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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. [ABSTRACT FROM AUTHOR]

In this paper, an accurate technique is used to find an approximate solution to the fractional-order Duffing-Van der Pol (DVP, for short) oscillators equation which is reproducing kernel Hilbert space (RKHS, for short) method. The numerical results show that the n-term approximation is a rapidly convergent series representation and they present also the high accuracy and effectiveness of this method. The efficiency of the proposed method has been proved by the theoretical predictions and confirmed by the numerical experiments. [ABSTRACT FROM AUTHOR]

The article discusses about the Equipartition theorem in thermodynamics with focus on general expression for the equipartition of energy and statistical physics. It also highlights the phenomena of quantum theory and quadratic Hamiltonian, concept of average energy for dimensional Hamiltonian and Van der pol oscillator in the polynomial form.

We investigate the dynamics of regular fractal-like networks of hierarchically coupled van der Pol oscillators. The hierarchy is imposed in terms of the coupling strengths or link weights. We study the low frequency modes, as well as frequency and phase synchronization, in the network by a process of repeated coarse-graining of oscillator units. At any given stage of this process, we sum over the signals from the oscillator units of a clique to obtain a new oscillating unit. The frequencies and the phases for the coarse-grained oscillators are found to progressively synchronize with the number of coarse-graining steps. Furthermore, the characteristic frequency is found to decrease and finally stabilize to a value that can be tuned via the parameters of the system. We compare our numerical results with those of an approximate analytic solution and find good qualitative agreement. Our study on this idealized model shows how oscillations with a precise frequency can be obtained in systems with heterogeneous couplings. It also demonstrates the effect of imposing a hierarchy in terms of link weights instead of one that is solely topological, where the connectivity between oscillators would be the determining factor, as is usually the case. [ABSTRACT FROM AUTHOR]

In this paper, an independent bifurcation tree of period-2 motions to chaos coexisting with period-1 motions in a periodically driven van der Pol–Duffing oscillator is presented semi-analytically. Symmetric and asymmetric period-1 motions without period-doubling are obtained first, and a bifurcation tree of independent period-2 to period-8 motions is presented. The bifurcations and stability of the corresponding periodic motions on the bifurcation tree are determined through eigenvalue analysis. The symmetry breaks of symmetric period-1 motions is determined by the saddle-node bifurcations, and the appearance of the independent bifurcation tree of period-2 motions to chaos is also due to the saddle-node bifurcations. Period-doubling cascaded scenario of period-2 to period-8 motions are predicted analytically, and unstable periodic motions are also obtained. Numerical simulations are performed to illustrate motion complexity in such a van der Pol–Duffing oscillator. Such nonlinear systems can be applied in nonlinear circuit design and fluid-induced oscillations. [ABSTRACT FROM AUTHOR]

Chimera states are peculiar spatiotemporal patterns of coupled oscillators composed of coherent and incoherent groups. In this paper, we study a network of non-identical coupled Van der Pol's oscillators and investigate the robustness of the chimera states against the oscillators frequency mismatches. The dynamical behaviors of the network are obtained for various coupling strengths and ranges, in different inhomogeneity levels, and the phase diagrams are presented. It is observed that the parameter inhomogeneity leads to the occurrence of synchronization in higher coupling strengths and expansion of the chimera region. In contrast to the identical network, in the non-identical network, a great deal of the chimera region belongs to the non-stationary chimera. [ABSTRACT FROM AUTHOR]

In this paper, we presented different behaviors such as chaotic and hyperchaotic of the generalized van der Pol oscillator with distributed order. We introduced the parameter intervals of these behaviors by computing the Lyapunov exponents of the oscillator, which is a good test for classifying the dynamical systems' solutions. The active control approach with the Laplace transform technique was used to realize the antisynchronization and control of the proposed oscillator. Finally, numerical investigations have been carried out on the dynamics of the proposed oscillator to verify the reliability of our analytical results. [ABSTRACT FROM AUTHOR]

