to 95% of the data. It is shown that the fixed-point ESN is able to reconstruct the complex dynamics from only 5∼10% of the data. For a relatively simple non-chaotic dynamical system, the numerical experiments on a forced van der Pol oscillator show that it is possible to reconstruct the nonlinear dynamics from only 1∼2% of the data.

Deep learning algorithm for data-driven simulation of noisy dynamical system

We present a deep learning model, DE-LSTM, for the simulation of a stochastic process with an underlying nonlinear dynamics. The deep learning model aims to approximate the probability density function of a stochastic process via numerical discretization and the underlying nonlinear dynamics is modeled by the Long Short-Term Memory (LSTM) network. It is shown that, when the numerical discretization is used, the function estimation problem can be solved by a multi-label classification problem. A penalized maximum log likelihood method is proposed to impose a smoothness condition in the prediction of the probability distribution. We show that the time evolution of the probability distribution can be computed by a high-dimensional integration of the transition probability of the LSTM internal states. A Monte Carlo algorithm to approximate the high-dimensional integration is outlined. The behavior of DE-LSTM is thoroughly investigated by using the Ornstein–Uhlenbeck process and noisy observations of nonlinear dynamical systems; Mackey–Glass time series and forced Van der Pol oscillator. It is shown that DE-LSTM makes a good prediction of the probability distribution without assuming any distributional properties of the stochastic process. For a multiple-step forecast of the Mackey–Glass time series, the prediction uncertainty, denoted by the 95% confidence interval, first grows, then dynamically adjusts following the evolution of the system, while in the simulation of the forced Van der Pol oscillator, the prediction uncertainty does not grow in time even for a 3,000-step forecast.

A novel reduced order model for vortex induced vibrations of long flexible cylinders

In this manuscript the development of a reduced order model for the analysis of long flexible cylinders in an offshore environment is proposed. In particular the focus is on the modelling of the vortex induced vibrations (VIV) and the aim is the development of a model capable of capturing both the in-line and cross-flow oscillations. The reduced order model is identified starting from the results of a high fidelity solver developed coupling together a Finite Element Solver (FEM) with a Computational Fluid Dynamics (CFD) solver. The high fidelity analyses are conducted on a reduced domain size representing a small section of the long cylinder, which is nevertheless, already flexible. The section is forced using a motion which matches the expected motion in full scale, and the results are used for the system-parameter identification of the reduced order model. The reduced order model is identified by using a system and parameter identification approach. The final proposed model consists in the combination of a forced van der Pol oscillator, to model the cross-flow forces, and a linear state-space model, to model the in-line forces. The model is applied to study a full scale flexible model and the results are validated by using experiments conducted on a flexible riser inside a towing tank.

Time analysis of forced variable-order fractional Van der Pol oscillator

This paper presents a new discretization technique for the forced Van der Pol oscillator with variable order derivatives. The study introduces the variable-order fractional time derivatives into the state-space model and investigates their influence upon the system dynamics. The resulting model of the variable-order fractional Van der Pol oscillator is solved and analyzed in the time domain.

On the limit cycles, period-doubling, and quasi-periodic solutions of the forced Van der Pol-Duffing oscillator

In this paper, the limit cycles, period-doubling, and quasi-periodic solutions of the forced Van der Pol oscillator and the forced Van der Pol-Duffing oscillator are studied by combining the homotopy analysis method (HAM) with the multi-scale method analytically. Comparisons of the obtained solutions and numerical results show that this method is effective and convenient even when t is large enough, and the convergence of the approximate solutions is discussed by the so-called discrete square residual error. This method is a capable tool for solving this kind of nonlinear problems.

LabVIEW implementation of chaotic masking with adaptively synchronised forced Van der Pol oscillators and its application in real-time image encryption

Chaotic oscillators find many applications in the field of cryptography and secure communication engineering. This paper first details one such approach of applying a forced Van der Pol chaotic oscillator and its synchronised slave model in chaotic masking and image encryption. This approach applies the chaotic masking to the secondary image and thus changing the image pixels and the masked image is used for encoding the original image. We have developed a real-time image encryption algorithm by capturing pictures through the NI IMAQdx tools. The proposed model is implemented in LabVIEW and experimentally verified for the efficiency of the encryption algorithm.

LabVIEW implementation of chaotic masking with adaptively synchronised forced Van der Pol oscillators and its application in real-time image encryption

Chaotic oscillators find many applications in the field of cryptography and secure communication engineering. This paper first details one such approach of applying a forced Van der Pol chaotic oscillator and its synchronised slave model in chaotic masking and image encryption. This approach applies the chaotic masking to the secondary image and thus changing the image pixels and the masked image is used for encoding the original image. We have developed a real-time image encryption algorithm by capturing pictures through the NI IMAQdx tools. The proposed model is implemented in LabVIEW and experimentally verified for the efficiency of the encryption algorithm.