Duffing Research Articles

A generalized Duffing equation with the Coulomb’s friction law and Signorini–type contact conditions

This work provides mathematical and numerical analyses for a spring–mass system, in which Signorini–type contact conditions and Coulomb’s friction law with thermal effects are taken into consideration. The motion of a mass attached to a viscoelastic (Kelvin–Voigt type) nonlinear spring is described by a generalized Duffing equation. Signorini contact conditions are understood as extended complementarity conditions (CCs), where convolution is incorporated, allowing to consider thermal aspects of an obstacle. We prove the existence of global weak solutions for the highly nonlinear differential equation system with all the conditions, based on the regularized differential equation and the normal compliance condition with the standard mollifier. In addition, we investigate what side effects produce higher singularities of contact forces in dynamic contact problems, which is also supported by numerical evidences. Numerical schemes are proposed and then several groups of data are selected for the display of our numerical simulations.

Dynamics of coexisting rotating waves in unidirectional rings of bistable Duffing oscillators

We study the dynamics of multistable coexisting rotating waves that propagate along a unidirectional ring consisting of coupled double-well Duffing oscillators with different numbers of oscillators. By employing time series analysis, phase portraits, bifurcation diagrams, and basins of attraction, we provide evidence of multistability on the route from coexisting stable equilibria to hyperchaos via a sequence of bifurcations, including the Hopf bifurcation, torus bifurcations, and crisis bifurcations, as the coupling strength is increased. The specific bifurcation route depends on whether the ring comprises an even or odd number of oscillators. In the case of an even number of oscillators, we observe the existence of up to 32 coexisting stable fixed points at relatively weak coupling strengths, while a ring with an odd number of oscillators exhibits 20 coexisting stable equilibria. As the coupling strength increases, a hidden amplitude death attractor is born in an inverse supercritical pitchfork bifurcation in the ring with an even number of oscillators, coexisting with various homoclinic and heteroclinic orbits. Additionally, for stronger coupling, amplitude death coexists with chaos. Notably, the rotating wave speed of all coexisting limit cycles remains approximately constant and undergoes an exponential decrease as the coupling strength is increased. At the same time, the wave frequency varies among different coexisting orbits, exhibiting an almost linear growth with the coupling strength. It is worth mentioning that orbits originating from stronger coupling strengths possess higher frequencies.

Non-trivial solutions and their stability in a two-degree-of-freedom Mathieu–Duffing system

The Mathieu–Duffing equation represents a basic form for a parametrically excited system with cubic nonlinearities. In multi-degree-of-freedom systems, parametric resonances and the associated limit cycles take place at both principal and combination resonance frequencies. Furthermore, using asynchronous parametric excitation of coupling terms leads to a broadband destabilization of the trivial solution and the appearance of limit cycles at non-resonant frequencies. Regarding applications, the utilization of this excitation method has its significant importance in micro- and nanosystems. On the one hand, cubic nonlinearities are found to be abundant in these systems. On the other hand, parametric excitation is preferably utilized in these systems for better amplification leading to an enhanced sensitivity and for squeezing thermal noise, and thus, proved to be significantly useful in mechanical, optical and microwave systems. Therefore, this theoretical investigation should be of relevant importance to those small-scaled systems. Accordingly, a general two-degree-of-freedom Mathieu–Duffing system is studied. The non-trivial solutions are obtained at different parametric resonance conditions. A bifurcation analysis is carried out using the multiple scales method, followed by investigating the effect of the asynchronous parametric excitation on the existence of limit cycles at resonant and non-resonant frequencies.

