Van Der Pol Oscillator Research Articles

Effect of fractional derivatives on amplitude chimeras and symmetry-breaking death states in networks of limit-cycle oscillators.

We study networks of coupled oscillators whose local dynamics are governed by the fractional-order versions of the paradigmatic van der Pol and Rayleigh oscillators. We show that the networks exhibit diverse amplitude chimeras and oscillation death patterns. The occurrence of amplitude chimeras in a network of van der Pol oscillators is observed for the first time. A form of amplitude chimera, namely, "damped amplitude chimera" is observed and characterized, where the size of the incoherent region(s) increases continuously in the course of time, and the oscillations of drifting units are damped continuously until they are quenched to steady state. It is found that as the order of the fractional derivative decreases, the lifetime of classical amplitude chimeras increases, and there is a critical point at which there is a transition to damped amplitude chimeras. Overall, a decrease in the order of fractional derivatives reduces the propensity to synchronization and promotes oscillation death phenomena including solitary oscillation death and chimera death patterns that were unobserved in networks of integer-order oscillators. This effect of the fractional derivatives is verified by the stability analysis based on the properties of the master stability function of some collective dynamical states calculated from the block-diagonalized variational equations of the coupled systems. The present study generalizes the results of our recently studied network of fractional-order Stuart-Landau oscillators.

Learning dynamics on invariant measures using PDE-constrained optimization.

We extend the methodology in Yang et al. [SIAM J. Appl. Dyn. Syst. 22, 269-310 (2023)] to learn autonomous continuous-time dynamical systems from invariant measures. The highlight of our approach is to reformulate the inverse problem of learning ODEs or SDEs from data as a PDE-constrained optimization problem. This shift in perspective allows us to learn from slowly sampled inference trajectories and perform uncertainty quantification for the forecasted dynamics. Our approach also yields a forward model with better stability than direct trajectory simulation in certain situations. We present numerical results for the Van der Pol oscillator and the Lorenz-63 system, together with real-world applications to Hall-effect thruster dynamics and temperature prediction, to demonstrate the effectiveness of the proposed approach.

Derivative-based SINDy (DSINDy): Addressing the challenge of discovering governing equations from noisy data

Recent advances in the field of data-driven dynamics allow for the discovery of ODE systems using state measurements. One approach, known as Sparse Identification of Nonlinear Dynamics (SINDy), assumes the dynamics are sparse within a predetermined basis in the states and finds the expansion coefficients through linear regression with sparsity constraints. This approach requires an accurate estimation of the state time derivatives, which is not necessarily possible in the high-noise regime without additional constraints. We present an approach called Derivative-based SINDy (DSINDy) that combines two novel methods to improve ODE recovery at high-noise levels. First, we denoise the state variables by applying a projection operator that leverages the assumed basis for the system dynamics. Second, we use a second order cone program (SOCP) to find the derivative and governing equations simultaneously. We derive theoretical results for the projection-based denoising step, which allow us to estimate the values of hyperparameters used in the SOCP formulation. This underlying theory helps limit the number of required user-specified parameters. We present results demonstrating that our approach leads to improved system recovery for the Van der Pol oscillator, the Duffing oscillator, the Rössler attractor, and the Lorenz 96 model.

P-Regularity Theory and the Existence of a Solution to a Boundary Value Problem Continuously Dependent on Boundary Conditions

For a given boundary value problem, the existence of a solution depending continuously on the boundary conditions is analyzed. Previously, such a fact has been known only for the Cauchy problem, which is a classical result in the theory of differential equations. We prove a similar result for boundary value problems in the case when they are p-regular. In the general case, this result does not hold. Several implicit function theorems are proved in the degenerate case, which is a development of p-regularity theory concerning the existence of a solution to nonlinear differential equations. The results are illustrated by an example of a classical boundary value problem, namely, a degenerate Van der Pol equation is considered, for which the existence of a solution depending continuously on the boundary conditions of the perturbed problem is proved.

Stationary random response evaluation of nonlinear oscillators with Preisach hysteresis discretized by RNN model

This manuscript is devoted to evaluating random stationary responses of nonlinear oscillators including Preisach hysteresis, which is modeled by a powerful tool, viz. Recurrent Neural Networks (RNN). First, a generalized harmonic transformation is introduced by adopting a geometric viewpoint of RNN-described hysteresis and intentionally separating the hysteretic force into conservative and dissipative components, to circumvent direct mathematical calculations of amounts of difference equations. The hysteretic force could be equivalently mimicked by a force that only depends on the present system states. Then, the system dimension is reduced by stochastic averaging, which is derived from the corresponding low-dimensional Fokker–Planck–Kolmogorov (FPK) equation associated with the probability density of slow-varying process. Additionally, solving the FPK equation yields the stationary probability density as well as various-order response statistics. Finally, two numerical examples, i.e., Duffing oscillator and van der Pol oscillator, are investigated to verify the efficacy of the proposed procedure.

