In this paper, we investigate two of the analytical approximate techniques, energy balance method and amplitude-frequency formulation, and these approximate techniques are applied to solve the strongly nonlinear differential equation of a mass attached to the center of a stretched elastic wire. We present a comparative study between the energy balance method and amplitude-frequency formulation with exact solution. The approximate results reveal that these methods are very effective and convenient for determining the frequencies of nonlinear dynamical systems.

We want to study the dynamics of a simple linear harmonic micro spring which is under the influence of the quantum Casimir force/pressure and thus behaves as a (an) nonlinear (anharmonic) Casimir oscillator. Generally, the equation of motion of this nonlinear micromechanical Casimir oscillator has no exact solvable (analytical) solution and the turning point(s) of the system has (have) no fixed position(s); however, for particular values of the stiffness of the micro spring and at appropriately well-chosen distance scales and conditions, there is (are) approximately sinusoidal solution(s) for the problem (the variable turning points are collected in a very small interval of positions). This, as a simple and elementary plan, may be useful in controlling the Casimir stiction problem in micromechanical devices.

In this paper, the efficient multi-step differential transform method (EMsDTM) is applied to get the accurate approximate solutions for strongly nonlinear duffing oscillator. The main improvement of EMsDTM which is to reduce the number of arithmetic operations, is thoroughly investigated and compared with the classic multi-step differential transform method (MsDTM). To illustrate the applicability and accuracy of the new method, six case studies of the free undamped and forced damped conditions are considered. The periodic response curves of both MsDTM and EMsDTM methods are obtained and contrasted with the exact solution or the numerical solution of Runge Kutta 4th order (RK4) method. This approach can be easily extended to other nonlinear systems and therefore is widely applicable in engineering and other sciences.

In this paper, we consider the nonlinear equations with the additive white noise, which are commonly impossible to be solved by an analytical procedure. The Block-Pulse functions as basic functions are proposed to solve these equations. In order to investigate the validity of this method, we used the Adomian decomposition method to approximate the solution of the stochastic Du ng equations. The results reveal that the proposed method is very e ective.

In the present study, some perturbation methods are applied to Duffing equations having a displacementtime-delayed variable to study the stability of such systems. Two approaches are considered to analyze Duffingoscillator having a strong delayed variable. The homotopy perturbation method is applied through the frequencyanalysis and nonlinear frequency is formulated as a function of all the problem’ s parameters. Based on the multiplescales homotopy perturbation method, a uniform second-order periodic solution having a damping part isformulated. Comparing these two approaches reveals the accuracy of using the second approach and further allowsstudying the stability behavior. Numerical simulations are carried out to validate the analytical finding.

Background and Objectives: The Differential transform method (DTM) is used in the analysis of ordinary, partial, and high-order differential equations. Recently, the DTM is used in the nonlinear analysis of physical nonlinear dynamic systems. Methods: The DTM method is used to analyze and analytically solve the nonlinear mathematical model of bias current-controlled Colpitts oscillator with variable coefficients. Intervals of the validity of the proposed method are evaluated by using the fourth order Runge-Kutta method (RK4M). In this note, the Lyapunov exponent (LE) can be used to analyze the Colpitts oscillator. By using DTM, the LEs are calculated analytically with unknown parameters in a short interval of time t [0, 3 Sec]. Results: In this paper, intervals of the validity of the proposed method are evaluated using RK4M. In addition, LEs are calculated using analytical and numerical methods based on DTM technique and Wolf method, respectively. LEs of the proposed system are presented as a function of the control parameter to confirm the applied technique’ s usefulness. Conclusion: By comparing these two methods, the proposed DTM analytical technique is relatively more precise. Simulation results confirmed the impact of different parameters on LEs with two different initial conditions. The results show good accuracy of the DTM in short time intervals t [0, 3 Sec].

Effect of nonlinear terms f (x, x&) , especially nonlinear damping on the periodic solutions of the harmonic oscillator is studied in this paper. Existence of limit cycle and periodic solutions for the perturbed oscillator are proved. Also stability of the solutions as well as Hopf bifurcation for the perturbed oscillator is verified. Suppose that the function f (x, x&) is analytic with respect to x and x& . Then the following equation is considered: x +ω2 x + έ (b0 x + a0 x2 + b2 x2 +b1xx +b2 x2 + b3 x 3)=0 where a0, bi, i = 0, 1 2...3 are real parameters and ε is small. We will show that if b0=0 and a0, bi, i=1,2...3 are sufficiently small, then the nonlinear terms do not change the phase space of the harmonic oscillator as long as the initial values are not large. In this case the Hopf bifurcation occurs. But if b0 ≠0, which means that the linear term is effected, then the periodic solutions will be destroyed. In this case a unique limit cycle exists.