Van Der Pol Oscillator Research Articles

Noise robust approach to reconstruction of van der Pol-like oscillators and its application to Granger causality

Van der Pol oscillators and their generalizations are known to be a fundamental model in the theory of oscillations and their applications. Many objects of a different nature can be described using van der Pol-like equations under some circumstances; therefore, methods of reconstruction of such equations from experimental data can be of significant importance for tasks of model verification, indirect parameter estimation, coupling analysis, system classification, etc. The previously reported techniques were not applicable to time series with large measurement noise, which is usual in biological, climatological, and many other experiments. Here, we present a new approach based on the use of numerical integration instead of the differentiation and implicit approximation of a nonlinear dissipation function. We show that this new technique can work for noise levels up to 30% by standard deviation from the signal for different types of autonomous van der Pol-like systems and for ensembles of such systems, providing a new approach to the realization of the Granger-causality idea.

A numerical-analytical iterative method for solving an electrical oscillator equation

Self-excited oscillation problem occurring from a triode electrical circuit has been modelled by van der Pol. Until now, the exact solution to the van der Pol equation is not available. This paper focuses on finding a new method for solving the van der Pol equation simply and accurately. There exists several approximate iterative methods available in the literature for solving the van der Pol equation, such as, the successive approximation method. The successive approximation method is simple, but inaccurate for large time values. In this paper, we propose a new variant of numerical-analytical method, which is simple but accurate, for solving the van der Pol equation. Our new variant of numerical-analytical method solves the van der Pol equation from its equivalent system of first order ordinary differential equations. Our strategy leads to a simple implementation of the numerical-analytical method in the multistage way. Furthermore, computational experiments show that our proposed method is accurate for large sizes of time domain in solving the van der Pol equation.

Modelling of wake-induced vibrations of tandem cylinders with a nonlinear wake-deficit oscillator

For a pair of elastically mounted circular cylinders in a tandem configuration, the downstream cylinder may experience the wake-induced vibration (WIV) due to the hydrodynamic excitation associated with the wake behind the upstream cylinder. In addition to WIV, the downstream cylinder may also be simultaneously subject to the vortex-induced vibration (VIV) due to the vortex shedding in its own wake. This nonlinear coupled wake-vortex interaction can result in a sustained large-amplitude response depending on the cylinder spacing. In this study, a new nonlinear wake-deficit oscillator model for predicting the combined WIV-VIV responses in the cross-flow direction of the downstream cylinder is presented. While the upstream stationary or oscillating cylinder is placed in a uniform steady flow, the downstream cylinder is subject to a non-uniform unsteady flow with a space–time varying velocity being modified by the wake deficit or shielding effect. The total cross-flow hydrodynamic force on the downstream cylinder is modelled as a combination of the wake-induced transverse force and the combined vortex-induced lift and drag forces. The wake-induced force is approximated using a wake deficit theory based on a linearised boundary layer equation. The vortex-induced force is modelled using the van der Pol oscillator which incorporates the wake-deficit flow velocity, relative flow-cylinder velocities and dynamically staggered positions between the two interfering cylinders. These empirical hydrodynamic force equations are nonlinearly coupled, simulating the wake-vortex interaction leading to the combined WIV-VIV responses depending on the fluid–structure parameters. By focusing on the wake interference regime, the wake profiles and hydrodynamic forces are calibrated and validated, introducing new empirical coefficients and functions. The downstream cylinder responses and oscillation frequencies are predicted and compared with relevant experimental data by accounting for the important effects of cylinder spacing ratio, reduced flow velocity and Reynolds number in a subcritical flow regime. The wake stiffness characterisation is also presented, providing an analytical formulation related to the dynamic wake profile. Several key features associated with WIV response amplitudes and frequencies are highlighted and discussed using the calibrated wake-deficit oscillator. The present concept and model can be further improved to account for the effects of in-line response and arbitrarily staggered arrangement as well as the application of multiple long flexible cylinders with multi modal responses.