Quantum synchronization effects induced by strong nonlinearities

A paradigm for quantum synchronization is the quantum analog of the Stuart--Landau oscillator, which corresponds to a van der Pol oscillator in the limit of weak (i.e. vanishingly small) nonlinearity. Due to this limitation, the quantum Stuart--Landau oscillator fails to capture interesting nonlinearity-induced phenomena such as relaxation oscillations. To overcome this deficiency we propose an alternative model which approximates the van der Pol oscillator to finitely large nonlinearities while remaining numerically tractable. This allows us to uncover interesting phenomena in the deep-quantum strongly-nonlinear regime with no classical analog, such as the persistence of amplitude death on resonance. We also report nonlinearity-induced position correlations in reactively coupled quantum oscillators. Such coupled oscillations become more and more correlated with increasing nonlinearity before reaching some maximum. Again, this behavior is absent classically. We also show how strong nonlinearity can enlarge the synchronization bandwidth in both single and coupled oscillators. This effect can be harnessed to induce mutual synchronization between two oscillators initially in amplitude death.

Parameter estimation of two coupled oscillator model for pure intrinsic thermo-acoustic instability

A nonlinear phenomenological model of two coupled oscillators is proposed, which is able to describe the rich nonlinear behaviour stemming from an unstable pure intrinsic thermo-acoustic (ITA) mode of a simple combustion system. In an experimental bifurcation analysis of a pure ITA mode, it was observed that for increasing mean upstream velocity the flames loose stability through a supercritical Hopf bifurcation and subsequently exhibit limit cycle, quasi-periodic and period-2 limit cycle oscillations. The quasi-periodic oscillations were characterised by low frequent amplitude and frequency modulation. It is shown that a phenomenological model consisting of two coupled oscillators is able to reproduce qualitatively all the different experimentally observed regimes. This model consists of a nonlinear Van der Pol oscillator and a linear damped oscillator, which are nonlinearly coupled to each other. Furthermore, a parameter estimation of the model parameters is conducted, which reveals a good quantitative match between the model response and the experimental data. Hence, the presented phenomenological dynamical model accurately describes the nonlinear self-excited acoustic behaviour of premixed flames and provides a promising model structure for nonlinear time-domain flame models.

Stochastic bifurcations induced by Lévy noise in a fractional trirhythmic van der Pol system

Trirhythmical nature has aroused considerable interest in describing dynamic behaviors of a fractional self-sustained system. In this paper, the fractional trirhythmic system is considered and a bifurcation analysis in a stochastic fractional trirhythmic self sustained system subjected to Lévy noise perturbation is presented. The fractional electronic circuit has been used to model the system and the oscillations are described by a nonlinear fractional differential equation to show a new bifurcation parameter. Then, based on the minimum mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of the damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Using the predictor–corrector simulation method, it is shown a good agreement between the fractional and the equivalent integer order system. The detailed parameter space study reveals that the multi-rhythmic properties of the oscillation of this system can be efficiently controlled by fractional order parameter and Lévy noise parameters. Finally, the analytical solutions are validated by numerical results of the Monte Carlo simulation of the original fractional modified van der Pol oscillator. The trirhythmic region can be detected in the parameter space (α,β,γ,δ) by choosing an approximate fractional order. The stability index and the noise intensity can in any case govern the dynamics of the self sustained system, but regulating the skewness parameter cannot bring about transitions between unimodal, bimodal and trimodal in this multi-rhythmic system. More bifurcations appear by adjusting the fractional order in this system. These results may be conducive to further exploring bifurcations in the real-world applications. This study sheds new insight lights on the understanding nontrivial effects of fractional derivative on a trirhythmic self sustained system.

The van der Pol physical reservoir computer

The van der Pol oscillator has historical and practical significance to spiking neural networks. It was proposed as one of the first models for heart oscillations, and it has been used as the building block for spiking neural networks. Furthermore, the van der Pol oscillator is also readily implemented as an electronic circuit. For these reasons, we chose to implement the van der Pol oscillator as a physical reservoir computer (PRC) to highlight its computational ability, even when it is not in an array. The van der Pol PRC is explored using various logical tasks with numerical simulations, and a field-programmable analog array circuit for the van der Pol system is constructed to verify its use as a reservoir computer. As the van der Pol oscillator can be easily constructed with commercial-off-the-shelf circuit components, this PRC could be a viable option for computing on edge devices. We believe this is the first time that the van der Pol oscillator has been demonstrated as a PRC.