We introduce the concept of degree of quantumness in quantum synchronization, a measure of the quantum nature of synchronization in quantum systems. Following techniques from quantum information, we propose the number of non-commuting observables that synchronize as a measure of quantumness. This figure of merit is compatible with already existing synchronization measurements, and it captures different physical properties. We illustrate it in a quantum system consisting of two weakly interacting cavity-qubit systems, which are coupled via the exchange of bosonic excitations between the cavities. Moreover, we study the synchronization of the expectation values of the Pauli operators and we propose a feasible superconducting circuit setup. Finally, we discuss the degree of quantumness in the synchronization between two quantum van der Pol oscillators. [ABSTRACT FROM AUTHOR]

An ensemble of van der Pol oscillators coupled in a ring has been numerically simulated. Peculiarities of chaotic dynamics have been studied by analysis of Fourier spectra and Lyapunov exponents. It is established that chaotic dynamics appearing upon the breakup of multifrequency tori is characterized by a broadband spectrum. [ABSTRACT FROM AUTHOR]

Dynamics of the Duffing–van der Pol driven oscillator is investigated. Periodic steady-state solutions of the corresponding equation are computed within the Krylov-Bogoliubov-Mitropolsky approach to yield dependence of amplitude A on forcing frequency Ω as an implicit function, F A , Ω = 0 , referred to as resonance curve or amplitude profile. In singular points of the amplitude curve the conditions ∂ F ∂ A = 0 , ∂ F ∂ Ω = 0 are fulfilled, i.e. in such points neither of the functions A = f Ω , Ω = g A , continuous with continuous first derivative, exists. Near such points metamorphoses of the dynamics can occur. In the present work the bifurcation set, i.e. the set in the parameter space, such that every point in this set corresponds to a singular point of the amplitude profile, is computed. Several examples of singular points and the corresponding metamorphoses of dynamics are presented. • The externally forced van der Pol oscillator is investigated. • Implicit equations for amplitude profiles of nonlinear resonances are derived. • The bifurcation manifold in the parameter space is computed. • Examples of dynamics near double singular points are computed. [ABSTRACT FROM AUTHOR]

We studied the coordination of arm movements in standing persons who performed an out-of-phase arm-swinging task while stepping in place or while standing. The subjects were instructed to stop one of the arms in response to an auditory signal while trying to keep the rest of the movement pattern unchanged. A significant increase was observed in the amplitude of the arm that continued swinging under both the stepping and standing conditions. This increase was similar between the right and left arms. A dynamic model was developed including two coupled nonlinear van der Pol oscillators. We assumed that stopping an arm did not eliminate the coupling but introduced a new constraint. Within the model, superposition of two factors, a command to stop the ongoing movement of one arm and the coupling between the two oscillators, has been able to account for the observed effects. The model makes predictions for future experiments. [ABSTRACT FROM AUTHOR]

Power combining with array configuration is an effective method for high output power in oscillators with resonant tunneling diodes (RTDs) in subterahertz and terahertz range. In this paper, output power and stability are analyzed for coupled oscillator array of RTDs. The coupled van der Pol equations are derived from the equivalent circuit for two-element oscillator arrays at first, and the condition in which only the fundamental oscillation mode stably oscillates is shown. The oscillation frequency of the array is single due to the mutual injection locking even if the individual frequencies of oscillator elements are different. Minimum difference between the individual frequencies to keep the mutual injection locking is analyzed. Experimental results are explained with theory. The analysis is extended to multielement oscillator arrays, and characteristics of oscillation modes, stability, combined output power, and locking range of the frequency difference between the elements are obtained. Theoretical combined output power increases with element number in proportion roughly to square of the element number if the frequency difference between the elements is small, and decreases with frequency difference. Different array configurations for stable and high-power operation are also discussed briefly. [ABSTRACT FROM AUTHOR]

