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International audience; We report on the mechanical excitation of a 220 μm thick thermoplastic film in its amorphous state by the radiation pressure of light. By modulating a low power visible laser (from 100 to 600 mW) at low frequencies (below 100 Hz), we observe a deformation of the film interfaces. The phenomenon, that is independent of the laser wavelength, is amplified at a resonant frequency and reaches 0.68 μm. The deformation is reversible and varies linearly with the optical power. Using the damped oscillator model, we show that the resonant frequency depends on the surface tension of the film. The associated free energy is then compared with the energy lost, taking into account the contribution of the damping corresponding to the imaginary part of the Young's modulus.

Networks of optical oscillators simulating coupled Ising spins have been recently proposed as a heuristic platform to solve hard optimization problems. These networks, called coherent Ising machines (CIMs), exploit the fact that the collective nonlinear dynamics of coupled oscillators can drive the system close to the global minimum of the classical Ising Hamiltonian, encoded in the coupling matrix of the network. To date, realizations of large-scale CIMs have been demonstrated using hybrid optical-electronic setups, where optical oscillators simulating different spins are subject to electronic feedback mechanisms emulating their mutual interaction. While the optical evolution ensures an ultrafast computation, the electronic coupling represents a bottleneck that causes the computational time to severely depend on the system size. Here, we propose an all-optical scalable CIM with fully-programmable coupling. Our setup consists of an optical parametric amplifier with a spatial light modulator (SLM) within the parametric cavity. The spin variables are encoded in the binary phases of the optical wavefront of the signal beam at different spatial points, defined by the pixels of the SLM. We first discuss how different coupling topologies can be achieved by different configurations of the SLM, and then benchmark our setup with a numerical simulation that mimics the dynamics of the proposed machine. In our proposal, both the spin dynamics and the coupling are fully performed in parallel, paving the way towards the realization of size-independent ultrafast optical hardware for large-scale computation purposes., Comment: 7 pages, 3 figures

Some predictions of quantum mechanics are in contrast with the macroscopic realm of everyday experience, in particular those originated by the Heisenberg uncertainty principle, encoded in the non-commutativity of some measurable operators. Nonetheless, in the last decade opto-mechanical experiments have actualized macroscopic mechanical oscillators exhibiting such non-classical properties. A key indicator is the asymmetry in the strength of the motional sidebands generated in an electromagnetic field that measures interferometrically the oscillator position. This asymmetry is a footprint of the quantum motion of the oscillator, being originated by the non-commutativity between its ladder operators. A further step on the path highlighting the quantum physics of macroscopic systems is the realization of strongly non-classical states and the consequent observation of a distinct quantum behavior. Here we extend indeed the analysis to a squeezed state of a macroscopic mechanical oscillator embedded in an optical cavity, produced by parametric effect originated by a suitable combination of optical fields. The motional sidebands assume a peculiar shape, related to the modified system dynamics, with asymmetric features revealing and quantifying the quantum component of the squeezed oscillator motion.

We study the effects of noise and diffusion in an excitable slow-fast population system of the Leslie-Gower type. The phenomenon of noise-induced excitement is investigated in the zone of stable equilibria near the Andronov-Hopf bifurcation with the Canard explosion. The stochastic generation of mixed-mode oscillations is studied by numerical simulation and stochastic sensitivity analysis. Effects of the diffusion are considered for the spatially distributed variant of this slow-fast population model. The phenomenon of the diffusion-induced generation of spatial patterns-attractors in the Turing instability zone is demonstrated. The multistability and variety of transient processes of the pattern formation are discussed. © 2020 The Author(s) Published by the Royal Society. All rights reserved. Russian Science Foundation, RSF: 16-11-10098 Data accessibility. This article does not contain any additional data. Authors’ contributions. All authors contributed equally to this study. Competing interests. We declare we have no competing interests. Funding. The work was supported by Russian Science Foundation (grant no. 16-11-10098).

