Showing papers on "Van der Pol oscillator published in 1990"

Comparison of Different Methods for Computing Lyapunov Exponents

Abstract: Different discrete and continuous methods for computing the Lyapunov exponents of dynamical systems are compared for their efficiency and accuracy. All methods are either based on the QR or the singular value decomposition. The relationship between the discrete methods is discussed in terms of the iteration algorithms and the decomposition procedures used. We give simple deriva­ tions of the differential equations for continuous methods proposed recently and show that they cannot be recommended because of their long computation time and numerical instabilities. The methods are tested with the damped and driven Toda chain and the driven van der Pol osciIlator.

Analysis of white and f−α noise in oscillators

Abstract: In this paper the general correlation spectrum of an oscillator with white and f -α noise sources is derived from the Langevin equations describing the deterministic and stochastic behaviour of an oscillator by perturbation methods. The treatment of f -α noise and the influence of the finite measuring time on the noise spectra are included in a time domain calculation. The theory is applied to a Van der Pol oscillator exhibiting parametric sideband amplification and to a realized oscillator demonstrating the applicability of the theory to technically relevant circuits

Mapping non-linear integro-differential equations into the frequency domain

Abstract: A recursive algorithm is derived to compute the generalized frequency response functions for a large class of non-linear integro-differential equations. Applications to Duffing's equation and a modified Van der Pol model are discussed.

A CMOS Implementation of Fitzhugh-Nagumo Neuron Model

Abstract: A CMOS circuit is proposed that emulates FitzHugh-Nagumo's differential equations using OTAs, diode connected MOSFETs and capacitors. These equations model the fundamental behavior of biological neuron cells. Fitz-Hugh-Nagumo's model is characterized by two threshold values. If the input to the neuron is between the two thresholds the output yields a sequence of firing pulses, if the input is outside this range, no output is observed. The resulting circuit due to the (voltage) programmability of the OTA allows one to easily vary parameters. Thus a large family of solutions can be obtained including the Van der Pol's equation. Experimental results from a CMOS prototype are given that show the suitability of the technique used, and their potential for biological CMOS system emulation.

Ito, Stratonovich and kinetic forms of stochastic equations

Abstract: The well known Ito and Stratonovich forms of presentation of stochastic equations are not, in general, physically equivalent. From the point of view of the statistical theory of nonequilibrium processes the third is most natural-the “kinetic form” of presentation of the Langevin and corresponding Fokker-Planck equations. Only in this case exist fluctuation- dissipation relations (the Einstein formula) for nonlinear systems. For the confirmation of this point of view the following different concrete systems are considered: Brownian motion of particles in a medium with nonlinear friction, of the Van der Pol oscillators and others. The connection between the master equation and the Fokker-Planck one is also considered.

Three identical oscillators with symmetric coupling

Abstract: This paper is theoretical and experimental investigation of three identical oscillators with weak symmetric coupling. The authors find that the dynamics is governed by an ODE on the 2-torus with S3 symmetry. A new type of codimension-one global bifurcation in such systems is predicted, and found experimentally in a network of three coupled van der Pol oscillators.

A period-adding phenomenon

Abstract: A simple geometrical explanation is given for the “period-adding” phenomenon observed in the 1927 experiment of van der Pol and van der Mark and refined recently by Chua and Kennedy. The problem is reduced to a simple circle map. In the second part of the paper some previous results are outlined to show how the same phenomenon arises in a class of van der Pol-type systems with periodic forcing, and in conclusion some recent topological results are applied to such systems.

Stability of limit cycles and bifurcations of generalized van der Pol oscillators

Abstract: The limit cycles of the van der Pol oscillator X + AX − 2BX 3 +e(z 3 +z 2 X 2 +z 1 X 4 ) X = 0 , for B > 0, are studied in first-order approximation, using the Jacobian elliptic functions with the method of harmonic balance. The transitory motion, and in consequence the limit cycles and their stability are also studied in an approximate quantitative way with a generalized method of the slowly varying amplitude and phase. The bifurcations of these non-linear oscillators are studied using the methods of differentiable dynamics to obtain the qualitative behaviour. Quantitative values for the radius, frequency and energy of the limit cycles are given. The presence and stability of zero, one, or two limit cycles depend on the parameters zi. The presence of bifurcations depends on zi and A.

