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Despite the simplicity of the Duffing oscillator, its dynamical behaviour is extremely rich. Hence, the Duffing equations are used to describe the dynamic behaviour of many real-world nonlinear systems for a wide range of frequency bands and amplitude of the excitation signal in basic sciences and engineering. For example, the Duffing oscillator has been successfully used to model a variety of physical processes such as stiffening springs, beam buckling, nonlinear electronic circuits, superconducting Josephson parametric amplifiers, and ionisation waves in plasmas etc. Therefore, the identification of the Duffing oscillator model/parameters directly from the measured input–output data is a topic of active research in many scientific fields In this paper, we use the concept of stochastic differential equations (SDEs) to identify a model of the Duffing oscillator. SDE-based grey-box models allow us to capture the underlying mathematical structure describing the physics of the system (e.g. the original Duffing equations) using the drift term and explicitly handling of model uncertainty (or the process noise) using the diffusion term whereas the measurement uncertainty is modelled using the measurement noise term respectively. In this paper, we propose a slight variation of the maximum likelihood estimation framework used for the identification of SDEs based grey-box models yielding improved performance for long-term predictions. The proposed framework is combined with an iterative residual analysis to develop a grey-box model of the forced Duffing oscillator. The benchmark data from the so-called Brussels “Silverbox system”, which is an electrical circuit mimicking the forced Duffing oscillator dynamics is used for the identification purpose. Finally, the identified model performance (the simulation errors) is compared with the existing results available in the literature.

In this paper, the coupled homotopy-variational approach (CHVA) based on combining homotopy with the variational approach is applied to solve the nonlinear Duffing equation, and new frequency- amplitude relationships are obtained. The coupled method works very well for the entire range of initial amplitudes and calculates higher-order approximations. The results are compared with those obtained using other known analytical and numerical methods to substantiate the accuracy and efficiency of the approximate analytical approach. The method provides better results than other existing methods. The advantage of the coupled method is that the current second order approximations almost coincide with the corresponding exact solutions. Thus, the presented CHVA could be applied to other strongly nonlinear differential equations.

On the basis of the double coupled Duffing oscillator, a differential double coupled Duffing oscillator method, called the proposed method 1, is proposed. The large scale state is judged by the differential timing sequence diagram, and its detection result is more intuitive. To overcome the frequency limitation of differential double coupled Duffing oscillator in detecting unknown frequency periodic signal, according to the proposed method 1, based on the intermittent chaos phenomenon, the differential double coupled Duffing oscillator variable step size detection method, called the proposed method 2, is proposed. This method can be used to detect Gaussian noise background periodic signal and K-distributed noise background periodic signal. Compared with the double Duffing method and the double coupled Duffing method, its average signal-to-noise ratio (SNR) threshold is reduced by 4.94 dB and 1.68 dB respectively, and the detection result is more intuitive to judge through the differential timing sequence diagram. The accuracy and universality of the proposed method 2 are proved by the detection of three measured underwater weak target signals. Experiments show that it has some characteristics such as intuitive detection result, high sensitivity, low SNR threshold, few oscillators, low system complexity and so on.

With both analytical and numerical methods, chaotic motions of the Rayleigh–Duffing oscillator with a non-smooth periodic perturbation and harmonic excitation are investigated in this paper. Chaos arising from homoclinic or heteroclinic intersections is studied with the Melnikov method. Chaos threshold is obtained and chaotic feature on the system parameters is investigated in detail. Some new interesting dynamic phenomena including “controllable frequency interval”, “chaotic band” and “uncontrollable parameters” are presented and proved rigorously. Numerical simulations are given to verify chaos threshold obtained by the analytical results.

The structure of solution branches of the equation e 2 u x x = u 3 − λ u + δ g ( x ) , for x ∈ R is studied, for real constants, e, λ and δ, where g ( x ) is periodic in x and u satisfies Neumann boundary conditions. For α : = u ( 0 ) bifurcation structure in the α–δ plane occurs that for small e leads to families of singular asymptotic solutions with interior transition layers between outer solutions. In contrast, bifurcation in the α–λ plane produces families of oscillatory solutions that for small e have large numbers, n, of oscillations between envelope outer solutions. The n label of the solution branch is defined by its local bifurcation at e 2 n 2 = λ > 0 . Further the bifurcation structure on the n branch is classified by the integer n being either even or odd. When n is even the pitchfork bifurcation in the α–δ plane leads to connected branches of solutions in the α–λ–δ space. But when n is odd the presence of g ( x ) leads to cusp catastrophes (for small e) in the α–δ plane and these give rise to disconnected (or isolated) branches of solution in the α–λ plane that appear to be torn from the main solution branch. The integer n acts as a count of the number of oscillations on these solution branches. As e 2 λ − 1 → 0 we see layer behaviour occurring about the top and bottom outer solutions and as e 2 λ − 1 → ∞ we find convergence to the middle outer solution.

In this paper, Hamiltonian approach for solving vibration in the first approximation is extended into a high order Hamiltonian approach method for n order approximation. The method is applied for computation of the frequency of doubly clamped beam type N/MEMS subjected to the van der Waals attraction and the Duffing oscillator with seventh order nonlinearity. The obtained results are compared with numerically and other analytically obtained results. It is concluded that the solution in the third approximation gives the most appropriate solution in comparison to other analytical methods. The time history diagrams and the phase plane diagrams calculated analytically are on the top of the curves obtained numerically.

