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In this paper, a criteria of suppressing chaos for a kind of nonlinear oscillators is established by the theory of the strange attractor. The oscillators considered include Duffing, van der Pol, Duffing–van der Pol and pendulum. According to this criteria, we analyze the phase effect using two methods, one by adding the second external force term and the other by adding parametric excitation, both of which may be used to suppress chaos in the systems. Some examples are used to illustrate the validity of the criteria and the importance of phase effect in suppressing chaos. [ABSTRACT FROM AUTHOR]

In this work it is shown that by a series of transformations the classical Van der Pol oscillator can be exactly reduced to Abel's equations of the second kind. The absence of exact analytic solutions in terms of known (tabulated) functions of the reduced equations leads to the conclusion that there are no exact solutions of the Van der Pol oscillator in terms of known (tabulated) functions. In the limits or small or large values of the parameter ε the reduced equations are amenable to asymptotic analysis. For the case of large values of the parameter (relaxation oscillations) an analytic solution to the problem is provided that is exact up to O(ε-2). [ABSTRACT FROM AUTHOR]

Random attractors allow one to classify qualitative and quantitative aspects of the long-time behaviour of stochastically forced systems viewed as random dynamical systems (RDS) in an analogous way to attractors for deterministic systems. We compare several notions of random attractor in RDS by examining convergence of trajectories to a random invariant set in different ways, including convergence of mean distance, convergence in probability and convergence on dense subsequences as well as pullback convergence. We give examples showing how these concepts of attraction are inequivalent. We also examine definitions for local attractors and possible methods of decomposition of random attractors into 'random Milnor' attractors. We point out some problems that remain in interpreting these. Finally, we examine numerical simulations of a stochastic van der Pol-Duffing equation and find cases where there appear to be Milnor attractors with positive top Lyapunov exponents. We explain bursting behaviour of the two-point motion for some parameter values in terms of the presence of non-trivial random Milnor attractors. [ABSTRACT FROM AUTHOR]

The forced van der Pol oscillator has been the focus of scientific scrutiny for almost a century, yet its global bifurcation structure is still poorly understood. In this paper, we present a hybrid system consisting of the dynamics of the trajectories on the slow manifold coupled with "jumps" at the folds in the critical manifold to approximate the fast subsystem. The global bifurcations of the fixed points and periodic points of this hybrid system lead to an understanding of the bifurcations in the periodic orbits (without canards) of the forced van der Pol system. [ABSTRACT FROM AUTHOR]

Examines the harmonic radiation from microstrip patch oscillators using a single and dual parallel-tuned circuit Van der Pol oscillator model. Analysis and simulation of active integrated antennas; Single resonator case; Dual-resonator case; Determination of the degree of harmonic suppression that could be achieved using microstip patch oscillators.

The Van der Pol equation is solved in the case where the parameter has a large value. [ABSTRACT FROM AUTHOR]

The limit cycle is associated in the Van der Pol oscillator with damping introduced to merely one of its state equations. In the present oscillator nonlinear damping is added to both equations. A unique behaviour is observed in spite of the smallness of the departure from the Van der Pol case. The present treatment is due to classical Poincaré-Bendixson and Liapunov methods. It illuminates in a surprisingly simple and elegant manner the great power of the methods. [ABSTRACT FROM AUTHOR]

The discrete BVP oscillator obtained through the Euler method is investigated, and also first proved that there exist chaotic phenomena in the sense of Marotto's definition of chaos and twoperiod cycles. And numerical simulations not only show the consistence with the theoretical analysis but also exhibit the complex dynamical behaviors, including the ten-periodic orbit, a cascade of period-doubling bifurcation, quasiperiodic orbits and the chaotic orbits in Marotto's chaos and intermitten's chaos. The computations of Lyapunov exponents confirm the existence of dynamical behaviors. [ABSTRACT FROM AUTHOR]

Presents information on a study which used a quantization process independent of Lagrangians and Hammiltonians for the quantization of a van der Pol system and Duffing system. Information on dynamic quantization; Limit of quantum dynamics.