Oscillatory Kalman filtering for Duffing, Coulomb, and Van der Pol oscillators

The popularly known Gaussian filtering witnesses intractable integrals numerically approximated during the filtering. However, the numerical approximation methods used in the existing Gaussian filters are generally inaccurate for oscillatory and chaotic (OC) systems, resulting in poor accuracy. In this paper, we propose a new Gaussian filter named oscillatory spherical-radial Kalman filter (OSRKF) to improve the accuracy of OC systems. The proposed OSRKF decomposes the intractable integral into spherical and radial integrals. The spherical integral is approximated using the higher-degree spherical cubature rule, while the radial integral is approximated using exponentially-fitted Gauss–Laguerre quadrature rule. We also formulate the state estimation problems for three OC dynamical systems: the Duffing, Coulomb, and Van-der Pol oscillators. Subsequently, we validate the improved accuracy of the proposed OSRKF for all three OC dynamical systems.

Effects of amplitude modulation on mixed-mode oscillations in the forced van der Pol equation

Effects of amplitude modulation on mixed-mode oscillations in the forced van der Pol equation

Master-slave synchronization in the van der Pol and Duffing systems via elastic, dissipative and a combination of both couplings

A modified master-slave scheme to look for synchronization, based on a general combination of elastic and dissipative couplings, is presented. We focus on solutions according the scheme presented, illustrating the method we use, by employing the van der Pol and Duffing oscillators and analyzing three different ways of coupling. We find synchronization in the oscillators.

Asymptotics of Regular and Irregular Solutions in Chains of Coupled van der Pol Equations

Chains of coupled van der Pol equations are considered. The main assumption that motivates the use of special asymptotic methods is that the number of elements in the chain is sufficiently large. This allows moving from a discrete system of equations to the use of a continuity argument and obtaining an integro-differential boundary value problem as the initial model. In the study of the behaviour of all its solutions in a neighbourhood of the equilibrium state, infinite-dimensional critical cases arise in the problem of the stability of solutions. The main results include the construction of special families of quasi-normal forms, namely non-linear boundary value problems of either Schrödinger or Ginzburg–Landau type. Their solutions make it possible to determine the main terms of the asymptotic expansion of both regular and irregular solutions to the original system. The main goal is the study of chains with diffusion- and advective-type couplings, as well as fully connected chains.

Topological synchronization of quantum van der Pol oscillators

To observe synchronization in a large network of classical or quantum systems demands both excellent control of the interactions between the nodes and very accurate preparation of the initial conditions due to the involved nonlinearities and dissipation. This limits the applicability of this phenomenon for future devices. Here, we demonstrate a route towards significantly enhancing the robustness of synchronized behavior in open nonlinear systems that utilizes the power of topology. In a lattice of quantum van der Pol oscillators with topologically motivated couplings, boundary synchronization emerges in the classical mean field as well as the quantum model. In addition to its robustness against disorder and initial state perturbations, the observed dynamics is independent of the underlying topological insulator model provided the existence of zero-energy modes. Our work extends the notion of topology to the general nonlinear dynamics and open quantum system realm with applications to networks where specific nodes need special protection like power grids or quantum networks.

Hopf bifurcation for a fractional van der Pol oscillator and applications to aerodynamics: implications in flutter

Flutter is an important instability in aeroelasticity. In this work,we derive a model for this phenomenon which naturally leads to an equation similar to a van der Pol oscillator in which the friction term is given by a fractional derivative. Motivated by these considerations,we study a fractional van der Pol oscillator and show that it exhibits a Hopf bifurcation. The model is based on a one-dimensional reduction where the instabilities associated with flutter are preserved. However, due to the fractional derivative, the bifurcation analysis differs from the standard case. We present both analytical and numerical results and discuss the implications to aerodynamics. Additionally, we contrast our qualitative results with experimental data.

Plug-and-Play Distributed Control of Large-Scale Nonlinear Systems.

A method to design plug-and-play (PnP) distributed controllers for large-scale nonlinear systems represented by interconnected Takagi-Sugeno fuzzy models with nonlinear consequent is presented in this article. From the combination of techniques to use multiple fuzzy summations and to explore the chordal decomposition of the interconnection graph associated with the large-scale nonlinear system, sufficient conditions for distributed stabilization are derived in terms of linear matrix inequalities (LMIs). Conditions specially designed to allow seamless subsystems plugging-in and unplugging operations from the large-scale system, without requiring the redesign of all previously tuned distributed controllers, are provided. The approach can be used together with fault detection and isolation (FDI) systems, and also in the context of mixed distributed and decentralized controllers operating in a network of interconnected systems. To illustrate the effectiveness of the proposed PnP approach, a network of nonlinearly coupled and heterogeneous Van der Pol oscillators is used in the numerical experiments.

