Nonlinear Duffing Research Articles

Enhancing entanglement and steering in a hybrid atom-optomechanical system via Duffing nonlinearity

Abstract We introduce a novel scheme for achieving quantum entanglement and Einstein-Podolsky-Rosen (EPR) steering between an atomic ensemble and a mechanical oscillator within a hybrid atom-optomechanical system. The system comprises an optical cavity, a two-level atomic ensemble, and a mechanical resonator that possess Duffing nonlinearity. The interaction between these components is mediated by the cavity mode, which is driven by an external laser. Our findings indicate that optimizing the coupling strengths between photons and phonons, as well as between atoms and cavity, leads to maximal entanglement and EPR steering. The amplitude of the driving laser plays a pivotal role in enhancing the coupling between photons and phonons, and the system maintains robust entanglement and EPR steering even under high dissipation case, thereby mitigating the constraints on initial conditions and parameter precision. Remarkably, the Duffing nonlinearity enhances the system’s resistance to thermal noise, ensuring its stability and entanglement protection. Our analysis of EPR steering conditions reveals that the party with lower dissipation exhibits superior stability and a propensity to steer the party with higher dissipation. These discoveries offer novel perspectives for advancing quantum information processing and communication technologies.

Quantum entanglement assisted via Duffing nonlinearity

We propose a scheme to enhance quantum entanglement in an optomechanical system by exploiting the so-called Duffing nonlinearity. Our model system consists of two mechanically coupled mechanical resonators, both driven by an optical field. One resonator supports Duffing nonlinearity, while the other does not. The resonators are coupled to each other via the so-called phonon-hopping mechanism. The hopping rate is θ-phase dependent that induces exceptional point singularities in the system. Interestingly, while the resonator with Duffing nonlinearity exhibits vanishing entanglement with light, we observe an increase in entanglement between light and the other mechanical resonator. This enhanced entanglement persists longer against thermal fluctuations compared to the one without the nonlinearity. Additionally, this entanglement features a sudden death and revival phenomenon, where the peaks happen at multiples of θ=π2. This work opens another avenue for exploiting nonlinear resources to generate strong quantum entanglement, paving the way for advancements in quantum information processing, quantum sensing, and quantum computing within complex systems. Published by the American Physical Society 2024

Combined Impact of Multidelay Feedback Number and Interval: A Novel Mechanism for Controlling the Stability of Stochastic Duffing Systems

In this paper, we study a class of nonlinear stochastic Duffing oscillators with multidelay feedback. We propose an effective reduction approach with the help of center manifold theory and stochastic averaging method. Taking the initial time-delay [Formula: see text] as the parameter, we reduce the original system to a one-dimensional averaged Itô equation. Our analysis reveals that the original system exhibits stochastic bifurcations, including stochastic D and P bifurcations. Once we have a clear understanding of the bifurcation structure, we can use this knowledge to choose appropriate system parameters and place the system in the desired state. For instance, by adjusting the initial time-delay [Formula: see text] of the control system, we can stabilize the system and achieve the desired outcome. Numerical simulations also verify the theoretical results. With appropriate parameter choices, multiple time delays can destabilize the equilibrium and promote chaotic behaviors, and can also lead to more stable dynamical behavior. Remarkably, we discovered that increasing the interval of time delays and feedback numbers can enhance system stability. It may potentially serve as a novel mechanism for stabilizing stochastic systems. The study provides a solid theoretical foundation for exploring stochastic systems subject to complex time-delay feedback control, and offers a valuable framework for related fields.

Numerical simulation of fractional-order Duffing system with extended Mittag-Leffler derivatives

In this paper, we studied the dynamics of a nonlinear fractional-order Duffing system combined with Mittag-Leffler derivatives in order to provide dynamic behaviors different from existing ones. The Mittag-Leffler derivative is a generalized version of the exponential kernel derivative. To achieve this goal, we introduced a modified extension to higher-order Mittag-Leffler derivatives to overcome the initialization problem. Moreover, we discussed some properties and relationships of the studied derivatives. Then we presented numerical schemes to handle fractional extensions of the considered oscillatory system including the Mittag-Leffler and the Caputo derivatives. Numerical simulations are carried out and the resulting simulation dynamics of the studied fractional oscillatory system are compared.

