Van Der Pol Oscillator Research Articles

A remark on renormalization group theoretical perturbation in a class of ordinary differential equations

Abstract We revisit the renormalization group (RG) theoretical perturbation theory on oscillator-type second-order ordinary differential equations. For a class of potentials, we show a simple functional relation among secular coefficients of the harmonics in the naive perturbation series. It leads to an inversion formula between bare and renormalized amplitudes and an elementary proof of the absence of secular terms in all orders of the RG series. The result covers nonautonomous as well as autonomous cases and refines earlier studies, including the classic examples of Van der Pol, Mathieu, Duffing, and Rayleigh equations.

Explosive and semi-explosive death in coupled oscillators

In this paper, we propose a new kind of transition process from oscillation to death state that is different from the traditional explosive death called semi-explosive death. In this process, two kinds of death arise in different ways. We investigate the occurrence of semi-explosive death transition for the first time on globally conjugate-coupled Van der Pol (VdP) oscillators with asymmetry factor. Semi-explosive death is an irreversible transition of half first order and half second order. That is, this process not only happens with an abrupt, irreversible transition that is a common feature of first order phase transition, but also includes a continuous second order change. Moreover, the forward and the backward second order transition points for this transition have been obtained theoretically, which is in complete agreement with the numerical results. Finally, the details of the transition mechanisms between semi-explosive death and explosive death along with dependence of asymmetry factor and damping coefficient are also discussed theoretically and numerically.

Class of Auto-Oscillating Systems

This article proposes a model that generalizes the well-known equations of the nonlinear theory of oscillations (Van der Pol and Rayleigh equations), the limit cycles of which are the curves of the phase plane, determined by the total energy of oscillations in the absence of dissipation/energy inflow into the system. By changing the parameters of the system and the type of force action, it is possible to set the oscillation characteristics that are resistant to disturbances. The phase portraits of the system are considered, including the case of multiply connected limit cycles.

Dynamics of hierarchical weighted networks of van der Pol oscillators.

We investigate the dynamics of regular fractal-like networks of hierarchically coupled van der Pol oscillators. The hierarchy is imposed in terms of the coupling strengths or link weights. We study the low frequency modes, as well as frequency and phase synchronization, in the network by a process of repeated coarse-graining of oscillator units. At any given stage of this process, we sum over the signals from the oscillator units of a clique to obtain a new oscillating unit. The frequencies and the phases for the coarse-grained oscillators are found to progressively synchronize with the number of coarse-graining steps. Furthermore, the characteristic frequency is found to decrease and finally stabilize to a value that can be tuned via the parameters of the system. We compare our numerical results with those of an approximate analytic solution and find good qualitative agreement. Our study on this idealized model shows how oscillations with a precise frequency can be obtained in systems with heterogeneous couplings. It also demonstrates the effect of imposing a hierarchy in terms of link weights instead of one that is solely topological, where the connectivity between oscillators would be the determining factor, as is usually the case.

Measurement of added mass for an object oscillating in viscous fluids using nonlinear self-excited oscillations

In this study, a method of measuring the added mass for an object oscillating in a viscous fluid using nonlinear self-excited oscillations was developed. The added mass produces an additional inertial effect on the vibrating object. In previous methods, the added mass is obtained experimentally from the response frequency and amplitude at the peak in the frequency response curve under a harmonically forced excitation. However, such methods cannot be utilized in highly viscous environments because the peak becomes ambiguous as a result of damping. Moreover, in very high-viscosity environments, the resonance peak ceases to exist in the frequency response curves. To solve this problem, self-excited oscillations were induced in the system using linear velocity feedback. Because linear velocity feedback can be used to eliminate the viscous damping effect when the linear feedback gain is set near the Hopf bifurcation point related to self-excited oscillations, the object becomes self-excited at a frequency equal to the natural frequency of the object oscillating in a vacuum, i.e., the undamped natural frequency. What distinguishes the proposed method from the abovementioned previous methods is that the value of the response is not needed; however, a reduction in the oscillation amplitude is required to obtain accurate measurement results. In the proposed method, to avoid an increase in the response amplitude under the applied linear velocity feedback, nonlinear feedback is also applied to produce a limit cycle similar to that produced in a van der Pol oscillator. A prototype of the measurement system was constructed based on the proposed method. A comparison of the experimentally and theoretically obtained added mass values confirmed the validity of the proposed method for added mass sensing.

High-order phase reduction for coupled oscillators

We explore the phase reduction in networks of coupled oscillators in the higher orders of the coupling parameter. For coupled Stuart–Landau oscillators, where the phase can be introduced explicitly, we develop an analytic perturbation procedure to explicitly obtain the higher-order approximation. We demonstrate this by deriving the second-order phase equations for a network of three Stuart–Landau oscillators. For systems where explicit expressions of the phase are not available, we present a numerical procedure that constructs the phase dynamics equations for a small network of coupled units. We apply this approach to a network of three van der Pol oscillators and reveal components in the coupling with different scaling in the interaction strength.

