Showing papers on "Van der Pol oscillator published in 1995"

Understanding Nonlinear Dynamics

Abstract: 1 Finite-Difference Equations.- 1.1 A Mythical Field.- 1.2 The Linear Finite-Difference Equation.- 1.3 Methods of Iteration.- 1.4 Nonlinear Finite-Difference Equations.- 1.5 Steady States and Their Stability.- 1.6 Cycles and Their Stability.- 1.7 Chaos.- 1.8 Quasiperiodicity.- 1 Chaos in Periodically Stimulated Heart Cells.- Sources and Notes.- Exercises.- Computer Projects.- 2 Boolean Networks and Cellular Automata.- 2.1 Elements and Networks.- 2.2 Boolean Variables, Functions, and Networks.- 2 A Lambda Bacteriophage Model.- 3 Locomotion in Salamanders.- 2.3 Boolean Functions and Biochemistry.- 2.4 Random Boolean Networks.- 2.5 Cellular Automata.- 4 Spiral Waves in Chemistry and Biology.- 2.6 Advanced Topic: Evolution and Computation.- Sources and Notes.- Exercises.- Computer Projects.- 3 Self-Similarity and Fractal Geometry.- 3.1 Describing a Tree.- 3.2 Fractals.- 3.3 Dimension.- 5 The Box-Counting Dimension.- 3.4 Statistical Self-Similarity.- 6 Self-Similarity in Time.- 3.5 Fractals and Dynamics.- 7 Random Walks and Levy Walks.- 8 Fractal Growth.- Sources and Notes.- Exercises.- Computer Projects.- 4 One-Dimensional Differential Equations.- 4.1 Basic Definitions.- 4.2 Growth and Decay.- 9 Traffic on the Internet.- 10 Open Time Histograms in Patch Clamp Experiments.- 11 Gompertz Growth of Tumors.- 4.3 Multiple Fixed Points.- 4.4 Geometrical Analysis of One-Dimensional Nonlinear Ordinary Differential Equations.- 4.5 Algebraic Analysis of Fixed Points.- 4.6 Differential Equations versus Finite-Difference Equations.- 4.7 Differential Equations with Inputs.- 12 Heart Rate Response to Sinusoid Inputs.- 4.8 Advanced Topic: Time Delays and Chaos.- 13 Nicholson's Blowflies.- Sources and Notes.- Exercises.- Computer Projects.- 5 Two-Dimensional Differential Equations.- 5.1 The Harmonic Oscillator.- 5.2 Solutions, Trajectories, and Flows.- 5.3 The Two-Dimensional Linear Ordinary Differential Equation.- 5.4 Coupled First-Order Linear Equations.- 14 Metastasis of Malignant Tumors.- 5.5 The Phase Plane.- 5.6 Local Stability Analysis of Two-Dimensional, Nonlinear Differential Equations.- 5.7 Limit Cycles and the van der Pol Oscillator.- 5.8 Finding Solutions to Nonlinear Differential Equations.- 15 Action Potentials in Nerve Cells.- 5.9 Advanced Topic: Dynamics in Three or More Dimensions.- 5.10 Advanced Topic: Poincare Index Theorem.- Sources and Notes.- Exercises.- Computer Projects.- 6 Time-Series Analysis.- 6.1 Starting with Data.- 6.2 Dynamics, Measurements, and Noise.- 16 Fluctuations in Marine Populations.- 6.3 The Mean and Standard Deviation.- 6.4 Linear Correlations.- 6.5 Power Spectrum Analysis.- 17 Daily Oscillations in Zooplankton.- 6.6 Nonlinear Dynamics and Data Analysis.- 18 Reconstructing Nerve Cell Dynamics.- 6.7 Characterizing Chaos.- 19 Predicting the Next Ice Age.- 6.8 Detecting Chaos and Nonlinearity.- 6.9 Algorithms and Answers.- Sources and Notes.- Exercises.- Computer Projects.- Appendix A A Multi-Functional Appendix.- A.1 The Straight Line.- A.2 The Quadratic Function.- A.3 The Cubic and Higher-Order Polynomials.- A.4 The Exponential Function.- A.5 Sigmoidal Functions.- A.6 The Sine and Cosine Functions.- A.7 The Gaussian (or "Normal") Distribution.- A.8 The Ellipse.- A.9 The Hyperbola.- Exercises.- Appendix B A Note on Computer Notation.- Solutions to Selected Exercises.