Long short-term memory (LSTM) neural networks for predicting dynamic responses and application in piezoelectric energy harvesting

Long Short-Time Memory (LSTM) deep neural networks are capable of learning order dependence in sequence problems and capturing long-term, non-linear temporal dependencies between the input and out of a system. With the long-term vision to model dynamical systems to which analytical or numerical methods are impossible or difficult to apply, this paper presents a study of modeling system dynamics and predicting responses using the LSTM networks, which have demonstrated excellent capability in predicting single-mode responses in a prior study. However, the LSTM network exhibits difficulties in modeling and predicting multi-mode responses accurately. To resolve the multi-mode issue, this paper presents an approach that obtains an equivalent network consisting of a set of sub-networks learned on isolated modes, and demonstrates its effectiveness on a simulated 2-degree-of-freedom mass-spring-damper system of nonlinear Duffing springs. The second part of the paper is focused on the application of the proposed approach in piezoelectric energy harvesting. Experiments are conducted on a harvester subjected to random base-motion excitation and exhibiting nonlinearity in its multi-mode response. Both the direct and mode-separation LSTM modeling approaches are applied to predict the output voltage given a random base-motion excitation. The mode-separation approach outperforms the direct approach significantly, and yields an excellent match between the actual and predicted responses. Specifically, for a test electrical voltage response of RMS value 0.2241 V, the difference between the actual test and predicted responses by using the mode-separation approach has an RMS value of 0.0504 V, compared to 0.1645 V obtained by using the direct LSTM approach. It is also much lower than the RMS value of 0.1835 V obtained by using the attention-based LSTM network, another comparison method. Leveraging a deep learning strategy, the validated approach opens up opportunities for accurately modeling energy harvesting systems of high complexities and/or strong nonlinearities.

Pseudo almost periodic solutions for a class of nonlinear Duffing equations on time scales

Pseudo almost periodic solutions for a class of nonlinear Duffing equations on time scales

The Presence of Chaos in a Viscoelastic Harmonically Forced Von Mises Truss

Abstract This work investigates how viscoelasticity affects the dynamic behavior of a lumped-parameter model of a bistable von Mises truss. The system is controlled by a linear first-order equation and a second-order nonlinear Duffing equation with a quadratic nonlinearity that governs mechanical behavior. The second-order equation controls mechanical oscillations, while the linear first-order equation controls viscoelastic force evolution. Combined, the two equations form a third-order jerk equation that controls system dynamics. Viscoelasticity adds time scales and degrees-of-freedom to material behavior, distinguishing it from viscosity-only systems. Due to harmonic excitation, the system exhibits varied dynamic responses, from periodic to quasi-periodic to chaotic. We explore the dynamics of a harmonically forced von Mises truss with viscous damping to address this purpose. We demonstrate this system's rich dynamic behavior due to driving amplitude changes. This helps explain viscoelastic system behavior. A viscoelastic unit replaces the viscous damper, and we show that, although viscous damping merely changes how fast the trajectory decays to an attractor, viscoelasticity modifies both the energy landscape and the rate of decay. In a conventional linear solid model, three viscoelastic parameters control the system's behavior instead of one, as in pure viscous damping. This adds degrees-of-freedom that affect system dynamics. We present the parameter space for chaotic behavior and the shift from regular to irregular motion. Finally, Melnikov's criteria identify the regular-chaotic threshold. The system's viscous and elastic components affect the chaotic threshold amplitude

Nonlinear dynamics and forced vibrations of simply-supported fractional viscoelastic microbeams using a fractional differential quadrature method

In this paper, free and forced nonlinear vibrations of fractional viscoelastic microbeams are modeled based on Euler-Bernoulli theory, the Von Karman’s nonlinear strain relations, the modified couple stress theory (MCST), and the fractional Kelvin-Voigt viscoelastic model. In the present work, the nonlinear-fractional order governing equations are discretized in the space domain by Galerkin’s method. Two different approaches are introduced to solve the resulting nonlinear fractional-order Duffing equation in the time domain. In the first approach, a time marching fractional finite difference method is presented to compute the transient time response starting from the initial conditions. This approach is time consuming if the frequency-amplitude curves of steady state response are required since it provides just one point on the frequency-amplitude curve. More importantly, this approach usually converges only for the stable branch with smallest amplitude of frequency-amplitude curve. In the second approach, we introduce a novel fractional differential quadrature method (FDQM) to discretize the fractional Duffing equation and apply a pseudo-arc length algorithm to directly construct the frequency-amplitude curves. Effects of the fractional-order, linear and nonlinear viscoelasticity coefficients, viscous damping parameter, microstructure parameters, and the micro-beam thickness on the nonlinear dynamics of the viscoelastic micro-beam are analyzed numerically. Numerical results show that each of these parameters can change natural frequency and/or the damping behavior of the structure. The present model can be used for designing and analyzing the nonlinear microstructure viscoelastic beam under dynamic loads.