Analyzing softening and hardening behavior in vibration of a thin incompressible hyperelastic cylindrical shell

This study focuses on nonlinear vibration of a thin incompressible hyperelastic cylindrical shell subjected to radial harmonic excitation. Mooney–Rivlin model is used to find the Strain Energy Density and model the material’s nonlinear behavior. The independent bending terms are considered for shape deformation of the middle surface. The nonlinear relevant equations of motion are obtained using Donnell’s nonlinear shallow-shell theory and Lagrange’s equation based on the small strain hypothesis. In this study, the linear and nonlinear models are verified by comparing the present theory’s natural frequency results with corresponding results obtained using a finite element method. Subsequently, the derived equations are solved under undamped and free vibration conditions, leading to an equation of motion in the form of a Duffing equation. This equation is then analyzed using the Lindstedt-Poincaré technique. Multiple Scale Method is used to address the system of coupled nonlinear equations in the presence of harmonic force and damping term. Backbone curves and amplitude response for both asymmetric and axisymmetric modes are presented by considering coupled and, non-coupled governed equations. The effects of the geometrical and material characteristics are studied on the free and forced vibration of shell structure as well as the impact of shape function, which describe the deformation of the middle surface, on both backbone and amplitude-response.

Higher-order averaging procedure for performance analysis of a mono-stable Duffing piezoelectric energy harvesting system under white noise excitation

This work is concerned with the energy harvesting of mono-stable Duffing oscillators with piezoelectric coupling under white noise excitation. This is the first time that the higher-order averaging procedure is extended to find the first-, second- and third-order approximate stationary joint probability density functions of amplitude and full phase by solving the corresponding Fokker-Planck-Kolmogorov (FPK) equation. Then the mean-square electric voltage and mean output power are derived through the approximate relation between the electric voltage and the mechanical states. The analytical results are verified by Monte Carlo simulation. The effects of system parameters, such as excitation intensity, mechanical damping term, nonlinear stiffness coefficients, and the resistance ratio of the resistive–capacitive circuit on the mean-square displacement and mean output power are discussed in detail. It has been shown that the averaging procedure of the averaging method of coefficients in FPK equation can be considered as a particular case of the proposed higher-order averaging procedure and the higher-order approximate solutions are much more accurate than the one obtained by the averaging method of coefficients in FPK equation.© 2022 The Authors. Published by Elsevier Ltd.

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.

From coexisting attractors to multi-spiral chaos in a ring of three coupled excitation-free Duffing oscillators

The dynamics of a ring of three unidirectional coupled excitation-free double well Duffing oscillators is considered. The coupling technique followed in this work consists in the amplitude modulation of an oscillator with a proportion of the amplitude of its neighboring counterpart. The rate equation of this model is a sixth-order autonomous nonlinear odd symmetric system endowed with twenty seven equilibrium points eight of which undergo Hopf bifurcation. A detailed analysis of the model based on classic nonlinear analysis tools (e.g. bifurcation diagrams, phase portraits, basins of attraction) discloses attractive and interesting dynamical features such as eight parallel bifurcation threes, Hopf bifurcations, and various types of coexisting modes of oscillations (e.g. eight coexisting limit cycles, four competing double spiral chaotic attractors, two coexisting four-spiral chaotic attractors), and eight-spiral chaotic attractor, just to cite a few. The electronic circuit implementation of the three coupled Duffing oscillators system was executed based on simple available electronic components. The simulation study of the proposed analog circuit in PSpice matches well with the findings the theoretical study. Our results yield the conclusion that coupling excitation-free oscillators of Duffing type in a ring topology can be seen as a useful approach to obtain multi-spiral chaotic attractors.

Quantum behavior of the Duffing oscillator at the dissipative phase transition

The non-deterministic behavior of the Duffing oscillator is classically attributed to the coexistence of two steady states in a double-well potential. However, this interpretation fails in the quantum-mechanical perspective which predicts a single unique steady state. Here, we measure the non-equilibrium dynamics of a superconducting Duffing oscillator and experimentally reconcile the classical and quantum descriptions as indicated by the Liouvillian spectral theory. We demonstrate that the two classically regarded steady states are in fact quantum metastable states. They have a remarkably long lifetime but must eventually relax into the single unique steady state allowed by quantum mechanics. By engineering their lifetime, we observe a first-order dissipative phase transition and reveal the two distinct phases by quantum state tomography. Our results reveal a smooth quantum state evolution behind a sudden dissipative phase transition and form an essential step towards understanding the intriguing phenomena in driven-dissipative systems.