Wavelet bicoherence: A new turbulence analysis tool

Abstract: A recently introduced tool for the analysis of turbulence, wavelet bicoherence [van Milligen, Hidalgo, and Sanchez, Phys. Rev. Lett. 16, 395 (1995)], is investigated. It is capable of detecting phase coupling—nonlinear interactions of the lowest (quadratic) order—with time resolution. To demonstrate its potential, it is applied to numerical models of chaos and turbulence and to real measurements. It detected the coupling interaction between two coupled van der Pol oscillators. When applied to a model of drift wave turbulence relevant to plasma physics, it detected a highly localized coherent structure. Analyzing reflectometry measurements made in fusion plasmas, it detected temporal intermittency and a strong increase in nonlinear phase coupling coinciding with the L/H (low‐to‐high confinement mode) transition.

Frequency Methods in Oscillation Theory

Abstract: Preface. 1. Classical Two-Dimensional Oscillating Systems and Their Multidimensional Analogues. 2. Frequency Criteria for Stability and Properties of Solutions of Special Matrix Inequalities. 3. Multidimensional Analogues of the Van der Pol Equation. 4. Yakubovich Auto-Oscillation. 5. Cycles in Systems with Cylindrical Phase Space. 6. The Barbashin-Ezeilo Problem. 7. Oscillations in Systems Satisfying Generalized Routh-Hurwitz Conditions. Aizerman Conjecture. 8. Frequency Estimates of the Hausdorff Dimension of Attractors and Orbital Stability of Cycles. Bibliography. Subject Index.

Computation of invariant tori by the method of characteristics

Abstract: In this paper we present a technique for the numerical approximation of a branch of invariant tori of finite-dimensional ordinary differential equations systems. Our approach is a discrete version of the graph transform technique used in analytical work by Fenichel [Indiana Univ. Math. J., 21 (1971), pp. 193–226]. In contrast to our previous work [L. Dieci, J. Lorenz, and R. D. Russell, SIAM J. Sci. Statist. Comput., 12 (1991), pp. 607–647], the method presented here does not require a priori knowledge of a suitable coordinate system for the branch of invariant tori, but determines and updates such a coordinate system during a continuation process. We give general convergence results for the method and present its algorithmic description. We also show how the method performs on two physically important nonlinear problems, a system of two coupled oscillators and the forced van der Pol oscillator. In the latter case, we discuss some modifications needed to approximate an invariant curve for the Poincare map.

Dynamics of two coupled van der Pol oscillators.

Abstract: A system of two coupled van der Pol oscillators showing multistable behavior for some control parameter ranges is studied. When several attractors coexist a rich fractal structure is found both on the border between basins and in extended zones of the phase space. In such zones strong mixing and self-similar structure of basins are manifest. A relationship is observed between the appearance of symmetric attractors and the fractal properties of the attraction basins. First return maps, Poincar\'e sections, and probability distribution functions have been computed for the model equations, indicating that the complex dynamics found in the system can be understood in terms of more simple discrete transformations related to the logistic map. A combined master-slave system based on the coupled oscillators studied is found to enter a chaotic synchronization regime for some values of the control parameters. The practical implications of the observed phenomena are discussed.

Controlling of chaotic motion by chaos and noise signals in a logistic map and a Bonhoeffer-van der Pol oscillator.

Abstract: The possibility of the conversion of a chaotic attractor to a strange but nonchaotic attractor is investigated numerically in both a discrete system, the logistic map, and in a continuous dynamical system, the Bonhoeffer--van der Pol oscillator. A suppression of the chaotic property, namely, the sensitive dependence on initial states, is found when an appropriate (i) chaotic signal and (ii) Gaussian white noise are added. A strange but nonchaotic attractor is shown to occur for some ranges of amplitude of the external perturbation. The controlled orbit is characterized by the Lyapunov exponent, correlation dimension, power spectrum, and return map.

