Nonlinear Van Der Pol Oscillator Research Articles

Two-dimensional Coupled Asymmetric van der Pol Oscillator

A reliable algorithm based on an adaptation of the standard differential transform method (DTM) is presented, which is the multi-step differential transform method (Ms-DTM) since it may be difficult to directly apply differential transform method (DTM) to obtain the series solutions for the present two- dimensional coupled asymmetric van der Pol oscillator. The solutions of a two dimensional coupled asymmetric van der Pol oscillator were obtained by Ms-DTM. Figurative comparisons between the Ms-DTM and the classical fourth order Runge-Kutta method (RK4) are given. The obtained results reveal that the proposed technique is a promising tool to solve the considered van der Pol oscillator and yield same information on the phase portrait confirming the stability of the system, effectively. It can be said that the considered approach can be easily extended to other nonlinear van der Pol oscillator systems and therefore is widely applicable in engineering and other sciences.

On polylinear differential realization of the determined dynamic chaos in the class of higher order equations with delay

The investigation has defined the characteristic criterion (and its modification) of solvability of the problem of differential realization of the bundle of controlled trajectory curves of determined chaotic dynamic processes in the class of bilinear non-autonomous ordinary second- and higher-order differential equations (with and without delay) in the separable Hilbert space. The problem statement under consideration belongs to the type of converse problems for the additive combination of nonstationary linear and bilinear operators of the evolution equation in the Hilbert space. The constructions of tensor products of the Hilbert spaces, structures of lattices with an orthocomplement, the theory of extension of M2 -operators and the functional apparatus of the entropy Relay Ritz operator represent the basis of this theory. It has been shown that in the case of the finite bundle of the controlled trajectory curves the existence of the property of sub-linearity of the given operator allows one to obtain sufficient conditions of existence of such realizations. Side by side with solving the main problems, grounded are topological-group conditions of continuity of projectivization of the Relay Ritz operator with computing the fundamental group (Poincare group) of its compact image. The results obtained give incentives for the development of the quantitative theory of converse problems of higher-order multilinear evolution equations with the operators of generalized delay describing, for example, differential modeling of nonlinear Van der Pol oscillators or Lorentz strange attractors.

An analytical study on vortex-induced vibrations of cylinders with variable cross-section in the range of low Reynolds numbers

In the current paper, vortex-induced oscillations of circular cylinders with variable have been investigated. Considering the von-Kármán nonlinear strain-displacement relations and using the displacement coupling model to simulate the system and fluid-structure interaction, the nonlinear differential equations governing the motion have been derived using Hamilton’s principle. To apply the effect of vortices on the system vibrations, the nonlinear Van der Pol oscillator equation has been used. Then, using the Galerkin method, differential equations with partial derivatives are discretized, and nonlinear differential relations are solved by the Runge-Kutta approach. Finally, the response of the system, the phase curves, and the changes in the maximum amplitude according to the fluid velocity are obtained for various parameters, and the results have been discussed. Regarding the obtained results, changes in the cross-section of the beam have a significant effect on the lock-in region and also the maximum deflection of the beam. The maximum dynamic deflection for θ = 2° occurs at the velocity u = 2 , which is equal to 0.927. Also, the lock-in phenomenon is created for the beam with θ = 2.5° and θ = 3° at the velocities u = 2.78 and u = 4.23 , respectively.

Parameter estimation of two coupled oscillator model for pure intrinsic thermo-acoustic instability

A nonlinear phenomenological model of two coupled oscillators is proposed, which is able to describe the rich nonlinear behaviour stemming from an unstable pure intrinsic thermo-acoustic (ITA) mode of a simple combustion system. In an experimental bifurcation analysis of a pure ITA mode, it was observed that for increasing mean upstream velocity the flames loose stability through a supercritical Hopf bifurcation and subsequently exhibit limit cycle, quasi-periodic and period-2 limit cycle oscillations. The quasi-periodic oscillations were characterised by low frequent amplitude and frequency modulation. It is shown that a phenomenological model consisting of two coupled oscillators is able to reproduce qualitatively all the different experimentally observed regimes. This model consists of a nonlinear Van der Pol oscillator and a linear damped oscillator, which are nonlinearly coupled to each other. Furthermore, a parameter estimation of the model parameters is conducted, which reveals a good quantitative match between the model response and the experimental data. Hence, the presented phenomenological dynamical model accurately describes the nonlinear self-excited acoustic behaviour of premixed flames and provides a promising model structure for nonlinear time-domain flame models.