Dancing synchronization in coupled spin-torque nano-oscillators

We are reporting a new type of synchronization, termed dancing synchronization, between two spin-torque nano-oscillators (STNOs) coupled through spin waves. Different from the known synchronizations in which two STNOs are locked with various fixed relative phases, in this new synchronized state two STNOs have the same frequency, but their relative phase varies periodically within the common period, resulting in a dynamic waving pattern. The amplitude of the oscillating relative phase depends on the coupling strength of two STNOs, as well as the driven currents. The dancing synchronization turns out to be universal, and can exist in two nonlinear Van der Pol oscillators coupled both reactively and dissipativly. Our findings open doors for new functional STNO-based devices.

Mixed-mode oscillations from a constrained extended Bonhoeffer-van der Pol oscillator with a diode.

An extended Bonhoeffer-van der Pol (BVP) oscillator is a circuit that is naturally extended to a three-variable system from a two-variable BVP oscillator. A BVP oscillator is known to exhibit a canard explosion, and the extended BVP oscillator generates mixed-mode oscillations (MMOs). In this work, we considered a case study where the nonlinear conductor in the extended BVP oscillator includes an idealized diode. The idealized case corresponds to a degenerate case where one of the parameters tends to infinity, and circuit dynamics are represented using a constrained equation, and at the expense of the model's naturalness, i.e., in a case in which the solutions of the dynamics are defined only forward in time, the Poincaré return maps are constructed as one-dimensional (1D). Using these 1D return maps, we explain various phenomena, such as simple MMOs and MMO-incrementing bifurcations. In this oscillator, there exists a small amplitude oscillation, which emerges as a consequence of supercritical Hopf bifurcation, and there exists large relaxation oscillation which appears via canard explosion by changing the bifurcation parameter. Between these small and large amplitude oscillations, the MMO bifurcations exhibit asymmetric Farey trees. Furthermore, these theoretical results were verified using laboratory measurements and experiments.

A Generalization of the Groszkowski’s Result in Differential Oscillator Topologies

The paper presents a generalization of the Groszkowski's result in differential oscillators, providing novel equations to describe the oscillation frequency dependence on the harmonic content. The effect of the common-mode oscillation is rigorously included. Moreover, an additional term, arising from the dependence of the transistor current on the drain voltage, which is dominant any time ohmic operation occurs, is disclosed here for the first time. This framework is applied to Van der Pol oscillators, both nMOS and CMOS, designed in a 28-nm bulk CMOS technology. The results correctly match the oscillation frequency dependence derived from detailed circuit simulations. The analysis shows that, when even harmonics are relevant, the classical Groszkowski's result is not able to account for close-in phase noise performance. The novel theoretical framework fully justifies, instead, the simulation results and sheds new light on the flicker noise up-conversion mechanisms in the considered oscillator structures.

Good practices for designing and experimental testing of dynamically excited jointed structures: The Orion beam

This paper proposes a new lap-joint configuration, the so-called Orion beam. The new setup is composed of two thin beams connected by three bolted joints with contact patches on each connecting bolt. The Orion beam suggests an assembly configuration, in which there are bolts dedicated to “static” functions to guarantee structural integrity and those that perform “damping” functions to increase energy dissipation due to frictional contact. In particular, this paper analyzes the influence of the Orion beam’s design in the repeatability of experimental measurements and the impact of the tightening torque and excitation amplitudes on the system’s response. A feedback controller is proposed to control the excitation amplitudes during step-sine tests in the vicinity of resonance frequencies to measure the frequency response curves used to perform such analysis. Additionally, a single-degree-of-freedom Duffing–Van der Pol oscillator is proposed to evaluate, through its calibrated parameters, the impact of several tightening torque conditions and multiple excitation amplitudes on the structural stiffness and damping considering the vibration mode that promotes pronounced deformation around the lap-joint. Due to the presence of contact patches, the results show that the structure maintained impressive repeatability after several experimental measurements. Besides, it is possible to increase the damping for amplitude attenuation without significant losses in the contact stiffness.