Resonance study of fractional-order strongly nonlinear duffing systems

Resonance study of fractional-order strongly nonlinear duffing systems

Nonlinearity enhanced mode localization in two coupled MEMS resonators

As an intriguing phenomenon in coupled dynamic systems, mode localization has garnered considerable attention in areas such as vibration control, sensing, etc. This paper investigates mode localization in two coupled MEMS resonators in both linear and nonlinear cases through experimental and theoretical analyses. The relationship of amplitude variation or energy transfer between two resonators in mode localization is unveiled. Subsequently, considering the influence of softening Duffing nonlinearity, we derive the scaling factor expression. For the first time, the experimental and theoretical results reveal that the presence of softening Duffing nonlinearity significantly enhances the sensitivity of the amplitude ratio compared to the linear case. Finally, a charge sensing scheme is proposed that utilizes nonlinear mode localization. This innovative scheme outperforms its linear counterpart, achieving an almost 5-fold increasement in charge sensing resolution. This advancement holds the potential to design high-performance sensors with significantly improved sensitivity.

One-to-two internal resonance in a micro-mechanical resonator with strong Duffing nonlinearity

This paper investigates the implementation of 1:2 internal resonance (InRes) in a clamped–clamped stepped beam resonator with a strong Duffing effect, focusing on its potential for frequency stabilization in micro-electro-mechanical systems (MEMS) resonators. InRes can arise in a nonlinear system of which mode frequencies are close to an integer ratio, facilitating the internal exchange of energy from an externally driven mode to an undriven mode. The presence of 1:2 InRes and Duffing hardening nonlinearity can result in frequency saturation phenomena, leading to a flat amplitude-frequency response range, which forms the basis for frequency stabilization. The stepped beam resonator design, combined with thermal frequency tuning, enables precise alteration of the frequency ratio between the second and third flexural modes required to achieve the desired 1:2 ratio for InRes. Experimental characterization and theoretical analysis revealed that frequency mismatch plays a significant role, with larger mismatch conditions leading to stronger energy exchange and a wider range of drive force for frequency saturation. The study highlights the frequency saturation mechanism utilizing 1:2 InRes and emphasizes the advantage of Duffing nonlinearity and larger intermodal frequency mismatch for broader frequency stabilization, providing valuable insights for the design and optimization of MEMS resonators.

Contact acoustic nonlinearity and local damage resonance for the detection of kissing bonds in structural adhesive joints

Adhesively bonded joints are susceptible to contamination of surfaces during manufacture and environmental deterioration in real operating conditions. These may cause the generation of so-called “kissing bonds” that can dramatically alter the strength of the joint leading to premature failure. Nonlinear acousto-ultrasonic (AU) techniques have shown great potential for monitoring kissing bonds with piezoelectric sensors permanently installed on the structural joint, thus enabling online and in-situ inspection. This paper investigated the combined contact acoustic nonlinearity (CAN) and local damage resonance (LDR) effects in the presence of kissing bonds in adhesive joints. Experimental nonlinear AU tests showed the formation of LDR frequency down-shifts and “jumps” of the fundamental damage resonance in adhesively bonded aluminium joints, with the kissing bond located internally to the overlapping region between the two adherends. These results were supported by a theoretical model based on the solution of the nonlinear Duffing's equation, under the assumption that the debonded region is a damped nonlinear harmonic oscillator subject to harmonic forcing. The Harmonic Balance Method was used to solve the nonlinear differential problem, showing the generation of frequency down-shifts via the dependence of the ratio between the excitation and LDR frequencies with the amplitude of the fundamental damage resonance. Additionally, two-dimensional finite element simulations using a reduced order model based on the Craig-Bampton technique were carried out to support experimental AU tests for the identification of the LDR frequency and the generation of nonlinear resonance effects. Good agreement between analytical, numerical, and experimental results revealed that a monitoring approach combining CAN and LDR is an extremely efficient and sensitive tool for ensuring integrity and safety of structural adhesive joints.