In this paper, nonlinear frequency-amplitude characteristics of periodic motions in a periodically forced van der Pol oscillator are studied systematically. The periodic motions of the van der Pol oscillator are determined by the semi-analytical method, and the corresponding stability and bifurcation analysis is completed through the eigenvalue analysis. From the finite Fourier series analysis, the nonlinear frequency-amplitude characteristics of periodic motions are analyzed. From the frequency-amplitude analysis, the limit cycle of the van der Pol oscillator can be obtained analytically as excitation amplitude approach to zero, rather than numerically. For the van der Pol oscillator, most of periodic motions in the van der pol oscillator are symmetric. However, an asymmetric period-1 motion in the van der Pol oscillator is discovered. Thus a bifurcation tree of period-1 motion to chaos can be found. [ABSTRACT FROM AUTHOR]

In this paper, we investigate on dynamics of a generalized Fermi accelerator model with a moving wall described by a nonlinear Van der Pol oscillator. Different acceleration behaviors appear due to the discontinuity brought by impact and the stationary periodic excitation of the moving wall. Utilizing the theory of discontinuous dynamical systems, acceleration mechanism of the particle is studied and the analytical conditions for stick motion and grazing motion are obtained. By the use of generic mappings between different boundaries including stick and non-stick motions, periodic motions of the Fermi accelerator can be constructed, and corresponding local stability and bifurcations can be discussed according to mapping dynamics. Finally, different acceleration behaviors including periodic, chatter and stick motions are presented and illustrated. [ABSTRACT FROM AUTHOR]

The dynamics of a two-dimensional ensemble of nonlocally coupled van der Pol oscillators is studied by numerical simulation. The possibility of realization of regular spiral and target waves is established. The emergence of a new type of chimera structure realized on the basis of target waves with nonlocal coupling is shown. [ABSTRACT FROM AUTHOR]

Mixed-mode oscillation-incrementing bifurcations (MMOIBs) are a universal phenomenon appearing in a large class of mixed-mode oscillation (MMO)-generating dynamics that appear to be governed by asymmetric Farey arithmetic. This report presents experimental and numerical observations of the simplest MMOIBs generated by an extended Bonhoeffer–van der Pol circuit. The results indicate that maintaining a constant temperature during the circuit experiments is required to yield stable MMOs and MMOIBs. We also clarify how the firing number variations related to MMOIB phenomena follow asymmetric Farey arithmetic. [ABSTRACT FROM AUTHOR]

Quasi-periodic vibrations in a delayed van der Pol oscillator with time-periodic delay amplitude is investigated in this paper. The case where the delay amplitude in the position is modulated with a certain amplitude and a resonant frequency is considered. Application of the double perturbation method enables approximation of the amplitudes of quasi-periodic vibrations of the oscillator near a parametric resonance for which the frequency of the delay amplitude modulation is near twice the natural frequency of the oscillator. Analytical approximations supported by numerical simulations provide the regions in parameter space where quasi-periodic vibrations exist. Results show that the modulation of the delay amplitude in the position not only gives birth to quasi-periodic vibrations in a large range of parameters, but also with broadband large-amplitude covering a wide range of excitation frequency. [ABSTRACT FROM AUTHOR]

Abstract The method for reconstruction of nonlinear potential, dissipation and coupling functions for all elements of ensembles of coupled generalized Van der Pol oscillators (including Rayleigh and Bonhoeffer–Van der Pol oscillators) from multivariate time series is proposed. It is based on the idea of using the smoothness of reconstructed potential function as a criterion for quality of reconstruction. Therefore, no parametrization for potential is required and an arbitrary potential function unique for each oscillator in the ensemble can be considered. Different approaches to reconstruction of dissipation are studied. Linear and nonlinear coupling between oscillators is considered, including coupling by derivatives. Effects of measurement noise, contaminating the series, are taken into account. Highlights • Ensembles of Van der Pol, Rayleigh and FitzHugh–Nagumo systems are reconstructed. • The method allows to reconstruct arbitrary nonlinear potential function. • Linear and nonlinear couplings by coordinate and by derivative were considered. • High order dissipation functions of individual nodes were approximated and fitted. • Ensembles up to 128 element were studied in chaotic and periodic regimes. [ABSTRACT FROM AUTHOR]