The dynamic transition from combustion noise to combustion instability was investigated experimentally in two laboratory-scale turbulent combustors (namely, swirl-stabilized and bluff-body-stabilized backward-facing-step combustors) by systematically varying the flow Reynolds number. We observe that the onset of combustion-driven oscillations is always presaged by intermittent bursts of high-amplitude periodic oscillations that appear in a near-random fashion amidst regions of aperiodic low-amplitude fluctuations. These excursions to periodic oscillations last longer in time as operating conditions approach instability and finally the system transitions completely into periodic oscillations. A continuous measure to quantify this bifurcation in dynamics can be obtained by defining an order parameter as the probability of the signal amplitude exceeding a predefined threshold. A hysteresis zone was observed in the bluff-body-stabilized configuration that was absent in the swirl-stabilized configuration. The recurrence properties of the dynamics of intermittent burst oscillations were quantified using recurrence plots and the distribution of the aperiodic phases was examined. From the statistics of these aperiodic phases, robust early-warning signals of an impending combustion instability may be obtained.

Earlier works on the smooth and discontinuous (SD) oscillator concentrated mainly on the time domain analysis using analytical, semi-analytical and numerical integration methods. In this paper, the frequency domain analysis of the SD oscillator subjected to harmonic excitation which is as important and giving further insight into the dynamics is carried out. Multi-Harmonic Balance Method (MHBM) in combination with arc length continuation is used to obtain the periodic solutions and their branches in the frequency domain for different values of the smoothing parameter α and exciting frequency ω. Stability of the periodic motions and bifurcation behavior are analyzed using the Floquet theory. For the discontinuous case, the oscillator is treated as a Filippov system and an event driven numerical integration method is used to obtain the response. For α > 1 , the dynamics of the SD oscillator is similar to that of the hardening Duffing oscillator, for α = 1 , it is like that of the Ueda oscillator and for 0 α 1 it is like that of the Duffing oscillator with double well potential. The SD oscillator exhibits period 1 solutions, higher order periodic solutions, chaotic solutions through symmetry breaking bifurcations, period doubling and boundary crises in different parameter ranges. Chaos is observed over a larger frequency range interspersed by narrow windows of higher order periodic solutions.

In this paper, we derive the exact solution of the cubic-quintic Duffing oscillator based on the use of Jacobi elliptic functions. We also showed that the exact angular frequency of this cubic-quintic Duffing equation is given in terms of the complete elliptic integral of the first kind. © 2012 Elsevier Inc.

In this work we consider the nonlinear equivalent representation form of oscillators that exhibit nonlinearities in both the elastic and the damping terms. The nonlinear damping effects are considered to be described by fractional power velocity terms which provide better predictions of the dissipative effects observed in some physical systems. It is shown that their effects on the system dynamics response are equivalent to a shift in the coefficient of the linear damping term of a Duffing oscillator. Then, its numerical integration predictions, based on its equivalent representation form given by the well-known forced, damped Duffing equation, are compared to the numerical integration values of its original equations of motion. The applicability of the proposed procedure is evaluated by studying the dynamics response of four nonlinear oscillators that arise in some engineering applications such as nanoresonators, microresonators, human wrist movements, structural engineering design, and chain dynamics of polymeric materials at high extensibility, among others. © 2013 Alex Elías-Zúñiga et al.

We report active electrical tuning of plasmon resonance of silver nanoprisms (Ag NPs) in the visible spectrum. Ag NPs are placed in close proximity to graphene which leads to additional tunable loss for the plasmon resonance. The ionic gating of graphene modifies its Fermi level from 0.2 to 1 eV, which then affects the absorption of graphene due to Pauli blocking. Plasmon resonance frequency and linewidth of Ag NPs can be reversibly shifted by 20 and 35 meV, respectively. The coupled graphene-Ag NPs system can be classically described by a damped harmonic oscillator model. Atomic layer deposition allows for controlling the graphene-Ag NP separation with atomic-level precision to optimize coupling between them. (C) 2016 Optical Society of America

We consider Liénard type and frequency-damped oscillator equations. Integrating factors and the associated first integrals are derived from the method to compute -symmetries and the associated reduction algorithm. The knowledge of a symmetry of the equation permits the determination of an integrating factor or a first integral by means of coupled first-order linear systems of partial differential equations. We will compare our results with those gained by the other methods.

