Duffing Van Der Pol Oscillator Research Articles

Chaotic Motions of the van der Pol-Duffing Oscillator Subjected to Periodic External and Parametric Excitations with Delayed Feedbacks

Chaotic Motions of the van der Pol-Duffing Oscillator Subjected to Periodic External and Parametric Excitations with Delayed Feedbacks

Dynamic analysis of the fractional-order Duffing-Van der Pol oscillator and its application extension

Dynamic analysis of the fractional-order Duffing-Van der Pol oscillator and its application extension

Nonlinear subband adaptive filter based on Andrew’s sine estimator for Van der Pol system identification

Echo cancellation has been effectively achieved through the successful application of subband adaptive filtering (SAF) algorithms. However, the performance of these algorithms may significantly deteriorate when the system exhibits nonlinear distortion. The Van der Pol–Duffing oscillator (VDPDO) system is widely recognized as a complex nonlinear system that displays intricate dynamical behaviors, emphasizing the importance of identifying this nonlinear model. This paper proposes a novel SAF, termed nonlinear SAF (NLSAF) for VDPDO system identification. The NLSAF is based on the functional link artificial neural network (FLANN) to achieve superior nonlinear modeling capability. In particular, the Andrew’s sine estimator is integrated into the NLSAF, generating NLSAF-ASE algorithm for performance improvement. The simulation results provide evidence of the accurate identification of the Van der Pol system through the effective utilization of the NLSAF-ASE algorithm.

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.

Tuning limit cycles with a noise: Survival and collapse.

We consider a general class of limit cycle oscillators driven by an additive Gaussian white noise. Based on the separation of timescales, we construct the equationof motion for slow dynamics after appropriate averaging over the fast motion. The equationfor slow motion whose coefficients are modified by noise characteristics is solved to obtain the analytic solution in the long time limit. We show that with increase of noise strength, the loop area of the limit cycle decreases until a critical value is reached, beyond which the limit cycle collapses. We determine the noise threshold from the condition for removal of secular divergence of the perturbation series and work out two explicit examples of Van der Pol and Duffing-Van der Pol oscillators for corroboration between the theory and numerics.

A dynamic study of a bead sliding on a wire in fractal space with the non-perturbative technique

Drawing on the principles of fractal properties and nonlinear vibration analysis, this paper delves into the investigation of a moving bead on a vertically rotated parabola. The dynamical nonlinear equation of motion, incorporating fractal derivatives, transforms traditional derivatives within continuous space. Consequently, the equation of motion takes the form of the Duffing-Van der Pol oscillator. Utilizing a non-perturbative approach, the nonlinear oscillator is systematically transformed into a linear one, boasting an exact solution. The analytical solution yields two valid formulas governing the frequency-amplitude relationships. Numerical solutions affirm that these proposed formulas offer highly satisfactory approximations to the analytical solution. Leveraging fractal properties through Galerkin’s method, the paper successfully determines the fractalness parameter of the medium, shedding light on the intricate dynamics of the system.

Multi-parametric optimization for controlling bifurcation structures

Bifurcations organize the dynamics of many natural and engineered systems. They induce qualitative and quantitative changes to a system’s dynamics, which can have catastrophic consequences if ignored during design. In this paper, we propose a general computational method to control the local bifurcations of dynamical systems by optimizing design parameters. We define an objective functional that enforces the appearance of local bifurcation points at targeted locations or even encourages their disappearance. The methodology is an efficient alternative to bifurcation tracking techniques capable of handling many design parameters ( > 10 2 ). The method is demonstrated on a Duffing oscillator featuring a hardening cubic nonlinearity and an autonomous van der Pol-Duffing oscillator coupled to a nonlinear tuned vibration absorber. The finite-element model of a clamped-free Euler–Bernoulli beam, coupled with a reduced-order modelling technique, is also used to show the extension to the shape optimization of more complicated structures. Results demonstrate that several local bifurcations of various types can be handled simultaneously by the bifurcation control framework, with both parameter and state target values.