The purpose of this work is to improve the understanding of the solution of the noisy Duffing-van der Pol equation. We achieve this by developing rigorous methods to replace, in some limiting regime, the original complicated system by a 'simpler, constructive, and rational' approximation - a low-dimensional model of the dynamical system. To this end, we study the equations as a random perturbation of a two-dimensional Hamiltonian system. We achieve the model-reduction through 'stochastic averaging' and the reduced Markov process takes its values on a graph with certain glueing conditions at the vertex of the graph. Examination of the reduced Markov process on the graph yields many important results, namely, mean exit times, probability density functions, and stochastic bifurcations. [ABSTRACT FROM AUTHOR]

Presents a study which discussed an analog simulation of the dynamics of a system consisting of a van der Pol oscillator coupled to a Duffing oscillator. Design of the analog simulator; Amplitude-response curves; Bifurcation and onset of chaos.

A forced van der Pol relaxation oscillator is studied experimentally in the regime of stable oscillations. The variable parameter is chosen to be the driving frequency. For a range of parameter values, we show that the rotation number varies continuously from 0 to 1. This work provides experimental evidence that period-adding bifurcations to chaos previously reported by Kennedy and Chua are intimately connected to the existence of a regime of stable oscillations where the rotation number shows a Devil's-staircase structure. [ABSTRACT FROM AUTHOR]

Presents information on a study which investigated the effect of element value deviation on the degenerate mode in rings of three and four coupled van der Pol oscillators. Fundamental equation; Derivation of the averaged equations; Computer simulation of the averaged equation; Verification of the computer simulation; Experiments.

In this paper, the strongly resonant bifurcations of a nonlinearly coupled Van der Pol-Duffing Oscillator by the classical multi-scale method are studied. It is shown that there exist periodic motions of a single oscillator, frequency-locking and quasi-periodic motions of two oscillators when the parameters vary. Meanwhile, some numerical results are given to test the theoretical ones. [ABSTRACT FROM AUTHOR]

Analysis of numerical solutions for a system of two van der Pol-Duffing oscillators with nonlinear coupling showed that there exist chaotic switchings (occurring at irregular time intervals) between two oscillatory regimes differing by nearly time-constant phase shifts between the coupled subsystems. The analysis includes the investigation of bifurcations of the periodic motions corresponding to synchronization of two subsystems, finding stability regions of synchronization regimes and scenarios of the transitions to chaos. [ABSTRACT FROM AUTHOR]

Offers insights into the structure of the Bonhoeffer-van der Pol (BVP) system, particularly into the problems of excitability and threshold for all three types of the system classified according to the existing attractors. T-repellers and T-attractors; BVP model; Isochrones.

Reports on oscillations that arise in a double plasma device when plasma production is restricted to the source chamber and the separating grid between the two chambers is biased negatively. Free oscillating system; Periodic pulling; Typical behavior of driven van der Pol type oscillators; Second interacting frequency; Virtual anode oscillations.

In analyzing differential equations, computer algebra or symbolic computation is useful when we want to know the relation between the solutions and the parameters at the point near bifurcation. The object of this paper is to show the fast computation method by means of computer algebra in analyzing the bifurcation problem from the steady state to the periodic oscillatory state. The proposed method of analysis depends on Lyapunov-Schmidt method. Instead of using trigonometric multiplication and sum formula in computation of nonlinear terms and their harmonics, both the basis vectors with polynomials in components and their inner products are utilized to reduce the computation time significantly. The example of a second-order Van der Pol equation and the example of the higher-order coupled system are used to give the bifurcation equation. Thus, the relation between the amplitude of oscillation and the parameter is given with the stability condition. By the proposed method with the aid of a microcomputer and symbolic computation language of interpreter type, the computation time is shorter. [ABSTRACT FROM AUTHOR]

Presents information on an analytical model for predicting gain and performance of CW singly resonant optical parametric oscillators. Inclusion of the pump and signal beam effects; Information on the benefits of an idler absorption-length product exceeding 10; Details on the model.