Parameter identification of two coupled oscillator model for pure intrinsic thermoacoustic instability

Thermo-acoustic instabilities are a well-known problem in combustions systems. In general, the type of these modes can be divided in two categories: (i) modes with an acoustic origin and (ii) modes with an intrinsic thermo-acoustic (ITA) origin. It is an challenging task to observe the ITA modes experimentally. Therefore, a setup with a burner in an acoustic embedding with close to anechoic up- and downstream terminations is required. In previous research we introduced such a setup and conducted an experimental bifurcation analysis on the dominant pure ITA mode. In this bifurcation analysis, we observed that for increasing upstream velocity the flames loose stability through a supercritical Hopf bifurcation and subsequently exhibit limit cycle, quasi-periodic and period-2 limit cycle oscillations. The quasi-periodic oscillations were characterized by low frequency amplitude and frequency modulation. In this research we present a phenomenological model consisting of two coupled oscillators that is able to reproduce all the different experimentally observed regimes. This model consists of a nonlinear Van der Pol oscillator and a linear mass-spring-damper oscillator, which are nonlinearly coupled to each other. Furthermore, we executed a parameter identification of the model and concluded that the model is able to describe the experimental data quantitatively well.

Approximate Solution of a Neutrosophic Nonlinear Van der Pol Oscillator Problem by Semi Analytical Method Using Thick Function

In this paper, an analytical method (Homotopy perturbation method HPM) is used for solving the initial value problem represented by a neutrosophic nonlinear Van der Pol oscillator equation (N-VDP) arising in applied dynamics using the thick function. We find the solutions of the (N-VDP) equation by HPM and then compare the numerical results with fourth order Runge-Kutta method (RK4). The results showed that the HPM lead to accurate and efficient results. Furthermore, these results of the HPM scheme and RK4 are implemented in Matlab.

Robust data-driven control for nonlinear systems using the Koopman operator*

Data-driven analysis and control of dynamical systems have gained a lot of interest in recent years. While the class of linear systems is well studied, theoretical results for nonlinear systems are still rare. In this paper, we present a data-driven controller design method for discrete-time control-affine nonlinear systems. Our approach relies on the Koopman operator, which is a linear but infinite-dimensional operator lifting the nonlinear system to a higher-dimensional space. Particularly, we derive a linear fractional representation of a lifted bilinear system representation based on measured data. Further, we restrict the lifting to finite dimensions, but account for the truncation error using a finite-gain argument. We derive a linear matrix inequality based design procedure to guarantee robust local stability for the resulting bilinear system for all error terms satisfying the finite-gain bound and, thus, also for the underlying nonlinear system. Finally, we apply the developed design method to the nonlinear Van der Pol oscillator.

Temporal Logic Control of Nonlinear Stochastic Systems Using a Piecewise-Affine Abstraction

Automatically synthesizing controllers for continuous-state nonlinear stochastic systems, while giving guarantees on the probability of satisfying (infinite-horizon) temporal logic specifications crucially depends on abstractions with a quantified accuracy. For this similarity quantification, approximate stochastic simulation relations are often used. To handle the nonlinearity of the system effectively, we use finite-state abstractions based on piecewise-affine approximations together with tailored simulation relations that leverage the local affine structure. In the end, we synthesize a robust controller for a nonlinear stochastic Van der Pol oscillator.

Nonlinear damping model for supersonic air-intake buzz

Air-intake buzz was initiated at Mach number of 3.0 on a two-dimensional isolated air-intake model in a wind tunnel by throttling the exit area. Schlieren images of shock oscillation around the intake entry during buzz were recorded using a high-speed camera. The recordings show a strong coupling between the bow-shock at the cowl lip and the oblique shock from the ramp. Image analyses were carried out, considering each image as a matrix of pixels and the change of intensity of light analyzed. A dominant frequency of 103.8 Hz (Strouhal no = 0.008 based on throat height) associated with the shock oscillations and harmonics of the shock oscillations are indicated. Phase-plots of the intensity and the rate of change of intensity show nearly perfect ellipse after filtering at the dominant frequency. Thus, the buzz phenomenon is associated with a limit-cycle oscillation as in a nonlinear Van-Der Pol oscillator. The computed damping factor synchronizes with the amplitude at all times, establishing that buzz is a stable and self-sustained oscillation with equilibrium between inertial and damping forces. It is proposed that a control scheme based on feed-forward control system using a suitable forcing function could be mathematically developed to suppress supersonic air-intake buzz.

Explosive synchronization induced by environmental coupling

The occurrence of explosive synchronization transition in a system of limit-cycle oscillators in the presence of two types of coupling; direct mean field diffusive and indirect environmental couplings, both operating simultaneously, is reported. The dynamics of coupled nonlinear Van der Pol and Rayleigh oscillators are studied in detail as a function of the distribution of intrinsic parameters of the oscillators. This explosive synchronization transition depends on the strength of indirect coupling and is irreversible giving rise to a characteristic hysteresis region. The different routes to synchronization observed in these coupled oscillators are studied in detail with the help of effective frequency and time series analysis. We have investigated the efficiency of the proposed scheme in various other topologies such as random, scale-free, and two–community networks as well.