Some typical bifurcation sets of a generalized autonomous Van der Pol-type model are discussed as archetypes of phase diagrams occurring in nonlinear dynamical systems. The relevance of the obtained bifurcation sets is exemplified by several experimental and numerical results from the literature of oscillating chemical reactions. © 1995 American Institute of Physics. [ABSTRACT FROM AUTHOR]

This article provides a comparative study of four different second-order sliding-mode (SOSM) differentiators proposed in the literature, namely, standard higher-order sliding-mode (HOSM) differentiator, non-homogeneous HOSM differentiator, uniform robust exact differentiator and hybrid fixed-time differentiator. Based on sliding-mode principles, these differentiators can provide robust exact differentiation with finite/fixed-time convergence. First, a comprehensive summary of the different methods is provided. Then, the differentiators are applied experimentally to estimate the states of a Van der Pol oscillator. Through experiments, it is shown that the different differentiators outperformed a Kalman-like observer, high-gain differentiator and extended Kalman filter. Finally, some suggestions are provided on the selection of SOSM differentiators for various applications. [ABSTRACT FROM AUTHOR]

Recycling noise is a kind of more common noise. The nonlinear dynamic system can be controlled by adjusting its parameters. However, so far, the effect of recycling noise on tri-stable dynamic system has not been reported. In this paper, stochastic P-bifurcations in tri-stable Duffing–Van der Pol oscillator induced by additive recycling noise are investigated. Firstly, the stationary probability density function is derived using stochastic averaging method. Then, the general expression of the critical parameter conditions of stochastic P-bifurcation is given by the singularity theory. The stationary probability density of response amplitude in different parameter areas are also shown, which is verified by Monte Carlo numerical simulation. Based on these results, the influence of related parameters of recycling noise and damping coefficient on the stochastic P-bifurcation is studied. The result shows that the critical parameter of bifurcation can be changed by adjusting the delay time and fraction coefficient of the recycling noise. It has also been found that the stationary probability density and stochastic bifurcation show a periodic dependence on the delay time. [ABSTRACT FROM AUTHOR]

Summary: In this paper, the homogeneous distributed control law is proposed such that the outputs of the networked Lur'e systems with disturbances are synchronized. A novel adaptive distributed state‐based disturbance compensator technique is utilized such that the disturbances can be offset asymptotically. The design of the distributed compensator is constructive in that some particular functions of the states of the Lur'e systems are included. The functions always exist for the considered Lur'e‐type system. Moreover, a theorem of networked Lur'e systems subject to disturbances is first presented to show the boundedness of the states of the closed‐loop system. Then, the synchronization is shown by the incremental passivity theory and graph theory. Finally, the effectiveness of the current method is illustrated by the networked Van der Pol oscillators with disturbances. [ABSTRACT FROM AUTHOR]

Summary: We present a comprehensive analysis of the asynchronous and synchronous operations of fourth‐order oscillators underlying dual‐band voltage‐controlled oscillators. We analyze the occurrence of the self‐synchronization phenomenon (internal resonance) if the ratio of normal frequencies is nearly a ratio of integers, which is 1:3 in the cubic approximation of the nonlinear oscillator characteristic. In this case, we have the simultaneous presence of 2 oscillations with a frequency ratio 1:3, which was demonstrated to be very effective in generating high‐frequency signals in mm‐wave range. The analysis is carried out by developing an analytical approach relying on the averaging principle, as it follows the van der Pol method. The averaging equations, derived simply by a quadrature, allow us to analyze easily the stationary and transient oscillations, and their stability, both in asynchronous and synchronous operations. Expressions for the amplitudes and phases were derived for a cubic nonlinearity and verified by Spice simulations. [ABSTRACT FROM AUTHOR]