The purpose of this paper is to present experimental results for jump resonance observed in a vibrating cantilever beam of a macro-scale atomic force microscope (AFM), and to connect these results with the well-known Duffing nonlinear oscillator dynamics. The paper gives a review of the theoretical analyses of such behaviors, as well as detailed experimental results for nonlinear frequency-domain jump resonance in the vibrating AFM probe. The experimental results clearly show the model-predicted input-dependent resonant frequency shift. A real-time simulation model of the experimental system is also presented as a useful tool to understand the nonlinear dynamic structure, as an aid in identifying the Duffing equation parameters, and allow simulation studies of ideas such as controlling the nonlinear system behavior. We also present a feedback linearization algorithm to compensate for the cubic stiffness internal to the Duffing nonlinear oscillator. The simple experimental system described herein can be easily and inexpensively constructed to allow students and researchers to conduct their own experiments with such nonlinear dynamics.

In this paper, we are interested in deriving a non-intrusive numerical approach to construct nonlinear and linear parameter varying reduced order models from data. More specifically, based on data collected from a stable non-linear time-domain simulator or experimental bench, we show how we can infer either a reduced order nonlinear or a (quasi) linear parameter dependent model. The proposed approach is based on a very recent procedure called MII for Mixed Interpolation Inference, involving three steps: pencil method, interpolation and model inference. The complete process is illustrated on a polynomial nonlinear Duffing oscillator use-case showing how a reduced either nonlinear or linear parameter varying model can be obtained from time-domain raw data.

This research presents a second-order non-linear chaotic system created using the Lienard’s theorem and combining trigonometric functions with the Duffing oscillator. The system’s parameters are properly chosen to assure a stable limit cycle with Lyapunov Exponents to generate chaotic and periodic trajectories in the phase space. Then the system is configured to operate in the state of intermittence for stationary frequency identification even at a signal-to-noise ratio of around -50 dB.

We report the full analytical solution of the large deformations of a cantilevered elastica loaded by a uniformly distributed follower pressure. We consider an unshearable, inextensible and linear elastic rod. We obtain a spatial nonlinear differential equation in the curvatures, analogous to the undamped Duffing oscillator with a constant driving force. We solve such differential equation, obtaining the curvature, although in implicit form, for arbitrarily large values of the load. We are then able to obtain the rotations owing to a change of variables from the curvilinear abscissa to the curvature. This step is somewhat mandatory due to the implicit nature of the solution. Finally, with the same change of variables, it is possible to obtain a closed-form solution for the deformation in Cartesian coordinates. The solutions show that the rod deforms into drop-like shapes. The number of drops is equal to the number of spatial periods of the solution, which goes with q ∗ 1 / 3 , with q ∗ a dimensionless load normalised to the bending stiffness. Interestingly, we find that for q ∗ ⩾ 3094.2 , the number of drop-like shapes does not increase, but remains three.

Functional link artificial neural network (FLANN) has received much attention due to its wide applicability. The Van der Pol-Duffing oscillator (VdPDO)-based nonlinear systems, which own complex dynamical behaviors, identification of such nonlinear model is vital. This paper exploits the nonlocality of fractional calculus, aiming to enhance the identification accuracy of the VdPDO-based nonlinear systems. The proposed combined FEM-LMS (CFEM-LMS) algorithm, which is based on the FLANN structure, convexly combines the least mean square (LMS) algorithm and the newly proposed fractional-order error modified LMS (FEM-LMS) algorithm. The CFEM-LMS algorithm has improved performance and can dynamically adapt to the nonlinearity of the system. As an added contribution, a novel mixing parameter adaptation criterion is proposed for performance improvement. Extensive simulation results in the context of VdPDO-based nonlinear system identification demonstrate the superiority of the proposed algorithm as compare to state-of-the-art approaches.

This paper proposes a slow flow model for a weakly nonlinear and parametrically driven Duffing oscillator and a complex valued state space model for the oscillator with noncollocated external disturbances. The combination of the parametrically driven duffing oscillator and noncollocated disturbances can be observed in resonant MEMS mirrors with a reinforcement structure to reduce dynamic mirror deformation. The model is based on a rational function approximation for the angular derivative of the out-of-plane comb drive capacitance, enabling qualitative analysis at large amplitudes while the stability analysis is maintained as the conventional cubic function approximation. The slow flow model is extended including a single tone noncollocated disturbance and is linearized at an equilibrium point for a small disturbance. The linearized disturbance model is reformulated by a complex valued state space model to cope with general wideband disturbances, allowing various analytic methods in traditional system theory. The simulation results demonstrate a good agreement between the proposed models and the ODE simulation, verifying the accuracy and benefits of the proposed models.

