Van Der Pol Oscillator Research Articles

Proportional and Derivative Coupling: a Way to Achieve Synchronization for Coupled Oscillators

In order to achieve synchronization, the concept of proportional coupling and derivative coupling, as analogue to PD control theory, is proposed. The author analyzes from the coupled simple harmonic oscillators to Kuromoto model, and to the coupled Van der Pol oscillators. It is conjectured that only proportional coupling for two or multiple linear and nonlinear coupled oscillators is not sufficient most of the time to achieve synchronization, whereas derivative coupling is the dominant factor for two or multiple linear and nonlinear coupled oscillators to get synchronization. In addition, it is noticed that if there is no coupling, there is no synchronization if starting from different initial conditions. In order to achieve synchronization, coupling is required. However, coupling does not necessary mean it will achieve synchronization, it has to be coupled in a particular way, and otherwise it will not get synchronization. As a result of this study, two major questions for the linear and nonlinear coupled oscillators to get synchronization are proposed. Generally, only proportional coupling (i.e. spring coupling) is earlier to analyze as it is a simple physics problem. Once we couple them using the derivative coupling approach, it will get major complications.

Temporal Logic Control of Nonlinear Stochastic Systems Using a Piecewise-Affine Abstraction

Automatically synthesizing controllers for continuous-state nonlinear stochastic systems, while giving guarantees on the probability of satisfying (infinite-horizon) temporal logic specifications crucially depends on abstractions with a quantified accuracy. For this similarity quantification, approximate stochastic simulation relations are often used. To handle the nonlinearity of the system effectively, we use finite-state abstractions based on piecewise-affine approximations together with tailored simulation relations that leverage the local affine structure. In the end, we synthesize a robust controller for a nonlinear stochastic Van der Pol oscillator.

Dynamics of Three Coupled Generators of Quasi-periodic Oscillations

The dynamics of three coupled generators capable of demonstrating autonomous quasi-periodic oscillations is studied. The complex structure of the Lyapunov charts of the system is discussed, which reveals invariant tori of different dimensions, quasi-periodic bifurcations of tori, and Arnold’s resonant web based on tori of different dimensions. Cases of different types of tuning of individual generators (periodic oscillations, quasi-periodic oscillations) are considered. A detailed numerical bifurcation analysis of the equilibrium state and limit cycles, which form a complex picture of dynamic regimes, is carried out.Common features and differences compared to the case of three coupled van der Pol oscillators are discussed.

Regularization for Distributionally Robust State Estimation and Prediction

The increasing availability of sensing techniques provides a great opportunity for engineers to design state estimation methods, which are optimal for the system under observation and the observed noise patterns. However, these patterns often do not fulfill the assumptions of existing approaches. We provide a direct method using samples of the noise to create a moving horizon observer for linear time-varying and nonlinear systems, which is optimal under the empirical noise distribution. Moreover, we show how to enhance the observer with distributional robustness properties in order to handle unmodeled components in the noise profile, as well as different noise realizations. We prove that, even though the design of distributionally robust estimators is a complex minmax problem over an infinite-dimensional space, it can be transformed into a regularized linear program using a system level synthesis approach. Numerical experiments with the Van der Pol oscillator show the benefits of not only using empirical samples of the noise to design the state estimator, but also of adding distributional robustness. We show that our method can significantly outperform state-of-the-art approaches under challenging noise distributions, including multi-modal and deterministic components.

Robust data-driven control for nonlinear systems using the Koopman operator*

Data-driven analysis and control of dynamical systems have gained a lot of interest in recent years. While the class of linear systems is well studied, theoretical results for nonlinear systems are still rare. In this paper, we present a data-driven controller design method for discrete-time control-affine nonlinear systems. Our approach relies on the Koopman operator, which is a linear but infinite-dimensional operator lifting the nonlinear system to a higher-dimensional space. Particularly, we derive a linear fractional representation of a lifted bilinear system representation based on measured data. Further, we restrict the lifting to finite dimensions, but account for the truncation error using a finite-gain argument. We derive a linear matrix inequality based design procedure to guarantee robust local stability for the resulting bilinear system for all error terms satisfying the finite-gain bound and, thus, also for the underlying nonlinear system. Finally, we apply the developed design method to the nonlinear Van der Pol oscillator.

