Large Uniform Expansions of Periodic Solutions to Strongly Non-Linear Evolution Equations with Odd Polynomial Non-Linearity

Abstract

A new method of uniform expansions of periodic solutions to ordinary differential equations with arbitrary odd polynomial non-linearity is constructed to study quasi-harmonic processes in non-linear dynamical systems, in particular when a small parameter of non-linearity is absent. The main idea of the method consists in using the ratio of the amplitudes of higher harmonics to the amplitude of the first harmonic of a periodic solution as a small formal parameter. In the particular case of a single-periodic solution, this small parameter appears due to descending the amplitudes of harmonics monotonically with increasing their number. Due to uniform expansion the amplitudes of higher harmonics turn out to be rational and fractional functions in the amplitude of the first harmonic and the frequency of oscillations. We show that the method of uniform expansions is an effective tool for obtaining convergent expansions of periodic solutions in explicit form all over the domain, where periodic solutions exist, independently of the magnitude of non-linearity. In each subsequent approximation, one more higher harmonic is taken into account, with all the other harmonics being corrected. We demonstrate the effectiveness of the method on the examples of the harmonically forced Duffing oscillator; free vibrations of the oscillator with fifth-power non-linearity and mathematical pendulum.