Showing papers on "Van der Pol oscillator published in 2022"

Forced nonlinear oscillator in a fractal space

Abstract: A critical hurdle of a nonlinear vibration system in a fractal space is the inefficiency in modelling the system. Specifically, the differential equation models cannot elucidate the effect of porosity size and distribution of the periodic property. This paper establishes a fractal-differential model for this purpose, and a fractal Duffing-Van der Pol oscillator (DVdP) with two-scale fractal derivatives and a forced term is considered as an example to reveal the basic properties of the fractal oscillator. Utilizing the two-scale transforms and He-Laplace method, an analytic approximate solution may be attained. Unfortunately, this solution is not physically preferred. It has to be modified along with the nonlinear frequency analysis, and the stability criterion for the equation under consideration is obtained. On the other hand, the linearized stability theory is employed in the autonomous arrangement. Consequently, the phase portraits around the equilibrium points are sketched. For the non-autonomous organization, the stability criteria are analyzed via the multiple time scales technique. Numerical estimations are designed to confirm graphically the analytical approximate solutions as well as the stability configuration. It is revealed that the exciting external force parameter plays a destabilizing role. Furthermore, both of the frequency of the excited force and the stiffness parameter, execute a dual role in the stability picture.

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

Abstract: 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.

Approximate solution to a generalized Van der Pol equation arising in plasma oscillations

Abstract: Motivated by some published theoretical investigations and based on the two-fluid model, nonlinear plasma oscillations are analyzed and discussed in the framework of the generalized Van der Pol equation. This equation is analyzed and solved using two different analytical approaches. In this first approach, the ansatz method is carried out for deriving an approximation in the form of a trigonometric function. In the second approach, the Krylov–Bogoliubov–Mitropolsky (KBM) technique is applied for obtaining a high-accurate approximation. The obtained approximations are compared with the numerical approximation using the Runge–Kutta (RK) method. Moreover, the distance error between the obtained approximations (using the ansatz method and the KBM technique) and the RK numerical approximation is estimated. In our investigation, both the proposed methods and obtained approximations can help many authors investigate several nonlinear oscillations in different plasma models and fluid mechanics.

Reversal of period doubling, multistability and symmetry breaking aspects for a system composed of a van der pol oscillator coupled to a duffing oscillator

Abstract: Owing to recently published works, the issues of multistability and symmetry breaking can be listed amongst the most followed ongoing research topics in nonlinear science. In this contribution, we consider the dynamics of a system composed of a van der Pol oscillator linearly coupled to a Duffing oscillator (Han, 2000; Kengne et al., 2012). Mention that coupled attractors of different types serve as convenient models for real world systems such as electromechanical, biological, physical, or economic systems. We analyze how the explicit symmetry break modifies the phase space location and nature of equilibrium points of the coupled system, the topology and number of competing attractors, the bifurcation modes, and the shape of the basins of attraction. These investigations are executed by resorting to classical nonlinear tools such as basins of attraction, phase portraits, plots of 1D and 2D largest Lyapunov exponent diagrams, and 1D bifurcation diagrams as well. We report intricate dynamical features such as critical transitions, hysteresis, the coexistence of (symmetric or asymmetric) bubbles of bifurcation and the occurrence of multiple coexisting dynamics (i.e. two, three, four or five coexisting attractors) resulting from the variation of both initial states and parameters of the coupled system.

Optimization of the Wake Oscillator for Transversal VIV

Abstract: Vibrations of slender structures associated with the external flow present a design challenge for the energy production systems placed in the marine environment. The current study explores the accuracy of the semi-empirical wake oscillator models for vortex-induced vibrations (VIV) based on the optimization of (a) the damping term and (b) empirical coefficients in the fluid equation. This work investigates the effect of ten fluid damping variations, from the classic van der Pol to more sophisticated fifth-order terms, and prediction of the simplified case of the VIV of transversally oscillating rigid structures provides an opportunity for an extended, comprehensive comparison of the performance of tuned models. A constrained nonlinear minimization algorithm in MATLAB is applied to calibrate considered models using the published experimental data, and the weighted objective function is formulated for three different mass ratios. Comparison with several sources of published experimental data for cross-flow oscillations confirms the model accuracy in the mass ratio range. The study indicates the advantageous performance of the models tuned with the medium mass ratio data and highlights some advantages of the Krenk–Nielsen wake oscillator.