How to build a MATLAB demonstrator solving dynamical systems in real-time, with audio output and MIDI control

This paper explains and provides code to synthesize and control, in real-time, the audio signals produced by a dynamical system. The code uses only the Matlab programming language. It can be controlled with an external MIDI (Musical Instrument Data Interface) device, such as a MIDI keyboard or wind controller, or with mouse-operated sliders. In addition to the audio output, the demonstrator computes and displays the amplitude and fundamental frequency of the signal, which is useful to quantify the dynamics of the model. For the sake of this example, it is a type of Van der Pol oscillator, but more complex systems can be handled. The demonstrator holds potential for pedagogical and preliminary research applications, for various topics related to dynamical systems: direct and inverse bifurcations, transient effects such as dynamical bifurcations, artifacts introduced by integration schemes, and above all, the dynamics of self-sustained musical instruments.

On Optimal Synchronization of Diffusively Coupled Heterogeneous Van der Pol Oscillators*

This paper proposes a novel method to achieve and preserve synchronization for a set of connected heterogeneous Van der Pol oscillators. Unlike the state-of-the-art synchronization methods, in which a constant and large coupling gain is applied to couple any pair of connected oscillators, the proposed method first divides the synchronization process into two phases. The first one considers the period from the beginning to the first instant of synchronization, while the second phase covers the following time in which synchronization must be preserved. It is shown that a large coupling gain is adopted for the first phase, while the averaged coupling gain to practically preserve the synchronization in the second phase can be reduced significantly by using an offline optimized coupling law. Efficiency and performance of this method are confirmed by a set of numerical tests with different graphs and system dynamics.

Moving Horizon Estimation for Digital Twins using Deep Autoencoders

Digital twins have emerged in recent years as black-box, software-based simulation tools that can mirror the behavior of complex dynamical systems. Digital twin simulations generate the same outputs as the target system using internal states; however, these states are not readily available online from the real system. In this paper, we develop a data-driven moving horizon estimation framework capable of using online noisy measurements of the real system in order to estimate digital twin states. Our framework combines the high expressiveness of deep autoencoders with a moving horizon state estimator that accurately predicts the internal state of the black-box digital twin without access to an analytical model of the system dynamics. We demonstrate that our approach outperforms extended and Koopman Kalman filter solutions on a benchmark reverse van der Pol oscillator example.

Global algebraic Poincaré–Bendixson annulus for the Rayleigh equation

We consider the Rayleigh equation x ¨ + λ ( x ˙ 2 / 3 − 1 ) x ˙ + x = 0 depending on the real parameter λ and construct a Poincaré–Bendixson annulus A λ in the phase plane containing the unique limit cycle Γ λ of the Rayleigh equation for all λ > 0 . The novelty of this annulus consists in the fact that its boundaries are algebraic curves depending on λ . The polynomial defining the interior boundary represents a special Dulac–Cherkas function for the Rayleigh equation which immediately implies that the Rayleigh equation has at most one limit cycle. The outer boundary is the diffeomorphic image of the corresponding boundary for the van der Pol equation. Additionally we present some equations which are linearly topologically equivalent to the Rayleigh equation and provide also for these equations global algebraic Poincaré–Bendixson annuli.

МОДЕЛИРОВАНИЕ ВОЗБУЖДЕНИЯ ПОВЕРХНОСТНЫХ ПЛАЗМОН-ПОЛЯРИТОНОВ В ПРЯМОУГОЛЬНОМ НАНОРЕЗОНАТОРЕ НА ОСНОВЕ ЗОЛОТА

This paper considers the modeling of surface plasmon polaritons excitation in a limited nanostructure on the basis of a two-dimensional area discrete model, the area interacts with the oscillator. The nanostructure is a rectangle defined on the metal surface at the interface of the gold–silicon oxide, the surface plasmon–polaritons are excited by an electromagnetic radiation point source located above the metal surface. The dynamics of radiation point source is described by a discrete version of the Van der Pol equation with a small nonlinearity of the source parameters. The parameters of the goldsilicon oxide structure (the wave phase speed, the radiation source frequency, the characteristic time in the system, etc.) will be received from the dispersion relation for surface plasmon–polaritons at a single metal–dielectric interface. The distributions of the wave field in the structure will be analyzed at different positions of the point oscillator and different coupling coefficients of the wave field with the oscillators. The resonant wave field mode composition at different positions of the point oscillator and different coupling coefficients will be found using the two-dimensional Fourier transform of the wave field, the time evolution of excited modes of the wave field amplitudes will be also analyzed. In conclusion, the paper gives applicability limits of the considered model for studying the surface plasmon polaritons excitation in a limited nanostructure.