Quantitative Evaluation of Limit-Cycle Oscillator Models of Spontaneous Otoacoustic Emissions

Abstract: Single limit-cycle-oscillator models of spontaneous otoacoustic emissions (SOAE’s) are based on the assumption that the pattern of interactions between spontaneous emissions and external tones in the ear canal may be partially described by the gross compaction of a full cochlear model to a single nonlinear differential equation such as that of a free (or driven) Van der Pol oscillator. Such an equation incorporates, in a highly idealized way, the type of nonlinear-active damping which, if assumed to be present over certain portions of the cochlear partition, would produce stabilized cochlear self-oscillations and lead to measurable spontaneous emissions in the ear-canal. These models have been used successfully by our group and by Wit and collaborators to account for a number of features of the emission data including: a) statistical properties of emissions (e.g. Bialek and Wit, 1984; Wit, 1986; van Dijk, 1990); b) suppression of emissions (Long and Tubis, 1990) and synchronization (phase locking) of emissions by external tones (e.g., van Dijk, 1990; Long, et al., 1990); and c) reduction of the level of emissions by aspirin consumption (e.g., Long and Tubis, 1988a,b).

Approximate analytical solutions for some strongly nonlinear problems

Abstract: In this paper, some problems of strongly nonlinear oscillations and waves are analyzed by using the modified method of full approximation developed by the authors. Firstly, a class of strongly nonlinear oscillation systems is studied and, in particular, for the modified van der Pol oscillator, the second-order expression of its limit cycle solution is quite straightforwardly found out, which agrees with the results obtained by way of the generalized method of averaging in Ref. [1]. Then for the modified KdV equation, the correct secondorder appoximate solution of solitary waves is given. Finally, for the generalized KdV equation with a fifth-order dispersion term, the third-order asymptotic solution of solitary waves is derived and the form of oscillatory solitary waves is analytically given. All the results show that the modified method of full approximation can be effectively applied to the investigation of some mathematical problems with strong nonlinearity.

Phase-stability analysis of the magnetron-driven vircator experiment

Abstract: A slow envelope approximate solution to the Van der Pol equation is evaluated as a model for a driven high-power oscillator experiment. It is shown that Adler's inequality gives a necessary but not sufficient condition for achieving phase-locking between the driving relativistic magnetron and the driven high-power cavity vircator oscillations. The amplitude of the entrained (phase-locked) oscillations is found as a function of the injected magnetron power, the initial frequency detuning, and other system parameters. The stability of these oscillations is examined. Not all entrained states are stable. Parameteric boundaries between stable and unstable states are given. Combination oscillations containing both magnetron and vircator frequency components are predicted and observed to occur when the initial detuning between the two sources is too large to allow entrainment. >

Chaotic Dynamics of Otoacoustic Emissions

Abstract: The discovery of otoacoustic emissions (OAE) has encourage? a re-asessment of active mechanisms in hearing. Nonlinear oscillators are attractive candidates for understanding active processes in hearing, and otoacoustic emissions in particular. Even the simplest nonlinear oscillators used in current theories shww hmlt cycle behavior but they are also capable of more complex, chaotic, behavior. For example, the forced van der Pol oscillator utilized in models of hearing (Duifhuis et al., 1986; Tubis et al., 1988) is also known to exhibit chaotic behavior. It is unknown whether OAEs actually exhibit chaotic behavior.

Chaos in Van der Pol's equation

Abstract: Van der Pol's Equation was first given in 1926. It gives limit cycles. The present paper reports the chaotic behavior of modified Van der Pol's Equation with forcing function. In three of six cases, CHAOS is found, while three other cases give limit cycles.

A Generalized Van Der Pol-Oscillator Cochlea Model

Abstract: Over the last decade it has become clear that active and nonlinear behavior of the cochlea is to a certain extent similar to that of a Van der Poloscillator. This was first proposed by Johannesma (1980). It was worked out in more detail by several groups (e.g., van Netten and Duifhuis, 1983, Duifhuis et al., 1985, Jones et al., 1986, Diependaal et al., 1987, van Dijk and Wit, 1988). We have been working on a cochlea model with many active components, i.e. self-sustained oscillators. Initially this was set up with spatial parameters that varied smoothly. The classical parabolic damping profile was used. Our major emphasis, however, shifted toward potentially more realistic biophysical models. For the damping term we now use a function in which two parts can be discerned. First, a passive (positive) part with exponential “tails”, which provides response behavior with a log-like characteristic for external stimuli. Secondly, there is an active (negative) part that, if sufficiently strong, produces net active (negative damping = energy production) behavior. In order to model spontaneous emissions, we also introduced spatial discontinuities in the damping parameters.