An Averaging Method for Phase Noise Studies of Quartz Phononic Frequency Combs.

A fast and accurate averaging method was derived and developed for the analysis and design of quartz phononic frequency combs. The phononic frequency combs were obtained from a pair of coupled nonlinear Duffing equations for quartz resonators by solving the equations in the time domain and performing a fast Fourier transformation (FFT) of the steady-state vibrations of the time series. Noise simulations were added to the drive frequency to study noise transfer characteristics between the drive signal and the resonances of the phononic frequency combs produced in 100-MHz quartz shear-mode resonators. Our new method averaged out the carrier frequency, thus allowing for a fast and efficient computation at parts per million accuracies of noise close to the carrier ( ∼ 10 Hz). The goal of our study was to develop methods and resonator requirements for engineering the properties of the phononic frequency combs for low-noise clock applications.

Programmable hardening and softening cubic inductive shunts for piezoelectric structures: Harmonic balance analysis and experiments

It has been well explored in the literature that intentionally introduced (designed) nonlinearities offer significant advantages in vibration absorption and isolation problems. In the context of using piezoelectric shunt damping for vibration attenuation in flexible structures, synthetic impedance circuits with digital control enable precise tuning capabilities and eliminate the need for bulky analog circuit components when targeting low frequencies, and they have been studied mostly for linear circuits. In this work, we explore nonlinear synthetic impedance-based Duffing shunts with hardening and softening nonlinearities, connected to a cantilever under base excitation in its geometrically and materially linear regime. We demonstrate a shunt of nonlinear impedance equivalent to a positive or negative cubic inductor connected in parallel to a linear inductor and a linear resistor. This shunt, along with the inherent piezoelectric capacitance, provides digital analog of a hardening- or softening-type nonlinear stiffness, linear stiffness, linear damping, and mass, as a synthetic Duffing circuit. Electromechanical modeling of a piezoelectric cantilevered beam shunted to the Duffing oscillator with hardening or softening inductance is performed. Numerical simulations for single-term and multi-term harmonic balance solutions are performed for periodic response. These approximate analytical predictions are compared against time-domain numerical simulations and are experimentally validated for various mechanical excitation amplitudes and nonlinear coefficients. Overall, a very good agreement is obtained between the model simulations and experimental results, in addition to demonstrating the unique digital programming capabilities of this nonlinear shunt.

Pell-Lucas polynomials for numerical treatment of the nonlinear fractional-order Duffing equation

Abstract The nonlinear fractional-order cubic-quintic-heptic Duffing problem will be solved through a new numerical approximation technique. The suggested method is based on the Pell-Lucas polynomials’ operational matrix in the fractional and integer orders. The studied problem will be transformed into a nonlinear system of algebraic equations. The numerical expansion containing unknown coefficients will be obtained numerically via applying Newton’s iteration method to the claimed system. Convergence analysis and error estimates for the introduced process will be discussed. Numerical applications will be given to illustrate the applicability and accuracy of the proposed method.

Bayesian Estimation of Experimental Parameters in Stochastic Inertial Systems: Theory, Simulations, and Experiments with Objects Levitated in Vacuum

High-quality nanomechanical oscillators can sensitively probe force, mass, or displacement in experiments bridging the gap between the classical and quantum domain. Dynamics of these stochastic systems is inherently determined by the interplay between acting external forces, viscous dissipation, and random driving by the thermal environment. The relevance of inertia then dictates that both position and momentum must, in principle, be known to fully describe the system, which makes its quantitative experimental characterization rather challenging. We introduce a general method of Bayesian inference of the force field and environmental parameters in stochastic inertial systems that operates solely on the time series of recorded noisy positions of the system. The method is first validated on simulated trajectories of model stochastic harmonic and anharmonic oscillators with damping. Subsequently, the method is applied to experimental trajectories of particles levitating in tailored optical fields and used to characterize the dynamics of particle motion in a nonlinear Duffing potential, a static or time-dependent double-well potential, and a nonconservative force field. The presented inference procedure does not make any simplifying assumptions about the nature or symmetry of the acting force field and provides robust results with trajectories 2 orders of magnitude shorter than those typically required by alternative inference schemes. In addition to being a powerful tool for quantitative data analysis, it can also guide experimentalists in choosing appropriate sampling frequency (at least 20 measured points per single characteristic period) and length of the measured trajectories (at least 10 periods) to estimate the force field and environmental characteristics with a desired accuracy and precision.