Using a singular perturbation technique, the non-sinusoidal (relaxation) oscillations of the Van der Pol non-linear oscillator are investigated. For high frequency and large transconductance and mutual inductance, a small parameter ∈ is found to be the multiplication factor to the highest order term in the non-linear equation. When &isin → 0 the equation and its solution become singular. The oscillation and its period can be calculated by proper matching at the voltage jump where the singularity occurs. The method is also applicable to relaxation problems of other non-linear oscillators. [ABSTRACT FROM AUTHOR]

Previously, a model for a microwave oscillator has been proposed which provides the oscillation load characteristics, unilateral injection locking characteristics and mutual locking characteristics that agree better with the experimental results than the van der Pol model often used in analysis. In this paper, the model is simplified and used to study an antiphase coupled system made of two oscillators with identical characteristics coupled by a magic T in terms of its zero input signal characteristics and the response with an external input signal. The results show good agreement with experimental data. It is found that a linear amplification is possible if the frequency of the external signal is equal to the antiphase locking frequency. Moreover, the antiphase locking is shifted to the quasi-in-phase or in-phase locking if the frequency is near the antiphase locking frequency. It is proposed that an application to digitally modulated signal amplification is possible. [ABSTRACT FROM AUTHOR]

Analytical expressions for the efficient computation of sensitivities with respect to fundamental periods in dynamic nonlinear characteristics under multitone excitation are presented. Almost-periodic oscillations in the forced Van der Pol oscillator are computed to illustrate the capability and efficiency of the resulting harmonic balance technique. [ABSTRACT FROM AUTHOR]

Presents a study which developed an experimental technique to determine the Van der Pol (VDP) parameters of a transistor oscillator. Behavior of the oscillator with three parameters; Number of parameters that can only be determined by spectrum analysis when operating in a steady state; Impact of VDP on the dynamics of the oscillator.

This communication contains a reply to a criticism of our previous work on the ‘improved’ van-der-Pol equation. After answering the main points expressed in the criticism, we discuss a generalized approach which may be helpful in treating the improved van-der-Pol equation and other related equations. [ABSTRACT FROM AUTHOR]

The Rayleigh-van der Pol oscillator, described by (* This character cannot be in ASCI text)-∈( 1-μx² — v(* This character cannot be in ASCI text)²) + 1 = 0 is analysed, in particular for the harmonic case μ = v, refuting claims made in a recent paper (Kaplan and Yaffe 1976). [ABSTRACT FROM AUTHOR]

The study of oscillators is complicated and investigators tend to resort to simplified models. One such model is that represented by the van-der-Pol equation. Another model, represented by an ‘improved’ van-der-Pol equation is suggested and investigated in this paper. The latter model is almost as simple as the original one and yet it possesses several advantages which are valuable for the design, investigation, simulation and presentation of real oscillator systems. [ABSTRACT FROM AUTHOR]

It is shown that the white-noise-induced transition between the limiting cycle and state of rest of a van der Pol oscillator has a threshold. The threshold value is directly proportional to the product of the characteristic energy of self-oscillations and the friction coefficient. It is shown that not only self-oscillations, but also external noise, disappear in the state of rest. [ABSTRACT FROM AUTHOR]

We have studied a system of coupled van der Pol oscillators under pulsed drive action. In the case where the coupled oscillators exhibit the death of oscillations, the introduction of a pulsed drive leads to the appearance of islands corresponding to quasi-periodic regimes in the drive period-amplitude plane, which are also dying with an increase in the frequency detuning. [ABSTRACT FROM AUTHOR]

A mathematical model of the new chaotic system based on a modified van der Pol equation featuring delayed parametric self-action is considered. The chaotization of oscillations in this “autoparametric” system is analyzed by numerical methods. [ABSTRACT FROM AUTHOR]