Symplectic calculation of Lyapunov exponents.

Abstract: The Lyapunov exponents of a chaotic system quantify the exponential divergence of initially nearby trajectories. For Hamiltonian systems the exponents are related to the eigenvalues of a symplectic matrix. We make use of this fact to develop a new method for the calculation of Lyapunov exponents of such systems. Our approach avoids the renormalization and reorthogonalization of usual techniques. It is also easily extendible to damped systems. We apply our method to two examples of physical interest: a model system that describes the beam halo in charged particle beams and the driven van der Pol oscillator.

Chaos engineering in Japan

Abstract: Since deterministic chaos is not only a profound concept in science but also a ubiquitous phenomenon in real-world nonlinear systems, extending to a variety of temporal and spatial scales, it can be naturally related to applications in science and technology [4]. In fact, it is not difficult to find the buds of such possible applications in historical papers by Van der Pol and Van der Mark [22], Ulam and von Neumann [21], and Kalman [12], although the term deterministic chaos was not used in those days.

Stochastic phase lockings in a relaxation oscillator forced by a periodic input with additive noise: A first-passage-time approach

Abstract: Noise effects on phase lockings in a system consisting of a piecewise-linear van der Pol relaxation oscillator driven by a periodic input are studied. The problem of finding the period of the oscillator is reduced to the first-passage-time problem of the Ornstein-Uhlenbeck process with time-varying boundary. The probability density functions of the first-passage time are used to define the operator which governs a transition of an input phase density after one cycle of the oscillator. Phase lockings in a stochastic sense are investigated on the basis of the density evolution by the operator.

Stability of mode locked states of coupled oscillator arrays

Abstract: A method for analyzing beat frequency entrained (or mode locked) systems of coupled nonlinear oscillators is presented. Stable mode locked states are almost periodic oscillations that occur outside of the fundamental entrainment region of the array, and generate a periodic pulse train when the oscillator outputs are summed. The analysis relates the stability of the states to the relationship between the frequency pullings and the time average phases of the oscillators. The method is applied to three mode locked Van der Pol oscillators with arbitrary coupling time delay, and shows that the mode locking bandwidth is maximized for specific values of coupling delay and oscillator nonlinearity parameter. >

Numerical shadowing near hyperbolic trajectories

Abstract: Shadowing is a means of characterizing global errors in the numerical solution of initial value ordinary differential equations by allowing for a small perturbation in the initial condition. The method presented in this paper allows for a perturbation in the initial condition and a reparameterization of time in order to compute the shadowing distance in the neighborhood of a periodic orbit or more generally in the neighborhood of an attractor. The method is formulated for one-step methods and both a serial and parallel implementation are applied to the forced van der Pol equation, the Lorenz equation and to the approximation of a periodic orbit.

Stochastic transient of a noisy van der pol oscillator

Abstract: Transient characteristics of a van der Pol oscillator perturbed by additive and multiplicative noises of different non-linearities are studied and compared systematically. We numerically determine the noise dependence of the period and of the life-time of a noisy oscillator. Attraction basins and stability are also investigated for the trivial fixed point and the limit cycle.

Inducing periodic oscillations from chaotic oscillations of a compound-cavity laser diode with sinusoidally modulated drive.

Abstract: Inducing periodicity from chaotic oscillations of a compound-cavity laser diode is numerically demonstrated through the Van der Pol model. The deep modulation beyond the limit of the perturbation approximation leads to both turbulence and periodicity. Bifurcation against the modulation frequencies and modulation depth occurs under proper conditions. Therefore it is found that the different periodicity can be induced from chaos by a small change in the frequency and the amplitude of the periodic modulations.

Bifurcations from periodic solution in a simplified model of two‐dimensional magnetoconvection

Abstract: A two‐dimensional Boussinesq fluid with nonlinear interaction between Rayleigh–Benard convection and an external magnetic field is investigated numerically and analytically. A simplified model consisting of a fifth‐order system of nonlinear ordinary differential equations with five parameters is introduced and integrated numerically in certain parameter regions. Various types of bifurcations from periodic solutions are found numerically: period‐doubling bifurcation, heteroclinic bifurcation, intermittency, and saddle‐node bifurcation. A normal form equation is also derived from the fifth‐order system, and center manifold theory is applied to it. An expression for the renormalized Holmes–Melnikov boundary for the evaluation of the numerical results is given. It is shown from the normal form equation that each property of the two phase portraits described by the Duffing equation and the van der Pol equation emanates from one common attractor in the five‐dimensional space of the fifth‐order system.