Justification of matching with the transition expansion of Van der Pol's equation

Abstract: The analysis of the relaxation oscillations of Van der Pol’s equation presents an especially challenging test of the formal techniques of the method of matched asymptotic expansions for solving singular perturbation problems. The formal analysis is described in Kevorkian and Cole’s monograph [J. Kevorkian and J. D. Cole, “Perturbation Methods in Applied Mathematics”, Springer-Verlag, Berlin, New York, 1981], which explains why the inner and outer expansions must necessarily be supplemented by a third “transition” expansion in order to obtain a uniformly valid approximation beyond $O(1)$ on a complete half-period. Kevorkian and Cole carry out the construction and delicate matching of several terms in the expansions. The present paper mathematically justifies their formal results to $O(\varepsilon ^{{1 / 3}} )$, and is the first such proof for any transition expansion. Partly for this reason, but also because the idea underlying the proof has been and will be applied to other singular perturbation problems,...

Non-stationary responses of self-excited driven system

Abstract: This is the first attempt to determine the effects of a broad range of linear non-stationary regimes related to the frequencies of external excitation v(t) ranging from evolving to robust, on dynamic responses of self-excited systems, governed by the Van der Pol differential equation. The results obtained, time histories and phase plane plots, are revealing. Because of continuous variations of v(t) , a representative point in the (v, K) plane, where K is the amplitude of excitation, continually crosses various regions and boundaries of periodic, aperiodic, stable, and unstable system motions, thus exhibiting a variety of new dynamic forms, which lead in some cases to a possible chaotic motion. The non-stationary responses are sensitive to the sweep rates α : the faster the sweep, the earlier are the appearances of new waveforms (patterns) and the shorter are the time intervals between the changed patterns. The responses are also sensitive to the variations of other system parameters: ϵ, which indicates the degree of non-linearity; K , amplitude of external excitation; σ, detuning, and σ 0 , initial ( t=0 ) detuning; and p 0 , initial amplitude of the response. The Rayleigh-Van der Pol oscillator is the basis of a model for a number of physical, biological, chemical, and engineering phenomena. This paper is an initial contribution to further theoretical and applied studies in non-stationary processes.

Models of driven relativistic magnetrons with nonlinear frequency-shift and growth-saturation effects

Abstract: The driven van der Pol equation is widely used to model the behavior of regenerative electronic and microwave oscillators driven by an external locking signal. In the case of high-power microwave oscillators such as relativistic magnetrons nonlinear frequency-shift effects are believed to be important. They have been modeled by inclusion in the van der Pol equation of an additional cubic restoring force (Duffing) term. Use of the slowlyvarying amplitude and phase approximation to study the behavior of the driven van der Pol-Duffing equation has predicted stable single-valued locked behavior within a skewed range of values of the frequency mismatch. For parameter values consistent with the slowly-varying amplitude and phase approximation numerical solutions of the van der Pol-Duffing equation confirm this prediction. For oscillators with high growth rate such as the relativistic magnetron however the slowly-varying amplitude and phase approximation may be unjustified. Furthermore regardless of the satisfaction of the assumptions on which the approximation is based it may fail to predict the occurrence of complicated dynamical behavior of coupled oscillators with important implications for phase locking. Numerical solutions of the driven van der Pol equation show that the time of transient evolution to phase-locked states instead of depending solely on the frequency mismatch as conventionally assumed is also a function of oscillator growth rate and injection power level. Recent work suggests that the form of model oscillator equation appropriate for the magnetron may differ in both nonlinear growth-saturation and frequency terms from the van der Pol-Duffing equation. 1.© (1990) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Application of Chaos Methods to Helicopter Vibration Reduction Using Higher Harmonic Control