On the geometric and analytical properties of the anharmonic oscillator

Here we consider the anharmonic oscillator that is a dynamical system given by yxx+δyn=0. We demonstrate that to this equation corresponds a new example of a superintegrable two-dimensional metric with a linear and a transcendental first integrals. Moreover, we show that for particular values of n the transcendental first integral degenerates into a polynomial one, which provides an example of a superintegrable metric with additional polynomial first integral of an arbitrary even degree. We also discuss a general procedure of how to construct a superintegrable metric with one linear first integral from an autonomous nonlinear oscillator that is cubic with respect to the first derivative. We classify all cubic oscillators that can be used in this construction. Furthermore, we study the Liénard equations that are equivalent to the anharmonic oscillator with respect to the point transformations. We show that there are nontrivial examples of the Liénard equations that belong to this equivalence class, like the generalized Duffing oscillator or the generalized Duffing–Van der Pol oscillator.

Resonance analysis and time-delay feedback controllability for a fractional horizontal nonlinear roller system

<p>In this paper, we investigated the nonlinear vibration characteristics and time-delay feedback controllability of a fractional horizontal roll system, which is described by a fractional Duffing-van der Pol oscillator under an external harmonic excitation. We focused on the resonance of fractional roller systems and conducted corresponding vibration control. The amplitude-frequency equations of primary resonance and superharmonic resonance were obtained using the multiple scale method. The amplitude-frequency characteristic curves of the system with different parameters were presented, and the influence of system parameters on the curves was analyzed. In addition, the time-delay feedback controller was designed to control the parameter excitation vibration. The numerical simulation results have verified the effectiveness of the time-delay controller in eliminating the jumping and hysteresis phenomena of the rolling system. The comparisons of approximate analytical solution and numerical solution was fulfilled, and the result certifies the correctness and satisfactory precision of the approximately analytical solution. The analysis results provide certain theoretical guidance for the vibration reduction of the horizontal nonlinear roller system.</p>

Solar magnetic cycles as a Van Der Pol-Duffing oscillator: new insights

ABSTRACT In this paper, we use an approximative stable limit cycle solution of the hybrid Van der Pol-Duffing differential equation, obtained by homotopy and Poincaré–Lindstedt perturbation methods, to describe the toroidal component of the solar magnetic field B(t). This analytic approach allows us to recover an explicit relationship between the parameter μ, which is related to the meridional circulation, and the period of the Hale’s magnetic cycle with a correlation coefficient of r = −0.58. Furthermore, assuming that the sunspot number (SN) is proportional to the square of the toroidal magnetic field (SN∝B2), our solution accurately predict the presence of an harmonic oscillation in the SN data, occurring at a period of T/4 = 5.52 ± 0.44 yr. This prediction has been validated through Lomb–Scargle analysis, with a high statistical significance. Additionally, we find that the ratio of spectral powers between the T/4 harmonic and the main T/2 oscillation is almost equal to the value obtained from our solution using the mean values of the parameters. Interestingly, this study also reveals a correlation between the intermittent 5.52-yr cycle and μ, the parameter associated with the meridional circulation of the Sun. Both follow a similar pattern, suggesting that the origin of the five-year cycle lies within the meridional circulation. Finally, we will see how, using this model, we can overcome the limitations of direct observations and reconstruct the variation profile of the meridional circulation over two centuries using a single observation (from the last magnetic cycle).

Planetary influences on the solar cycle: A nonlinear dynamics approach.

We explore the effect of some simple perturbations on three nonlinear models proposed to describe large-scale solar behavior via the solar dynamo theory: the Lorenz and Rikitake systems and a Van der Pol-Duffing oscillator. Planetary magnetic fields affecting the solar dynamo activity have been simulated by using harmonic perturbations. These perturbations introduce cycle intermittency and amplitude irregularities revealed by the frequency spectra of the nonlinear signals. Furthermore, we have found that the perturbative intensity acts as an order parameter in the correlations between the system and the external forcing. Our findings suggest a promising avenue to study the sunspot activity by using nonlinear dynamics methods.