The Duffing equation has been extensively studied in the last decades, especially in systems of vibrating micro-beams subjected to electro-mechanical fluctuating fields. However, mechanical systems are often subjected to uncertainties coming from the excitation and/or the design parameters. These uncertainties can then be considered as stationary or non-stationary stochastic processes. In many engineering applications, we consider stationary stochastic processes as they are easier to model and simplify our problems. However, some of them do not present stationarity properties and it is desirable to know what are the consequences of leaving the non-stationarity aside and consider a stationary case. For this purpose, two stochastic processes are applied to a Duffing oscillator and the results are analyzed when its stiffness modeling is based on (i) a stationary stochastic process generated with Langevin’s equation; and when it is based on (ii) a non-stationary stochastic process known as Brownian Bridge. A methodology is proposed to modify these stochastic processes in order to establish a lower limit for their support. In fact, this methodology can be applied to stationary and non-stationary stochastic processes. The stochastic results showed that the case with uncertainties in the stiffness coefficient of the non-linear term presented the highest variation on system’s response. In addition, the non-stationary case presented a result much closer to the deterministic one for the set of parameters chosen. The smooth variation of the second moment with time might explain it. There are cases in which the assumption of a stationary process might be not appropriate and non-stationarity must be assessed.

In this paper, a design of Poincare maps and time–piecewise–constant state–feedback control laws for the stabilization of limit cycles in periodically–forced, non–autonomous, nonlinear dynamical systems is achieved. Our methodology is based mainly on the linearization of the nonlinear dynamics around a desired period–m unstable limit cycle. Thus, this strategy permits to construct an explicit mathematical expression of a controlled Poincare map from the structure of several local maps. An expression of the generalized controlled Poincare map is also developed. To make comparisons, we design three different time–piecewise–constant control laws: an mT–piecewise–constant control law, a T–piecewise–constant control law and a T n –piecewise–constant control law. As an illustrative application, we adopt the chaotic Duffing oscillator. By applying the designed piecewise–constant control laws to the Duffing oscillator, the system is stabilized on its desired period–m limit cycle and hence the chaotic motion is controlled.

In this paper, the classical Duffing oscillator under dual frequency excitations is studied by the homotopy analysis method(HAM). Analytical study of the low-order approximations is firstly conducted and the saddle node(SN) bifurcation boundary for the initial guess solution is obtained. The maximum value bifurcation plot of the high order approximations with the bifurcation parameters f1 and λ1 are obtained and compared with the numerical solutions based on the Runge–Kutta method. The results show that the initial guess solution can qualitatively reflect the trend of the numerical solution, and the high order approximations agree well with the numerical solutions. The maximum value bifurcation plots of high order approximations show periodic and quasi-periodic solutions, which agree well with the numerical ones.

This paper investigates the potential of using a nonlinear positive position feedback controller for vibration mitigation of a Duffing oscillator. The proposed controller is designed based on the principle of similarity which states that anti-vibration devices should be governed by the same equations as those of the host structure. Closed-form expressions for the H∞ optimal control parameters that minimise the maximal response of the structure are firstly derived for the linear positive position feedback controller and then extended to the nonlinear counterpart. The harmonic balance method is employed to approximate the analytical solutions. Both numerical simulations and experimental validations are performed to demonstrate the proposed control strategy., SCOPUS: ar.j, info:eu-repo/semantics/published

Nonlinear systems are more attractive for the wide bandwidth in contrast to linear vibration energy harvesters. Most available generators belong to a certain type of nonlinear systems and the Duffing oscillator is a most popular kind. Once the system type is determined with a specific equation form, the performance optimization is limited in a certain range, generally related to the parameter study of the system within a given frame. In this paper, a novel generator is proposed by merging the stiffness curves of two different source generators through utilization of the nonlinear cantilever with a curve fixture. The special merged stiffness characteristics break the response limitation of the specific stiffness profile such as the Duffing equation and bring new benefits with the generator bandwidth further broadened by integrating the advantages of the source generator. In order to verify the stiffness-merging concept, two Duffing generators and a proposed generator with stiffness fitted from the former two are fabricated using the nonlinear cantilever with specially designed fixture curves. Experimental and numerical investigations reveal that the proposed generator with the merged stiffness has better performance than the generators with the typical stiffness characteristics of Duffing systems. The effectiveness of the stiffness-merging method is well demonstrated.

Assembled structures are nonlinear. The sources of this nonlinearity could include the jointed interfaces, damage and wear, non-idealized boundary conditions, or other features inherent in real parts. To study these systems and to ascertain if they will be operating in a regime in which the nonlinearity is prominent, nonlinear system identification techniques are needed to assess and characterize the nature of the nonlinearity in the structure. Significant progress over the last few years has focused on using nonlinear system identification to identify damage and other deviations from idealized structures. This research reviews nine different methods for nonlinear system identification (restoring force surface, Hilbert transform, direct quadrature, zero-crossing, short-time Fourier transform, Gabor wavelet, Morlet wavelet, Morse wavelet, and a neural network-based algorithm) in order to assess their accuracy. The methods are compared by identifying characteristics of two systems: a single degree of freedom model of a Duffing oscillator and measured data from a jointed structure. As neural networks are not commonly used for system identification, multiple variations of the method are investigated to study its effectiveness. Perturbation analysis is conducted to see the efficacy of the different methods for identifying parameters across a large range of design spaces, and the advantages and disadvantages of each method are discussed. The primary contribution of this paper is a comparison on both analytical and experimental data of multiple widely used system identification methods, and an assessment of when each method is most and least applicable, specifically in the context of jointed structures.

