The Duffing Oscillator Equation and Its Applications in Physics

In this paper, we solve the Duffing equation for given initial conditions. We introduce the concept of the discriminant for the Duffing equation and we solve it in three cases depending on sign of the discriminant. We also show the way the Duffing equation is applied in soliton theory.

Positive periodic solutions to the forced non-autonomous Duffing equations

We study the existence and multiplicity of positive solutions to the periodic problem for a forced non-autonomous Duffing equation u ′′ = p ⁢ ( t ) ⁢ u - h ⁢ ( t ) ⁢ | u | λ ⁢ sgn ⁡ u + f ⁢ ( t ) ; u ⁢ ( 0 ) = u ⁢ ( ω ) , u ′ ⁢ ( 0 ) = u ′ ⁢ ( ω ) , u^{\prime\prime}=p(t)u-h(t)\lvert u\rvert^{\lambda}\operatorname{sgn}u+f(t);\quad u(0)=u(\omega),\ u^{\prime}(0)=u^{\prime}(\omega), where p , h , f ∈ L ⁢ ( [ 0 , ω ] ) p,h,f\in L([0,\omega]) and λ > 1 \lambda>1 . The obtained results are compared with the results known for the equations with constant coefficients.

Linearized Homotopy Perturbation Method for Two Nonlinear Problems of Duffing Equations

In the paper, we solve two nonlinear problems related to the Duffing equations in space and in time. The first problem is the bifurcation of Duffing equation in space, wherein a critical value of the parameter initiates the bifurcation from a trivial solution to a non-trivial solution. The second problem is an unconventional periodic problem of Duffing equation in time to determine period and periodic solution. To save computational cost and even enhance the accuracy in seeking higher order analytic solutions of these two problems, a modified homotopy perturbation method is invoked after a linearization technique being exerted on the Duffing equation, whose nonlinear cubic term is decomposed at two sides via a weight factor, such that the Duffing equation is linearized as the Mathieu type differential equation. The constant preceding the displacement is expanded in powers of homotopy parameter and the coefficients are determined to avoid secular solutions appeared in the derived sequence of linear differential equations. Consequently, after setting the homotopy parameter equal to unity and solving the amplitude equation, the higher order bifurcated solutions can be derived explicitly. For the second problem, we can determine the period and periodic solution in closed-form, which are very accurate. For the sake of comparison the results obtained from the fourth-order Runge-Kutta numerical method are used to evaluate the presented analytic solutions.

Parametric Frequency Analysis of Mathieu–Duffing Equation

The classic linear Mathieu equation is one of the archetypical differential equations which has been studied frequently by employing different analytical and numerical methods. The Mathieu equation with cubic nonlinear term, also known as Mathieu–Duffing equation, is one of the many extensions of the classic Mathieu equation. Nonlinear characteristics of such equation have been investigated in many papers. Specifically, the method of multiple scale has been used to demonstrate the pitchfork bifurcation associated with stability change around the first unstable tongue and Lie transform has been used to demonstrate the subharmonic bifurcation for relatively small values of the undamped natural frequency. In these works, the resulting bifurcation diagram is represented in the parameter space of the undamped natural frequency where a constant value is allocated to the parametric frequency. Alternatively, this paper demonstrates how the Poincaré–Lindstedt method can be used to formulate pitchfork bifurcation around the first unstable tongue. Further, it is shown how higher order terms can be included in the perturbation analysis to formulate pitchfork bifurcation around the second tongue, and also subharmonic bifurcations for relatively high values of parametric frequency. This approach enables us to demonstrate the resulting global bifurcation diagram in the parameter space of parametric frequency, which is beneficial in the bifurcation analysis of systems with constant undamped natural frequency, when the frequency of the parametric force can vary. Finally, the analytical approximations are verified by employing the numerical integration along with Poincaré map and phase portraits.

Application of EEG Signal Recognition Method Based on Duffing Equation in Psychological Stress Analysis

Based on the study of the feature extraction algorithm based on the multiple empirical mode decomposition of the Duffing equation, this paper proposes a corresponding improved algorithm, completes the identification and analysis of the psychological pressure dimension space under the audiovisual induction method, and designs two typical psychological types of music and pictures. Based on the stress induction experiment, an audiovisual-induced psychological stress recognition system based on EEG (electroencephalogram) signals was built. Aiming at the problem that the spatial uniform sampling method cannot well reflect the dynamic characteristics of the multivariate EEG signal, based on the Duffing equation, a nonuniform sampling algorithm that adaptively selects the projection direction is proposed. At present, the use of the Duffing equation to detect weak unknown signals is to select a set of fixed parameters. Analysis of these two aspects to determine the parameters of the system is based on the parameter analysis of the Duffing equation oscillator. Due to the sensitivity of the Duffing equation to the initial value, the choice of parameters has a great influence on the detection effect. In response to this situation, the relationship between the parameters and initial values of the Duffing equation is analyzed. From the relationship between the parameters and the initial values, the influence of different parameters on the detection effect is analyzed to verify the superiority of the current equation parameters. First, the multichannel EEG signal is nonuniformly sampled multiempirical modal decomposition, and an effective intrinsic modal function is selected to extract the mental stress EEG characteristics. Experimental results show that the EEG signal recognition algorithm based on the Duffing equation effectively extracts EEG signal features and improves the classification accuracy of mental stress EEG signals.

On the development of a nonlinear time-domain numerical method for describing vortex-induced vibration and wake interference of two cylinders using experimental results

A nonlinear mathematical model is developed in the time domain to simulate the behaviour of two identical flexibly mounted cylinders in tandem while undergoing vortex-induced vibration (VIV). Subsequently, the model is validated and modified against experimental results. Placing an array of bluff bodies in proximity frequently happens in different engineering fields. Chimney stacks, power transmission lines and oil production risers are few engineering structures that may be impacted by VIV. The coinciding of the vibration frequency with the structure natural frequency could have destructive consequences. The main objective of this study is to provide a symplectic and reliable model capable of capturing the wake interference phenomenon. This study shows the influence of the leading cylinder on the trailing body and attempts to capture the change in added mass and damping coefficients due to the upstream wake. The model is using two coupled equations to simulate the structural response and hydrodynamic force in each of cross-flow and stream-wise directions. Thus, four equations describe the fluid–structure interaction of each cylinder. A Duffing equation describes the structural motion, and the van der Pol wake oscillator defines the hydrodynamic force. The system of equations is solved analytically. Two modification terms are added to the excitation side of the Duffing equation to adjust the hydrodynamic force and incorporate the effect of upstream wake on the trailing cylinder. Both terms are functions of upstream shedding frequency (Strouhal number). Additionally, the added mass modification coefficient is a function of structural acceleration and the damping modification coefficient is a function of velocity. The modification coefficients values are determined by curve fitting to the difference between upstream and downstream wake forces, obtained from experiments. The damping modification coefficient is determined by optimizing the model against the same set of experiments. Values of the coefficients at seven different spacings are used to define a universal function of spacing for each modification coefficient so that they can be obtained for any given distance between two cylinders. The model is capable of capturing lock-in range and maximum amplitude.

Modified Hybrid Method with Four Stages for Second Order Ordinary Differential Equations

A modified explicit hybrid method with four stages is presented, with the first stage exactly integrating exp(wx), while the remaining stages exactly integrate sin(wx) and cos(wx). Special attention is paid to the phase properties of the method during the process of parameter selection. Numerical comparisons of the proposed and existing hybrid methods for several second-order problems show that the proposed method gives high accuracy in solving the Duffing equation and Kramarz’s system.