Derivation of the slow mechanical system

This chapter proves various normal form results and formulates the coordinate changes that are used to derive the slow system at the double resonance. The discussions here apply to arbitrary degrees of freedom. The results also apply to the proof of the main theorem by restricting to the case n = 2. First, the chapter reduces the system near an n-resonance to a normal form. After that, it performs a coordinate change on the extended phase space, and an energy reduction to reveal the slow system. The chapter then describes a resonant normal form, before explaining the affine coordinate change and the rescaling, revealing the slow system. Finally, it discusses variational properties of these coordinate changes.

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.