The possibility of reducing and even suppressing chaos in dissipative nonautonomous systems by additional incommensurate chaos-suppressing excitations is theoretically demonstrated through the universal example of a perturbed Duffing oscillator by considering rational approximations to the irrational ratio Ω/ω between the chaos-suppressing and chaos-inducing frequencies. For each chosen rational approximation, analytical predictions for the suitable amplitudes and initial phases of the chaos-suppressing excitation are numerically confirmed for small amplitudes and at the predicted suitable initial phases. For the significant case of the golden ratio Φ ≡ ( 1 + 5 ) / 2 , we study the structural stability of the suppressory scenario as the respective convergents approximate the irrational ratio Ω / ω = 1 / Φ . Our theory predicts and numerical simulations confirm that the values of the suitable amplitudes are rather insensitive to high-order convergents. On the contrary, the number and values of the suitable initial phases critically depend on each particular convergent in order to satisfy two requirements: (1) Maximum approximation to the frustration of the homoclinic bifurcation existing in the absence of the chaos-suppressing excitation and (2) maximum survival of a relevant spatio-temporal symmetry of the dynamical equation.

A new class of two-step linear symmetric methods is introduced for the numerical solution of second-order initial value problems that have highly oscillatory solutions. In this class, for the first time in the literature, we calculate the coefficients of the method by a combination of a trigonometrically fitted (TF) method and a vanishing of the phase lag and some of its derivatives (VSDPL) method, and we construct a new class of methods that has the properties of TF methods and VSDPL methods, which we call the TFPL method. This method is of algebraic order 8, and has an important P-stability property. The main structure of the method is multiderivative, and the combined phases were applied to expand the stability interval and to achieve P-stability. The advantage of the method in comparison with similar methods in terms of efficiency, accuracy, and stability is shown by its implementation in some important problems, the undamped Duffing equation, etc.

Unwanted vibrations caused by the commanded motions lower the positioning accuracy and degrade the control performance of flexible link manipulators. Much work has been devoted to the dynamics and control of flexible link manipulators. However, there are few studies dedicated to the flexible link manipulators having Duffing oscillator dynamics. This paper develops a new model for a flexible single-link manipulator by assuming large mechanical impedance in the drives or large inertia of the motor hub and by considering the flexibility of the single-link manipulator. The derived model includes an infinite number of uncoupled Duffing oscillators by ignoring the modal coupling effects. Two new methods are presented for controlling the vibrations of the flexible manipulator governed by Duffing oscillators. One is designed for the single-mode Duffing oscillator, and the other is for the multi-mode Duffing oscillators. A comparison of these two methods is made using the results of numerical simulations and experimental measurements. Experimental investigations are also performed on a flexible single-link manipulator to validate the dynamic behavior of Duffing oscillators and the effectiveness of the new control methods.

The present paper proposes a novel adaptive passive stiffness (APS) device which can produce variable stiffness in an adaptive fashion. This APS device consists of four springs arranged in a rhombus configuration and a guidance template. The stiffness of the device is controlled through varying the angle of the rhombus configuration. The function of the guidance template is to relate the angle of the rhombus to the targeted system response such that the stiffness can be varied in an adaptive manner, without the need for feedback or external energy. Two analytical models are developed to capture the behavior of the APS device which shows good agreement with the experimental data. Experimental result indicates that the proposed APS device can produce adaptive stiffness as expected. A cubic nonlinearity is produced and deployed in a single-degree-of-freedom (SDOF) system to form a hardening Duffing oscillator. Dynamic tests show that the experimental characteristics of the Duffing system agrees well with the numerical results, thereby demonstrating that the APS device is effective in producing smoothly and continuously adaptive stiffness in an adaptive fashion. In addition, a fourth order even function is used as the guidance template to produce a softening-hardening adaptive stiffness. The performance of this adaptive stiffness is numerically evaluated for seismic protection. It is found that the structural responses can be reduced effectively when using this softening-hardening adaptive stiffness. The key value of this APS device is that the guidance template can be easily changed to any desired continuous function such that the targeted adaptive stiffness can be obtained. This provides possibility for controlling the vibrations of structures in a passively adaptive manner.

We give a complete algebraic characterization of the first integrals of the Rayleigh–Duffing oscillator. We prove the non existence of centers of such system and we study the form of the singular first integrals at the origin.

An improved method is proposed to combine the incremental harmonic balance method (IHBM) with the equivalent piecewise linearization for solving the dynamics of complicated nonlinear stiffness systems. Compared with the traditional IHBM, the proposed method has a wider application range and can provide a standard and unified procedure to deal with various forms of nonlinear stiffness systems. Firstly, the equivalent piecewise linearization is adopted to approximate the nonlinear restoring force of the system to the piecewise-linear restoring force and the linearized error is put forward to evaluate the computation error of this approximation. The procedure is suitable for any form of nonlinear stiffness expression that can be linearized into piecewise-linear stiffness expression. Then the IHBM procedure for the general piecewise-linear stiffness system is derived considering the sub-harmonic responses. The Duffing oscillator is presented to validate the proposed method. Finally, three cases of nonlinear stiffness systems are carried out. The results of these cases show that the proposed method is capable of analyzing the dynamics of these complicated nonlinear stiffness systems.