Reachable set estimation and H∞$H_\infty$ performance for delayed fuzzy multi‐agent systems under false data injection attacks

AbstractAddressed in this paper is the reachable set estimation (RSE) problem for fuzzy‐model‐based leader‐follower multi‐agent systems with time‐varying delays and false data injection attacks. First, the aperiodic sampled‐data control is designed for the follower agents with randomly occurring false data injection attacks. Then, using the Kronecker product, the error system between the leader and the follower is obtained in a compact general form. Next, a novel Lyapunov‐Krasovskii functional is constructed with the knowledge of sampling patterns and time‐varying delays. In the framework of linear matrix inequalities, sufficient consensus conditions are determined from the performance index and Lyapunov theory to guarantee that its reachable set is enclosed by an ellipsoid in the existence of bounded perturbations. In the end, the Duffing Van der Pol oscillator and the single‐link robot arm models are employed to validate the derived theoretical results.

Finding nonlocal Lie symmetries algorithmically

Here we present a new approach to compute symmetries of rational second order ordinary differential equations (rational 2ODEs). This method can compute Lie symmetries (point symmetries, dynamical symmetries and nonlocal symmetries) admitted by a rational 2ODE if it presents a Liouvillian first integral. The procedure is based on an idea arising from the formal equivalence between the total derivative operator and the vector field associated with the 2ODE over its solutions (Cartan vector field). Basically, from the formal representation of a Lie symmetry it is possible to extract information that allows to use this symmetry practically (in the 2ODE integration process) even in cases where the formal operation cannot be performed, i.e., in cases where the symmetry is nonlocal. Furthermore, when the 2ODE in question depends on parameters, the procedure allows an analysis that determines the regions of the parameter space in which the integrable cases are located. We finish by applying the method to the 2ODE that represents a forced modified Duffing–Van der Pol oscillator (which presents chaos for arbitrary values of the parameters) and determining a region of the parameter space that leads to integrable cases, some not previously known.

Bifurcation and chaos detection of a fractional Duffing–van der Pol oscillator with two periodic excitations and distributed time delay

This paper analytically and numerically investigates the dynamical characteristics of a fractional Duffing–van der Pol oscillator with two periodic excitations and the distributed time delay. First, we consider the pitchfork bifurcation of the system driven by both a high-frequency parametric excitation and a low-frequency external excitation. Utilizing the method of direct partition of motion, the original system is transformed into an effective integer-order slow system, and the supercritical and subcritical pitchfork bifurcations are observed in this case. Then, we study the chaotic behavior of the system when the two excitation frequencies are equal. The necessary condition for the existence of the horseshoe chaos from the homoclinic bifurcation is obtained based on the Melnikov method. Besides, the parameters effects on the routes to chaos of the system are detected by bifurcation diagrams, largest Lyapunov exponents, phase portraits, and Poincaré maps. It has been confirmed that the theoretical predictions achieve a high coincidence with the numerical results. The techniques in this paper can be applied to explore the underlying bifurcation and chaotic dynamics of fractional-order models.

Effect of an amplitude modulated force on vibrational resonance, chaos, and multistability in a modified Van der Pol-Duffing oscillator

Effect of an amplitude modulated force on vibrational resonance, chaos, and multistability in a modified Van der Pol-Duffing oscillator

Numerical Solutions of Duffing Van der Pol Equations on the Basis of Hybrid Functions

In the present work, a new approximated method for solving the nonlinear Duffing-Van der Pol (D-VdP) oscillator equation is suggested. The approximate solution of this equation is introduced with two separate techniques. First, we convert nonlinear D-VdP equation to a nonlinear Volterra integral equation of the second kind (VIESK) using integration, and then, we approximate it with the hybrid Legendre polynomials and block-pulse function (HLBPFs). The next technique is to convert this equation into a system of ordinary differential equation of the first order (SODE) and solve it according to the proposed approximate method. The main goal of the presented technique is to transform these problems into a nonlinear system of algebraic equations using the operational matrix obtained from the integration, which can be solved by a proper numerical method; thus, the solution procedures are either reduced or simplified accordingly. The benefit of the hybrid functions is that they can be adjusted for different values of n and m , in addition to being capable of yield greater correct numerical answers than the piecewise constant orthogonal function, for the results of integral equations. Resolved governance equation using the Runge-Kutta fourth order algorithm with the stepping time 0.01 s via numerical solution. The approximate results obtained from the proposed method show that this method is effective. The evaluation has been proven that the proposed technique is in good agreement with the numerical results of other methods.