One THz harmonic oscillation was observed in a sub-THz oscillating GaInAs/AlAs resonant tunneling diode integrated with a slot antenna. The fundamental and third-harmonic frequencies were 342 GHz and 1.02 THz, respectively, for a 50 μm long antenna. The maximum output power of the fundamental mode was around 23 μW, and that of the third-harmonic component was 2.6% of the fundamental. Theoretical analysis with the van der Pole equation qualitatively explained the measured results. [ABSTRACT FROM AUTHOR]

The connection between self-sustained current oscillations in superlattices and the famous van der Pol (vdP) equation is established by mapping a widely used model of self-sustained current oscillations to the vdP equation. Since this equation can describe an inductor-capacitor-resistor circuit with a nonlinear resistor of negative differential conductance, we have obtained an equivalent electric circuit for superlattices in the self-sustained current oscillation regime. The origin of the equivalent inductance of a superlattice is revealed. [ABSTRACT FROM AUTHOR]

Taking coupled non-smooth Bohoffer-Van der Pol (BVP) system with periodic excitation as an example, the difference of magnitude between periodic excitation frequency and natural frequency of system was investigated. The difference of magnitude was the coupling effect of two time scales in frequency domain. Because the excitation frequency was much lower than the natural frequency of system, the whole excitation term was considered as a slow-varying parameter. The generalized equilibrium points and stabilities of system were obtained when the slow-varying parameter varied. The system had periodic bursting behavior when the value of parameters were appropriate. By combining the fast-slow analysis method and transformed phase portraits, the complex dynamical behavior as well as related mechanisms were analyzed when the slow-varying parameter crossed different bifurcation points and non-smooth boundary. [ABSTRACT FROM AUTHOR]

The dynamical properties of fractional-order Duffing–van der Pol oscillator are studied, and the amplitude–frequency response equation of primary resonance is obtained by the harmonic balance method. The stability condition for steady-state solution is obtained based on Lyapunov theory. The comparison of the approximate analytical results with the numerical results is fulfilled, and the approximations obtained are in good agreement with the numerical solutions. The bifurcations of primary resonance for system parameters are analyzed. The results show that the harmonic balance method is effective and convenient for solving this problem, and it provides a reference for the dynamical analysis of similar nonlinear systems. [ABSTRACT FROM AUTHOR]

We analyse the properties of the synchronisation transition in a many-body system consisting of quantum van der Pol oscillators with all-to-all coupling using a self-consistent mean-field method. We find that the synchronised state, which the system can access for oscillator couplings above a critical value, is characterised not just by a lower phase uncertainty than the corresponding unsynchronised state, but also a higher number uncertainty. Just below the critical coupling the system can evolve to the unsynchronised steady state via a long-lived transient synchronised state. We investigate the way in which this transient state eventually decays and show that the critical scaling of its lifetime is consistent with a simple classical model. [ABSTRACT FROM AUTHOR]