In nature many interactions of different oscillatory systems can be described by the phenomenon of synchronization. This phenomenon can be widely used for engineering applications. In this contribution, the synchronization between a self-sustained oscillator and linear multibody systems with harmonic force excitation is used successfully to damp unwanted vibrations of the host structure and additionally to harvest energy from the load. To this end, the parameters of the Rayleigh–van der Pol oscillator are adjusted to a given harmonic base excitation in such a way that the response is harmonic as well. Thereby the amplitude of the response and the phase-shift between the excitation and the oscillator response remain free. Afterwards the oscillator is attached to a forced multibody system and the amplitude and phase-shift are adjusted for the tasks of vibration absorption and energy harvesting. [ABSTRACT FROM AUTHOR]

In this work, we study the multistability and chaos phenomena in a classical two-mode van der Pol generator which consists of a nonlinear element and two linear oscillators. We show that the configuration of the connections of the linear oscillators in the two-mode self-oscillating system significantly affects its oscillation regimes and bifurcational transitions. In the case of the feedback loop including one oscillator, the two-mode system demonstrates the well-known effect of frequency entrainment, including bistability and hysteresis phenomena. If the feedback loop involves both linear oscillators, the entrainment effect disappears; however, two new complex regimes of quasi-periodicity and chaotic self-oscillations emerge. We present here the results of the bifurcation analysis of the multistability formation and transition to chaos. [ABSTRACT FROM AUTHOR]

In this article, we present a microcontroller implementation of the synchronization of two Van der Pol oscillators submitted to disturbances of the pulse-like type. Three coupling schemes are used: the classical linear proportional coupling, a power order coupling and an adaptive coupling. After obtaining the coupling coefficients for synchronization through numerical simulation, the microcontroller implementation is carried out using simulation based on Euler algorithm. Agreement is found between both simulation strategies. [ABSTRACT FROM AUTHOR]

Robust synchronization of master slave chaotic systems are considered in this work. First an approximate model of the error system is obtained using the ultra-local model concept. Then a Continuous Singular Terminal Sliding-Mode (CSTSM) Controller is designed for the purpose of synchronization. The proposed approach is output feedback-based and uses fixed-time higher order sliding-mode (HOSM) differentiator for state estimation. Numerical simulation and experimental results are given to show the effectiveness of the proposed technique. [ABSTRACT FROM AUTHOR]

The van der Pol(VDP) oscillator is an essential nonlinear oscillator, which has been extensively studied. However, the results of delayed generalized VDP oscillator are relatively less. The issue of bifurcation control for a delayed extended VDP oscillator is fully considered in this paper. Firstly, a nonlinear time-delayed feedback controller is carefully designed to control the dynamics for the proposed oscillator, and the results of stability of such oscillator are obtained by means of stability switches theory. Secondly, a general explicit formula is derived to establish the properties of Hopf bifurcation by utilizing the method of multiple scales. Moreover, the impact of feedback gain on the bifurcation point for the controlled oscillator is demonstrated numerically. Finally, the effectiveness of theoretical results is verified via a numerical example. [ABSTRACT FROM AUTHOR]

A real-time modeling and control architecture for coupled shear layer von Kármán instabilities on a wing section wake is developed. Previous wind tunnel experiments highlighted that these wake instabilities give rise to limit cycles on both local and resulting airloads. The two-state Van der Pol system is employed to model the self-sustained oscillations exhibited by the actual flow. Schemes featuring either one or two coupled Van der Pol oscillators in parallel are taken under consideration. Open-loop and closed-loop control laws to manipulate and reduce the self-sustained oscillations are extensively studied. An adaptive control scheme is developed to optimize the performance of the controller, according to the flow conditions. For the open-loop control scheme, the consistency with experimental tests performed on a wind tunnel model is shown. [ABSTRACT FROM AUTHOR]