In this study, multiple analytical mode decompositions (M-AMD) are proposed to identify the parameters of nonlinear oscillators from free vibration. The time-varying damping (stiffness) coefficient of an oscillator is divided into slow- and fast-varying components by a bisecting frequency in reference to velocity (displacement). The slow-varying damping and stiffness components are estimated from the oscillator’s responses and their Hilbert transforms, and filtered with the adaptive low-pass filter AMD. Each fast-varying component is estimated from the oscillator’s responses and the determined slow-varying components, and corrected with AMD using an approximate bisecting frequency of the two fast-varying components. The computational efficiency and accuracy of the proposed M-AMD are illustrated with three characteristic oscillators described by Duffing, Bouc-Wen, and spherical bearing models. The errors in estimation of all model parameters of the representative nonlinear oscillators are less than 3% from uncontaminated displacement responses and 9% when the root-mean-square value of displacement noises correspond to 0.05% of the peak displacement. In the case of Duffing oscillator, the error of the proposed method is less than 1.2% unless the ratio between the nonlinear and linear terms of the restoring force (degree of nonlinearity) exceeds 25. In the case of spherical bearing, the error of the proposed method is less than 1.2% when the coefficient of friction is less than 0.017. The instantaneous stiffness determined from Hilbert spectral analysis differs from the system stiffness by amount that rapidly increases with the degree of nonlinearity.

This paper aims at exposing a nonlinear piezoelectric vibration energy harvester which can achieve low-cost orbit jump and therefore improve its energy harvesting capability through energy-efficient transient stiffness modulation using Synchronized Switch Stiffness Control (SSSC) technique. The proposed orbit jump concept is numerically and experimentally studied with a softening harvester. Corresponding experimental results validate the jump orbit concept and confirm the significant enhancement of the harvested energy. While the presented softening Duffing system is used as demonstration example, the proposed approach can be implemented to hardening Duffing oscillator as well.

In the present paper, the primary resonance of a special type of nonlinear Duffing oscillator with fractional-order derivative is studied by the averaging method. First, the parametric amplitude-frequency equation is obtained, and then, the effects of the some parameters such as fractional order, nonlinear coefficients and force amplitude on the system dynamics are investigated. Moreover, experimental test were performed on the case study and a suitable model is identified. The obtained results are very useful in the nonlinear identification field. Keywords: Fractional-order derivative, Averaging method, Amplitude-frequency curve, Nonlinear identification, Inertial and geometrical nonlinearity

Harmonic Balance is a very popular semi-analytic method in nonlinear dynamics. It is easy to apply and is known to produce good results for numerous examples. Adding an error criterion taking into account the neglected terms allows an evaluation of the results. Looking on the therefore determined error for increasing ansatz orders, it can be evaluated whether a solution really exists or is an artifact. For the low-error solutions additionally a stability analysis is performed which allows the classification of the solutions in three types, namely in large error solutions, low error stable solutions and low error unstable solution. Examples considered in this paper are the classical Duffing oscillator and an extended Duffing oscillator with nonlinear damping and excitation. Compared to numerical integration, the proposed procedure offers a faster calculation of existing multiple solutions and their character.

Dynamical systems play a fundamental role related understanding phenomena inherent to several fields of science. Technological advances over the previous several decades have generated a large amount of data that might be used in the inference of dynamical systems. Regardless of the sensor types adopted to perform the data acquisition procedure, it is useful to verify the existence of certain noise corruption in data. Generically, system identification is directly affected by noisy scenarios, which result in the false discovery of non-spurious models. In this work, we demonstrate how the hybridization of several machine learning techniques improves the robustness to noise with respect to the system identification assignment, advancing a pioneer methodology known as Sparse Identification of Nonlinear Dynamics (SINDy). Specifically, in the current work, we show the success of the proposed strategy from numerical examples, such as a logistic equation with forcing, Duffing oscillator, FitzHugh–Nagumo model, Lorenz attractor and a Susceptible–Infectious–Recovered modeling of SARS-CoV-2.

In the present study, the influence of noise on the responses of continuous-time dynamical systems are considered. In particular, the influence of white Gaussian noise on the frequency responses of monostable and bistable, softening Duffing oscillators are studied. A combination of experimental, analytical, and numerical studies are undertaken to understand the shifting of jump-up and jump-down frequencies and an eventual collapse of upper and lower responses branches of into one response curve with no jump instabilities for the considered Duffing oscillator. It is noted that with noise, the hysteresis observed in the response of the nonlinear oscillator without noise can be destroyed.