A Microcontroller-based Liénard Oscillator

Van der Pol Equation is a special case of Liénard equations. Both oscillators have significant historical importance. There are lots of different circuit topologies for Van der Pol and Liénard oscillators. Such oscillators can be made using vacuum tubes, diodes, etc. Some oscillators are made using microcontrollers, which are cheap and easy-to-use devices. They provide accurate adjustability of the frequency and magnitude of the waveforms. Arduino Nano Klon V3.0 microcontroller is a commonly used microcontroller. In this study, to the best of our knowledge, for the first time in literature, a Liénard Oscillator has been made with the Direct digital synthesis (DDS) method using the Arduino Nano Klon V3.0 microcontroller and two DACs. The experimental results of the oscillator are given. The circuit is able to produce the state variables of the oscillator, the effect of quantization can be seen on the waveforms, and it is shown that it performs well. The two variable outputs of the system let its phase portrait be examined easily. Also, using a microcontroller helps to design the oscillator in mere a few days.

On Certain Noise Filtering Techniques in Fixed Point Iteration-based Adaptive Control

In control applications the use of noise-burdened sensor signals cannot be evaded. Also, certain signals can be lost. This problem traditionally is tackled by the use of Kalman filters that provide some “optimal solution” to these problems based on reasonable assumptions that are not always well underpinned in the practice. These are assumptions are made with regard to the system model and the statistical distribution of the noise signals. The Fixed Point Iteration-based adaptive controller is applicable for various strongly nonlinear models. Because feeding back the order of time-derivative of the system’s variable that immediately can be varied by the control signal, its noise sensitivity can be considerable. In this paper the operation of an unscented Kalman filter-based technique is compared with that of a simple moving window with affine signal approximation, and the use of a third order low pass filter in the control of a modified van der Pol oscillator. In this model a quadratic drag term is added to the original model to describe the motion of the system in turbulent fluid environment. According to the numerical simulations it can be stated that the simpler methods can replace the more complicated Kalman filter.

Application of Heavy and Underestimated Dynamic Models in Adaptive Receding Horizon Control Without Constraints

In the heuristic “Adaptive Receding Horizon Controller” (ARHC) the available dynamic model of the controlled system usually is placed in the role of a constraint under which various cost functions can be minimized over a horizon. A possible secure design can be making calculations for a “heavy dynamic model” that may produce high dynamical burden that is efficiently penalized by the cost functions and instead of the original nominal trajectory results a “deformed” one that can be realized by the controlled system of “less heavy dynamics”. In the lack of accurate system model a fixed point iterationbased adaptive approach is suggested for the precise realization of this deformed trajectory. To reduce the computational burden of the control the usual approach in which the dynamic model is considered as constraint and Lagrange-multipliers are introduced as co-state variables is evaded. The heavy dynamic model is directly built in the cost and the computationally greedy Reduced Gradient Algorithm is replaced by a transition between the simple and fast Newton-Raphson and the slower Gradient Descent algorithms (GDA). In the paper simulation examples are presented for two dynamically coupled van der Pol oscillators as a strongly nonlinear system. The comparative use of simple nondifferentiable and differentiable cost functions is considered, too.

Theoretical investigation of injection-locked differential oscillator

A preliminary analysis of published works on this topic showed that at present there is no sufficiently substantiated theory of such devices, and the approximate approaches used are rough and do not always meet the requirements of practice. The proposed transition from a differential self-oscillator to an equivalent single-circuit oscillator has not received a convincing justification.
 This article presents a methodology for studying a synchronized differential oscillator using rigorous methods. A mathematical model of such oscillator is presented in the form of nonlinear differential equations obtained on the basis of Kirchhoff's laws. Their analysis made it possible to substantiate the transition to the model of a single circuit LC oscillator, equivalent to a differential one. A technique for such a transition is proposed, including the procedure for determining the nonlinear characteristics of the amplifying element of this self-oscillator, based on the nonlinear characteristics of two amplifying elements of the differential oscillator.
 The mathematical model of an equivalent oscillator is represented by a non-linear differential Van der Pol equation in a dimensionless form, it is simple and accurate. This form of representation made it possible to single out a small parameter and estimate its value. In the case of small values of the small parameter, as is known, traditional methods can be used for its analysis. Thus, the task of studying the synchronization process of a differential oscillator is reduced to the study of the synchronization process of a Van der Pol oscillator. The presented results can be useful in the development of various devices based on synchronized differential oscillators.

