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Abstract In this article, we focus on studying the Duffing problem with the time delay of pantograph type via the Hilfer fractional derivatives on the infinite interval ( 0 , ∞ ) $(0,\infty )$ . An appropriate Banach space supported with the Bielecki norm in the Mittag–Leffler function sense is introduced for new and convenient analysis. The existence and uniqueness ( E & U $\mathbf{E\&U}$ ) of the solutions are proved by utilizing the classical fixed point theorems (FPT s). Moreover, the Hyers–Ulam (HU) stability is discussed for our Hilfer fractional Duffing pantograph system (HFDPS). Ultimately, our results are enhanced by providing numerical examples with graphics simulations to check the validity of the main outcomes.

In this article, we focus on studying the Duffing problem with the time delay of pantograph type via the Hilfer fractional derivatives on the infinite interval (0 , ∞) . An appropriate Banach space supported with the Bielecki norm in the Mittag–Leffler function sense is introduced for new and convenient analysis. The existence and uniqueness ( E & U ) of the solutions are proved by utilizing the classical fixed point theorems (FPTs). Moreover, the Hyers–Ulam (HU) stability is discussed for our Hilfer fractional Duffing pantograph system (HFDPS). Ultimately, our results are enhanced by providing numerical examples with graphics simulations to check the validity of the main outcomes. [ABSTRACT FROM AUTHOR]

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In this paper, we study the pseudo almost periodic solutions for a class of nonlinear Duffing equations with S p -pseudo almost periodic coefficients and delays on time scales. For this purpose, we establish a result of the existence and uniqueness of pseudo almost periodic solution for an abstract linear equation with S p -almost periodic coefficients and S p -pseudo almost periodic forcing term. Meanwhile, to deal with the delay, we extend some concepts of functions from T → R to T → Π , where T is a time scale with translation set Π , and give some basic properties for these concepts. Then, applying these results, we obtain some results on the existence and uniqueness of pseudo almost periodic solutions for the Duffing equation. Moreover, some examples are given to illustrate our main results. [ABSTRACT FROM AUTHOR]

Using the Duffing equation to model body temperature's circadian rhythms, we simulated four cases with varying driving force parameters γ (0.32, 0.33, 0.34, and 0.35). We compared these simulations to published experimental data using Fourier and entropy analyses along Poincaré sections. For normal work-sleep conditions (γ = 0.35), the system showed precise resetting at specific times. Under shift work conditions (γ = 0.34), resetting behavior was reduced, causing a phase shift in the rhythm. Long-term constant temperature conditions (γ = 0.33) displayed a reset at the period's start, similar to normal conditions. With no external driving forces (γ = 0.32), multiple resets occurred within the period, emphasizing the need for frequent resets to maintain stability. The simulation results align partially with experimental data, highlighting the system's adaptability to external changes. [ABSTRACT FROM AUTHOR]

Employing variable horizontal and vertical shifts for the unperturbed solution, a new perturbation method is proposed. The new method is named as the shift perturbation method. The existing perturbation methods are interpreted with respect to the horizontal and vertical shifts for the unperturbed solution. The new method is applied to some well-known vibrational problems. The method is capable of producing admissible approximate solutions for a wide range of problems such as the free and forced vibrations of the Duffing equation, equations with quadratic and cubic nonlinearities, variable amplitude nonlinear problems, and boundary layer problems. A variant of the method also produced valid solutions for large perturbation parameters. [ABSTRACT FROM AUTHOR]

A local fractional modification of the Duffing equation is considered, and the homotopy perturbation method is employed to reveal its frequency-amplitude relationship, which is of paramount importance in the optimal design of the energy harvesting devices and chatter detection. Effects of the initial conditions on the periodic property is also discussed. [ABSTRACT FROM AUTHOR]