Benchmarking Nonlinear Oscillators for Grid-Forming Inverter Control

Abstract: 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.

Nonlinear suppression using time-delayed controller to excited Van der Pol–Duffing oscillator: analytical solution techniques

Abstract: Abstract To suppress the nonlinearity of an excited Van der Pol–Duffing oscillator (VdPD), time-delayed position and velocity are used throughout this study. The time delay is supplemental to prevent the nonlinear vibration of the considered system. The topic of this work is extremely current because technologies with a time delay have been the subject of several studies in the latest days. The classical homotopy perturbation method (HPM) is utilized to extract an approximate systematic explanation for the system at hand. Furthermore, a modification of the HPM reveals a more accurate approximate solution. This accuracy is tested through a comparison with the numerical solution. The practical approximate analytical methodology makes the work possible to qualitatively evaluate the results. The time histories of the obtained solutions are drawn for various values of the natural frequency and the time delay parameters. Discussion of the results is presented in light of the plotted curves. On the other hand, the multiple scale procedure examines the organized nonlinear prototypical approach. The influence of the diverse regulatory restrictions on the organization’s vibration performances is explored. Two important cases of resonance, the sub-harmonic and super-harmonic, are examined according to the cubic nonlinearity. The modulation equations achieved for these cases are examined graphically according to the impact of the used parameters.

Dynamics and Synchronization of Complex-Valued Ring Networks

Abstract: Networks of coupled oscillators have been used to model various real-world self-organizing systems. However, the dynamics, especially chaos and bifurcation, of complex-valued networks are rarely investigated. In this paper, a ring network of interacting complex-valued van der Pol oscillators is studied to model the formation of ring dynamics. Although there are only stable limit cycles in a complex-valued van der Pol oscillator, chaos, hyperchaos, and coexisting chaotic attractors are observed from the ring network, which are analyzed by using the Lyapunov exponent spectrum, bifurcation diagram and 0–1 test. In addition, complexity analysis on nonlinear coefficients and coupling strengths illustrates that the range of parameters within the chaotic (hyperchaotic) region has positive correlation with the number of oscillators. It is shown that the chaotic bifurcation path is highly robust against the size variation of the ring network, which always evolves to chaos directly from period-1 and quasi-periodic states, respectively. Moreover, it is demonstrated that complete synchronization and phase synchronization of oscillations are stable in a large-scale ring network, while chaotic phase synchronization is unstable in a small-scale network.

Effects of damping on the dynamics of an electromechanical system consisting of mechanical network of discontinuous coupled system oscillators with irrational nonlinearities: Application to sand sieves

Abstract: The electromechanical system consisting of an electrical part which is the forced Vander Pol oscillator coupled magnetically to a mechanical part is investigated. The mechanical part is the network consisting of discontinuous elastically coupled system oscillators with strong irrational nonlinearities in which the damping is introduced. This coupled electromechanical system is connected at the output to movable sieve for the industrial applications, it can be used for the filtering of different types of building materials. By using then the Newton’s second law and Kirchhoff’s law, the set of model damped equations governing the dynamics of the system are established. These set of equations have strong irrational nonlinearities, with smooth or discontinuous characteristics depending just to the inclination angles of strings. Then the resonance phenomenon showing the appearing of hysteresis as the frequency shift increases is found and is more and more complex as the cell number increases. By solving numerically the set of equations of the system, one obtains the oscillatory bursting in the electrical part, and impulse bursting in the mechanical part, with their widths which decrease as the excitation frequency increases. It is also found that chaotic bursting appears in the mechanical part when the electric part exhibits periodic bursting oscillations.

On stochastic response of fractional-order generalized birhythmic van der Pol oscillator subjected to delayed feedback displacement and Gaussian white noise excitation

Abstract: This paper addresses the occurrence of P-bifurcation in a fractional-delay modified birhythmic van der Pol (BVDP) oscillator, for enzyme-substrate reactions in brain waves, under Gaussian white noise excitation. The minimum mean-square error is used to reduce the system to its equivalent integer-order nonlinear stochastic equation. An averaged Itô equation is obtained via the stochastic averaging method, with the amplitude being the solution of the Fokker-Planck-Kolmogorov equation. From the latter, the stationary density functions are found analytically. This helps to predict the appearance of birhythmicity theoretically and shows itself to respond to parameter changes, namely, fractional-orders, fractional coefficient, and noise intensity. There is an agreement between the theoretical solutions and the numerical solution, which confirms the accuracy of our predictions. In general, bifurcation scenarios are dominated by the changes in fractional orders, as strongly supported by the behaviors of the calculated potential barriers.

