Van Der Pol Type Equation Research Articles

Unusual Nonpolynomial Van der Pol Oscillator Equations With Exact Harmonic and Isochronous Solutions

We do not know Van der Pol-type equations with nonlinear restoring force having explicitly an exact periodic solution. We present, for the first time, nonpolynomial Van der Pol oscillator equations that do not satisfy the classical existence theorems. We exhibit their exact harmonic and isochronous solutions and prove the existence of limit cycles by using averaging theory. We also present first integrals and exact solutions of polynomial Van der Pol-Duffing equations to show that they do not have any limit cycle. Additionally, we prove that the damped Duffing-type equations are equivalent to the conservative Duffing equations exhibiting nonoscillatory solutions.

Coherence and Stochastic Resonances in a Noisy van der Pol-Type Circadian Pacemaker Model Driven by Light

Daylight plays a major role in the wake/sleep cycle in humans. Indeed, the wake/sleep system stems from biological systems that follow a circadian rhythm determined by the light/dark alternation. The oscillations can be modeled by the higher-order nonlinear van der Pol-type equation driven by a term that mimics the light cycle. In this work, noise in the illumination is introduced to investigate its effect on the human circadian cycle. It is found that the presence of noise is detrimental for the sleep/wake rhythm, except for some special values for which it may favor regular oscillations. Depending on system parameters, noise induces regularities, such as stochastic resonance; if the natural light is turned off, it emerges that there is an optimal value of intensity noise which most deteriorates the regularity of the cycle; it is the phenomenon of anti-coherent resonance. Also, the phenomenon of stochastic resonance occurs; in the presence of natural light and there is an optimal noise intensity which improves the evolution of the wake/sleep system. However, there is a critical value of the noise beyond which the system becomes chaotic; indeed, for sufficiently high noise levels (how high depends upon the parameter of the system), the sleep/wake cycle evolves in a random and unpredictable manner, for whatever parameters of the external light.

Synchronization of quasiperiodic oscillations in nearly Hamiltonian systems: The degenerate case.

Quasiperiodic perturbations of two-dimensional nearly Hamiltonian systems with a limit cycle are considered. The behavior of solutions in a small neighborhood of a degenerate resonance is studied. Special attention is paid to the synchronization problem. Bifurcations of quasiperiodic solutions that arise when the limit cycle passes through the neighborhood of a resonance phase curve are investigated. The study is based on an analysis of an autonomous pendulum-type system, which is obtained by the method of averaging and determines the dynamics in the resonance zone. Two possible topological structures of the unperturbed averaged system are distinguished. For each case, the intervals of a control parameter that correspond to oscillatory synchronization are found. The results are applied to a Duffing-Van der Pol-type equation.

System identification and early warning detection of thermoacoustic oscillations in a turbulent combustor using its noise-induced dynamics

Despite significant research, self-excited thermoacoustic oscillations continue to hinder the development of lean-premixed gas turbines, making the early detection of such oscillations a key priority. We perform output-only system identification of a turbulent lean-premixed combustor near a Hopf bifurcation using the noise-induced dynamics generated by inherent turbulence in the fixed-point regime, prior to the Hopf point itself. We model the pressure fluctuations in the combustor with a van der Pol-type equation and its corresponding Stuart–Landau equation. We extract the drift and diffusion terms of the equivalent Fokker–Planck equation via the transitional probability density function of the pressure amplitude. We then optimize the extracted terms with the adjoint Fokker–Planck equation. Through comparisons with experimental data, we show that this approach can enable prediction of (i) the location of the Hopf point and (ii) the limit-cycle amplitude after the Hopf point. This study shows that output-only system identification can be performed on a turbulent combustor using only pre-bifurcation data, opening up new pathways to the development of early warning indicators of thermoacoustic instability in practical combustion systems.

Input-output system identification of a thermoacoustic oscillator near a Hopf bifurcation using only fixed-point data.