The consequences of introducing the fourth order orientation tensor as an independent variable in addition to the second order one are investigated. In the first part consequences of the Second Law of Thermodynamics are exploited. The cases with the second order alignment tensor in the state space on one hand and with the second and fourth order alignment tensors on the other hand are analogous. In the latter case differential equations for the second and fourth order tensors result from linear force-flux relations with a coupling arising due to coupling terms in the free energy. In the second part the differential equations for the second order orientation tensor or the second and fourth order orientation tensors, respectively are given explicitly in the special case of a rotation symmetric orientation distribution. The Folgar-Tucker equation with a quadratic closure relation leads to a Riccati equation for the second order parameter. In comparison the Folgar-Tucker equation and the differential equation for the fourth order parameter are considered. The fourth order parameter is eliminated later. The resulting equation for the second order parameter is a Duffing equation with a behavior of solutions completely different from the solutions of the Riccati equation. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence, uniqueness, and the Mittag–Leffler–Ulam stability of solutions for fractional Duffing equations involving three fractional derivatives. Uniqueness result for solution of the underlying Duffing problem is given with the help of Banach's fixed point theorem, where the existence result is computed using Leray–Schauder's alternative. Also, the Mittag–Leffler–Ulam stability results are computed by employing generalized singular Gronwall's inequality. An illustrative example is also given. [ABSTRACT FROM AUTHOR]

In recent years, the development of quick and streamlined methods for the detection and localization of structural damage has been achieved by analysing key dynamic parameters before and after significant events or as a result of aging. Many Structural Health Monitoring (SHM) systems rely on the relationship between occurred damage and variations in eigenfrequencies. While it is acknowledged that damage can affect eigenfrequencies, the reverse is not necessarily true, particularly for minor frequency variations. Thus, reducing false positives is essential for the effectiveness of SHM systems. The aim of this paper is to identify scenarios where observed changes in eigenfrequencies are not caused by structural damage, but rather by non-stationary combinations of input and system response (e.g., wind effects, traffic vibrations), or by stochastic variations in mass, damping, and stiffness (e.g., environmental variations). To achieve this, statistical variations of thresholds were established to separate linear non-stationary behaviour from nonlinear structural behaviour. The Duffing oscillator was employed in this study to perform various nonlinear analyses via Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

In this paper, the variational iterative method (VIM) with the Laplace transform is utilized to solve the nonlinear problems of a simple pendulum and mass spring oscillator, which corresponds to the Duffing equation. Finding the Lagrange multiplier (LM) is a significant phase in the VIM, and variational theory is frequently employed for this purpose. This paper demonstrates how the Laplace transform can be utilized to locate the LM in a more efficient manner. The frequency obtained by Laplace-based VIM is the same as that defined in the already existing methods in the literature in order to ensure the clarity of the results. Numerous analytical techniques can be used to solve the Duffing equation, but we are the first to do it using a Laplace-based VIM and a distinctive LM. The fundamental results of my paper are that LM is also the same in the Elzaki transformation. In the vast majority of instances, Laplace-based VIM only requires one iteration to arrive at an answer with high precision and linearization, discretization or intensive computational work is required for this purpose. Comparing analytical results of VIM by Laplace transform to the built-in Simulink command in MATLAB which gives us the surety about the method's applicability for solving nonlinear problems. Future work on the basic pendulum may examine the effects of nonlinearities and damping on its motion and the application of advanced control mechanisms to regulate its behavior. Future research on mass spring oscillators could examine the system's response to random or harmonic input. The mass spring oscillator could also be used in vibration isolation to minimize vibrations from one building to another. [ABSTRACT FROM AUTHOR]

