Ciclos limites e a equação de van der Pol

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

A Van der Pol-Mathieu equation for the dynamics of dust grain charge in dusty plasmas

The chaotic profile of dust grain dynamics associated with dust-acoustic oscillations in a dusty plasma is considered. The collective behaviour of the dust plasma component is described via a multi-fluid model, comprising Boltzmann distributed electrons and ions, as well as an equation of continuity possessing a source term for the dust grains, the dust momentum and Poisson's equations. A Van der Pol–Mathieu-type nonlinear ordinary differential equation for the dust grain density dynamics is derived. The dynamical system is cast into an autonomous form by employing an averaging method. Critical stability boundaries for a particular trivial solution of the governing equation with varying parameters are specified. The equation is analysed to determine the resonance region, and finally numerically solved by using a fourth-order Runge–Kutta method. The presence of chaotic limit cycles is pointed out.

Effective Fokker-Planck equation for birhythmic modified van der Pol oscillator

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Modified Adomian Decomposition Method for Van der Pol equations

In the paper, the well known Adomian Decomposition Method (ADM) is modified to solve the parabolic equations. The present method is quite different than the numerical method. The results are compared with the existing exact or analytical method. The already known existing Adomian Decomposition Method is modified to improve the accuracy and convergence. Thus, the modified method is named as Modified Adomian Decomposition Method (MADM). The Modified Adomian Decomposition Method results are found to converge very quickly and are more accurate compared to ADM and numerical methods. MADM is quite efficient and is practically well suited for use in these problems. Several examples are given to check the reliability of the present method. Modified Adomian Decomposition Method is a non-numerical method which can be adapted for solving parabolic equations. In the current paper, the principle of the decomposition method is described, and its advantages are shown in the form of parabolic equations. (C) 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

Complex-order forced van der Pol oscillator

In this paper we consider a complex-order forced van der Pol oscillator. The complex derivative Dα1jβ, with α, β ∈ ℝ+, is a generalization of the concept of an integer derivative, where α = 1, β = 0. The Fourier transforms of the periodic solutions of the complex-order forced van der Pol oscillator are computed for various values of parameters such as frequency ω and amplitude b of the external forcing, the damping μ, and parameters α and β. Moreover, we consider two cases: (i) b = 1, μ = {1.0, 5.0, 10.0}, and ω = {0.5, 2.46, 5.0, 20.0}; (ii) ω = 20.0, μ = {1.0, 5.0, 10.0}, and b = {1.0, 5.0, 10.0}. We verified that most of the signal energy is concentrated in the fundamental harmonic ω0. We also observed that the fundamental frequency of the oscillations ω0 varies with α and μ. For the range of tested values, the numerical fitting led to logarithmic approximations for system (7) in the two cases (i) and (ii). In conclusion, we verify that by varying the parameter values α and β of the complex-order derivative in expression (7), we accomplished a very effective way of perturbing the dynamical behavior of the forced van der Pol oscillator, which is no longer limited to parameters b and ω.

Complex order van der Pol oscillator

In this paper a complex-order van der Pol oscillator is considered. The complex derivative Dα±ȷβ , with α,β∈R + is a generalization of the concept of integer derivative, where α=1, β=0. By applying the concept of complex derivative, we obtain a high-dimensional parameter space. Amplitude and period values of the periodic solutions of the two versions of the complex-order van der Pol oscillator are studied for variation of these parameters. Fourier transforms of the periodic solutions of the two oscillators are also analyzed.

Bifurcation Analysis of a Van der Pol-Duffing Circuit with Parallel Resistor

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One-dimensional Fermi accelerator model with moving wall described by a nonlinear van der Pol oscillator

A modification of the one-dimensional Fermi accelerator model is considered in this work. The dynamics of a classical particle of mass m, confined to bounce elastically between two rigid walls where one is described by a nonlinear van der Pol type oscillator while the other one is fixed, working as a reinjection mechanism of the particle for a next collision, is carefully made by the use of a two-dimensional nonlinear mapping. Two cases are considered: (i) the situation where the particle has mass negligible as compared to the mass of the moving wall and does not affect the motion of it; and (ii) the case where collisions of the particle do affect the movement of the moving wall. For case (i) the phase space is of mixed type leading us to observe a scaling of the average velocity as a function of the parameter (χ) controlling the nonlinearity of the moving wall. For large χ, a diffusion on the velocity is observed leading to the conclusion that Fermi acceleration is taking place. On the other hand, for case (ii), the motion of the moving wall is affected by collisions with the particle. However, due to the properties of the van der Pol oscillator, the moving wall relaxes again to a limit cycle. Such kind of motion absorbs part of the energy of the particle leading to a suppression of the unlimited energy gain as observed in case (i). The phase space shows a set of attractors of different periods whose basin of attraction has a complicated organization. © 2013 American Physical Society.