Dancing synchronization in coupled spin-torque nano-oscillators

We are reporting a new type of synchronization, termed dancing synchronization, between two spin-torque nano-oscillators (STNOs) coupled through spin waves. Different from the known synchronizations in which two STNOs are locked with various fixed relative phases, in this new synchronized state two STNOs have the same frequency, but their relative phase varies periodically within the common period, resulting in a dynamic waving pattern. The amplitude of the oscillating relative phase depends on the coupling strength of two STNOs, as well as the driven currents. The dancing synchronization turns out to be universal, and can exist in two nonlinear Van der Pol oscillators coupled both reactively and dissipativly. Our findings open doors for new functional STNO-based devices.

Vibration analysis of fluid-structure interaction using tuned mass dampers

This paper has investigated the semi-analytical analysis of the solid-fluid interaction vibration in the presence of concentrated mass-spring-damper vibration absorber. The nonlinear partial differential equations of motion are derived by considering von Karman-type large deformations and viscoelastic behaviour. Fluid-structure interaction is modelled by using an acceleration coupling model in which a nonlinear Van der Pol oscillator simulates fluctuating nature of the vortex street. The nonlinear equations are discretized via the Galerkin approach, and the obtained equations are numerically solved by applying the Runge-Kutta method. Eventually, the dynamic response, phase plane plots, and variations of maximum amplitude in terms of fluid velocity for different parameters are extracted. The results reveal that utilizing vibration absorber leads to a significant effect on the dynamic characteristics of the system, displaces the lock-in phenomenon, and remarkably reduce the amplitude of the system oscillations.

Acceleration behaviors of Fermi accelerator model excited by Van der Pol oscillator

In this paper, we investigate on dynamics of a generalized Fermi accelerator model with a moving wall described by a nonlinear Van der Pol oscillator. Different acceleration behaviors appear due to the discontinuity brought by impact and the stationary periodic excitation of the moving wall. Utilizing the theory of discontinuous dynamical systems, acceleration mechanism of the particle is studied and the analytical conditions for stick motion and grazing motion are obtained. By the use of generic mappings between different boundaries including stick and non-stick motions, periodic motions of the Fermi accelerator can be constructed, and corresponding local stability and bifurcations can be discussed according to mapping dynamics. Finally, different acceleration behaviors including periodic, chatter and stick motions are presented and illustrated.

Spectral functions and negative density of states of a driven-dissipative nonlinear quantum resonator

We study the spectral properties of Markovian driven-dissipative quantum systems, focusing on the nonlinear quantum van der Pol oscillator as a paradigmatic example. We discuss a generalized Lehmann representation, in which single-particle Green's functions are expressed in terms of the eigenstates and eigenvalues of the Liouvillian. Applying it to the quantum van der Pol oscillator, we find a wealth of phenomena that are not apparent in the steady-state density matrix alone. Unlike the steady state, the photonic spectral function has a strong dependence on interaction strength. Further, we find that the interplay of interaction and non-equilibrium effects can result in a surprising ‘negative density of states’, associated with a negative temperature, even in absence of steady state population inversion.

Analysis and prediction for structural responses and hydrodynamic properties of two-degree-of-freedom vortex induced vibration

The objective of this article is to introduce a modified wake-oscillator model to systematically analyze and predict response characteristics and hydrodynamic coefficients of vortex-induced vibration (VIV) of an elastically mounted cylinder with the low mass ratio in two-degree-of-freedom (2-DOF). According to two-dimensional potential flow theory and discrete vortices, the in-line and cross-flow fluctuating vortex forces are simplified and quantified. The model is based on a nonlinear van der Pol oscillator coupled with the cylinder motion and solved by numerical method. The dynamical behavior of the coupled model is investigated and the empirical parameters in the fluid oscillator are calibrated against experimental results. Direct comparison of numerical and experimental results shows that the model qualitatively and quantitatively captures the main dynamic features in 2-DOF VIV, such as oscillation amplitudes, lock-in ranges, hydrodynamic coefficients, phase angles, and X-Y trajectories.