The type of symmetry exhibited by a travelling wave can have important implications for its behaviour and properties, such as its polarization, dispersion, and interactions with other waves or boundaries. The fractional differential Duffing problem refers to the mathematical modelling of nonlinear, damped oscillations of a system with fractional derivatives. It is a generalization of the classical Duffing equation, which describes the behaviour of a nonlinear, damped oscillator (the equation becomes symmetric under time-reversal). The fractional derivatives allow for a more accurate description of the system's memory and hereditary properties. The solution of the fractional Duffing equation can provide insight into the complex dynamic behaviour of various physical, biological, and engineering systems. We are concerned with studying a new differential Duffing fractional problem. It involves some sequential Caputo derivatives with an infinite series of Riemann-Liouville integrals and some other functions. We begin by proving a first existence and uniqueness result, then we discuss two types of stability for the obtained uniqueness result. An illustrative examples is given to show the applicability of the result. We are also concerned with applying the Tanh method to obtain new classes of travelling wave solutions for three important classes of (Khalil) fractional conformable problems; the generalized equation of Duffing, the Landau-Ginzburg-Higgs equation and the Sine-Gordon one. Some numerical simulations are plotted and a conclusion is given at the end. [ABSTRACT FROM AUTHOR]

The article considers the dynamics of a nonlinear mechanical system under the action of a kinematic perturbation on it. The object's vibration isolation system is described by a rigid cubic power characteristic and is based on compensation of external perturbations — the introduction of an additional elastic element with negative stiffness into the suspension. Numerical modeling of the system is performed, the results of which are analyzed by the method of spectral analysis, based on the representation of the correlation function on a small time interval by a square polynomial. As a result of the analysis, it is found that in the pre-resonant and resonant regions, the general solution should consist of three components: a subharmonic of the order of 1/3, the fundamental harmonic, and the third harmonic. It is noted that only the subharmonic of the order of 1/3 and the fundamental harmonic are important in the resonant zone. It is also noted that even simple nonlinear mechanical systems in the study of dynamics should use approximate analytical and numerical methods in combination with spectral analysis, since traditional methods of nonlinear mechanics are not adapted to solving problems taking into account a relatively large number of harmonic components that appear due to nonlinearity.

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.

The type of symmetry exhibited by a travelling wave can have important implications for its behaviour and properties, such as its polarization, dispersion, and interactions with other waves or boundaries. The fractional differential Duffing problem refers to the mathematical modelling of nonlinear, damped oscillations of a system with fractional derivatives. It is a generalization of the classical Duffing equation, which describes the behaviour of a nonlinear, damped oscillator (the equation becomes symmetric under time-reversal). The fractional derivatives allow for a more accurate description of the system's memory and hereditary properties. The solution of the fractional Duffing equation can provide insight into the complex dynamic behaviour of various physical, biological, and engineering systems. We are concerned with studying a new differential Duffing fractional problem. It involves some sequential Caputo derivatives with an infinite series of Riemann–Liouville integrals and some other functions. We begin by proving a first existence and uniqueness result, then we discuss two types of stability for the obtained uniqueness result. An illustrative examples is given to show the applicability of the result. We are also concerned with applying the Tanh method to obtain new classes of travelling wave solutions for three important classes of (Khalil) fractional conformable problems; the generalized equation of Duffing, the Landau–Ginzburg–Higgs equation and the Sine–Gordon one. Some numerical simulations are plotted and a conclusion is given at the end.

Various types of interleavers in turbo codes and their parameters affecting the efficiency of turbo codes are considered. It is noted that the type of interleaver directly affects the efficiency and error correcting of turbo codes. Also, the efficiency of turbo codes is influenced by the parameters of the minimum distance, the length of interleaver, and the distance spectrum of the code. A modification of the chaotic interleaver of turbo codes is proposed with the change of the equation to the Duffing and with examining the code's distance spectrum with the condition of increasing the code's distance between the code words with a small weight. The algorithm for modifying the chaotic interleaver with the Duffing equation and with examining the code's distance spectrum of turbo codes is presented. The characteristics of the modified chaotic interleaver with the Duffing equation and with examining the code's distance spectrum of turbo codes according to various parameters of turbo codes are given. This modification of the interleaver of turbo codes increased the minimum distance between elements for different lengths of the interleaver and polynomials of the turbo code by 10 % …33 %. Given this, there was an increase in the energy efficiency of the turbo codes by 0,05, …, 0,25 dB in comparison with a chaotic interleaver without modification at the same value of the bit error probability. When increasing the length of the modified chaotic interleaver with the Duffing equation and applying distance spectrum of the code the increasing the energy efficiency of the turbo code slows down compared to the chaotic interleaver without modification. The application scope of the modified chaotic interleaving with the Duffing equation and with examining the code's distance spectrum of turbo codes is the infocommunication channels for mobile, wired, and satellite communications. [ABSTRACT FROM AUTHOR]