We present a framework for performing input-output system identification near a Hopf bifurcation using data from only the fixed-point branch, prior to the Hopf point itself. The framework models the system with a van der Pol-type equation perturbed by additive noise, and identifies the system parameters via the corresponding Fokker-Planck equation. We demonstrate the framework on a prototypical thermoacoustic oscillator (a flame-driven Rijke tube) undergoing a supercritical Hopf bifurcation. We find that the framework can accurately predict the properties of the Hopf bifurcation and the limit cycle beyond it. This study constitutes an experimental demonstration of system identification on a reacting flow using only prebifurcation data, opening up pathways to the development of early warning indicators for nonlinear dynamical systems near a Hopf bifurcation.

Numerical simulation of a wind excited quasi-periodic regime of a stochastic van der Pol-type equation

The contribution regards a mathematical single-degree-of-freedom model of a slender structure vibrating in an air flow. Based on an experimental investigation, movement of such structures can be expressed by van der PolDuffing-type equations. Several particular configuration parameter settings for a white and non-white Gaussian random excitation together with deterministic harmonic forcing are considered and numerically analysed. The results support recently published analytic formulas.

Role of spontaneous emission in the formation of the steady-state optical spectrum of a diode laser

Based on Maxwell’s equations, we derive a van der Pol-type equation for self-oscillations in the cavity of a diode laser. Its solution is self-oscillations (lasing) even in the absence of spontaneous emission. This solution differs fundamentally from solutions obtained using rate equations containing an additional term in the form of spontaneous emission intensity. Taking into account spontaneous emission in the van der Pol model makes calculation results more realistic and allows one to find parameters characterising diode laser output coherence, including the spectral densities of laser light amplitude and phase fluctuations.

Dynamics of mutual harmonic synchronization between two coupled glow discharge plasma systems

The dynamics of harmonic synchronization between the anode glow oscillations of two mutually coupled glow discharge plasma systems are studied. The individual systems are operated in the regime where the frequencies of the anode glow oscillations increase linearly with an increase in the discharge voltage. In the experiments, the discharge voltage of one of the systems is kept fixed and that of the other increased progressively. A host of mutual synchronization phenomena is observed which includes a Devil's staircase progression of harmonic synchronization and frequency pulling and chaotic states. A systematic delineation of these experimental observations is presented, and some theoretical understanding is provided by the numerical simulation results of two coupled Van der Pol type equations.

Synchronization between two coupled direct current glow discharge plasma sources

Experimental results on the nonlinear dynamics of two coupled glow discharge plasma sources are presented. A variety of nonlinear phenomena including frequency synchronization and frequency pulling are observed as the coupling strength is varied. Numerical solutions of a model representation of the experiment consisting of two coupled asymmetric Van der Pol type equations are found to be in good agreement with the observed results.

A class of exact solutions of the Liénard-type ordinary nonlinear differential equation

A class of exact solutions is obtained for the Li\'{e}nard type ordinary non-linear differential equation. As a first step in our study the second order Li\'{e}nard type equation is transformed into a second kind Abel type first order differential equation. With the use of an exact integrability condition for the Abel equation (the Chiellini lemma), the exact general solution of the Abel equation can be obtained, thus leading to a class of exact solutions of the Li\'{e}nard equation, expressed in a parametric form. We also extend the Chiellini integrability condition to the case of the general Abel equation. As an application of the integrability condition the exact solutions of some particular Li\'{e}nard type equations, including a generalized van der Pol type equation, are explicitly obtained.

First Integrals and Solutions of Duffing–Van der Pol Type Equations

A simple transformation is used to obtain the first integrals and the solutions of the Duffing–van der Pol type equation under certain conditions. It is shown that the system can be totally integrable and this total integrability admits new solutions. The new solutions require weaker conditions on the system's parameters than hereto known.