Control and synchronization in the Duffing-van der Pol and $$\Phi ^6$$ Duffing oscillators

Control and synchronization in the Duffing-van der Pol and $$\Phi ^6$$ Duffing oscillators

Novel Approximations to the Damped Parametric Driven Pendulum Oscillators

The damped parametric driven nonlinear pendulum equation/oscillator (NPE), also known as the damped disturbed NPE, is examined, along with some associated oscillators for arbitrary angles with the vertical pivot. For analyzing and solving the current pendulum equation, we reduce this equation to the damped Duffing equation (DDE) with variable coefficients. After that, the DDE with variable coefficients is divided into two cases. In the first case, two analytical approximations to the damped undisturbed NPE are obtained. The first approximation is determined using the ansatz method while the second one is derived using He’s frequency formulation. In the second case, i.e., the damped disturbed NPE, three analytical approximations in terms of the trigonometric and Jacobi elliptic functions are derived and discussed using the ansatz method. The semianalytical solutions of the two mentioned cases are graphically compared with the 4th-order Runge–Kutta (RK4) approximations. In addition, the maximum error for all the derived approximations is estimated as compared with the RK4 approximation. The proposed approaches as well as the obtained solutions may greatly help in understanding the mysteries of various nonlinear phenomena that arise in different scientific fields such as fluid mechanics, plasma physics, engineering, and electronic chips.

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

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

Control of chaos with time-delayed feedback based on deep reinforcement learning

The time-delayed feedback control method, as one of popular methods for chaos control, is noninvasive and flexible for various dynamical systems from different fields of science and technology. In the method, however, an appropriate choice of the feedback gain is challenging, which requires the explicit mathematical model of the controlled system for stability analysis. Additionally, another limitation of the method is the so-called odd number limitation. The two problems restrict its application to practical situations. Fortunately, a technique called deep reinforcement learning is capable of learning the controlled environment (the dynamical system), by continually interacting with the controlled system, which makes it possible to obtain the right feedback gain without the requirement of the accurate mathematical model of the system. Hence, in this work, a time-delayed feedback control method based on deep reinforcement learning is put forward to solve the two problems. Compared with the traditional time-delayed feedback control, the proposed method, as a data-driven method due to the combination with deep reinforcement learning, offers a time-varying feedback gain according to the well-trained policy learned by the deep reinforcement learning algorithm. It maintains the non-invasive property of the time-delayed feedback control, but expands the operating range due to the time-varying feedback gain overcoming the odd number limitation. With numerical simulations, the proposed method is successfully applied to three different kinds of systems, the discrete logistic map, the non-autonomous Duffing oscillator and the autonomous Lorenz system.

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

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

Ansatz and Averaging Methods for Modeling the (Un)Conserved Complex Duffing Oscillators

In this study, both the ansatz and averaging methods are carried out for analyzing the complex Duffing oscillators including the undamped/conserved complex Duffing oscillator (CDO) and the damped/unconserved CDO to obtain some approximate analytical solutions. To analyze the conserved CDO, it is reduced to two decoupled conserved Duffing oscillators. After that, the exact solution of the conserved Duffing oscillator is employed to derive an approximation of the conserved CDO in terms of the Jacobi elliptic function. To analyze the damped CDO, two methodologies are considered. For the first methodology, the damped CDO is reduced to two decoupled damped Duffing oscillators, and the ansatz method is devoted to analyzing the damped Duffing oscillator. Accordingly, an approximation of the damped CDO in terms of trigonometric functions is obtained. In the second methodology, the averaging method is applied directly to the damped CDO to derive an approximation in terms of trigonometric functions. All the obtained solutions are compared with the fourth-order Runge–Kutta (RK4) numerical approximations. This study may help many researchers interested in the field of plasma physics to interpret their laboratory and observations results.

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.