This paper presents an alternative representation of a system of differential equations qualitatively showing the behavior of the biological rhythm of a crayfish during their transition from juvenile to adult stages. The model focuses on the interaction of four cellular oscillators coupled by diffusion of a hormone, a parameter is used to simulate the quality of communication among the oscillators, in biological terms, it measures developmental maturity of the crayfish. Since some quorum-sensing mechanism is assumed to be responsible for the synchronization of the biological oscillators, it is natural to investigate the possibility that the underlying diffusion process is not standard, i.e. it may be a so-called anomalous diffusion. In this case, it is well understood that diffusion equations with fractional derivatives describe these processes in a more realistic way. The alternative formulation of these equations contains fractional operators of Liouville-Caputo and Caputo-Fabrizio type. The numerical simulations of the equations reflect synchronization of ultradian rhythms leading to a circadian rhythm. The classical behavior is recovered when the order of the fractional derivative is . We discuss possible biological implications. [ABSTRACT FROM AUTHOR]

The primary resonance of van der Pol oscillator under fractional-order delayed negative feedback and forced excitation is studied. Firstly, the approximate analytical solution is obtained based on the averaging method, and it could be found that the fractional-order delayed feedback has not only the property of delayed velocity feedback but also that of delayed displacement feedback. Moreover, the amplitude-frequency equation for the steady-state solution is established, and its stability conditions are also obtained. Then, the results of the approximate analytical solution and numerical integration are compared and analyzed. The agreement between the two methods is very high, so that the correctness and accuracy of the approximate analytical solution are verified. Finally, the effects of all the parameters in the fractional-order delayed feedback on the amplitude-frequency curves are analyzed. It could be concluded that fractional-order delayed feedback has important influences on the dynamical behavior of van der Pol oscillator, which is very significant to the optimization and control of a similar system. [ABSTRACT FROM AUTHOR]

We aim to investigate bifurcation behaviors in a stochastic birhythmic van der Pol (BVDP) system subjected to delay self-control feedback. First, the harmonic approximation is adopted to drive the delay self-control feedback to state variables without delay. Then, Fokker-Planck-Kolmogorov (FPK) equation and stationary probability density function (SPDF) for amplitude are obtained by applying stochastic averaging method. Finally, dynamical scenarios of the change of delay self-control feedback as well as noise that markedly influence bifurcation performance are observed. It is found that: the big feedback strength and delay will suppress the large amplitude limit cycle (LC) while the relatively big noise strength facilitates the large amplitude LC, which imply the proposed regulation strategies are feasible. Interestingly enough, the inner LC is never destroyed due to noise. Furthermore, the validity of analytical results was verified by Monte Carlo simulation of the dynamics. [ABSTRACT FROM AUTHOR]

The phenomenon of synchronization (trapping) of coming waves by the resonant water area in a coastal zone of the sea found from the observed data is considered in the paper. Edge waves with the period of about 10.7 minutes are visually observed in sea level fluctuations near the village of Okhotskoye and the cape Ostri on the southeast coast of Sakhalin Island. These waves are synchronized with the resonance water area. It becomes apparent from the unlimited increase of a phase between the bottom stations installed at distance of about 7.5km. In relation to the phenomenon found, the problem of weak and periodic impact on regular self-oscillatory system - Van der Paul's oscillator - is considered. Good compliance between theoretical model and data of experiments is obtained. [ABSTRACT FROM AUTHOR]

This paper presents a new discretization technique for the forced Van der Pol oscillator with variable order derivatives. The study introduces the variable-order fractional time derivatives into the state-space model and investigates their influence upon the system dynamics. The resulting model of the variable-order fractional Van der Pol oscillator is solved and analyzed in the time domain. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a method to prove existence of several rapidly and slowly oscillating periodic solutions of a delayed Van der Pol oscillator. The proof is a combination of pen and paper analytic estimates, the contraction mapping theorem and a computer program using interval arithmetic. Using this approach we extend some existence results obtained by Nussbaum (Ann Mat Pura Appl 4(101):263-306, 1974). [ABSTRACT FROM AUTHOR]