This paper focuses on the vibration periodic property of Duffing oscillator with generalized initial conditions. Firstly, the undamped case is solved by Ji-Huan He’s frequency formulation; Secondly, the formulation is extended to the damped case. Numerical verification shows that the frequency formulation is mathematically simple and physically insightful and practically applicable. This paper paves a novel way for engineers to use the formulation to study nonlinear vibration system with ease and reliability. [ABSTRACT FROM AUTHOR]

This investigation takes an analytical approach to the oscillations of a cantilevered plate immersed in a hypersonic flow with shock impingement. In particular, we derive a mathematical model keyed in on the fact that (1) the shock impingement point moves along the structure as it oscillates and (2) the local curvature of the structure changes the shock reflection angle and thus the compressible flow properties. For cantilever boundary conditions, large motion at the free end my render both effects significant. We show that the movement of the shock impingement point varies approximately with the cotangent of the oblique shock angle and this implicit relationship generates surprisingly strong nonlinear effects in the equation governing the first mode of vibration. A new, geometrically modified third-order Piston Theory is adapted to accurately capture structural curvature induced changes in flow properties as well as possible expansion wave interaction. Our model takes the form of a nonlinearly damped Duffing oscillator with quadratic and cubic nonlinearities stemming from the nonlinear aerodynamic generalized forces as well as geometric and inertial structural nonlinearities. Despite inviscid flow modeling, a perturbation solution to the governing equation shows good agreement with structural oscillations predicted by high-fidelity turbulent computational fluid dynamic simulations. [ABSTRACT FROM AUTHOR]

Let A be the generator of a strongly continuous periodic one-parameter group of bounded linear operators in a Banach space H, and let Rλ be the resolvent of A at the point λ and Φ(A) is a polynomial of A with complex coefficients. We have obtained conditions for compactness be the operator Rλ. Using this result, conditions of the existence of a solution to the nonlinear equation Φ(A)x = f(x) are obtained, where f is a continuous mapping of H into itself. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Sciences, Islamic Republic of Iran is the property of University of Tehran and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Featured Application: This work applies machine learning, specifically LSTM and LSTM-NN models, to predict the Duffing equation's convergence outcomes. It has potential applications in predicting vibration patterns in real-world systems like aircraft wings or automobile components to prevent adverse effects. This study addresses the problem of predicting convergence outcomes in the Duffing equation, a nonlinear second-order differential equation. The Duffing equation exhibits intriguing behavior in both undamped free vibration and forced vibration with damping, making it a subject of significant interest. In undamped free vibration, the convergence result oscillates randomly between 1 and −1, contingent upon initial conditions. For forced vibration with damping, multiple variables, including initial conditions and external forces, influence the vibration patterns, leading to diverse outcomes. To tackle this complex problem, we employ the fourth-order Runge–Kutta method to gather convergence results for both scenarios. Our approach leverages machine learning techniques, specifically the Long Short-Term Memory (LSTM) model and the LSTM-Neural Network (LSTM-NN) hybrid model. The LSTM-NN model, featuring additional hidden layers of neurons, offers enhanced predictive capabilities, achieving an impressive 98% accuracy on binary datasets. However, when predicting multiple solutions, the traditional LSTM method excels. The research encompasses three critical stages: data preprocessing, model training, and verification. Our findings demonstrate that while the LSTM-NN model performs exceptionally well in predicting binary outcomes, the LSTM model surpasses it in predicting multiple solutions. [ABSTRACT FROM AUTHOR]

