Van Der Pol-Duffing 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

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.

Novel approach for solving higher-order differential equations with applications to the Van der Pol and Van der Pol–Duffing equations

BackgroundNumerical methods are used to solve differential equations, but few are effective for nonlinear ordinary differential equations (ODEs) of order higher than one. This paper proposes a new method for such ODEs, based on Taylor series expansion. The new method is a second-order method for second-order ODEs, and it is equivalent to the central difference method, a well-known method for solving differential equations. The new method is also simple to implement for higher-order differential equations. The proposed technique was applied to solve the Van der Pol and Van der Pol–Duffing equations. It is stable over a wide range of nonlinearity and produces accurate and reliable results. For the self-excitation Van der Pol equation, the proposed technique was applied with different values of nonlinear damping.ResultsThe results were compared with those obtained using the ODE15s solver in MATLAB. The two sets of results showed excellent agreement. For the forced Van der Pol–Duffing equation, the proposed technique was applied with different values of exciting force amplitude and frequency. It was found that for certain conditions, the solution obtained using the proposed technique differed from that obtained using ODE15s.ConclusionsThe solution obtained using the proposed technique showed good agreement with the solutions obtained using ODE45 and Runge–Kutta fourth order. The results show that the proposed approach is very simple to apply and produces acceptable error. It is a powerful and versatile tool for solving of high-order nonlinear differential equations accurately.

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.

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.

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

Time-frequency high-resolution for weak signal detection using chaotic intermittence

BackgroundA non-linear Liènard-type system based on trigonometric functions is designed and used as a highly sensitive signal detector. Based on the intermittence mechanism presented in some chaotic systems, an adaptive detection array built by five chaotic oscillators is fully tuned and tested. In the array, each oscillator identifies changes in frequency for non-stationary weak signals and gives an estimate of the timing at which intermittence occurred, allowing the representation of a time-frequency spectrogram even when high-level noise is present. MethodsThis work shows that the trigonometric-based system has an asymptotically stable limit cycle centered at the origin through two methods: first under the fixed-point analysis and later by the Melnikov method, which uses a family of periodic orbits. Once the limit cycle is assured, through an iterative method for searching and tuning parameters, the system is configured to drive itself into the route to chaos named intermittence. Then, the new system can be used as a non-stationary signal detector that simultaneously provides high-resolution time-frequency representation. ResultsComputer experiments showed that our trigonometric oscillator when working in the chaotic intermittence regime and configured as an adaptive array, overcomes the SNR threshold present in other second-order non-linear oscillators such as Duffing, Van der Pol, and Van der Pol-Duffing, achieving high detection rates under noisy conditions, and maintaining a relative error of around 1% with -35 dB SNR. ConclusionsOur oscillator provides better sensitivity to small signals as well as noise rejection, in comparison with traditional oscillators. Furthermore, our entire system formed by a Liènard-type five-oscillator array promises to detect low-power signals and their TF reconstruction with a minimum error.

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.

Existence of Self-Excited and Hidden Attractors in the Modified Autonomous Van Der Pol-Duffing Systems

This study investigates the multistability phenomenon and coexisting attractors in the modified Autonomous Van der Pol-Duffing (MAVPD) system and its fractional-order form. The analytical conditions for existence of periodic solutions in the integer-order system via Hopf bifurcation are discussed. In addition, conditions for approximating the solutions of the fractional version to periodic solutions are obtained via the Hopf bifurcation theory in fractional-order systems. Moreover, the technique for hidden attractors localization in the integer-order MAVPD is provided. Therefore, motivated by the previous discussion, the appearances of self-excited and hidden attractors are explained in the integer- and fractional-order MAVPD systems. Phase transition of quasi-periodic hidden attractors between the integer- and fractional-order MAVPD systems is observed. Throughout this study, the existence of complex dynamics is also justified using some effective numerical measures such as Lyapunov exponents, bifurcation diagrams and basin sets of attraction.