Bifurcation Structures of Nested Mixed-Mode Oscillations

In recently published work [Inaba & Kousaka, 2020a; Inaba & Tsubone, 2020b], we discovered significant mixed-mode oscillation (MMO) bifurcation structures in which MMOs are nested. Simple mixed-mode oscillation-incrementing bifurcations (MMOIBs) are known to generate [Formula: see text] oscillations for successive [Formula: see text] between regions of [Formula: see text]- and [Formula: see text]-oscillations, where [Formula: see text] and [Formula: see text] are adjacent simple MMOs, e.g. [Formula: see text] and [Formula: see text], where [Formula: see text] is an integer. MMOIBs are universal phenomena of evidently strong order and have been studied extensively in chemistry, physics, and engineering. Nested MMOIBs are phenomena that are more complex, but have an even stronger order, generating chaotic MMO windows that include sequences [Formula: see text] for successive [Formula: see text], where [Formula: see text] and [Formula: see text] are adjacent MMOIB-generated MMOs, i.e. [Formula: see text] and [Formula: see text] for integer [Formula: see text]. Herein, we investigate the bifurcation structures of nested MMOIB-generated MMOs exhibited by a classical forced Bonhoeffer–van der Pol oscillator. We use numerical methods to prepare two- and one-parameter bifurcation diagrams of the system with [Formula: see text], and 3 for successive [Formula: see text] for the case [Formula: see text]. Our analysis suggests that nested MMOs could be widely observed and are clearly ordered phenomena. We then define the first return maps for nested MMOs, which elucidate the appearance of successively nested MMOIBs.

Controlling Canard Cycles

Canard cycles are periodic orbits that appear as special solutions of fast-slow systems (or singularly perturbed ordinary differential equations). It is well known that canard cycles are difficult to detect, hard to reproduce numerically, and that they are sensible to exponentially small changes in parameters. In this paper, we combine techniques from geometric singular perturbation theory, the blow-up method, and control theory, to design controllers that stabilize canard cycles of planar fast-slow systems with a folded critical manifold. As an application, we propose a controller that produces stable mixed-mode oscillations in the van der Pol oscillator.

Viability, viscosity, and storage functions in model-predictive control with terminal constraints

Under terminal constraints, the set of states such that a model-predictive control strategy is feasible, and thus stable, is equivalent to the viable-reachable set of the controlled system. We prove that viscosity and storage function approaches to viability theory are equivalent to each other. Based on these results we propose inner and outer bounds of the feasible set by storage functions, which for polynomial systems can be computed using sum-of-squares programmes. Numerical, discrete-time and continuous-time examples are given for lateral car dynamics and the forced Van der Pol oscillator, respectively.

Modelling oscillations in the supply chain: the case of a just-in-sequence supply process from the automotive industry

Pervasive and ubiquitous oscillations, mapping the repetitive variation in time of a specific state, are well known as abundant phenomena in research and practice. Motivated by the success of oscillators in the modelling, analysis and control of dynamical systems, we developed a related approach for the dynamic description of supply chains. This paper aims to introduce a generic oscillator model for supply chains by the original application of oscillator equations. Therefore an established oscillator model for deductive modelling of supply chain echelons is used. With the help of coupled van der Pol oscillators, the dynamical interaction of an inventory system is described and applied to a real-life supply process in automotive industry. According to its reductionist approach only two differential equations are used to analyse a Just-in-Sequence supply process in car industry. Based on the fact that any oscillatory state can be reduced to the phase of the oscillation (phase reduction), a phase space map is generated. This compact visual reference of the supply process can act as the quantitative basis for an adaptive control mechanism during its operation. By delaying or accelerating the inventory oscillations of the supplier stock a detuned coupled supply process can be re-synchronised without changing the amplitude. An additional analysis of Hilbert transform is applied to determine the boundaries of phase-locking between the inventory oscillation phases, where the instantaneous phases are bounded. Furthermore parameters of the synchronisation threshold and the transient phases between synchronous and non-synchronous regimes have been investigated, supported by an Arnold tongue representation. The investigations show that with the help of a generic oscillatory model it is possible to measure and quantify phenomena of inventory dynamics in supply chains. Especially the analysis of synchronisation phenomena with the help of phase space and Arnold tongue representations foster developments of performance measurement in supply chain management.