In recent years, the development of quick and streamlined methods for the detection and localization of structural damage has been achieved by analysing key dynamic parameters before and after significant events or as a result of aging. Many Structural Health Monitoring (SHM) systems rely on the relationship between occurred damage and variations in eigenfrequencies. While it is acknowledged that damage can affect eigenfrequencies, the reverse is not necessarily true, particularly for minor frequency variations. Thus, reducing false positives is essential for the effectiveness of SHM systems. The aim of this paper is to identify scenarios where observed changes in eigenfrequencies are not caused by structural damage, but rather by non-stationary combinations of input and system response (e.g., wind effects, traffic vibrations), or by stochastic variations in mass, damping, and stiffness (e.g., environmental variations). To achieve this, statistical variations of thresholds were established to separate linear non-stationary behaviour from nonlinear structural behaviour. The Duffing oscillator was employed in this study to perform various nonlinear analyses via Monte Carlo simulations.

Heartbeat rhythms are related to a complex dynamical system based on electrical activity of the cardiac cells usually measured by the electrocardiogram (ECG). This paper presents a mathematical model to describe the electrical activity of the heart that consists of three nonlinear oscillators coupled by delayed Duffing-type connections. Coupling alterations and external stimuli are responsible for different cardiac rhythms. The proposed model is employed to build synthetic ECGs representing a variety of responses including normal and pathological rhythms: ventricular flutter, torsade de pointes, atrial flutter, atrial fibrillation, ventricular fibrillation, polymorphic ventricular tachycardia and supraventricular extrasystole. Moreover, the sinoatrial rhythm variations are described by time-dependent frequency, representing transient disturbances. This kind of situation can represent transitions between different pathological behaviors or between normal and pathological physiologies. In this regard, a nonlinear dynamics perspective is employed to describe cardiac rhythms, being able to represent either normal or pathological behaviors. [ABSTRACT FROM AUTHOR]

We study the existence, exact multiplicity, and structure of the set of positive solutions to the periodic problem u'= p (t)u + h (t)| u | λ sgn u + μ f (t); u (0)= u (ω), u'(0)= u'(ω), where μ ∈ R is a parameter. We assume that p, h, f ∈ L ([ 0, ω ]), λ > 1, and the function h is non-negative. The results obtained extend the results known in the existing literature. We do not require that the Green's function of the corresponding linear problem be positive and we allow the forcing term f to change its sign. [ABSTRACT FROM AUTHOR]

Purpose: Under certain conditions, the governing equation of motion of magnetic spherical pendulum results in a cubic-quintic Duffing equation. The current work aims to achieve an analytical bounded procedure of this equation. Methods: This may be accomplished by grouping nonlinear expanded frequency, Homotopy perturbation method (HPM), and Laplace transforms. Therefore, this technique helps disregard the appearance of the source of secular terms. Results: To validate the obtained explanation, based on the method of Runge–Kutta of the fourth order (RK4), the numerical calculation is performed. On the other hand, the linearized stability analysis is carried out to explore stability neighbouring the fixed points. Moreover, the time history of the attained solution and the corresponding phase plane plots are obtained to expose the influence of the affecting factors in the behavior of motion. Conclusions: A comparison between both solutions gives a good matching between them, which explores the worthy accuracy of the approach in question. Several phase portraits are planned toward illustrating the different types of stability and instability near the equilibrium points, where the relation between the expanded and the cyclotron frequency (that are generated by the magnetic field) is characterized for diverse standards of the azimuthal angular velocity. [ABSTRACT FROM AUTHOR]

Ultradiscretization with parity variables facilitates the ultradiscretization of even a difference equation with subtraction. However, the uniqueness of the solution may be lost under specific conditions, and an "indeterminate solution" may be obtained. In this study, the origin of the indeterminate solution is investigated from the perspective of the approximation of solutions for the difference equation by its leading term. [ABSTRACT FROM AUTHOR]

This paper presents an efficient numerical method to solve two versions of the Duffing equation by the hybrid functions based on the combination of Block-pulse functions and Legendre polynomials. This method reduces the solution of the considered problem to the solution of a system of algebraic equations. Moreover, the convergence of the method is studied. Some examples are given to demonstrate the applicability and effectiveness of the proposed method. Also, the obtained results are compared with some other results. [ABSTRACT FROM AUTHOR]