Solving Perturbed Dynamic Systems Using Schur Decomposition

Abstract: 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.

A definition of the asymptotic phase for quantum nonlinear oscillators from the Koopman operator viewpoint.

Abstract: We propose a definition of the asymptotic phase for quantum nonlinear oscillators from the viewpoint of the Koopman operator theory. The asymptotic phase is a fundamental quantity for the analysis of classical limit-cycle oscillators, but it has not been defined explicitly for quantum nonlinear oscillators. In this study, we define the asymptotic phase for quantum oscillatory systems by using the eigenoperator of the backward Liouville operator associated with the fundamental oscillation frequency. By using the quantum van der Pol oscillator with a Kerr effect as an example, we illustrate that the proposed asymptotic phase appropriately yields isochronous phase values in both semiclassical and strong quantum regimes.

Some Novel Approaches for Analyzing the Unforced and Forced Duffing–Van der Pol Oscillators

Abstract: In the current investigation, both unforced and forced Duffing–Van der Pol oscillator (DVdPV) oscillators with a strong nonlinearity and external periodic excitations are analyzed and investigated analytically and numerically using some new and improved approaches. The new approach is constructed based on Krylov–Bogoliubov–Metroolsky method (KBMM). One of the most important features of this approach is that we do not need to solve a system of differential equations, but only solve a system of algebraic equations. Moreover, the ease and faster of applying this method gives high-accurate results and this approach is better than many approaches in the literature. This approach is applied for analyzing (un)forced DVdP oscillators. Also, some improvements are made to He’s frequency-amplitude formulation in order to solve unforced DVdP oscillator to obtain high-accurate results. Furthermore, the He’s homotopy perturbation method (He’s HPM) is employed for analyzing unforced DVdP oscillator. The comparison between all mentioned approaches is carried out. The application of our approach is not limited to (un)forced DVdPV oscillators only but can be applied to analyze many higher-order nonlinearity oscillators for any odd power and it gives more accurate results than other approaches. Both used methods and obtained approximations will help many researchers in general and plasma physicists in particular in the interpretation of their results.

Dynamical states and bifurcations in coupled thermoacoustic oscillators

Abstract: The emergence of rich dynamical phenomena in coupled self-sustained oscillators, primarily synchronization and amplitude death, has attracted considerable interest in several fields of science and engineering. Here, we present a comprehensive theoretical study on the manifestation of these exquisite phenomena in a reduced-order model of two coupled Rijke tube oscillators, which are prototypical thermoacoustic oscillators. We characterize the dynamical behaviors of two such identical and non-identical oscillators by varying both system parameters (such as the uncoupled amplitudes and the natural frequencies of the oscillators) and coupling parameters (such as the coupling strength and the coupling delay). The present model captures all the dynamical phenomena-namely, synchronization, phase-flip bifurcation, amplitude death, and partial amplitude death-observed previously in experiments on coupled Rijke tubes. By performing numerical simulations and deriving approximate analytical solutions, we systematically decipher the conditions and the bifurcations underlying the aforementioned phenomena. The insights provided by this study can be used to understand the interactions between multiple cans in gas turbine combustors and develop control strategies to avert undesirable thermoacoustic oscillations in them.

On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified Van der Pol–Duffing system with slow-varying periodic excitation