Collective behaviour in a square lattice of driven Duffing resonators coupled to van der Pol oscillators

The global and local synchronisation of a square lattice composed of alternating Duffing resonators and van der Pol oscillators coupled through displacement is studied. The lattice acts as a sensing device in which the input signal is characterised by an external driving force that is injected into the system through a subset of the Duffing resonators. The parameters of the system are taken from MEMS devices. The effects of the system parameters, the lattice architecture and size are discussed.

Nonlinear dynamics of a periodically driven Duffing resonator coupled to a Van der Pol oscillator

We explore the dynamics of a periodically driven Duffing resonator coupled elastically to a van der Pol oscillator in the case of 1?:?1 internal resonance in the cases of weak and strong coupling. Whilst strong coupling leads to dominating synchronization, the weak coupling case leads to a multitude of complex behaviours. A two-time scales method is used to obtain the frequency-amplitude modulation. The internal resonance leads to an antiresonance response of the Duffing resonator and a stagnant response (a small shoulder in the curve) of the van der Pol oscillator. The stability of the dynamic motions is also analyzed. The coupled system shows a hysteretic response pattern and symmetry-breaking facets. Chaotic behaviour of the coupled system is also observed and the dependence of the system dynamics on the parameters are also studied using bifurcation analysis.

Sur l’existence de solutions pour l’équation de van der Pol et pour certaines équations différentielles du second ordre, en présence d’impulsions ; sur la moyennisation pour les équations différentielles floues

Cette thèse est constituée de deux parties : Dans la première partie nous étudions l’existence de solutions périodiques, de periode donnée, et à variations bornées, de l’équation de van der Pol en présence d’impulsions. Nous étudions, en premier, le cas où les impulsions ne dépendent pas de l’état. Ensuite, nous considèrons le cas où les impulsions dépendent de la moyenne de l’état et enfin, nous traitons le cas général où les impulsions dépendent de l’état. La méthode de résolution est basée sur le principe de point fixe de type contraction. Nous nous intéressons ensuite à l’étude d’un problème avec trois points aux limites, associé à certaines équations différentielles impulsives du second ordre. Nous obtenons un premier résultat d’existence de solutions en appliquant le théorème de point fixe de Schaefer. Un deuxième résultat est obtenu en utilisant le théorème de point fixe de Sadovskii. Pour le résultat d’unicité des solutions nous appliquons, enfin, un théorème de point fixe de type contraction. La deuxième partie est consacrée à la justification de la technique de moyennisation dans le cadre des équations différentielles floues. Les conditions sur les données que nous imposons sont moins restrictives que celles de la littérature.

High Aspect Ratio (L/D) Riser Viv Prediction Using Wake Oscillator Model

A two-dimensional (2-D) vortex-induced vibration (VIV) prediction model for high aspect ratio (LID) riser subjected to uniform and sheared flow is studied in this paper. The nonlinear structure equations are considered. The near wake dynamics describing the fluctuating nature of vortex shedding is modeled using classical van der Pol equation. A new approach was applied to calibrate the empirical parameters in the wake oscillator model. Compared the predicted results with the experimental data and computational fluid dynamic (CFD) results. Good agreements are observed. It can be concluded that the present model can be used as simple computational tool in predicting some aspects of VIV of long flexible structures. (C) 2008 Elsevier Ltd. All rights reserved.

A NEW WAKE OSCILLATOR MODEL FOR PREDICTING VORTEX INDUCED VIBRATION OF A CIRCULAR CYLINDER

This article proposes a new wake oscillator model for vortex induced vibrations of an elastically supported rigid circular cylinder in a uniform current. The near wake dynamics related with the fluctuating nature of vortex shedding is modeled based on the classical van der Pol equation, combined with the equation for the oscillatory motion of the body. An appropriate approach is developed to estimate the empirical parameters in the wake oscillator model. The present predicted results are compared to the experimental data and previous wake oscillator Model results. Good agreement with experimental results is found.