Continuous Lyapunov Controlled Non-linear System Optimization Using Deep Learning with Memory

The introduction of unexpected system disturbances and new system dynamics does not allow guaranteed continuous system stability. In this research we present a novel approach for detecting early failure indicators of non-linear highly chaotic system and accordingly predict the best parameter calibrations to offset such instability using deep machine learning regression model. The approach proposed continuously monitors the system and controller signals. The Re-calibration of the system and controller parameters is triggered according to a set of conditions designed to maintain system stability without compromise to the system speed, intended outcome or required processing power. The deep neural model predicts the parameter values that would best counteract the expected system in-stability. To demonstrate the effectiveness of the proposed approach, it is applied to the non-linear complex combination of Duffing Van der pol oscillators. The approach is also tested under different scenarios the system and controller parameters are initially chosen incorrectly or the system parameters are changed while running or new system dynamics are introduced while running to measure effectiveness and reaction time.

Suppressing birhythmicity by parametrically modulating nonlinearity in limit cycle oscillators

Multirhythmicity, a form of multistability, in an oscillator is an intriguing phenomenon found across many branches of science. From an application point of view, while the multirhythmicity is sometimes desirable as it presents us with many possible coexisting stable oscillatory states to tap into, it can also be a nuisance because a random perturbation may make the system settle onto an unwanted stable state. Consequently, it is not surprising that there are many natural and artificial mechanisms available that can control the multirhythmicity. We propose in this paper the idea of incorporating parametric (periodic) modulation of the nonlinear damping in the limit cycle oscillators with a view to exciting resonance and antiresonance responses at particular angular driving frequencies, and controlling the resulting birhythmicity by changing the amplitude of the modulation. To this end, we employ analytical (perturbative) and numerical techniques on the van der Pol oscillator—a paradigmatic limit cycle system—having additional position dependent time delay term and its modified autonomous birhythmic version. We also bring the fact to the fore that introduction of delay—a commonly adopted method of controlling multirhythmicity—in such a system can sometimes bring forth unwanted birhythmicity; and interestingly, our method of controlling birhythmicity through periodic modulation can suppress such a delay induced birhythmic response.

Quantifying the Transition from Spiral Waves to Spiral Wave Chimeras in a Lattice of Self-sustained Oscillators

The present work is devoted to the detailed quantification of the transition from spiral waves to spiral wave chimeras in a network of self-sustained oscillators with two-dimensional geometry. The basic elements of the networks are the van der Pol oscillator and the FitzHugh-Nagumo neuron. Both models are in the regime of relaxation oscillations. We analyze the regime by using the indices of local sensitivity which enables us to evaluate the sensitivity of each individual oscillator at finite time. Spi-ral waves are observed in both lattices when the interaction between elements have the local character. The dynamics of all the elements is regular. There are no high-sensitive regions. We have discovered that when the coupling becomes nonlocal, the features of the systems significantly changes. The oscillation regime of the spiral wave center element switches to chaotic one. Besides this, a region with high sensitivity occurs around this oscillator. Moreover, we show that the latter expands in space with elongation of the coupling range. As a result, an incoherence cluster of the spiral wave chimera is formed exactly within this high-sensitive area. Formation of this cluster is accompanied by the sharp increase in values of the maximal Lyapunov exponent to the positive region. Furthermore, we explore that the system can even switch to hyperchaotic regime, when several Lyapunov exponents becomes positive.

Numerical verification of the orders of accuracy of truncated asymptotic expansion solutions to the van der Pol equation

We propose a numerical method to verify the orders of accuracy of truncated asymptotic expansion solutions to the van der Pol equation. Our proposed method is simpler than an existing method (Deeba and Xie in J Comput Anal Appl 3(2): 165–171, 2001). Our proposed method does not need any technique of a least-squares fit of error data, whereas the existing one does. Numerical simulations confirm that our proposed method works effectively.

Estimating the region of attraction based on a polynomial lyapunov function

This paper deals with the problem of obtaining estimates of the domains of attraction and stability in nonlinear dynamical systems. It is based on a special recursive procedure for constructing polynomial Lyapunov functions. The procedure is tested on several examples used in previous studies by other authors, with a comparison of the results. The approach is also applied to estimate the domain of attraction for single DoF mechanical systems with uncertain parameters (Van der Pol and Duffing oscillators).