Repulsive inter-layer coupling induces anti-phase synchronization

We present numerical results for the synchronization phenomena in a bilayer network of repulsively coupled 2D lattices of van der Pol oscillators. We consider the cases when the network layers have either different or the same types of intra-layer coupling topology. When the layers are uncoupled, the lattice of van der Pol oscillators with a repulsive interaction typically demonstrates a labyrinth-like pattern, while the lattice with attractively coupled van der Pol oscillators shows a regular spiral wave structure. We reveal for the first time that repulsive inter-layer coupling leads to anti-phase synchronization of spatiotemporal structures for all considered combinations of intra-layer coupling. As a synchronization measure, we use the correlation coefficient between the symmetrical pairs of network nodes, which is always close to -1 in the case of anti-phase synchronization. We also study how the form of synchronous structures depends on the intra-layer coupling strengths when the repulsive inter-layer coupling is varied.

A three-time-level a posteriori error estimator for GS4-2 framework: Adaptive time stepping for second-order transient systems

The most commonly used adaptive time stepping technique is mainly based on the simple notion of a local error estimation. Accordingly, a variety of error estimators have mostly been designed for particular algorithms on an algorithm by algorithm selection basis. In this paper, a novel design of a three-time-level general purpose a posteriori error estimator is demonstrated for second-order transient systems, providing features of: (a) being universal for the most popular and entire class of second-order accurate unconditionally stable implicit LMS methods with/without numerical dissipation developed over the past few decades and on newer and optimal designs (the focus is on unconditionally stable implicit LMS methods and not explicit methods); (b) being simple for numerical implementation in the general purpose GS4-2 framework encompassing the entire class of LMS methods that are second order time accurate such that only the acceleration at three time levels is required; (c) preserving the third-order convergence in time for the local error; (d) predicting accurate local error such that the effectivity index is close to unity. In contrast to the existing a posteriori error estimators which are designed for certain algorithms and on algorithm by algorithm basis, the newly proposed approach can be directly applied to all the known and existing and new second-order accurate time integration algorithms in the entire class of the unconditionally stable implicit LMS family under the umbrella of the GS4-2 framework; thereby, the cumbersome work of designing different error estimators for different algorithms is avoided. In particular, the proposed error estimator can be directly applied also to the novel V0 family of algorithms without any limitation. Besides the limited selection of estimators existing within the U0 family of algorithms where most of the traditional methods belong to, the remainder of the algorithms within the U0 family and several others such as the V0 family do not have estimators that can be readily designed as the V0 family of algorithms only exist in the GS4-2 framework, which are entirely novel and new and are highly competitive. The numerical examples of stiff nonlinear spring pendulum, van der Pol equation, wave propagation, and phase-field modeling are described here to show the benefits to the computational cost whilst preserving generality, accuracy, and the like when the adaptive time stepping approach coupled with the proposed error estimator is applied to the transient simulations.

Limit cycle periods in damage detection

This paper introduces a damage detection scheme based on changes in the period of limit cycles (LCs) realized by nonlinear output feedback. The parameterized feedback law is selected to make the dynamic equilibrium of the actuated coordinate approximate that of a generalized van der Pol oscillator and potential model-related difficulties are avoided by fixing the parameters from experimental results. It is found that not all sensor/actuator arrangements are suitable for generating a LC in closed loop, so a methodology to determine which arrangements are viable is presented. Numerical results show that LC periods are adequately stable with respect to noise and system changes from environmental effects and have sensitivity to localized damage that is often orders of magnitude larger than those of the modal frequencies.