Coplanar superconducting resonators with nonlinear kinetic inductance bridge

We present a model and experimental realization of coplanar superconducting resonators terminated by a shunting kinetic inductance bridge made of ultra-thin Al films. The fabrication process that we propose allows us to create very homogeneous films, which makes them suitable for many applications in quantum devices. Due to the specific properties of the films, the resonators exhibit a Duffing oscillator behavior resulting in bifurcations and interactions between different power sources, which was previously observed in similar systems. Moreover, since the nonlinearity of such a system is concentrated at the bridge, while the wave propagates in a linear environment, it is possible to propose a simple model that accurately describes its behavior. We show that, when resonators are operated within a notch-port architecture, our model has a closed-form solution for the transmission coefficient and allows one to accurately extract parameters of the system, including the kinetic inductance of the bridge and its depairing current. Potential applications of such systems include tunable resonators, photon detectors, bifurcation and parametric amplifiers, as well as a measurement device for studying the properties of thin films.

Numerical solutions of Troesch and Duffing equations by Taylor wavelets

The aim of this study is to obtain accurate numerical results for the Troesch and Duffing equations by using Taylor wavelets. Important features of the method include easy implementation and simple calculation. The effectiveness and accuracy of the applied method is illustrated by solving these problems for several variables. One of the important variable is the resolution parameter which enables to use low degree polynomials and decrease the computational cost. Results show that the proposed method yields highly accurate solutions by using quite low degree polynomials.

Asymptotic equivalence of impulsive dynamic equations on time scales

The asymptotic equivalence of linear and quasilinear impulsive dynamic equations on time scales, as well as two types of linear equations, are proven under mild conditions. To establish the asymptotic equivalence of two impulsive dynamic equations a method has been developed that does not require restrictive conditions, such as the boundedness of the solutions. Not only the time scale extensions of former results have been obtained, but also improved for impulsive differential equations defined on the real line. Some illustrative examples are also provided, including an application to a generalized Duffing equation.

The 0:1 resonance bifurcation associated with the supercritical Hamiltonian pitchfork bifurcation

We consider the non-semisimple 0:1 resonance (i.e., the unperturbed equilibrium has two purely imaginary eigenvalues ( and ) and a non-semisimple double-zero one) Hamiltonian bifurcation with one distinguished parameter, which corresponds to the supercritical Hamiltonian pitchfork bifurcation. Based on BCKV singularity theory established by Broer et al. (Z. Angew. Math. Phys. 44:389-432, 1993), this bifurcation essentially triggered by the reversible universal unfolding with respect to BCKV-restricted morphisms of the planar non-semisimple singularity (the is regarded as distinguished parameter with respect to the external parameter λ). We first give the plane bifurcation diagram of the integrable Hamiltonian on each level of integral in detail, which is related to the usual supercritical Hamiltonian pitchfork bifurcation. Then, we use the -symmetry generated by the additional pair of imaginary eigenvalues to reconstruct the above plane bifurcation phenomenon caused by the zero eigenvalue pair into the case with two degrees of freedom. Finally, we prove the persistence of typical bifurcation scenarios (e.g., 2-dimensional invariant tori and the symmetric homoclinic orbit) under the small Hamiltonian perturbations, as proposed by Broer et al. (Z Angew Math Phys 44: 389-432, 1993). An example system (the coupled Duffing oscillator) with strong linear coupling and weak local nonlinearity is given for this bifurcation.

Evaluating route to impact convergence of the harmonic balance method for piecewise-smooth systems

In this work, we investigate the applicability of the harmonic balance method (HBM) to predict periodic solutions of a single degree-of-freedom forced Duffing oscillator with freeplay nonlinearity. By studying the route to impact, which refers to a parametric study as the contact stiffness increases from soft to hard, the convergence behavior of the HBM can be understood in terms of the strength of the non-smooth forcing term. HBM results are compared to time-integration results to facilitate an evaluation of the accuracy of nonlinear periodic responses. An additional contribution of this study is to perform convergence and stability analysis specifically for isolas generated by the non-smooth nonlinearity. Residual error analysis is used to determine the approximate number of harmonics required to get results accurate to a given error tolerance. Hill’s method and Floquet theory are employed to compute the stability of periodic solutions and identify the types of bifurcations in the system.