Inverter Parallelization for an Islanded Microgrid Using the Hopf Oscillator Controller Approach With Self-Synchronization Capabilities

Nonlinear dynamical systems such as weakly coupled oscillators are an interesting approach to be adopted for the regulation of power inverters inside microgrids. Aiming at the synchronization and load sharing in islanded mirogrid, this article is inspired by oscillator synchronization property to propose a Hopf oscillator controller for the single-phase inverters. The Hopf oscillator dynamic equations are used for providing the inverter's frequency and amplitude voltage references, which lead to a robust nonlinear droop behavior for driving the system without using communications. The Hopf oscillator provides better sharing of the load between inverters with higher robustness, less harmonic distortion, and faster time response of the associated limit cycle than the achieved by the other approach made with a Van der Pol oscillator. In addition, global asymptotic synchronization of system is proven by Lyapunov approach. Simulation results of a system composed by paralleled inverters are provided and compared with a Van der Pol oscillator approach reported in literature. Experimental results are also provided to prove the Hopf oscillator based controller under different circumstances.

Vortex-induced swing (VIS) motion for energy harvesters and flowmeters

Vortex-induced vibration (VIV) has been widely studied in the fields of vibration control, building construction, and underwater vehicles. Recently, researchers began utilizing the VIV phenomenon for flow energy harvesting. Here, we describe that vortex shedding causes periodic rotational motions and explore these vortex-induced swing (VIS) motions for harvesting flow energy and measuring flow speed. An arc-bluff structure was constructed to enlarge the VIS motions, and a phenomenological model was developed using the Van der Pol equation. Swing characteristics when flow velocities were in the range of (0.15, 0.45) m/s were assessed. Experiments showed that the maximum swing amplitude of the device is 120°, and it converges to 80° as the water velocity increases. The frequency, amplitude, and initial angle curve of the VIS device can be used to represent the water speed vector. The proposed flowmeter showed a flow rate sensitivity of 7∼10 Hz/(m·s−1) in the experimental range. Energy harvester prototypes demonstrated a peak-peak output of 3.28 V in water with a velocity of 0.45 m/s. The present work provides an approach for the flow measurement and energy harvesting under low-speed and low-frequency conditions.

Design of evolutionary optimized finite difference based numerical computing for dust density model of nonlinear Van-der Pol Mathieu’s oscillatory systems

In this study, a new evolutionary optimized finite difference based computing paradigm is presented for dynamical analysis of dust density model for the ensemble of electrical charges and dust particles represented with nonlinear oscillatory system based on hybridization of Van-der Pol and Mathieu equation (VDP-ME). Strength of accurate and effective discretization ability of finite difference method (FDM) is exploited to transform VDP-ME to equivalent nonlinear system of algebraic equations. The residual error based fitness function of the transformed model is constructed by the competency of approximation theory in mean square sense. The optimization of the residual error of the system through hybrid meta-heuristic computing paradigm GA-SQP; genetic algorithm (GA) for viable global search aided with rapid fine tuning of sequential quadratic programming (SQP). The proposed GA-SQP-FDM is applied on variants of dust density model of VDP-ME by varying the rate of charged dust grain production, as well as, loss and comparison of results with state of art numerical procedure established the worth of the scheme in terms of accuracy and convergence measures endorsed through statistical observations on large dataset.

Tuning coupling rate to control oscillation quenching in fractional-order coupled oscillators.

Introducing the fractional-order derivative into the coupled dynamical systems intrigues gradually the researchers from diverse fields. In this work, taking Stuart-Landau and Van der Pol oscillators as examples, we compare the difference between fractional-order and integer-order derivatives and further analyze their influences on oscillation quenching behaviors. Through tuning the coupling rate, as an asymmetric parameter to achieve the change from scalar coupling to non-scalar coupling, we observe that the onset of fractional-order not only enlarges the range of oscillation death, but attributes to the transition from fake amplitude death to oscillation death for coupled Stuart-Landau oscillators. We go on to show that for a coupled Van der Pol system only in the presence of a fractional-order derivative, oscillation quenching behaviors will occur. The results pave a way for revealing the control mechanism of oscillation quenching, which is critical for further understanding the function of fractional-order in a coupled nonlinear model.

Abnormal route to aging transition in a network of coupled oscillators.

In this article, we investigate the dynamical robustness in a network of Van der Pol oscillators. In particular, we consider a network of diffusively coupled Van der Pol oscillators to explore the aging transition phenomena. Our investigation reveals that the route to aging transition in a network of Van der Pol oscillator is different from that of typical sinusoidal oscillators such as Stuart-Landau oscillators. Unlike sinusoidal oscillators, the order parameter does not follow smooth second-order phase transition. Rather, we observe an abnormal phase transition of the order parameter due to the sudden appearance of unbounded trajectories at a critical point. We provide detailed bifurcation analysis of such an abnormal phase transition. We show that the boundary crisis of a limit-cycle oscillator is at the helm of such an unusual discontinuous path of aging transition.