Abstract: : Chaos is used to understand complex nonlinear dynamics. The geometric and topological methods of Chaos theory are applied, for the first time, to the study of flight test data. Data analyzed is from the OH-6A Higher Harmonic Control (HHC) test aircraft. HHC is an active control system used to suppress helicopter vibrations. Some of the first practical applications of Chaos methods are demonstrated with the HHC data. although helicopter vibrations are mostly periodic, evidence of chaos was found. The presence of a strange attractor was shown by computing a positive Lyapunov exponent and computing a non integer fractal correlation dimension. A broad band Fourier spectrum and a well defined attractor in pseudo phase space are observed. A limit exists to HHC vibration reduction due to the presence of chaos. A new technique based on a relationship between the Chaos methods (the Poincare section and Van der Pol plane) and the vibration amplitude and phase was discovered. This newly introduced technique results in the following: (1) it gives the limits of HHC vibration reduction, (2) it allows rapid determination of best phase for a HHC controller, (3) it determines the minimum HHC controller requirement for any helicopter from a few minutes duration of flight test data, (4) it shows that the HHC controller transfer matrix is linear and repeatable when the vibrations are defined in the Rotor Time Domain and that the matrix is nonlinear and nonrepeatable when the vibrations are defined in the Clock Time Domain. (jhd)

Resonances and power series solutions of the forced Van Der Pol oscillator

Abstract: To the first approximation, the nonresonant steady-state motion of the forced van der Pol oscillator is a combination of a free and a forced response. When the forcing amplitude increases beyond a certain critical value, which depends on the forcing frequency, the free response will decay and the motion is represented by just a particular solution that has the same frequency as the forcing term. In the first part of this paper, a general expression is determined for all of the higher-order resonant frequencies associated with a formal multiple timescale perturbation expansion of the solution. These frequencies are shown to be dense on the real axis. Then a formal power series expansion is developed for a particular solution in terms of the small damping parameter $\varepsilon $, which is shown to be valid when the forcing frequency is not close to a certain subset of the resonant frequencies. Using Pade approximants, information is obtained about the location and nature of the singularities in the complex...

Nonlinear two-mode coupling models of radial stellar oscillations

Abstract: Les oscillateurs de van der Pol couples sont analyses numeriquement avec des coefficients de couplage determines a partir d'un modele realiste de Cepheide. Il est montre que des oscillations synchronisees et que le developpement d'une bifurcation a doublement de periode chevauchant leur battement peuvent etre realises. Le verrouillage de phase entre les differents modes est propose pour les Cepheides a mode double.

Inhibition of chaos in bistable Hamiltonian systems by a critical external resonance.

Abstract: We have shown that, depending on some critical values of the phase and amplitude, an external periodic resonant driving field can inhibit classical deterministic stochasticity in a bistable Hamiltonian system. Our analysis is based on a coupled oscillator model describing a harmonic oscillator quadratically coupled to a nonlinear oscillator with a double-well potential.

a Perturbation Method and Some Applications

Abstract: For differential equations with one fast variable, a perturbation method is introduced that transforms a solution valid over only a short time interval to a new solution composed of averaged variables plus a periodic function of the averaged variables The averaged variables are governed by a set of differential equations where the fast variable has been removed and thus can be numerically integrated quickly or solved directly This method is applied to a perturbed harmonic oscillator with a cubic perturbation, van der Pol's equation, coorbital motion in the restricted three-body problem, and to nearly circular motion of a particle near one of the primaries in the restricted three-body problem

Identification Techniques for Nonlinear Differential Equations of Motion

Abstract: : This report develops and systematically demonstrates by computer simulations new nonlinear system stochastic techniques to determine the amplitude-domain and frequency-domain properties of nonlinear systems as described in proposed nonlinear differential equations of motion. From measurements of input excitation data and output response data, it is shown that this new method, based upon multiple-input/single-output (MI/SO) linear analysis of reverse dynamic systems, allows for the efficient identification of different nonlinear systems. Nonlinear systems simulated here include Duffing, Van der Pol, Mathieu, and Dead-Band systems. Keywords: Degrees of freedom, Civil engineering.

Chaos via torus breakdown in the forced van der Pol oscillator

Abstract: Chaos in the forced van der Pol oscillator including a pair of diodes is investigated. First, the authors show that the normal forced van der Pol equation exhibits chaos. The generation of chaos is explained by a Lorenz plot. Secondly, they analyze chaos rigorously by using a piecewise-linear technique and a degeneration technique which corresponds to an idealized case where the diode is assumed to operate as an ideal switch. In this case, the Poincare map is derived strictly as one-dimensional mapping which is a noninvertible return mapping of a circle. This mapping clarifies the mechanism of chaos. >