The amplitudes of nonlinear oscillations

Consider the nonlinear, nonautonomous differential equationy″(t)+p(t) f(y(t))=0,t0foru≠0, and∫0±∞f(u) du=∞. We obtain some results about the asymptotic behavior of the amplitudes of all oscillatory solutions of this equation. Combining this result with some known oscillation results, we obtain the global results for all solutions of our equation. © 2004 Elsevier Ltd. All rights reserved

NONLINEAR OSCILLATIONS IN ION CYCLOTRON RESONANCE

Nonlinear equations of motion of an ion in a quartic electrostatic potential are solved using a perturbation method. A valid region of the quartic potential in an cubic ion trap is determined from the numerical comparison with the exact potentials. The quartic potential introduces the coupled cubic nonlinear forces to the equations of motion. Cubic nonlinearities that couple the axial and radial motions yield amplitude-dependent frequency shifts. Nonlinear solutions are derived to first order. The ion trajectory within the valid region of the quartic potential reproduces the essential features of the exact numerical results. The amplitudes of nonlinear oscillations are derived and their mass dependences are compared. The Fourier transform ion cyclotron resonance spectra are compared for the linear and nonlinear oscillations. The collisional damping is included as a perturbation in the nonlinear equations of motion. Collisions damp the cyclotron and axial oscillations but amplify the magnetron motion. The amplitude-dependent frequencies of nonlinear oscillations become time dependent due to the collisional damping of oscillation amplitudes. The collisional line broadenings of the nonlinear oscillation trajectories are compared with those of the linear oscillation trajectories.X119sciescopu