Stabilization of Highly Nonlinear Stochastic Time-Varying Coupled Systems via Aperiodically Intermittent Control

Highly non-linear stochastic time-varying coupled systems are first considered in which the coupling structure is time-variant. It is worth pointing out that aperiodically intermittent control (AIC) is considered to achieve the stability of highly non-linear stochastic time-varying coupled systems. Since the existing research methods for processing AIC are not suitable for stochastic highly non-linear systems, a new Halanay-type differential inequality with higher-order nonlinear terms that extends the existing Halanay-type differential inequalities is established. Then, with the help of the Lyapunov method, some techniques of inequalities and the graph theory, two stabilization criteria are presented to guarantee the exponential stabilization for highly non-linear stochastic time-varying coupled systems. Finally, as an application of our results, the modified time-varying coupled Van der Pol-Duffing oscillators are studied via AIC with numerical simulations provided.

Separation of the fetal heart signal in a synchronous network consisting of maternal and fetal hearts

This paper studies the heart's oscillation model to separate fetal and maternal ECGs from abdominal recordings. To this aim, two phases are designed. A Modified version of the Duffing-Van der Pol oscillator is considered a computational heart model in the modeling phase. To evaluate the interaction effects of the maternal and fetal heart and the differences and features of the fetal heart structure, the fetal heart model is Modified based on the maternal heart model. A non-identical network is employed as an interactive network of the mother and the fetus's heart. Then the degree of the network synchronization is measured with the help of a pattern synchronization index of the non-identical network. An attempt is made to separate the fetal signal from the mother's in abdominal signals in the separation phase. Two problem-solving approaches are explained; the step-by-step mode that calculates the signal at any given moment and the construction of general equations. These approaches end up calculating the variables, which stand for maternal and fetal signal. That makes it possible to achieve the separation of maternal and fetal ECGs.

Linearizabiliy and Lax representations for cubic autonomous and non-autonomous nonlinear oscillators

In this work we consider a family of cubic, with respect to the first derivative, second-order ordinary differential equations. We study linearizability conditions for this family of equations via generalized nonlocal transformations. We construct linearizability criteria in the explicit form for some particular cases of these transformations. We show that each linearizable equation admits a quadratic rational first integral. Moreover, we demonstrate that generalized nonlocal transformations under certain conditions preserve Lax integrability for equations from the considered family. Consequently, we find that any linearizable equation from the considered family possesses a Lax representation. This provides a connection between two different approaches for studying integrability and demonstrates that the Lax technique can be effectively applied to finding and classifying dissipative integrable autonomous and non-autonomous nonlinear oscillators. We illustrate our results by several examples of linearizable equations, including a non-autonomous generalization of the Rayleigh–Duffing oscillator, a family of chemical oscillators and a generalized Duffing–Van der Pol oscillator.

Application of Orthogonal Functions to Equivalent Linearization Method for MDOF Duffing–Van der Pol Systems under Nonstationary Random Excitations

Many mechanical systems manifest nonlinear behavior under nonstationary random excitations. Neglecting this nonlinearity in the modeling of a dynamic system would result in unacceptable results. However, it is challenging to find exact solutions to nonlinear problems. Therefore, equivalent linearization methods are often used to seek approximate solutions for this kind of problem. To overcome the limitations of the existing equivalent linearization methods, an orthogonal-function-based equivalent linearization method in the time domain is proposed for nonlinear systems subjected to nonstationary random excitations. The proposed method is first applied to a single-degree-of-freedom (SDOF) Duffing–Van der Pol oscillator subjected to stationary and nonstationary excitations to validate its accuracy. Then, its applicability to nonlinear MDOF systems is depicted by a 5DOF Duffing–Van der Pol system subjected to nonstationary excitation, with different levels of system nonlinearity strength considered in the analysis. Results show that the proposed method has the merit of predicting the nonlinear system response with high accuracy and computation efficiency. In addition, it is applicable to any general type of nonstationary random excitation.