This paper introduces a new technique for using chaotic Duffing oscillators in secure communication. The secret message is encrypted using the parameters of the Duffing oscillator that indirectly affect the generated chaotic orbits. The mathematical model of the chaotic transmitter uses three parameters that can be altered between two levels, depending on the binary sequence of the digital secret message and the corresponding cipher. Consequently, a total of eight chaotic attractors are used to scramble the binary-based signal. Two different structures, for the synchronization and encryption mechanisms, between the transmitter and the receiver, are introduced. The first one uses the time series of the two chaotic signals generated by the Duffing transmitter, while implementing a cipher-less encryption mechanism. The second one uses a single time series of the Duffing transmitter, while allowing one of its parameters to act as a digital cipher. Better utilization of the communication channel is proved for the second method, while using a new adaptive Lyapunov-based technique to implement both the state synchronization and the parameters estimation at the receiver side. Both simulation and experimental case studies are presented to illustrate the proposed techniques and their applicability to securely transmit various multimedia digital signals.

This paper presents a practical implementation of a memristor based Duffing oscillator. We replace a diode based nonlinear element by a simple flux-controlled memristor in an existing circuit early proposed.We model the current-voltage characteristic and show experimentally that our circuit presents, for the same forcing amplitude, a route to chaos on the falling branch of the hysteresis loop, while its dynamics remains regular on the raising branch. This observation clearly highlights the memory effect of the memristor.

This paper deals with a particular arrangement of a statically balanced system using 3 springs of prescribed material stiffness and critical geometrical parameter. The dynamics is described by a nonlinear differential equation upto septic power following odd nonlinearity for small disturbance from static equilibrium position. The governing differential equation is solved analytically by the combination of the linearisation of the equation with the method of Harmonic Balancing to observe the low natural frequency at fixed point and a finite displacement range in the neighbourhood of the equilibrium point where the dynamic stiffness is low. By this approximation method, the behaviour of the displacement with increase in time as well as the phase-plot of Cubic Quintic Septic Duffing equation for a set of parameter values is studied at the equilibrium position and its neighbourhood.

Predictions of multistable structural dynamics are paramount to the development and deployment of air vehicles operating under extreme loading conditions. Although time-stepping numerical techniques may capture the multi-physics interactions that occur in these environments, the generalized insight on parameters that predominantly govern the system behaviors may remain clouded while a large computational expense may be incurred to obtain response predictions. Alternatively, analytical methods may be employed to streamline the prediction process, yet current theoretical approaches do not facilitate such opportunity for multistable structures. Although a recently developed analytical formulation has enabled the prediction of near- and far-from-equilibrium responses for a simplified multistable structure, the preliminary formulation does not illuminate the underlying aspects of modal response and intricate nonlinear coupling manifest in myriad multistable systems. This research rectifies these limitations by a broad expansion of the analytical framework that empowers a new modal perspective of multistable structural dynamics and enables the study of such dynamic systems governed by reduced order models. This new modal analysis indicates that the characteristic frequency response of a single degree-of-freedom Duffing oscillator is preserved in the fundamental equivalent nonlinear mode of a multistable structure. The new analytical formulation is also shown to accurately predict the near- and far-from-equilibrium dynamics of equation systems containing global nonlinear coupling consistent with reduced order models. The advancements achieved in this work contribute to the suite of techniques available to researchers to characterize the near-to- and far-from equilibrium behaviors of nonlinear dynamic systems consisting of many degrees-of-freedom.

In this study, multiple analytical mode decompositions (M-AMD) are proposed to identify the parameters of nonlinear structures from forced vibration. For the time-varying damping (or stiffness) coefficient of a weakly-to-moderately nonlinear system, the slow-varying part is first estimated from the system responses and their Hilbert transforms, which is corrected with an adaptive low-pass filter referred to as analytical mode decomposition (AMD). The fast-varying part can then be identified from the responses together with the estimated slow-varying part, which is again corrected with the AMD. The computational efficiency and accuracy of the proposed M-AMD are demonstrated with a Duffing oscillator subjected to harmonic loading. The errors in estimation of all model parameters are less than 3% from uncontaminated displacement responses, which is more accurate compared with the results from Hilbert spectral analysis. Changes of the fast-varying stiffness part have been taken into account with high accuracy. The M-AMD algorithm is then validated with a ¼-scale, 3-story building with one piezoelectric friction damper under earthquake excitations. The parameters of such a semi-active damper are identified with less than 1% error on average.

This paper proposed a highly accurate nonlinear dynamic model and parameters identification method to predict the shock pulses generated by rubber waveform generator (RWG) used in shock test. The establishment of the model is based on the primary results of the Duffing equation, Pan-Yang model and empirical stiffness formulae of cylindrical rubber isolator. We proposed a series of measures to eliminate the interferences in the drop shock test experiment, which are also of benefit to design, regulate and calibrate drop shock testers. The overall nonlinear system is solved in time domain with Runge–Kutta. And the parameters of the nonlinear dynamic model are identified by minimizing the holistic shape difference between experimental and predicted shock pulses with multiple population genetic algorithm. Meanwhile, the capacity of this model in characterizing the dynamic behavior of RWG under shock excitations is also investigated. Compared with previous work, our model considers nonlinear viscoelasticity of RWG, the geometric factor of RWG and the effect of initial shock velocity, which makes our model more accurate and general. Experimental results shown great agreements with the predictions and that the proposed method can determine the parameters conveniently and synergistically.