A rational elliptic balance method is introduced to obtain exact and approximate solutions of nonlinear oscillators by using Jacobi elliptic functions. To illustrate the applicability of the proposed rational elliptic forms in the solution of nonlinear oscillators, we first investigate the exact solution of the non-homogenous, undamped Duffing equation. Then, we introduce first and second order rational elliptic form solutions to obtain approximate solutions of two nonlinear oscillators. At the end of the paper, we compare the numerical integration values of the angular frequencies with approximate solution results, based on the proposed rational elliptic balance method. © 2010 Elsevier Ltd. All rights reserved.

The probability structure of the response and energy harvested from a nonlinear oscillator subjected to white noise excitation is investigated by solution of the corresponding Fokker-Planck (FP) equation. The nonlinear oscillator is the classical double well potential Duffing oscillator corresponding to the first mode vibration of a cantilever beam suspended between permanent magnets and with bonded piezoelectric patches for purposes of energy harvesting. The FP equation of the coupled electromechanical system of equations is derived. The finite element method is used to solve the FP equation giving the joint probability density functions of the response as well as the voltage generated from the piezoelectric patches. The FE method is also applied to the nonlinear inductive energy harvester of Daqaq and the results are compared. The mean square response and voltage are obtained for different white noise intensities. The effects of the system parameters on the mean square voltage are studied. It is observed that the energy harvested can be enhanced by suitable choice of the excitation intensity and the parameters. The results of the FP approach agree very well with Monte Carlo Simulation (MCS) results. � 2013 Elsevier Ltd.

Nonlinear oscillators subjected to colored Gaussian/non-Gaussian excitations are modelled through a set of three coupled first-order stochastic differential equations by representing the excitation as a first-order filtered white noise. A 3-D finite element (FE) formulation is developed to solve the corresponding 3-D Fokker Planck (FP) equations. The joint probability density functions of the state variables, obtained as a solution of the FP equation, are typically non-Gaussian and are used for computing the crossing statistics of the response - an essential metric for time variant reliability analysis. The method is illustrated through a noisy Lorenz attractor and a Duffing oscillator subjected to additive colored noise. The increase in state-space dimension when the Duffing oscillator is additionally excited with a parametric Gaussian noise is effectively handled by using stochastic averaging to reduce the state-space dimension. Investigations are carried out to examine the accuracy of the FE method vis-a-vis Monte Carlo simulations. The proposed method is observed to be computationally significantly cheaper for these three problems. � 2014 Elsevier Ltd. All rights reserved.

The standard approach to non-equilibrium thermodynamics describes transport in terms of generalised forces and coupled currents, a typical example being the Fourier law that relates temperature gradient to the heat flux. Here we demonstrate the presence of persistent energy currents in a lattice of classical nonlinear oscillators with uniform temperature and chemical potential. In strong analogy with the well known Josephson effect, the currents are generated only by the phase differences between the oscillators. The phases play the role of additional thermodynamical forces, that drive the system out of equilibrium. Our results apply to a large class of oscillators and indicate novel ways to practically control the propagation of coupled currents and rectification effects in many different devices. They also suggest a simple, macroscopic setup for studying a phenomenon which hitherto have only been observed in microscopic, quantum-mechanical systems., Comment: 5 pages, 4 figures

We demonstrate theoretically and experimentally that Fano resonances can be obtained in terahertz metamaterials that are composed of periodic continuous metallic wires dressed with periodic split ring resonators. An asymmetric Fano lineshape has been found in a narrow frequency range of the transmission curve. By using a transmission line combined with lumped element model, we are able to not only fit the transmission spectra of Fano resonance which is attributed to the coupling and interference between the transmission continuum of continuous metallic wires and the bright resonant mode of split ring resonators, but also reveal the capacitance change of the split ring resonators induced frequency shift of the Fano resonance. Therefore, the proposed theoretical model shows more capabilities than conventional coupled oscillator model in the design of Fano structures. The effective parameters of group refractive index of the Fano structure are retrieved, and a large group index more than 800 is obtained at the Fano resonance, which could be used for slow light devices. (C) 2014 Optical Society of America