On linearizability via nonlocal transformations and first integrals for second-order ordinary differential equations

Nonlinear second-order ordinary differential equations are common in various fields of science, such as physics, mechanics and biology. Here we provide a new family of integrable second-order ordinary differential equations by considering the general case of a linearization problem via certain nonlocal transformations. In addition, we show that each equation from the linearizable family admits a transcendental first integral and study particular cases when this first integral is autonomous or rational. Thus, as a byproduct of solving this linearization problem we obtain a classification of second-order differential equations admitting a certain transcendental first integral. To demonstrate effectiveness of our approach, we consider several examples of autonomous and non-autonomous second order differential equations, including generalizations of the Duffing and Van der Pol oscillators, and construct their first integrals and general solutions. We also show that the corresponding first integrals can be used for finding periodic solutions, including limit cycles, of the considered equations.

Identification method for backbone curve of cantilever beam using van der Pol-type self-excited oscillation

This study presents an experimental method for identification of the backbone curves of cantilevers using the nonlinear dynamics of a van der Pol oscillator. The backbone curve characterizes the nonlinear stiffness and nonlinear inertia of the resonator, so it is important to identify this curve experimentally to realize high-sensitivity and high-accuracy sensing resonators. Unlike the conventional method based on the frequency response under external excitation, the proposed method based on self-excited oscillation enables direct backbone curve identification, because the effect of the viscous environment is eliminated under the linear velocity feedback condition. In this research, the method proposed for discrete systems is extended to give an identification method for continuum systems such as cantilever beams. The actuation is given with respect to both the linear and nonlinear feedbacks so that the system behaves as a van der Pol oscillator with a stable steady-state amplitude. By varying the nonlinear feedback gain, we can produce the self-excited oscillation experimentally with various steady-state amplitudes. Then, using the relationship between these steady-state amplitudes and the corresponding experimentally measured response frequencies, we can detect the backbone curve while varying the nonlinear feedback gain. The efficiency of the proposed method is determined by identifying the backbone curves of a macrocantilever with a tip mass and a macrocantilever subjected to atomic forces, which are representative sources of hardening and softening cubic nonlinearities, respectively.

Derivation of a slow phase model of vortex-induced vibrations for smooth and turbulent oncoming flows

This paper analyzes the influence of turbulence on a wake-oscillator model. Turbulence is introduced by randomizing the model proposed by Facchinetti et al. under the quasi-steady assumption. A multiple scale analysis of the deterministic model shows that the response is governed by a dimensionless group D, expressed as a function of the amplitudes of the forcing terms in the two governing equations, the total (aerodynamic plus structural) damping and the parameter ε of the fluid Van der Pol oscillator. The influence of turbulence is interpreted as a stochastic noise of small intensity and with a slower timescale than the (fast) oscillations, which is typical of wind engineering applications. A slow phase model of the problem is then derived by assuming that the small turbulence drives the system only slightly away from its limit cycle in smooth flow conditions. Standard modeling techniques borrowed from other fields of physics, in particular the observation of phase shifts and their accumulation, are used to highlight conditions under which the turbulence of the oncoming flow might reduce the amplitudes of vibrations of the body. The slow phase model is derived in smooth flow conditions, then extended to turbulent flow. It recalls that the phase plays a central role in synchronization problems, and that the response amplitude should only be considered as a sub-product of the slow phase. The slow phase model is expressed by means of a first order nonlinear differential equation for the phase and a memoryless transformation for the response amplitudes. Its solution is explicit and simple in some limiting cases. In particular, for small turbulence intensity, the response is shown to be insensitive to turbulence when its frequency content is not low enough. This major dependence upon the frequency content of the turbulence explains that the reduction of VIV due to turbulence cannot be explained by the turbulence intensity only, as usually considered today. The required relative smallnesses of the turbulence and its frequency content naturally appear in the derivation, which is led in a dimensionless manner. Finally, the present study constitutes an analysis of a phenomenological model which could be used in a much wider concept than of the elastically-mounted circular cylinder.

Synchronization boost with single-photon dissipation in the deep quantum regime

Synchronization occurs ubiquitously in nature. The van der Pol oscillator has been a favorite model to investigate synchronization. Here we study the oscillator in the deep quantum regime, where nonclassical effects dominate the dynamics. Our results show: (i) squeezed driving loses its effect, (ii) noise boosts synchronization, (iii) synchronization is bounded, and (iv) the limit-cycle is insensitive to strong driving. We propose a synchronization measure and analytically calculate it. These results reflect intrinsic differences between synchronization in the quantum and deep quantum regimes.