Dynamics of the Anharmonic Oscillators

Abstract

Most of the physical systems executing oscillations are anharmonic in nature. Anharmonic oscillators (AHO) are the deviation from the ideal case of the harmonic oscillator and are mostly not exactly solvable because of the presence of the nonlinear terms in the governing equations of the systems. Therefore, it is very difficult to find an approximate analytical or numerical solutions to them. Wide applications of AHO in describing the physical systems is the main reason for attracting the attention of researchers in science and engineering. Still, one strives for finding an accurate solution to the governing equation of AHO in an easy way. newlineThe perturbation method is a widely used approximate method for getting the solution of complex nonlinear systems, especially with the nonlinear terms appearing as the additional term of small order to an exactly solvable problem. This technique is based on the assumption of the existence of a small parameter (perturbation parameter) so that the approximate solutions can be expressed as a power series in terms of the small parameter. Making a proper choice of the perturbation parameter is a tricky job. Moreover, many nonlinear systems have no small parameter. newlineMany new techniques have been proposed to eliminate the small-parameter assumption so that the major limitation of the perturbation method may be avoided. Energy balance method (EBM), based on the variational principle, is one of the commonly used non-perturbative techniques. Recently, it has been shown that in some cases the application of frequency-amplitude-formulation (FAF) followed by EBM to the systems of strongly nonlinear vibration yields improved results in comparison to EBM alone.