We obtain improved approximate fully analytical steady state solutions for the ordinary differential equation of the forced and damped anharmonic oscillator with quadratic and cubic nonlinearities. The solutions, are obtained in the form of a truncated Fourier series using the harmonic balance method (HBM). In order to obtain explicit analytical expressions for the Fourier coefficients we introduce the hypothesis of weak interdependence between the equations in the system of nonlinear algebraic equations produced by the HBM. Comparison with the numerical solution shows that the analytical results for the first harmonics which we investigated are valid in comparatively strong nonlinear regimes, where previously obtained expressions are completely inaccurate. Our expressions are correct over a wide range of the physical parameters, such as excitation frequency, nonlinearity and forcing. The proposed method of solution is expected to be suitable for other nonlinear ordinary differential equations.

In this paper, a bi-stable piezoelectric energy harvester based on the bi-stable hybrid composite laminate with a new stacking sequence design is proposed. The new stacking sequence enables this bi-stable energy harvester to have some unique features, such as uniform strains of piezoelectric elements and symmetric stable configurations. Meanwhile, the new stacking sequence design enhances the room to adjust its natural frequency such that this proposed harvester allows for adjusting to lower frequency range and increasing the inertia by a tip mass to lower the demand of excitation level in comparison to previous settings. A finite element analysis is developed to investigate the static and dynamic characteristics of this new bi-stable energy harvester. A simple numerical model with a modified version of the Duffing equation is developed based on the unique nonlinear restoring force obtained from finite element analysis to describe the fundamental response characteristics of this nonlinear bi-stable energy harvester. Numerical simulations and experiments are carried out at different harmonic excitation levels ranging from 10 to 40 Hz. The results verify that the proposed model can reasonably capture the dynamic characteristics of this broadband bi-stable energy harvester. The output powers were measured under different vibration patterns. Maximum power of 31.1 mW was generated under large-amplitude and high-energy orbits cross-well vibration pattern at an excitation frequency of 22 Hz and acceleration of 3g (g = 9.8 m/s2).

We examine how insertion of a delay in a system of ordinary differential equations can change the asymptotic dynamics and stabilize the structure of the system under further change.

The stochastic stability of a Duffing oscillator with fractional derivative damping under combined harmonic and Poisson white noise parametric excitations is investigated. A stochastic averaging method and the Khasminskii’s procedure are applied to evaluate the largest Lyapunov exponent. Then the asymptotic Lyapunov stability with probability one of the original system is determined approximately by using the largest Lyapunov exponent. Finally, the analytical results are verified by those from the Monte Carlo simulation of the original system.

This paper addresses the issues of nonlinear chemical dynamics modeled by a modified Van der Pol-Duffing oscillator with asymmetric potential. The Melnikov method is utilized to analytically determine the domains boundaries where Melnikov’s chaos appears in chemical oscillations. Routes to chaos are investigated through bifurcations structures, Lyapunov exponent, phase portraits and Poincare section. The effects of parameters in general and in particular the effect of the constraint parameter β which shows the difference between a nonlinear chemical dynamics order two differential equation and ordinary Van der Pol-Duffing equation are analyzed. Results of analytical investigations are validated and complemented by numerical simulations.

In this article, we address the model identification of nonlinear vibratory systems, with a specific focus on systems modeled with distributed nonlinearities, such as geometrically nonlinear mechanical structures. The proposed strategy theoretically relies on the concept of nonlinear modes of the underlying conservative unforced system and the use of normal forms. Within this framework, it is shown that without internal resonance, a valid reduced order model for a nonlinear mode is a single Duffing oscillator. We then propose an efficient experimental strategy to measure the backbone curve of a particular nonlinear mode and we use it to identify the free parameters of the reduced order model. The experimental part relies on a Phase-Locked Loop (PLL) and enables a robust and automatic measurement of backbone curves as well as forced responses. It is theoretically and experimentally shown that the PLL is able to stabilize the unstable part of Duffing-like frequency responses, thus enabling its robust experimental measurement. Finally, the whole procedure is tested on three experimental systems: a circular plate, a chinese gong and a piezoelectric cantilever beam. It enable to validate the procedure by comparison to available theoretical models as well as to other experimental identification methods.

In this paper, we adapt a parallel time integration scheme to track the trajectories of noisy non-linear dynamical systems. Specifically, we formulate a parallel algorithm to generate the sample path of nonlinear oscillator defined by stochastic differential equations (SDEs) using the so-called parareal method for ordinary differential equations (ODEs). The presence of Wiener process in SDEs causes difficulties in the direct application of any numerical integration techniques of ODEs including the parareal algorithm. The parallel implementation of the algorithm involves two SDEs solvers, namely a fine-level scheme to integrate the system in parallel and a coarse-level scheme to generate and correct the required initial conditions to start the fine-level integrators. For the numerical illustration, a randomly excited Duffing oscillator is investigated in order to study the performance of the stochastic parallel algorithm with respect to a range of system parameters. The distributed implementation of the algorithm exploits Massage Passing Interface (MPI).