In this paper, a Van der Pol-Duffing (VdPD) jerk oscillator is designed. The proposed VdPD jerk oscillator is built by converting the autonomous two-dimensional VdPD oscillator to a jerk oscillator. Dynamical behaviours of the proposed VdPD jerk oscillator are investigated analytically, numerically and analogically. The numerical results indicate that the proposed VdPD jerk oscillator displays chaotic oscillations, symmetrical bifurcations and coexisting attractors. The physical existence of the chaotic behaviour found in the proposed VdPD jerk oscillator is verified by using Orcad-PSpice software. A good qualitative agreement is shown between the numerical simulations and the PSpice results. Moreover, the fractional-order form of the proposed VdPD jerk oscillator is studied using stability theorem of fractional-order systems and numerical simulations. It is found that chaos, periodic oscillations and coexistence of attractors exist in the fractional-order form of the proposed jerk oscillator with order less than three. The effect of fractional-order derivative on controlling chaos is illustrated. It is shown that chaos control is achieved in fractional-order form of the proposed VdPD jerk oscillator only for the values of linear controller used. Finally, the problem of drive-response synchronisation of the fractional-order form of the chaotic proposed VdPD jerk oscillators is considered using active control technique. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the stochastic response of the van der Pol (VDP) oscillator with two kinds of fractional derivatives under Gaussian white noise excitation. First, the fractional VDP oscillator is replaced by an equivalent VDP oscillator without fractional derivative terms by using the generalized harmonic balance technique. Then, the stochastic averaging method is applied to the equivalent VDP oscillator to obtain the analytical solution. Finally, the analytical solutions are validated by numerical results from the Monte Carlo simulation of the original fractional VDP oscillator. The numerical results not only demonstrate the accuracy of the proposed approach but also show that the fractional order, the fractional coefficient and the intensity of Gaussian white noise play important roles in the responses of the fractional VDP oscillator. An interesting phenomenon we found is that the effects of the fractional order of two kinds of fractional derivative items on the fractional stochastic systems are totally contrary. [ABSTRACT FROM AUTHOR]

We study specific changes in repetitive firing in the two-dimensional Hindmarsh—Rose (2dHR) oscillatory system that undergoes a bifurcation transition from the supercritical Andronov—Hopf (AH) type to the subcritical Andronov—Hopf (SAH) type. We identify dynamical mechanisms which are responsible for changes of the repetitive firing rate during the AH to SAH bifurcation transitions. These include frequency-shift functions in response to small perturbations of a timescale parameter, its multiplicative parameter, and an external input current in the 2dHR oscillatory system. The frequency-shift functions are explicitly represented as functions relating to the phase response curves (PRCs). Then, we demonstrate that when the timescale is normal and relatively fast, the repetitive firing rate slightly increases and decreases respectively during the AH to SAH bifurcation transition with a change of the intrinsic parameter, whereas it decreases during the SAH to AH bifurcation transition with an increase in the timescale. By analyzing the three different frequency-shift functions, we show that such changes of the repetitive firing rate depend largely on changes of the PRC size. The PRC size for the SAH bifurcation shrinks to the PRC size for the AH bifurcation. [ABSTRACT FROM AUTHOR]

We present an explicit solution based on the phase-amplitude approximation of the Fokker-Planck equation associated with the Langevin equation of the birhythmic modified van der Pol system. The solution enables us to derive probability distributions analytically as well as the activation energies associated with switching between the coexisting different attractors that characterize the birhythmic system. Comparing analytical and numerical results we find good agreement when the frequencies of both attractors are equal, while the predictions of the analytic estimates deteriorate when the two frequencies depart. Under the effect of noise, the two states that characterize the birhythmic system can merge, inasmuch as the parameter plane of the birhythmic solutions is found to shrink when the noise intensity increases. The solution of the Fokker-Planck equation shows that in the birhythmic region, the two attractors are characterized by very different probabilities of finding the system in such a state. The probability becomes comparable only for a narrow range of the control parameters, thus the two limit cycles have properties in close analogy with the thermodynamic phases. [ABSTRACT FROM AUTHOR]

This paper contains a numerical study of the periodically forced van der Pol system. Our aim is to determine the extent to which chaotic behavior occurs in this system as well as the nature of the chaos. Unlike previous studies, which used continuous forcing, we work with instantaneous kicks, for which the geometry is simpler. Our study covers a range of parameters describing nonlinearity, kick sizes, and kick periods. We show that horseshoes are abundant whenever the limit cycle is kicked to a specific region of the phase space and offer a geometric explanation for the stretch-and-fold behavior which ensues. [ABSTRACT FROM AUTHOR]