The paper proposes a single degree of freedom oscillator in order to accurately represent the lateral force acting on a rigid floor due to human walking. As a pedestrian produces itself the energy required to maintain its motion, it can be modelled as a self-sustained oscillator that is able to produce: (i) self-sustained motion; (ii) a lateral periodic force signal; and (iii) a stable limit cycle. The proposed oscillator is a modification of hybrid Van der Pol–Duffing–Rayleigh oscillator, by introducing an additional nonlinear hardening term. Stability analysis of the proposed oscillator has been performed by using the energy balance method and the Lindstedt–Poincare perturbation technique. Model parameters were identified from the experimental force signals of ten pedestrians using the least squares identification technique. The experimental and the model generated lateral forces show a good agreement. [ABSTRACT FROM AUTHOR]

The fluid-structure interaction in vortex-induced vibrations of circular structures is modelled as a non-linear damping force. The Van der Pol and Rayleigh oscillators are classic examples of physical systems, which attain a limit cycle under self-induced vibrations. Their application for the case of vortex-induced vibrations has a theoretical justification in Marris, who found an analogy with the Magnus effect for a rotating cylinder, because the lift force on a vibrating cylinder is produced by the non-symmetric boundary layer separation from the top and bottom surfaces. In accordance with the Rayleigh formulation, Vickery&Basu propose for the aerodynamic damping a parabolic relationship function of the rms oscillation. This approach is recommended in many international codes (e.g. Eurocode) for the design of circular structures with regard to cross-wind vibrations, even though the predictions result extremely conservative. Recent wind tunnel experiments performed at WISt (Ruhr-Universität Bochum, Germany) on a circular cylinder in forced-oscillations evidenced a behaviour of the aerodynamic damping with positive curvature, which is in contrast with the negative curvature in Rayleigh oscillator. These experimental results require a critical analysis of the Vickery formulation and its theoretical background. [ABSTRACT FROM AUTHOR]

This paper employs a new analytical approach to model the influence of aerodynamic excitation on the dynamics of a bladed cascade at the flutter state. The flutter is an aero-elastic phenomenon that is linked to the interaction of the flow and the traveling deformation wave in the cascade when only the damping of the cascade changes. As a case study the dynamic properties of the five-blade-bunch excited by the running harmonic external forces and aerodynamic self-excited forces are investigated. This blade-bunch is linked in the shroud by means of the viscous-elastic damping elements. The external running excitation depends on the ratio of stator and rotor blade numbers and corresponds to the real type of excitation in the steam turbine. The aerodynamic self-excited forces are modeled by two types of Van der Pol nonlinear models. The influence of the interaction of both types of self-excitation with the external running excitation is investigated on the response curves. [ABSTRACT FROM AUTHOR]

Vortex-induced vibrations at lock-in conditions are modeled through generalized van der Pol-Duffing oscillators endowed with frequency-dependent coefficients, taking inspiration from fluid-elastic models. Accordingly, it is found that the limit-cycle amplitude and the non-linear frequency are mutually dependent (feedback effect), differently from the classic oscillator behavior. Consequently, the mechanical non-linearities, which are often believed to be unimportant, do affect the amplitude of motion. Examples concerning an ideal one degree-of-freedom van der Pol-Duffing oscillator and a two degree-of-freedom model, coarsely representative of a tower building, confirm the importance of this approach also from a technical point of view. Thus, non-linear geometric terms and modal interaction (even in non-resonant cases) can lead to non-negligible modifications of purely aeroelastic problems. [ABSTRACT FROM AUTHOR]