We study the dynamical response of an asymmetric forced, damped Helmholtz-Duffing oscillator by using Jacobi elliptic functions, the method of elliptic balance, and Fourier series. By assuming that the modulus of the elliptic functions is slowly varying as a function of time and by considering the primary resonance response of the Helmholtz-Duffing oscillator, we derived an approximate solution that provides the time-dependent amplitude-frequency response curves. The accuracy of the derived approximate solution is evaluated by studying the evolution of the response curves of an asymmetric Duffing oscillator that describes the motion of a damped, forced system supported symmetrically by simple shear springs on a smooth inclined bearing surface. We also use the percentage overshoot value to study the influence of damping and nonlinearity on the transient and steady-state oscillatory amplitudes. © 2013 Alex Elías-Zúñiga and Oscar Martínez-Romero.

We introduce a nonlinearization procedure that replaces the system potential energy by an equivalent representation form that is used to derive analytical solutions of strongly nonlinear conservative oscillators. We illustrate the applicability of this method by finding the approximate solutions of two strongly nonlinear oscillators and show that this procedure provides solutions that follow well the numerical integration solutions of the corresponding equations of motion. © 2013 Alex Elías- Zúñiga and Oscar Martínez-Romero.

The enhanced cubication method is applied to develop approximate solutions for the most common nonlinear oscillators found in the literature. It is shown that this procedure leads to amplitude-time response curves and angular frequency values with maximum relative errors lower than those found by previously developed approximate solutions. © 2013 Alex Elías-Zúñiga and Oscar Martínez-Romero.

The cubication and the equivalent nonlinearization methods are used to replace the original Duffing-harmonic oscillator by an approximate Duffing equation in which the coefficients for the linear and cubic terms depend on the initial oscillation amplitude. It is shown that this procedure leads to angular frequency values with a maximum relative error of 0.055. This value is 21 lower than the relative errors attained by previously developed approximate solutions. © 2012 Alex Elías-Zúiga et al.

Recently, necessary and sufficient conditions for output synchronization of linear systems via diffusive couplings have been reported. In this paper, we study the case when such conditions are not satisfied and exact synchronization is impossible. In particular, we study two kinds of heterogeneous linear networks: (i) non-identical harmonic oscillators and (ii) double-integrators. We show that static diffusive couplings render heterogeneous networks of harmonic oscillators asymptotically stable. Networks of non-identical double-integrators, in contrast, are not asymptotically stable but synchronize with bounded synchronization error depending on the network topology and the heterogeneity in the agent dynamics. Numerical examples illustrate the results. QC 20130215

This paper proposes a derivative-free two-stage extended Kalman filter (2-EKF) especially suited for state and parameter identification of mechanical oscillators under Gaussian white noise. Two sources of modeling uncertainties are considered: (1) errors in linearization, and (2) an inadequate system model. The state vector is presently composed of the original dynamical/parameter states plus the so-called bias states accounting for the unmodeled dynamics. An extended Kalman estimation concept is applied within a framework predicated on explicit and derivative-free local linearizations (DLL) of nonlinear drift terms in the governing stochastic differential equations (SDEs). The original and bias states are estimated by two separate filters; the bias filter improves the estimates of the original states. Measurements are artificially generated by corrupting the numerical solutions of the SDEs with noise through an implicit form of a higher-order linearization. Numerical illustrations are provided for a few single- and multidegree-of-freedom nonlinear oscillators, demonstrating the remarkable promise that 2-EKF holds over its more conventional EKF-based counterparts. DOI: 10.1061/(ASCE)EM.1943-7889.0000255. (C) 2011 American Society of Civil Engineers.

The sensitivity of laser interferometers can be pushed into regimes that enable the direct observation of the quantum behaviour of mechanical oscillators. In the past, membranes with subwavelength thickness (thin films) have been proposed to be high-mechanical-quality, low-thermal-noise oscillators. Thin films from a homogeneous material, however, generally show considerable light transmission accompanied by heating due to light absorption, which potentially limits quantum opto-mechanical experiments, in particular at low temperatures. In this paper, we experimentally analyse a Michelson-Sagnac interferometer including a translucent silicon nitride (SiN) membrane with subwavelength thickness. We found that such an interferometer provides an operational point that is optimally suited for quantum opto-mechanical experiments with translucent oscillators. In the case of a balanced beam splitter of the interferometer, the membrane can be placed at a node of the electromagnetic field, which simultaneously provides lowest absorption and optimum laser noise rejection at the signal port. We compare the optical and mechanical models of our interferometer with experimental data and confirm that the SiN membrane can be coupled to a laser power of the order of 1Wat 1064 nm without significantly degrading the membrane's quality factor of the order of 106, at room temperature. DFG/EXC/QUEST IMPRS Alexander von Humboldt Foundation