A new operational matrix of fractional order integration for the Chebyshev wavelets and its application for nonlinear fractional Van der Pol oscillator equation

In this paper, an efficient and accurate computational method based on the Chebyshev wavelets (CWs) together with spectral Galerkin method is proposed for solving a class of nonlinear multi-order fractional differential equations (NMFDEs). To do this, a new operational matrix of fractional order integration in the Riemann–Liouville sense for the CWs is derived. Hat functions (HFs) and the collocation method are employed to derive a general procedure for forming this matrix. By using the CWs and their operational matrix of fractional order integration and Galerkin method, the problems under consideration are transformed into corresponding nonlinear systems of algebraic equations, which can be simply solved. Moreover, a new technique for computing nonlinear terms in such problems is presented. Convergence of the CWs expansion in one dimension is investigated. Furthermore, the efficiency and accuracy of the proposed method are shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient. As a useful application, the proposed method is applied to obtain an approximate solution for the fractional order Van der Pol oscillator (VPO) equation.

An equivalent linearization method for nonlinear Van der Pol oscillator subjected to random vibration using orthogonal functions

This paper deals with the idea of the orthogonal functions in the equivalent linearization of the nonlinear systems. Block Pulse (BP) function gives effective tools to approximate complex problems. The aim of this work is on using properties of the BP function as an orthogonal function in process of linearization. The BP functions have been used to propose an equivalent linearization method in the time domain to determine the unknown linearization coefficients. The accuracy of the proposed method compared with the other equivalent linearization approaches, including the regulation linearization and the dual criterion linearization methods. This study exploited the nonlinear Van der Pol oscillator system under stationary random excitation to demonstrate the feasibility of the proposed method. The validity of the analytical method is verified by applying different values of nonlinearity and intensity of excitation. Besides, by comparing the mean-square responses and frequency response functions of the linearized systems for a wide range of nonlinearity depicted the present method is in agreement with other methods.

Hopf bifurcation of hybrid Van der Pol oscillators

The nonlinear Van der Pol oscillator is long recognized for the generation of a stable limit cycle, independent of initial conditions. This feature explains its popular use in the model reference control. A specific modification of the Van der Pol oscillator is motivated by the practical need to reshape the limit cycle to the one with harmonic behavior whose amplitude and frequency would straightforwardly rely on the oscillator parameters. In this work, such a modification is numerically analyzed in the hybrid setting under unilateral constraints. An equilibrium of the constrained modification of the Van der Pol oscillator is revealed to bifurcate to a stable limit cycle under certain parameter variations. This phenomenon, otherwise known as a Hopf bifurcation, is further studied via the corresponding Poincaré map to numerically carry out the set of the bifurcation parameter values and to verify the asymptotic stability of the generated hybrid limit cycle. The knowledge of the bifurcation parameter values is then utilized to illustrate the capability of the constrained oscillator to degenerate its limit cycle into an asymptotically stable equilibrium once the oscillator parameters are properly modified online.

Nonlinear dynamics of a fluid-conveying pipe under the combined action of cross-flow and top-end excitations

We report a theoretical investigation of an elastic and slender fluid-conveying pipe with a top-end excitation subjected to uniform cross flows. Considering the mean drag force and the time varying vortex-induced lift force which is modeled using a nonlinear van der Pol oscillator, the nonlinear partial differential equations of the motion of coupled fluid-structure system are constructed and simplified to a reduced-order model through the Galerkin-type discretization. By virtue of quasi-static displacement conditions, the characteristics of vortex-induced vibration of the pipe are evaluated for the first two lock-in modes. The results show that the top-end excitation can increase the vibration amplitude of the pipe when the cross-flow speed is out of the lock-in regions. When the cross-flow speed is within the lock-in region, however, the top-end oscillation causes a transition between quasi-periodic and periodic in the responses of the pipe, significantly reducing or increasing the vibration amplitudes depending on the excitation acceleration and frequency. This finding has an important guidance in suppressing vortex-induced vibrations by balancing the internal fluid velocity and the top-end excitation.

The Closed-Form Steady-State Probability Density Function of van der Pol Oscillator under Random Excitations

The strongly nonlinear van der Pol oscillator represents a special challenge, which has prevented many methods from obtaining the closed-form solutions of the steady-state probability density functions (PDFs) in the literature. In this paper, we apply our recently developed method called the iterative method of weighted residue to analytically construct steady-state PDFs of the van der Pol oscillator under external Gaussian white noise excitation. The steady-state PDF is assumed to be an exponential function of polynomials in the state variables. The iterative method of weighted residue is used to compute the PDF. The iterative procedure that makes use of the obtained closed-form solutions of steady-state PDFs as the weighting function for the method improves the accuracy of the solution and the convergence of the solution process. The closed-form steady-state PDFs of strongly nonlinear van der Pol oscillator are presented in this paper, which were not available in the literature before, and are compared with those from the Monte Carlo simulations. The analytical and simulation results are in excellent agreement over a wide range of damping coefficients.