This paper presents the experimental investigation on synchronization of two Van der Pol oscillators with polynomial and delay unidirectional couplings. The intervals of coupling coefficients and delay leading to synchronization are determined experimentally using analog electronic circuits. Three cases are considered: autonomous Van der Pol oscillators, sinusoidally exited Van der Pol oscillators in the chaotic state and Van der Pol oscillators with two slowly sinusoidal excitations delivering periodic patterns of periodic pulses. It is found that the degree of the polynomial coupling reduces the intervals of coupling coefficients leading to synchronization and the delay affects the coupling intervals in a periodic way. The experimental results agree well with the results of the theoretical (mathematical and numerical) investigation. [ABSTRACT FROM AUTHOR]

We prove sharp, computable error estimates for the propagation of errors in the numerical solution of ordinary differential equations. The new estimates extend previous estimates of the influence of data errors and discretization errors with a new term accounting for the propagation of numerical round-off errors, showing that the accumulated round-off error is inversely proportional to the square root of the step size. As a consequence, the numeric precision eventually sets the limit for the pointwise computability of accurate solutions of any ODE. The theoretical results are supported by numerically computed solutions and error estimates for the Lorenz system and the van der Pol oscillator. [ABSTRACT FROM AUTHOR]

The behaviour of a space-modulated, so-called “argumental” oscillator, is studied. The oscillator is submitted to an external harmonic force, which is amplitude-modulated by the oscillator's position in space. An analytic expression of a stability criterion is given. Using the averaging method, an integrating factor and a Van der Pol representation in the (amplitude, phase)-space, an exact implicit analytic solution is given when there is no damping, and an approximate implicit analytic solution is given when there is damping, allowing the plotting of the separatrix curve. An attractor is identified. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the emergence of bursting dynamics with complex waveforms and their relation to periodic behavior in typical Van der pol-Duffing equation with fifth order polynomial stiffness nonlinearity, when the external force changes slowly with the variation of time. We exploit bifurcation characteristics of the fast subsystem using the slowly changing periodic excitation as a bifurcation parameter to show how the bursting oscillations are created in this model. We also identify that some regimes of bursting patterns are related to codimension two bifurcation type over a wide range of parameters. A subsequent two-parameter continuation reveals a transition in the bursting behavior from fold/fold hysteresis cycle to sup-Hopf/sup-Hopf or limit point cycle/sub-Hopf bursting type. Furthermore, the effects of external forcing item on bursting oscillations are investigated. For instance, the time interval between two adjacent spikes of bursting oscillations is dependent on the forcing frequency. Some numerical simulations are included to illustrate the validity of our study. [ABSTRACT FROM AUTHOR]

In this paper, the Bogdanov-Takens (B-T) and triple zero bifurcations are investigated for coupled van der Pol-Duffing oscillators with symmetry, in the presence of time delays due to the intrinsic response and coupling. Different from previous works, third order unfolding normal forms associated with B-T and triple zero bifurcations are needed, which are obtained by using the normal form theory of delay differential equations. Numerical simulations are also presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

We consider the response to uncorrelated noise and harmonic excitation of a birhythmic van der Pol-type oscillator. This system, as opposed to the standard van der Pol oscillator, is characterized by two stable orbits. The noisy oscillator can be analytically mapped, with the technique of stochastic averaging, onto an ordinary bistable system with a bistable (quasi)potential. The birhythmic oscillator can also be numerically characterized through the diagnostics of coherent resonance and the signal-to-noise-ratio. The analysis shows the presence of noise-induced coherent states, influenced by the different time scales of the oscillator. [ABSTRACT FROM AUTHOR]

This paper proposes a procedure for designing a fast terminal sliding-mode control involving an integral filter that guarantees a finite time of convergence to zero dynamics, robustness against polynomial bounded uncertainties, and external disturbances. It reduces the chattering effect by employing a state-dependent gain and adds an integral filter to smooth the input control signal. We suggest a simplified algorithm to design the controller. To test the performance of the algorithm, we use a Van der Pol chaotic oscillator synchronization with perturbations in the states. Finally, we compare the performance of the method by using the proposed controller with the filter turned-on and turned-off. [ABSTRACT FROM PUBLISHER]