[EN] We consider the numerical integration of the Gross-Pitaevskii equation with a potential trap given by a time-dependent harmonic potential or a small perturbation thereof. Splitting methods are frequently used with Fourier techniques since the system can be split into the kinetic and remaining part, and each part can be solved efficiently using fast Fourier transforms. Splitting the system into the quantum harmonic-oscillator problem and the remaining part allows us to get higher accuracies in many cases, but it requires us to change between Hermite basis functions and the coordinate space, and this is not efficient for time-dependent frequencies or strong nonlinearities. We show how to build methods that combine the advantages of using Fourier methods while solving the time-dependent harmonic oscillator exactly (or with a high accuracy by using a Magnus integrator and an appropriate decomposition). © 2011 American Physical Society., We would like to acknowledge the referees for their careful reading and suggestions. The authors acknowledge the support of the Generalitat Valenciana through the project GV/2009/032. The work of S. B. has also been partially supported by Ministerio de Ciencia e Innovacion (Spain) under the coordinated project MTM2010-18246-C03 (cofinanced by the ERDF of the European Union).

In this paper, an order reduction technique for higher-dimensional nonlinear oscillator models, based on a center manifold approach, is presented. By modeling the oscillator circuit in the higher-dimensional state space, influences of parasitic elements and of structural extensions of the oscillator architecture on the dynamical system behavior can be examined. Using the proposed order reduction technique, a generalized second order model will be derived, which includes selected design parameters of the higher order model. By using an Andronov-Hopf bifurcation analysis, the reduced system can be studied with respect to stability as well as the amplitude and frequency of the individual state variables. The concept is applied to the design of LC-tank VCOs., Im Rahmen dieser Arbeit wird ein Konzept zur Ordnungsreduktion von höherdimensionalen nichtlinearen Oszillatormodellen vorgestellt. Hierbei werden zwei wesentliche Ziele verfolgt. Zum einen wird eine höherdimensionale Modellierung der Oszillatorschaltung verwendet. Hierdurch lassen sich die Einflüsse parasitärer Effekte sowie struktureller Erweiterungen auf das dynamische Verhalten des Systems berücksichtigen. Zum anderen wird durch eine anschließende Ordnungsreduktion über die Methode der Zentrumsmannigfaltigkeit eine zweidimensionale Systembeschreibung erzeugt, deren wesentliche Dynamik derjenigen des höherdimensionalen Systems entspricht. Durch diese, in der Ordnung reduzierte, nichtlineare und parameterabhängige Systembeschreibung wird die Anwendbarkeit nichtlinearer Analysemethoden ermöglicht bzw. vereinfacht. Mit der Anwendung der Andronov-Hopf-Bifurkationsanalyse auf das reduzierte System lässt sich eine Stabilitätsuntersuchung durchführen sowie die Amplitude und Frequenz aller Zustandsgrößen approximieren. Das vorgestellte Konzept wird anhand des Beispielsystems eines LC-Tank-VCOs durchgeführt.

A crucial insight into handwriting dynamics is embodied in the idea that stable, robust handwriting movements correspond to attractors of an oscillatory dynamical system. We present a phase dynamic model of visuomotor performance involved in copying simple oriented lines. Our studies on human performance in copying oriented lines revealed a systematic error pattern in orientation of drawn lines, i.e., lines at certain orientation are drawn more accurately than at other values. Furthermore, human subjects exhibit "flips" in direction at certain characteristic orientations. It is argued that this flipping behavior has its roots in the fact that copying process is inherently ambiguous-a line of given orientation may be drawn in two different (mutually opposite) directions producing the same end result. The systematic error patterns seen in human copying performance is probably a result of the attempt of our visuomotor system to cope with this ambiguity and still be able to produce accurate copying movements. The proposed nonlinear phase-dynamic model explains the experimentally observed copying error pattern and also the flipping behavior with remarkable accuracy. � 2009 Springer-Verlag.