Tipping induced by multiplexing on two-layer networks

We report the study of sudden transitions or tipping in a collection of systems induced due to multiplexing with another network of systems. The emergent dynamics of oscillators on one layer can undergo a sudden transition to steady state due to indirect coupling with a shared environment, mean field couplings and conjugate couplings among them. In all these cases, when multiplexed with another set of similar systems, the tipping phenomena are induced on the second layer also with a similar pattern of behaviour. We consider van der Pol oscillator as nodal dynamics with various network topologies like scale free and regular networks with local and nonlocal couplings. We also report how the coupling topology influences the nature of transitions on both layers, under multiplexing.

Resonance Oscillation and Transition to Chaos in $$\phi ^8$$-Duffing–Van der Pol Oscillator

Resonance Oscillation and Transition to Chaos in $$\phi ^8$$-Duffing–Van der Pol Oscillator

Identification of van der Pol oscillator parameters

An invariant relations method for parameters estimation is used for the wellknown van der Pol oscillator. The approach is based on dynamical extension of original system and synthesis of appropriate invariant relations, from which the unknowns can be expressed as a functions of the known quantities on the trajectories of extended system. The stability property is formally checked taking into account the oscillatory behavior of the system. The simulation results confirm efficiency of the proposed scheme of nonlinear identificator design.

Adaptive Stepsize Control for Extrapolation Semi-Implicit Multistep ODE Solvers

Developing new and efficient numerical integration techniques is of great importance in applied mathematics and computer science. Among the variety of available methods, multistep ODE solvers are broadly used in simulation software. Recently, semi-implicit integration proved to be an efficient compromise between implicit and explicit ODE solvers, and multiple high-performance semi-implicit methods were proposed. However, the computational efficiency of any ODE solver can be significantly increased through the introduction of an adaptive integration stepsize, but it requires the estimation of local truncation error. It is known that recently proposed extrapolation semi-implicit multistep methods (ESIMM) cannot operate with existing local truncation error (LTE) estimators, e.g., embedded methods approach, due to their specific right-hand side calculation algorithm. In this paper, we propose two different techniques for local truncation error estimation and study the performance of ESIMM methods with adaptive stepsize control. The first considered approach is based on two parallel semi-implicit solutions with different commutation orders. The second estimator, called the “double extrapolation” method, is a modification of the embedded method approach. The introduction of the double extrapolation LTE estimator allowed us to additionally increase the precision of the ESIMM solver. Using several known nonlinear systems, including stiff van der Pol oscillator, as the testbench, we explicitly show that ESIMM solvers can outperform both implicit and explicit linear multistep methods when implemented with an adaptive stepsize.

Quantum Zeno effect in self-sustaining systems: Suppressing phase diffusion via repeated measurements

We study the effect of frequent projective measurements on the dynamics of quantum self-sustaining systems by considering the prototypical example of the quantum van der Pol oscillator. Quantum fluctuations are responsible for phase diffusion which progressively blurs the semiclassical limit-cycle dynamics and synchronization, either to an external driving or between two coupled self-sustained oscillators. We show that by subjecting the system to repeated measurements of heterodyne type at an appropriate repetition frequency one can significantly suppress phase diffusion without spoiling the semiclassical dynamics. This quantum Zeno-like effect may be effective in the case of either one or two coupled van der Pol oscillators, and we discuss its possible implementation in the case of trapped ions.

Smooth Inertial Manifolds for Neutral Differential Equations with Small Delays

In this paper, we study the dynamical behaviors of neutral differential equations with small delays. We first establish the existence and smoothness of the global inertial manifolds for these equations. Then we further prove the smoothness of inertial manifolds with respect to small delays for a certain class of neutral differential equations. The method can be also applied to deal with more general small-delay systems. Finally, we apply our main results to the van der Pol oscillator model with small delay.