Velocity-Dependent Conservative Nonlinear Oscillators With Exact Harmonic Solutions

Abstract

Conservative oscillator equations which have quadratic non-linearities in both velocity and displacement and which possess an exact harmonic solution are investigated. The conserved quantity is constructed, and its zero value corresponds to the harmonic solution. The further signi"cance of the harmonic solution as corresponding to a bifurcation is revealed.

Modified Van der Pol-Helmholtz oscillator equation with exact harmonic solutions

“…known as the conservative quadratic nonlinear Helmholtz oscillator or quadratic anharmonic oscillator [8,9,10]. When γ = 0 and p = 1, equation (1.3) becomes the damped anharmonic Helmholtz oscillator ẍ + λ ẋ + a 1 x + a 2 x 2 + a 3 = 0, (…”

Harmonic and non-periodic solutions of velocity-dependent conservative equations

“…The equation (20) is amplitude-dependent frequency due to the term = ac , the formula (20) becomes isochronous solutions. In this context we have successfully proved that exact harmonic solutions according to the form (2) can exist under arbitrary initial conditions where the first integral of the system (1) is not identically zero, contrary to the results of Gottlieb[1]. Let us consider now 0 the solution (20) gives the general harmonic solutions of the equation (21) as (22) shows also the existence of exact harmonic solutions according to the form (3) under arbitrary initial conditions without taking the first integral of the equation (21) equal identically to zero, contrary to the existence conditions determined by Gottlieb[1].…”

“…situation the general solution (18) is a complex-valued formula. This solution (18) satisfies the Gottlieb conditions[1] to obtain the exact harmonic solution(2), that is (14) leads to the general solution of the equation (9) in the form…”

“…where overdot means derivative with respect to time, and 1 a , 2 a , 3 a and β are constants, presumed to be conservative nonlinear oscillator differential equation by the author of [1]. It is interesting to notice that when 0…”

Haar wavelet–quasilinearization technique for fractional nonlinear differential equations

“…Consider the ath order nonlinear oscillator ordinary differential equation [41]: c D a yðxÞ À yðxÞ þ y 2 ðxÞ þ y 02 ðxÞ À 1 ¼ 0; 1 < a 6 2;…”