Nonlinear nanomechanical resonators approaching the quantum ground state

It is an open question whether mechanical resonators can be made nonlinear with vibrations approaching the quantum ground state. This requires the engineering of a mechanical nonlinearity far beyond what has been realized so far. Here we discover a mechanism to boost the Duffing nonlinearity by coupling the vibrations of a nanotube resonator to single-electron tunnelling and by operating the system in the ultrastrong-coupling regime. We find that thermal vibrations become highly nonlinear when lowering the temperature. The average vibration amplitude at the lowest temperature is 13 times the zero-point motion, with approximately 42% of the thermal energy stored in the anharmonic part of the potential. Our work may enable the realization of mechanical Schrödinger cat states, mechanical qubits and quantum simulators emulating the electron–phonon coupling.

Research on Wireless Energy Transmission Systems of Coupled Nonlinear Resonator

Due to environmental interference, device aging, and the strict requirements for the relative position of the transceiver coil during charging, we always hope that the output voltage and power can remain stable when the transmission distance of the transmitting and receiving coils changes in the largest range possible. However, once the relative position of the transceiver coil of the traditional linear magnetic coupling resonance WPT system changes, that is, when the transmission distance is not optimal, the transmission performance of the system will be seriously degraded. In practical applications, the charging equipment inevitably moves, and the harsh transmission conditions limit the wide application of WPT technology. To solve this problem, based on the nonlinear characteristics presented by the nonlinear Duffing resonance model, this paper proposes to use a reverse series varactor at the receiving end by adjusting the reverse bias voltage of the varactor. This method compensates for the transmission performance attenuation caused by the change of the coupling coefficient of the transceiver coil to maintain a high level of transmission performance when the relative distance between the transceiver coils is as large as possible.

Dynamical analysis of a damped harmonic forced duffing oscillator with time delay

This paper is concerned with a time-delayed controller of a damped nonlinear excited Duffing oscillator (DO). Since time-delayed techniques have recently been the focus of numerous studies, the topic of this investigation is quite contemporary. Therefore, time delays of position and velocity are utilized to reduce the nonlinear oscillation of the model under consideration. A much supplementary precise approximate solution is achieved using an advanced Homotopy perturbation method (HPM). The temporal variation of this solution is graphed for different amounts of the employed factors. The organization of the model is verified through a comparison between the plots of the estimated solution and the numerical one which is obtained utilizing the fourth order Runge–Kutta technique (RK4). The outcomes show that the improved HPM is appropriate for a variety of damped nonlinear oscillators since it minimizes the error of the solution while increasing the validation variety. Furthermore, it presents a potential model that deals with a diversity of nonlinear problems. The multiple scales homotopy technique is used to achieve an estimated formula for the suggested time-delayed structure. The controlling nonlinear algebraic equation for the amplitude oscillation at the steady state is gained. The effectiveness of the proposed controller, the time delays impact, controller gains, and feedback gains have been investigated. The realized outcomes show that the controller performance is influenced by the total of the product of the control and feedback gains, in addition to the time delays in the control loop. The analytical and numerical calculations reveal that for certain amounts of the control and feedback signal improvement, the suggested controller could completely reduce the system vibrations. The obtained outcomes are considered novel, in which the used methods are applied on the DO with time-delay. The increase of the time delay parameter leads to a stable case for the DO, which is in harmony with the influence of this parameter. This drawn curves show that the system reaches a stable fixed point which assert the presented discussion.

CHEBYSHEV PSEUDOSPECTRAL METHOD FOR DUFFING NONLINEAR DIFFERENTIAL EQUATIONS

The system of Duffing nonlinear differential equations is often used in dynamics, which are known to describe many important oscillating phenomena in nonlinear engineering systems. This article presents the pseudospectral method to calculate numerical solutions for nonlinear Duffing differential equations on the interval [–1, 1]. This method is based on the differentiation matrix using the Chebyshev Gauss – Lobatto points. To find numerical solutions of the nonlinear Duffing differential equations, we have built an iterative procedure. The software used for the calculations in this study was Mathematica 10.4. The numerical results of the comparison show that this solution had a high degree of accuracy and very small errors.