The response of a non-linear mechanical oscillator, depending on the type of non-linearity, to an excitation of particular harmonic might contain sub- and super-harmonics, transients and chaotic components. The conventional frequency response function, used to completely characterise the input-output behaviour of a linear mechanical oscillator, not only does not capture the above characteristics but also, in case of broadband excitation, fails to attribute them to individual frequencies. In this paper we apply the Choi-Williams time frequency distribution to simulated data of a single and three degree of freedom Duffing oscillator in order to assess its input-output relationship. © (2006) by the Katholieke Universiteit Leuven Department of Mechanical Engineering All rights reserved. 5 2863 2872 2863-2872

For an attractive trapped Bose-Einstein condensate an imaginary three-body recombination loss term and an imaginary linear source term are usually included in the Gross-Pitaevskii (GP) equation for a proper account of dynamics. Under the action of an impulsive force, generated by suddenly changing the atomic interaction or the trapping potential, the solution of this complex GP equation for attractive interaction is found to lead to a very long term chaotic oscillation., Comment: RevTeX 4, 5 pages with 4 figures

We determine the quantitative effect, in the maximum number of particles and other static observables, due to small anharmonic terms added to the confining potential of an atomic condensed system with negative two-body interaction. As an example of how a cubic or quartic anharmonic term can affect the maximum number of particles, we consider the trap parameters and the results given by Roberts et al. [Phys. Rev. Lett. 86, 4211 (2001)]. However, this study can be easily transferred to other trap geometries to estimate anharmonic effects., Comment: Total of 5 pages, 3 figures and 1 table. To appear in Phys. Rev. A

Date of Conference: 27-29 Aug. 1997 Conference name: 1st International Conference, Control of Oscillations and Chaos Proceedings, 1997 In this work we consider the control of forced chaotic oscillators. To obtain any desirable behavior, the system parameters are effectively modified using state feedback. The system states used in the feedback are estimated through a nonlinear observer. The application of the proposed method is illustrated for Duffing and Van der Pol oscillators.

Chaotic motion of a harmonically excited square prism modelled as a Duffing oscillator and kept in fluid flow is considered. The fluid dynamic forces contribute additional non-linear terms to the inherent non-linearity of the system. The harmonically excited oscillator without the fluid flow exhibits three types of steady state periodic motions, dual period 1 motions and period 3 motion without dual. The flow changes the nature of motion of the oscillator. With the flow velocity as bifurcation parameter, the system exhibits period 1 motion without dual, bifurcating into dual period 1 motions which period double into dual period 2, 4, 8 motions, etc., leading to chaos as the flow velocity is increased. Lyapunov exponents for different flow velocities are computed which show the bifurcation points. The harmonic balance method is used to obtain approximate solutions for the periodic motions and to predict the period doubling bifurcations by a stability analysis.

The active control of flow-induced oscillations, specifically, vortex-induced oscillations of circular cylinders and galloping oscillations of circular cylinders and galloping oscillations of a square prism, are considered. In the case of vortex-induced oscillations, the vibrating cylinder is modeled as a single-degree-of-freedom (sdof) linear damped oscillator and the fluid motion as a Rayleigh oscillator. Galloping oscillations are modeled as a sdof limit-cycle system with cubic and higher order non-linearities in the velocity term. In formulating the control method it is assumed that the fluid interaction is an external excitation/disturbance on the structure. A suitable disturbance model is chosen based on the spectral content of the disturbance. For the problems considered here, the disturbance model assumes the form of a harmonic oscillator the frequency of which, in the case of vortex-induced oscillations, is equal to the natural frequency of the structure or the "lock-in" frequency, and in the case of galloping oscillations, is equal to the limit-cycle frequency. A disturbance counteracting control law is synthesized by using a Luenberger observer incorporating the structure and disturbance dynamics. It is shown that using this approach one is able to completely suppress the fluid-induced oscillations of the structure. ? 1993 Academic Press. All rights reserved.