Relaxation dynamics of spontaneous otoacoustic emissions perturbed by external tones. III. Response to a single tone at multiple suppression levels

Abstract: The relaxation dynamics of spontaneous otoacoustic emissions (SOAEs) interacting with an external tone have been successfully described using a van der Pol limit cycle oscillator model [Murphy et al., J. Acoust. Soc. Am. 97, 3702–3710 (1995) and Murphy et al., J. Acoust. Soc. Am. 97, 3711–3720 (1995)]. Data were collected for an SOAE interacting with a single‐frequency ipsilateral suppressor. Transitions between different suppressed states were achieved by adding or removing signal at the suppressor frequency. The relaxation dynamics of the van der Pol model provided a good fit to the data.

Secure communications via chaotic synchronization in Chua's circuit and Bonhoeffer-Van der Pol equation: numerical analysis of the errors of the recovered signal

Abstract: The signal recovered from the first reported experimental secure communication system via chaotic synchronization contains some inevitable noise which degrades the fidelity of the original message. By cascading the output of the receiver in the original system into an identical copy of this receiver, it had be shown by computer experiments that this noise can be significantly reduced. We discuss the heuristic laws governing the errors of the recovered signal which are observed in a new series of very careful computer experiments. This discussion is done comparing these laws to the results obtained using the linear filtering theory. Some discrepancy appears. In order to understand the origin of the discrepancy we consider another simpler model based on the Bonhoeffer-Van der Pol equation where no chaos occurs. In this case both two heuristic laws are in good agreement with the linear filtering theory.

Some simple bifurcation sets of an extended Van der Pol model and their relation to chemical oscillators

Abstract: Some typical bifurcation sets of a generalized autonomous Van der Pol‐type model are discussed as archetypes of phase diagrams occurring in nonlinear dynamical systems. The relevance of the obtained bifurcation sets is exemplified by several experimental and numerical results from the literature of oscillating chemical reactions.

Constructing Galerkin's Approximations of Invariant Tori Using MACSYMA*

Abstract: Invariant tori of solutions for nonlinearly coupled oscillators are generalizations of limit cycles in the phase plane. They are surfaces of aperiodic solutions of the coupled oscillators with the property that once a solution is on the surface it remains on the surface. Invariant tori satisfy a defining system of nonlinear partial differential equations. This case study shows that with the help of a symbolic manipulation package, such as MACSYMA, approximations to the invariant tori can be developed by using Galerkin's variational method. The resulting series must be manipulated efficiently, however, by using the Poisson series representation for multiply periodic functions, which makes maximum use of the list processing techniques of MACSYMA. Three cases are studied for the single van der Pol oscillator with forcing parameter e=0.5, 1.0, 1.5, and three cases are studied for a pair of nonlinearly coupled van der Pol oscillators with forcing parameters e=0.005, 0.5, 1.0. The approximate tori exhibit good agreement with direct numerical integrations of the systems.

Closed-loop suppression of chaos in nonlinear driven oscillators

Abstract: This paper discusses the suppression of chaos in nonlinear driven oscillators via the addition of a periodic perturbation. Given a system originally undergoing chaotic motions, it is desired that such a system be driven to some periodic orbit. This can be achieved by the addition of a weak periodic signal to the oscillator input. This is usually accomplished in open loop, but this procedure presents some difficulties which are discussed in the paper. To ensure that this is attained despite uncertainties and possible disturbances on the system, a procedure is suggested to perform control in closed loop. In addition, it is illustrated how a model, estimated from input/output data, can be used in the design. Numerical examples which use the Duffing-Ueda and modified van der Pol oscillators are included to illustrate some of the properties of the new approach.