Most of the practical inductive power transfer (IPT) systems are the ones with variable coupling coefficients and loads. The output voltage, current and power are affected by the variation in coupling coefficient and load. In this paper, a novel approach based on a nonlinear resonator is proposed to obtain stable output voltage, which is independent of coupling coefficient and load variation. First, the theory and properties of nonlinear resonators are analyzed by Duffing equation. Then, a nonlinear IPT system with a magnetic saturation inductor is proposed, and the saturable inductor modeling and its effect on system performance are further studied. Finally, the experimental prototype is built to validate the effectiveness of the nonlinear IPT system. The experimental results show that when the coupling coefficient varies from 0.32 to 0.24 and the load resistance varies from 80 Ω to 120 Ω , the system works in a nonlinear state, the output voltage ripple is 1.77%, and the overall efficiency of the system is not less than 82.60%. The experimental results are basically consistent with the theoretical analysis. The novel design approach improves the output voltage stability with respect to position misalignment and load variation, and the bandwidth of the system is also enhanced. [ABSTRACT FROM AUTHOR]

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.

In this paper, nonlinear electrostatic structures on the ion time scale in plasma consisting of two populations of electrons (cold and hot), positrons, and warm adiabatic ions are investigated. The multiple scale method is used to derive the modified Korteweg–de Vries equation (mKdVE). The Jacobi elliptic function expansion method (JEFEM) is employed to find some exact analytical solutions such as periodic, solitonic, and shock solutions. It is shown that the variation in the plasma parameters of interest, for our model, allows the existence of solitary and periodic structures and no shocks. It is also shown that the most important plasma parameters for the plasma model under consideration are positron concentration, α, and the percentage of cold and hot electrons, represented by the parameters μ and ν, respectively. Additionally, the qualitative behavior of the mKdVE is studied using dynamical system theory. The topological structure of the solution is discussed in the phase plane. In this work, the phase plane analysis, which is restricted to the discrete values of the parameter, is extended to the continuous range of the parameter using a bifurcation diagram. Bifurcation diagrams are drawn to forecast the behavior of the solution for specifically chosen essential plasma parameters. The analytical solution and the qualitative behavior of the solution presented in this paper are shown to be compatible with each other. The results presented here are general and can be gainfully employed to study a variety of nonlinear waves in space, laboratory plasmas, and astrophysical plasmas.

Mechanical systems with inherent chaotic behavior are of notable practical interest due to their applicability in many fields of technology, from industrial mills and concrete mixers to microscopic micromechanical random bit generators. One of the most generic mathematical models for designing chaotic mechanical systems is the Duffing oscillator, which demonstrates chaotic motion under periodic excitation. The mechanical implementation of Duffing oscillator requires nonlinear springs, which can be implemented using different physical principles. In the current study, we propose the mechanical Duffing oscillator with magnetic springs as a low-wear, robust and easy-to-implement solution. We show by simulation and experimentation that the developed mechanical system performs chaotic oscillations in a wide range of parameters. The proposed design can be revised in a problem-specific manner and achieve many practical applications. [ABSTRACT FROM AUTHOR]

We found constructive conditions of solvability and the scheme for constructing solutions of the nonlinear autonomous boundary-value problem in the case of multiple solutions of the equation for generating constants. The convergent iterative scheme was constructed for finding approximations to solutions of the nonlinear autonomous boundary-value problem for the system of ordinary differential equations in the case of multiple solutions of the equation for generating constants. We found the approximations to solutions of the periodic boundary-value problem for autonomous equation of Duffing type as an example of applying the constructed iterative scheme. We applied residuals in the original equation for controlling the accuracy of the found approximations to solutions of the periodic boundary-value problem for the autonomous equation of Duffing type. [ABSTRACT FROM AUTHOR]

