Obtenção e análise de modelos discretizados dinamicamente válidos para sistemas conservativos: dois estudos de caso

Abstract

When dealing with the solution of conservative nonlinear dierential equations, several problems such as energy loss and symmetry break can occur. In order to avoid such problems as much as possible, several numerical integration methods can be found in the literature. Although the primary objective of these methods is the solution itself, a possible and interesting by-product is a difference equation (a discretized model) thathopefully reproduces the same behavior as the one generated by the original differential equations. Discretization methods, not only provide discretized models which may result in valid solutions, but also give the possibility to use models for analysis, modelling and control. The purpose of this work is to find and analyse valid discretized models using several discretization schemes for conservative systems. These discretized models must exhibit the same behavior as the original counterpart and therefore conserve the energy and symmetry of the solution even for large discretization steps. In order to obtain dynamically valid models, two discretization methods are investigated and used to generate conservative discretized models. For each model, energy andsymmetry are analysed simultaneously to the increase of the discretization step. Dynamical invariants, such as the Lyapunov exponents, are also shown. Results show that is possible to maintain the stability and the simmetry, characteristic of conservative continuous systems, even for higher discretization steps.