Abstract: The main purpose of the paper is to reveal the mechanism of certain special phenomena in bursting oscillations such as the sudden increase of the spiking amplitude. When multiple equilibrium points coexist in a dynamical system, several types of stable attractors via different bifurcations from these points may be observed with the variation of parameters, which may interact with each other to form other types of bifurcations. Here we take the modified van der Pol–Duffing system as an example, in which periodic parametric excitation is introduced. When the exciting frequency is far less than the natural frequency, bursting oscillations may appear. By regarding the exciting term as a slow-varying parameter, the number of the equilibrium branches in the fast generalized autonomous subsystem varies from one to five with the variation of the slow-varying parameter. The equilibrium branches may undergo different types of bifurcations, such as Hopf and pitchfork bifurcations. The limit cycles, including the cycles via Hopf bifurcations and the cycles near the homo-clinic orbit, may interact with each other to form the fold limit cycle bifurcations. With the increase of the exciting amplitude, different stable attractors and bifurcations of the generalized autonomous fast subsystem involve the full system, leading to different types of bursting oscillations. Fold limit cycle bifurcations may cause the sudden change of the spiking amplitude, since at the bifurcation points, the trajectory may oscillate according to different stable limit cycles with obviously different amplitudes. At the pitchfork bifurcation point, two possible jumping ways may result in two coexisted asymmetric bursting attractors, which may expand in the phase space to interact with each other to form an enlarged symmetric bursting attractor with doubled period. The inertia of the movement along the stable equilibrium may cause the trajectory to pass across the related bifurcations, leading to the delay effect of the bifurcations. Not only the large exciting amplitude, but also the large value of the exciting frequency may increase inertia of the movement, since in both the two cases, the change rate of the slow-varying parameter may increase. Therefore, a relative small exciting frequency may be taken in order to show the possible influence of all the equilibrium branches and their bifurcations on the dynamics of the full system.

Analytical and Numerical Study to a Forced Van der Pol Oscillator

Abstract: 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.

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

Abstract: 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.

Central pattern generator based on self-sustained oscillator coupled to a chain of oscillatory circuits.

Abstract: We have proposed and studied both numerically and experimentally a multistable system based on a self-sustained Van der Pol oscillator coupled to passive oscillatory circuits. The number of passive oscillators determines the number of multistable oscillatory regimes coexisting in the proposed system. It is shown that our system can be used in robotics applications as a simple model for a central pattern generator (CPG). In this case, the amplitude and phase relations between the active and passive oscillators control a gait, which can be adjusted by changing the system control parameters. Variation of the active oscillator's natural frequency leads to hard switching between the regimes characterized by different phase shifts between the oscillators. In contrast, the external forcing can change the frequency and amplitudes of oscillations, preserving the phase shifts. Therefore, the frequency of the external signal can serve as a control parameter of the model regime and realize a feedback in the proposed CPG depending on the environmental conditions. In particular, it allows one to switch the regime and change the velocity of the robot's gate and tune the gait to the environment. We have also shown that the studied oscillatory regimes in the proposed system are robust and not affected by external noise or fluctuations of the system parameters. Moreover, using the proposed scheme, we simulated the type of bipedal locomotion, including walking and running.

Explosive Transition in Coupled Oscillators Through Mixed Attractive-Repulsive Interactions

Abstract: The emergence of many fascinating dynamic behaviors is affected by more than one interaction among the elements or cells in a network. In fact, the concurrence and competition of different types of effects among subsystems show a strong connection to the dynamic transition process between oscillation patterns. Here, a network of generic oscillators with mixed attractive-repulsive couplings is introduced to demonstrate the transition from oscillatory states to stationary equilibria, specifically for van der Pol oscillators and Lorenz oscillators. Through the observation of the normalized amplitude changing with the coupling strength, the sudden and irreversible transition appears in both systems, which has a close relation to the mutual repulsion on coupled oscillators. Whereas, for coupled van der Pol oscillators, three typical transition scenarios are found by varying the weight ratio of these two couplings, while the Lorenz system shows only one transition mode no matter how the weight ratio changes. Besides, in the cases of explosive transitions, the coexistence areas of oscillatory and death states also reveal a distinct manifestation for periodic and chaotic systems. The details of theoretical critical transition points on the first-order phase transition are also obtained. Our results pave a new way to control the explosive phenomenon, which is crucial to explain the sudden oscillation quenching and the coexistence of oscillatory and stationary states in biological as well as chemical systems.

Behavior of a Hybrid Rayleigh-Van der Pol- Duffing Oscillator with a PD Controller.

Abstract: In our consideration, we studied the vibration analysis and dynamic responses of aHybrid Rayleigh-Van der Pol- Duffing oscillator. This system presented by one –degree-of - freedom containing the fifth order of nonlinear terms and an external force. To control the vibration of the main system, we applied the proportional derivative controller (PD). The average method used to obtain the approximate solution of the vibrating system. The stability of the system is investigated at the primary resonance case (   ). Numerically, we used Runge–Kutta of the fourth order to scrutinize the time histories of the system before and after using PD. For response curves, we investigated the performance of some chosen parameters of studied system. Finally, there is a good agreement between the approximate solution which obtained from the average method and the numerical one.