This paper provides mathematical and numerical analyses for a dynamic frictionless contact problem in which both of Signorini's condition and the normal compliance condition are used. The contact problem is considered by employing two viscoelastic (Kelvin-Voigt type) objects: a linear Timoshenko beam and a nonlinear spring. In addition, a transmission condition is imposed on one end of the beam and the top of the spring so that they can touch and vibrate together. We prove the existence of solutions satisfying all the conditions. Time discretizations and finite element methods are utilized to propose the fully discrete numerical schemes. We select several groups of data to present and discuss numerical simulations. [ABSTRACT FROM AUTHOR]

In this manuscript, we study the uniqueness and Ulam-stability type of solutions for nonlinear sequential Duffing problem with two Caputo-Hadamardtype fractional derivatives. The uniqueness of solutions is derived by Banach's fixed point theorem. Also, we prove the Ulam-Hyers stability and the Ulam-Hyers-Rassias stability of considered problem. An example is provided to illustrate our results. [ABSTRACT FROM AUTHOR]

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.

This study addresses the problem of predicting convergence outcomes in the Duffing equation, a nonlinear second-order differential equation. The Duffing equation exhibits intriguing behavior in both undamped free vibration and forced vibration with damping, making it a subject of significant interest. In undamped free vibration, the convergence result oscillates randomly between 1 and −1, contingent upon initial conditions. For forced vibration with damping, multiple variables, including initial conditions and external forces, influence the vibration patterns, leading to diverse outcomes. To tackle this complex problem, we employ the fourth-order Runge–Kutta method to gather convergence results for both scenarios. Our approach leverages machine learning techniques, specifically the Long Short-Term Memory (LSTM) model and the LSTM-Neural Network (LSTM-NN) hybrid model. The LSTM-NN model, featuring additional hidden layers of neurons, offers enhanced predictive capabilities, achieving an impressive 98% accuracy on binary datasets. However, when predicting multiple solutions, the traditional LSTM method excels. The research encompasses three critical stages: data preprocessing, model training, and verification. Our findings demonstrate that while the LSTM-NN model performs exceptionally well in predicting binary outcomes, the LSTM model surpasses it in predicting multiple solutions.

This paper is concerned with the classical Duffing equation which describes the motion of a nonlinear oscillator with an elastic force that is odd with respect to the value of deviation from its equilibrium position, and in the presence of an external periodic force. The equation depends on three dimensionless parameters. When they satisfy some relation, the equation admits exact periodic solutions with a period that is a multiple of the period of external forcing. These solutions can be written in explicit form without using series. The paper studies the nonlinear problem of the stability of these periodic solutions. The study is based on the classical Lyapunov methods, methods of KAM theory for Hamiltonian systems and the computer algorithms for analysis of area-preserving maps. None of the parameters of the Duffing equation is assumed to be small. [ABSTRACT FROM AUTHOR]

The aim of the present work is to solve the cubic-quintic Duffing Equation by using elementary tools. The solution obtained is in terms of trigonometric functions and it is compared with the solution obtained numerically. [ABSTRACT FROM AUTHOR]

Problems of non-linear equations to model real-life phenomena have a long history in science and engineering. One of the popular of such non-linear equations is the Duffing equation. An adapted block hybrid numerical integrator that is dependent on a fixed frequency and fixed step length is proposed for the integration of Duffing equations. The stability and convergence of the method are demonstrated; its accuracy and efficiency are also established.

The paper studies the existence, exact multiplicity, and a structure of the set of positive solutions to the periodic problem $$ u''=p(t)u+q(t,u)u+f(t);\quad u(0)=u(\omega),\ u'(0)=u'(\omega), $$ where $p,f\in L([0,\omega])$ and $q\colon[0,\omega]\times\mathbb{R}\to\mathbb{R}$ is Carathéodory function. The general results obtained are applied to the forced non-autonomous Duffing equation $$ u''=p(t)u+h(t)|u|^{\lambda}\operatorname{sgn} u+f(t), $$ with $\lambda>1$ and a~non-negative $h\in L([0,\omega])$. We allow the coefficient $p$ and the forcing term $f$ to change their signs.