Controlling unstable periodic orbits in a Bonhoeffer-van der Pol equation

Abstract: In this paper, controlling of chaotic dynamics by stabilizing desired unstable periodic orbits embedded in a chaotic attractor of a Bonhoeffer-van der Pol (BVP) equation is studied using recently proposed three control algorithms [1–4]. A comparative study of these methods is made. Variation of correction signal with time, dependence of recovery time on the stiffness ϵ of the control signal and stability region show different characteristic features in the three control schemes.

Heart muscle contraction oscillation

Abstract: van der Pol developed a mathematical model for self-sustained radio oscillations described by his non-linear differential equation D2X + epsilon(X2-1)DX + X = 0 in which X is a function of time T and D/DT the differential operator to T. For epsilon = 0, this is the differential equation for the harmonic oscillator which has sinusoidal solutions. For epsilon not equal to 0 the equation is non-linear. If epsilon > 1 van der Pol coined the name relaxation oscillations for its solutions. These are non-linear and quite different from simple sinusoidal oscillations. They are mathematical models for many physical and biological phenomena. van der Pol suggested that his equation is also a model for the heartbeat. However, biomedical oscillations, including the heartbeat, have a threshold which the mathematical model described by van der Pol's equation does not possess. It has, in addition to an unstable origin, only a stable limit cycle of Poincare. In this paper, van der Pol's equation is extended in such a way that it has in addition to a stable origin and a stable limit cycle, an unstable limit cycle. Because it possesses such an unstable limit cycle, the extension obtained is a mathematical model for a threshold oscillation. It is also shown that an asymmetric analogy of the extended equation is a mathematical model for an isometric contraction of the mammalian cardiac muscle.

Global bifurcations and chaos in a Van der Pol-Duffing-Mathieu system with three-well potential oscillator

Abstract: Semi-analytical and semi-numerical method is used to investigate the global bifurcations and chaos in the nonlinear system of a Van der Pol-Duffing-Mathieu oscillator. Semi-analytical and semi-numerical method means that the autonomous system, called Van der Pol-Duffing system, is analytically studied to draw all global bifurcations diagrams in parameter space. These diagrams are called basic bifurcation diagrams. Then fixing parameter in every space and taking parametrically excited amplitude as a bifurcation parameter, we can observe the evolution from a basic bifurcation diagram to chaotic pattern by numerical methods.

A Nonlinear Oscillator Model For Vortex Shedding From a Forced Cylinder. Part 1: Uniform Flow And Model Parameters