Nonlinear dynamics of viscoelastic fluid-conveying pipe installed within uniform external cross flow by pipe clamps

Nonlinear dynamic response of a viscoelastic fluid-conveying pipe excited by uniform external cross flow is investigated, which is installed by pipe clamps. A pipe clamp theoretical model is proposed for the actual pipe clamp instead of the commonly used simply supported or clamped boundary conditions. Based on the model, equivalent support stiffness and torsional stiffness are obtained from structure and material parameters of the pipe clamp. The vortex-induced lift force and the mean drag force are described by van der Pol oscillator model. The nonlinear equations of motion with the boundary conditions of different constraint stiffness are discretized by Differential Quadrature Method (DQM) and then solved by Runge-Kutta algorithm. The results are verified by Galerkin method for the pinned-pinned pipe, and by ANSYS for the pipe with two arbitrary constraint stiffnesses. The dependence of critical external fluid velocity upon constraint stiffness and pipe length is investigated. Moreover, the effects of support stiffness, torsional stiffness, internal fluid velocity, and viscoelastic coefficient on the amplitude response of the pipe are studied in detail. The influence of constraint stiffness on characteristics of lock-in region is elucidated. Consequently, pipe clamp spacing, material and structure parameters of the rubber mat can be designed to ensure the requirement of economy and safety, which are helpful to guide the layout and design of pipe clamps.

Chaotic van der Pol Oscillator Control Algorithm Comparison

The damped van der Pol oscillator is a chaotic non-linear system. Small perturbations in initial conditions may result in wildly different trajectories. Controlling, or forcing, the behavior of a van der Pol oscillator is difficult to achieve through traditional adaptive control methods. Connecting two van der Pol oscillators together where the output of one oscillator, the driver, drives the behavior of its partner, the responder, is a proven technique for controlling the van der Pol oscillator. Deterministic artificial intelligence is a feedforward and feedback control method that leverages the known physics of the van der Pol system to learn optimal system parameters for the forcing function. We assessed the performance of deterministic artificial intelligence employing three different online parameter estimation algorithms. Our evaluation criteria include mean absolute error between the target trajectory and the response oscillator trajectory over time. Two algorithms performed better than the benchmark with necessary discussion of the conditions under which they perform best. Recursive least squares with exponential forgetting had the lowest mean absolute error overall, with a 2.46% reduction in error compared to the baseline, feedforward without deterministic artificial intelligence. While least mean squares with normalized gradient adaptation had worse initial error in the first 10% of the simulation, after that point it exhibited consistently lower error. Over the last 90% of the simulation, deterministic artificial intelligence with least mean squares with normalized gradient adaptation achieved a 48.7% reduction in mean absolute error compared to baseline.

The Darboux Polynomials and Integrability of Polynomial Levinson–Smith Differential Equations

We provide the necessary and sufficient conditions of Liouvillian integrability for nondegenerate near infinity polynomial Levinson–Smith differential equations. These equations generalize Liénard equations and are used to describe self-sustained oscillations. Our results are valid for arbitrary degrees of the polynomials arising in the equations. We find a number of novel Liouvillian integrable subfamilies. We derive an upper bound with respect to one of the variables on the degrees of irreducible Darboux polynomials in the case of nondegenerate or algebraically degenerate near infinity polynomial Levinson–Smith equations. We perform the complete classification of Liouvillian first integrals for the nondegenerate or algebraically degenerate near infinity Rayleigh–Duffing–van der Pol equation that is a cubic Levinson–Smith equation.

Solutions of the Van der Pol Equation

Summary Presented in this paper are various solutions to the Van der Pol equation. Numerical solutions are utilized as an independent means of validating the various solutions discussed. A new solution in the form of a power series has been found. Although this solution is exact, its interval of convergence can only be estimated for a special case. Numerical experiments reveal that the power series solution can provide an exact solution over intervals where other approximate solutions are not valid. Thus, the new solution represents an additional solution that can complement other existing solutions. This work also emphasizes the importance and the role of computation. Although the power series solution along with the other approximate solutions mentioned are of theoretical interest, their restrictions limit their usefulness in real applications, and therefore numerical methods should also be considered.