The concept of isochrons is crucial for the analysis of asymptotically periodic systems. Roughly, isochrons are sets of points that partition the basin of attraction of a limit cycle according to the asymptotic behavior of the trajectories. The computation of global isochrons (in the whole basin of attraction) is however difficult, and the existing methods are inefficient in high-dimensional spaces. In this context, we present a novel (forward integration) algorithm for computing the global isochrons of high-dimensional dynamics, which is based on the notion of Fourier time averages evaluated along the trajectories. Such Fourier averages in fact produce eigenfunctions of the Koopman semigroup associated with the system, and isochrons are obtained as level sets of those eigenfunctions. The method is supported by theoretical results and validated by several examples of increasing complexity, including the 4-dimensional Hodgkin-Huxley model. In addition, the framework is naturally extended to the study of quasiperiodic systems and motivates the definition of generalized isochrons of the torus. This situation is illustrated in the case of two coupled Van der Pol oscillators. [ABSTRACT FROM AUTHOR]

Propagating waves in a ring of unidirectionally coupled symmetric Bonhoeffer-van der Pol (BVP) oscillators were studied. The parameter values of the BVP oscillators were near a codimension-two bifurcation point around which oscillatory, monostable, and bistable states coexist. Bifurcations of periodic, quasiperiodic, and chaotic rotating waves were found in a ring of three oscillators. In rings of large numbers of oscillators with small coupling strength, transient chaotic waves were found and their duration increased exponentially with the number of oscillators. These exponential chaotic transients could be described by a coupled map model derived from the Poincaré map of a ring of three oscillators. The quasiperiodic rotating waves due to the mode interaction near the codimension-two bifurcation point were evidently responsible for the emergence of the transient chaotic rotating waves. [ABSTRACT FROM AUTHOR]

Relaxation oscillations are commonly associated with the name of Balthazar van der Pol via his paper (Philosophical Magazine, 1926) in which he apparently introduced this terminology to describe the nonlinear oscillations produced by self-sustained oscillating systems such as a triode circuit. Our aim is to investigate how relaxation oscillations were actually discovered. Browsing the literature from the late 19th century, we identified four self-oscillating systems in which relaxation oscillations have been observed: (i) the series dynamo machine conducted by Gérard-Lescuyer (1880), (ii) the musical arc discovered by Duddell (1901) and investigated by Blondel (1905), (iii) the triode invented by de Forest (1907), and (iv) the multivibrator elaborated by Abraham and Bloch (1917). The differential equation describing such a self-oscillating system was proposed by Poincaré for the musical arc (1908), by Janet for the series dynamo machine (1919), and by Blondel for the triode (1919). Once Janet (1919) established that these three self-oscillating systems can be described by the same equation, van der Pol proposed (1926) a generic dimensionless equation which captures the relevant dynamical properties shared by these systems. Van der Pol's contributions during the period of 1926-1930 were investigated to show how, with Le Corbeiller's help, he popularized the 'relaxation oscillations' using the previous experiments as examples and, turned them into a concept. [ABSTRACT FROM AUTHOR]

The aim of this work is to establish that the bifurcation parameter value leading to a canard explosion in dimension 2 obtained by the so-called geometric singular perturbation method can be found according to the flow curvature method. This result will be then exemplified with the classical Van der Pol oscillator. [ABSTRACT FROM AUTHOR]

We investigate the continuous time domain numerical treatment of a van der Pol oscillator, applying the trial solution as an artificial feed-forward neural network model containing unknown adjustable parameters. The optimization of the network is performed by simulated annealing in an unsupervised method. The proposed scheme is tested successfully by its application in both non-stiff and stiff conditions. Its reliability and effectiveness is validated through comprehensive statistical analyses. The obtained results are in good agreement with the classical RK45 method. [ABSTRACT FROM AUTHOR]

Discusses the use of the van der Pol (VDP) differential equation driven by a periodic force to model macroeconomic dynamics. Use of the models of a forced oscillator and coupled oscillators to study complex nonlinear phenomena in economic and physical systems; Understanding of stable, unstable and saddle paths in nonlinear business cycles.