Abstract: A van der Pol equation with parametric forcing is introduced as a model for vortex shedding from a forced cylinder in a uniform flow. The parametric forcing function is developed so that the boundaries of the synchronization region correspond to the boundaries found experimentally by Williamson and Roshko (1988). A scaling constant is also developed so that the response magnification factor corresponds with the lift magnification factor measured by Bishop and Hassan (1964). VORTEX SHEDDING AND SYNCHRONIZATION Billah, these models fail to provide a good representation to the width and asymmetry of the synchronization region as measured by Williamson and Roshko (1988) as a function of the forcing amplitude. (3) (2) d2u (2 ) du (dY) -+E u -1 -+u=eu! y,r2 r dr where aI' a:z, and r are constants. This "quadratic" form for f, when multiplied by u as in Eq. 2, provides that no even ordered subharmonics (1/2, 1/4, etc.) arise in the solution of Eq. 2 for harmonic forcing. It also provides, via the IdyldrfY term, that the We introduce the dimensionless time r= OJ,t, the dimensionless axial coordinate {;= zlD where D is the diameter of the cylinder, the dimensionless near wake fluid quantity u = 2qlQo, and the dimensionless cylinder displacement y = Y/D. In terms of these dimensionless quantities, Eq. 1 becomes: For harmonic forcing at frequency OJ, that is for y = Asinnr where A is the dimensionless forcing amplitude and n = aiOJs is the dimensionless forcing frequency, a notable feature of the experimentally determined fluid response is the suppression of the natural shedding frequency and the entrainment or synchronization of the response frequency to the forcing frequency. Williamson and Roshko (1988) have conducted an extensive investigation of synchronized vortex shedding from an oscillating cylinder. For the fundamental synchronization defined by n"" 1, they find that the width of the synchronization region increases with increasing A and that the boundaries of the synchronization region are not symmetric about n= 1. They also find a secondary synchronization defined by n"" 1/3 (i.e., a 1/3 subharmonic), but no secondary synchronization defined by n"" 1/2 (a 1/2 subharmonic). Using analytical experimentation, we have developed a form for the coupling function that allows us to replicate closely the synchronization findings of Williamson and Roshko. Specifically, we takefiy, dyldr) as (1) During the past several years, investigators have demonstrated that the Ginzburg-Landau equation (Albarede and Monkewitz, 1992) or its close relative, the van der Pol equation (Noack et aI., 1991), arises as the leading order approximation for the vortex shedding instability from a stationary cylinder in a uniform flow. In this paper, we extend the van der Pol equation to the description of the vortex shedding instability from a forced cylinder in a uniform flow. The model equation is given by: INTRODUCTION where t is time and z is the axial coordinate along the cylinder. Also, q is any near wake fluid quantity associated with the vortex shedding instability, Qo is the amplitude of q for flow over an unforced cylinder, OJs is the shedding frequency, and E is a scaling constant. The cylinder displacement normal to the flow is denoted by Y. The parametric form of the forcing function, qf(Y,dYldt), is based on an observation by Billah (1989) concerning the nature of the feedback between forced cylinder motions and the global vortex shedding instability. For harmonic forcing, Eq. 1 can be solved by approximation techniques for the synchronized shedding response. This solution allows us to formulate the coupling functionfiY,dYldt) so that the boundaries of the synchronization region correspond to the boundaries found experimentally by Williamson and Roshko (1988). The value of E is determined by matching the magnification factor for q with the lift magnification measured by Bishop and Hassan (1964). A number of papers have appeared, over the past 20 years, using the van der Pol equation to model vortex shedding from a forced cylinder. An in-depth review can be found in Billah (1989). In all cases, the forcing function has been assumed to be nonparametric; that is, the right-hand side of Eq. 1 has been taken only as a function of Yand its time derivatives. As discussed by Received March 3, 1995: revised manuscript received by the editors September 20, 1995. The original version (prior to the final revised manuscript) was presented at the Fifth International Offshore and Polar Engineering Conference (lSOPE-95), The Hague, The Netherlands, June 11-16, 1995.

Ducks and rivers: three existence results

Abstract: The beginnings of nonstandard analysis in Prance are strongly related to the study of singular perturbations of the Van der Pol equation [110] and its ducks [29, 12]. This equation is an interesting and simple example of a slow-fast equation, in other words, an equation of the form: where e > 0 is a fixed i-small number, f a near-standard function, and f0 := 0f. We will study here a typical equation of such a kind and this will provide us with the opportunity to look at some of the tools that have been developed for their study.

Chaos in mechanical systems — A review

Abstract: In this paper, a review of the various developments in the field of chaotic dynamics with specific emphasis on chaos in structural and mechanical systems is presented. The paper discusses some known chaotic systems such as the Lorenz, Rossler, Ueda and Henon attractors as well as chaos in Duffing and Van der Pol oscillators. The paper also covers chaos in piecewise linear systems, impacting oscillators and flow induced vibrating systems. Topics such as bifurcations and routes to chaos, different ways of characterising chaos, domains of attraction and control of chaos are also discussed.

A Numerical Investigation of Phase-Locked and Chaotic Behavior of Coupled Van Der Pol Oscillators

Abstract: Limit cycle oscillators arise in a wide variety of mechanical, electrical and biological systems. Recently, emphasis has been placed on the study of systems of coupled limit cycles, such as cardiac oscillations. Synchronization criteria have remained a focus of most investigations. One area of investigation in the eld of coupled limit cycles is studying the behavior of a pair of linearly coupled van der Pol oscillators [32, 46, 50]. Previous investigations [57, 58] found the stability regions of the coupled oscillators for their in-phase and out-of-phase modes numerically. This research presents results obtained from numerical analysis of a pair of coupled van der Pol oscillators describing new dynamic behavior; phase-locked trajectories and non-periodic behavior. Also presented are the stability regions and a description of new dynamic behavior of a pair of coupled van der Pol oscillators with detuning.