Van der Pol-type equation for modeling vortex-induced oscillations of bridge decks

A careful modeling of vortex-induced vibration of a bridge deck allows to avoid fatigue damages and the reduction of travel safety and/or comfort levels for the users. The existing empirical models do not seem to be fully reliable because they are not able to predict the response oscillation amplitude for values of the mass and damping away from those at which their aeroelastic parameters were estimated. A van der Pol-type equation is used in the Ehsan–Scanlan's model, which for practical reasons seems to be the more suitable model for vortex-induced vibration of bridge decks. In this work this model was deeply studied and wind-tunnel tests on a 4:1 rectangular cylinder were performed to apply it to a realistic case study. Numerical integrations of the van der Pol-type equation highlighted its physical coherence to model vortex-induced vibrations of structures prone to the wind action. Moreover, it was verified that the assumptions standing behind the Ehsan–Scanlan's identification procedure of the aeroelastic parameters are justified. Finally, an alternative identification procedure was proposed, consisting in a numerical estimation of the aeroelastic parameters directly from the nonlinear differential equation.

Approximate transformations for van der Pol-type equations

Classification of van der Pol-type equations with respect to admitted approximate transformation groups transforming a small parameter is given. It is shown that approximate symmetries transforming the small parameter as well as the usual approximate symmetries can be used for approximate integration (e.g., by method of successive reduction of order) of ordinary differential equations with a small parameter.

The bifurcation structure of periodically forced current disruptions

The bifurcation structure of periodically driven current oscillations in the central chamber of a magnetized triple plasma device is investigated experimentally. The target chamber of the triple plasma device is positively biased with respect to the source chamber and the bias voltage mainly drops in a strong double layer formed in the central chamber. At the low potential side of the double layer, a variable negative potential forms that gives rise to a region of negative resistance in the static current-voltage characteristic of the device. In this regime, if a sufficiently high inductance is added to the external circuit, strong nonlinear low-frequency oscillations both in the plasma current and the voltage drop over the plasma occur. These oscillations are interpreted as the periodically repeated exchange between magnetic energy and particle motion in the double layer. The dynamics of the current circuit is described by a van der Pol-type equation where the nonlinearity is given by the derivative of the current-voltage characteristic of the plasma. An additional periodic driver signal, added to the bias voltage, gives rise to a considerably enriched dynamical behaviour as predicted by the theory of driven nonlinear oscillators, including frequency entrainment, quasiperiodicity, periodic pulling and period doubling bifurcations. The experimental observations are well explained by means of the known bifurcation structure of the the periodically driven van der Pol oscillator model.

Time dependence of the solutions of van der Pol type equation with periodic forcing term

In this paper we discuss the time dependence of solutions of the equation (2.3). From it we will get the global qualitative analysis of the equation.

Qualitative analysis of van der pol type equation with periodic forcing term

In this paper we analyze the qualitative behaviour of the equation $$\varepsilon \ddot X + q(X) \dot X + \varepsilon X = b p (t)$$ , wheree is a small parameter. We divide the interval of parameterb into four sets of subintervals.A, B, C andD. Forb∈A, B orD, we discuss the different structures of the attractors of the equation and their stabilities. When chaotic phenomena appear, we also estimate the entropy. Forb∈C, the set of bifurcation intervals, we analyze the bifurcating type and get a series of consequences from the results of Newhouse and Palis.

Numerical analysis of quasiperiodic solutions to nonlinear differential equations

The paper first generalizes the definition of exponential dichotomy and gives another simple proof of Urabe’s Existence Theorem to nonlinear quasiperiodic differential equations. Next, a useful numerical analysis of quasiperiodic solutions to second order differential equations is presented. The paper extends the results given in previous papers [3], [9], [16] which dealt with weakly nonlinear differential equations. A few examples of numerical analysis of Van der Pol type equations are illustrated.

Synergetic Approach to the Phenomena of Mode-Locking in Nonlinear Systems

The computer experiments are carried out on the phenomena of self-synchronization in a many-mode system described by the van der Pol type equation. The results are success­ fully explained in terms of the perturbation theory based on a mean field approximation proposed in a previous article. The extension of the theory to further complicated pheno­ mena of mode-locking is briefly cliscussed.