This paper presents a novel numerical procedure to predict nonlinear free and steady state forced vibrations of clamped–clamped curved beam in the vicinity of postbuckling configuration. Nonlinear Euler–Bernoulli kinematics assumptions including mid-plane stretching are proposed to exhibit a large deformation but a small strain of von Karman. To simulate the interaction of beam with the surrounding elastic medium, nonlinear elastic foundation with cubic nonlinearity and shearing layer are employed. The nonlinear integro-differential equation that governs the buckling of beam is discretized using the differential-integral quadrature method (DIQM) and then is solved using Newton’s method. The problem of linear vibration is discretized using DIQM and then is solved as a linear eigenvalue problem. Afterwards, a single-mode Galerkin discretization is used to reduce the nonlinear governing equation into a time-varying Duffing equation. The Spectral differentiation matrix operators are exploited to discretize the Duffing equation. The discretized Duffing equation is a nonlinear eigenvalue problem which is directly solved using pseudo arc length continuation method. Results obtained by the proposed numerical solution are compared with analytical solutions available in the literature and good agreement is obtained. Parametric studies are carried out to show the effects of applied axial load, imperfection and nonlinear elastic foundations on the natural frequency as well as forced damped vibration behavior of the beam. The above mention effects play very important role on the dynamic behavior of buckled curved beam.

The general structure of irreducible invariant algebraic curves for a polynomial dynamical system in C 2 is found. Necessary conditions for existence of exponential factors related to an invariant algebraic curve are derived. As a consequence, all the cases when the classical force-free Duffing and Duffing–van der Pol oscillators possess Liouvillian first integrals are obtained. New exact solutions for the force-free Duffing–van der Pol system are constructed.

Time delays are many sources of complex behavior in dynamical systems. Yet its relationship with bursting dynamics needs to be further explored, particularly when the strength of feedback is a nonlinear function of delay. In this paper, we analyze the dynamics of the van der Pol–Duffing fast-slow oscillator controlled by the parametric delay feedback, where the strength of feedback control is a function exponential varying with the time delay. The system may exhibit a unique equilibrium point and three ones for the different parameters by employing the pitchfork bifurcation. Next, the stability-switches and the Hopf bifurcation curves are presented as the delay varies, which leads to the occurrence of novel bursting phenomena. Some weak resonant or non-resonant double Hopf bursting oscillations are presented due to the vanishing of real parts of two pairs of characteristic roots. Not only the magnitude of the time delay itself but also the strength of feedback control may influence the dynamical evolution process of bursting behaviors in the delayed system. Such fast-slow forms about bursting dynamics, as well as classifications about local dynamics are investigated. Furthermore, periodic and quasi-periodic bursting motions are verified in both theoretical and numerical ways.

Sufficient and necessary conditions are found for the existence of a positive periodic solution to the Duffing type equation u ′ ′ = p ( t ) u + q ( t , u ) u . The results obtained are compared with facts well known for the autonomous Duffing equation y ′ ′ + a y − b y 3 = 0 . Uniqueness of solutions and possible generalizations are discussed, as well.

Anisometric integrity measures defined as improvement and generalization of two existing measures (LIM, local integrity measure, and IF, integrity factor) of the extent and compactness of basins of attraction are introduced. Non-equidistant measures make it possible to account for inhomogeneous sensitivities of the state space variables to perturbations, thus permitting a more confident and targeted identification of the safe regions. All four measures are used for a global dynamics analysis of the twin-well Duffing oscillator, which is performed by considering a nearly continuous variation of a governing control parameter, thanks to the use of parallel computation allowing reasonable CPU time. This improves literature results based on finite (and commonly large) variations of the parameter, due to computational constraints. The seamless evolution of key integrity measures highlights the fine aspects of the erosion of the safe domain with respect to the increasing forcing amplitude.

The existence of a closed micro-crack in plates can be detected by using the nonlinear harmonic characteristics of the Lamb wave. However, its location identification is difficult. By considering the transient nonlinear Lamb under the noise interference, we proposed a location identification method for the closed crack based on the quantitative measurement of Duffing oscillator transient transfer in the phase space. The sliding short-time window was used to create a window truncation of to-be-detected signal. And then, the periodic extension processing for transient nonlinear Lamb wave was performed to ensure that the Duffing oscillator has adequate response time to reach a steady state. The transient autocorrelation method was used to reduce the occurrence of missed harmonic detection due to the random variable phase of nonlinear Lamb wave. Moreover, to overcome the deficiency in the quantitative analysis of Duffing system state by phase trajectory diagram and eliminate the misjudgment caused by harmonic frequency component contained in broadband noise, logic operation method of oscillator state transition function based on circular zone partition was adopted to establish the mapping relation between the oscillator transition state and the nonlinear harmonic time domain information. Final state transition discriminant function of Duffing oscillator was used as basis for identifying the reflected and transmitted harmonics from the crack. Chirplet time-frequency analysis was conducted to identify the mode of generated harmonics and determine the propagation speed. Through these steps, accurate position identification of the closed crack was achieved.

In this paper, a joint state and parameter estimation problem of Duffing oscillator is explored using Bayesian filters, where the parameter to be identified is considered as an additional state variable. From a variety of Bayesian filters, the unscented Kalman filter (UKF), cubature Kalman filter (CKF) and Gauss-Hermite filter (GHF) are chosen for solving this problem. The performance of these filters are compared in terms of the root mean square error (RMSE) calculated over a specified number of Monte-Carlo runs. From simulation results, it is found that the accuracy of CKF and GHF are almost same while the computational time for GHF is almost three times higher.