Multi-Frequency Homotopy Analysis Method for Coupled Van der Pol-Duffing System with Time Delay

This paper mainly studied the analytical solutions of three types of Van der Pol-Duffing equations. For a system with parametric excitation frequency, we knew that the ordinary homotopy analysis method would be unable to find the analytical solution. Thus, we primarily used the multi-frequency homotopy analysis method (MFHAM). First, the MFHAM was introduced, and the solution of the system was expressed by constructing auxiliary linear operators. Then, the method was applied to three specific systems. We compared the numerical solution obtained using the Runge–Kutta method with the analytical solution to verify the correctness of the latter. Periodic solutions, with and without time delay, were also compared under the same parameters. The results demonstrated that it was both effective and correct to use the MFHAM to find analytical solutions to Van der Pol-Duffing systems, which were classical systems. By comparison, the MFHAM proved to be effective for time delay systems.

Features of the problem on synchronization of two van der Pol-Duffing oscillators in the case of a direct connection and the presence of symmetry

Features of the problem on synchronization of two van der Pol-Duffing oscillators in the case of a direct connection and the presence of symmetry

Unusual Nonpolynomial Van der Pol Oscillator Equations With Exact Harmonic and Isochronous Solutions

We do not know Van der Pol-type equations with nonlinear restoring force having explicitly an exact periodic solution. We present, for the first time, nonpolynomial Van der Pol oscillator equations that do not satisfy the classical existence theorems. We exhibit their exact harmonic and isochronous solutions and prove the existence of limit cycles by using averaging theory. We also present first integrals and exact solutions of polynomial Van der Pol-Duffing equations to show that they do not have any limit cycle. Additionally, we prove that the damped Duffing-type equations are equivalent to the conservative Duffing equations exhibiting nonoscillatory solutions.

Efficient Implementation of Mixing Sequence-Based Van der Pol–Duffing System on the Modulated Wideband Converter Compressed Sensing Scheme

Efficient Implementation of Mixing Sequence-Based Van der Pol–Duffing System on the Modulated Wideband Converter Compressed Sensing Scheme

New Van der Pol–Duffing Jerk Fractional Differential Oscillator of Sequential Type

The subject of this paper is the existence, uniqueness and stability of solutions for a new sequential Van der Pol–Duffing (VdPD) jerk fractional differential oscillator with Caputo–Hadamard derivatives. The arguments are based upon the Banach contraction principle, Krasnoselskii fixed-point theorem and Ulam–Hyers stabilities. As applications, one illustrative example is included to show the applicability of our results.

An efficient approach to solving fractional Van der Pol–Duffing jerk oscillator

The motive behind the current work is to perform the solution of the Van der Pol–Duffing jerk oscillator, involving fractional-order by the simplest method. An effective procedure has been introduced for executing the fractional-order by utilizing a new method without the perturbative approach. The approach depends on converting the fractional nonlinear oscillator to a linear oscillator with an integer order. A detailed solving process is given for the obtained oscillator with the traditional system.

Stabilisation of stochastic hybrid high-order nonlinear coupled systems by intermittent control

This paper considers the stabilisation of high-order nonlinear stochastic delayed coupled systems with Markovian switching (HNSMS) by periodically intermittent control (PIC). Notably, the coefficients in HNSMS no longer satisfy the linear growth condition. In the mean time, PIC is first introduced to ensure the stabilisation of HNSMS. Since current Halanay-type differential inequalities (H-type inequalities) are not applicable to high-order nonlinear systems with PIC, a generalised H-type inequality with high-order terms for such systems is established, in which the order of high-order terms can be different. Together with graph theory and Lyapunov method, some sufficient conditions reaching the stability are presented. Then, our results are used to study the stability of modified Van der Pol–Duffing oscillators with Markovian switching. Ultimately, the effectiveness of theoretical results is checked by a numerical test.

Bursting oscillations induced by multiple coexisting attractors in a modified 3D van der Pol-Duffing system

In this paper, a novel three-dimensional modified van der Pol-Duffing circuit with a quintic nonlinear resistor is proposed, and the multiple stable attractors are observed. The complex bursting patterns are investigated, various patterns of bursting oscillations are obtained under slowly changing frequency, and the transition mechanism of which can be revealed via the modified slow–fast analysis as well as the transformed phase portrait. It can be found that for the system with mono-stability, only one bursting pattern exists, while for the case with multi-stability, the trajectory can enter into different attraction areas in the transition region, leading to complex forms of bursting oscillations. Furthermore, the multi-valued characteristic of the system are observed by the domain of attraction.