On Perturbative Methods for Analyzing Third-Order Forced Van-der Pol Oscillators

In this investigation, an (un)forced third-order/jerk Van-der Pol oscillatory equation is solved using two perturbative methods called the Krylov–Bogoliúbov–Mitropólsky method and the multiple scales method. Both the first- and second-order approximations for the unforced and forced jerk Van-der Pol oscillatory equations are derived in detail using the proposed methods. Comparative analysis is performed between the analytical approximations using the proposed methods and the numerical approximations using the fourth-order Runge–Kutta scheme. Additionally, the global maximum error to the analytical approximations compared to the Runge–Kutta numerical approximation is estimated.

К вопросу о периодических решениях системы дифференциальных уравнений, описывающих колебания двух слабосвязанных осцилляторов Ван дер Поля

Изучается система двух слабосвязанных полностью идентичных осцилляторов Ван дер Поля в случае диффузионной связи.В работе изучен в полном объеме вопрос о существовании и устойчивости периодических решений рассматриваемой системы. Показано, что у нее могут быть периодические решения трех типов, которые порождают циклы Андронова-Хопфа, противофазный, и третий тип циклов синхронизации: асимметричные циклы. Анализ задачи использовал метод нормальных форм Пуанкаре-Дюлака, а также метод интегральных многообразий. We study a system of two weakly coupled completely identical van der Pol oscillators in the case of diffusion coupling. the question of the existence and stability of periodic solutions of the system under consideration. It is shown that it can have periodic solutions of three types, which generate Andronov-Hopf cycles, antiphase, and the third type of synchronization cycles: asymmetric cycles. The analysis of the problem used the Poincare-Dulac method of normal forms, as well as the method of integral manifolds.

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.

A novel mean square formulation of stochastic nonlinear dynamic systems based on Adomian decomposition

An Adomian decomposition based mathematical framework to derive the mean square responses of nonlinear structural systems subjected to stochastic excitation is presented. The exact mean square response estimation of certain class of nonlinear stochastic systems is achieved using Fokker–Planck–Kolmogorov (FPK) equations resulting in analytical expressions or using Monte Carlo simulations. However, for most of the nonlinear systems, the response estimation using Monte Carlo simulations is computationally expensive, and, also, obtaining solution of FPK equation is mathematically exhaustive owing to the requirement to solve a stochastic partial differential equation. In this context, the present work proposes an Adomian decomposition based formalism to derive semi-analytical expressions for the second order response statistics. Further, a derivative matching based moment approximation technique is employed to reduce the higher order moments in nonlinear systems into functions of lower order moments without resorting to any sort of linearization. Three case studies consisting of Duffing oscillator with negative stiffness, Rayleigh Van-der Pol oscillator and a Pendulum tuned mass damper inerter system with linear auxiliary spring–damper arrangement subjected to white noise excitation are undertaken. The accuracy of the closed form expressions derived using the proposed framework is established by comparing the mean square responses of the systems with the exact solutions. The results demonstrate the robustness of the proposed framework for accurate statistical analysis of nonlinear systems under stochastic excitation.

Resonances of a forced van der Pol equation with parametric damping

Resonances of a forced van der Pol equation with parametric damping

Central pattern generator network model for the alternating hind limb gait of rats based on the modified Van der Pol equation.

Herein, we employed a central pattern generator (CPG), a spinal cord neural network that regulates lower-limb gait during intra-spinal micro-stimulation (ISMS). Particularly, ISMS was used to determine the spatial distribution pattern of CPG sites in the spinal cord and the signal regulation pattern that induced the CPG network to produce coordinated actions. Based on the oscillation phenomenon of the single CPG neurons of Van der Pol (VDP) oscillators, a double-cell CPG neural network model was constructed to realise double lower limbs, six-joint modelling, the simulation of 12 neural circuits, the CPG loci characterising stimuli-inducing alternating movements and changes in polarity stimulation signals in rat hindlimbs, and leg-state change movements. The feasibility and effectiveness of the CPG neural network were verified by recording the electromyographic burst-release mode of the flexor and extensor muscles of the knee joints during CPG electrical stimulation. The results revealed that the output pattern of the CPG presented stable rhythm and coordination characteristics. The 12-neuron CPG model based on the improved VDP equation realised single-point control while significantly reducing the number of stimulation electrodes in the gait training of spinal cord injury patients. We believe that this study advances motor function recovery in rehabilitation medicine.