学位論文: [徳島大学.先端技術科学教育部.システム創生工学専攻.知能情報システム工学コース]/[課程博士]/Numerical Bifurcation Analysis Method for Hybrid Systems/20090930/[上田 哲史]
(英) Numerical Bifurcation Analysis Method for Hybrid Systems
(英) In the field of dynamical system analysis, piecewise-smooth models have grown in popularity due to there greater flexibility and accuracy in representing some hybrid systems in applications such as electronics or mechanics. Hybrid dynamical systems have two sets of variables, one which evolve in a continuous space, and the other in a discrete one. Most analytical methods require the orbit to be smooth during objective intervals, so that some special treatments are inevitable to study the existence and stability of solutions in hybrid dynamical systems. Based on a piecewise-smooth model, where the orbit of the system is broken down into locally smooth pieces, and a hybrid bifurcation analysis method, using a Poincare map with sections ruled by the switching conditions of the system, we review the analysis process in details. Then we apply it to various extensions of the Alpazur oscillator, originally a nonsmooth 2-dimension switching oscillator. The original Alpazur oscillator, as a simple nonlinear switching system, was a perfect candidate to prove the efficiency of the approach. Each of its extensions shows a new scenario and how it can be handled, in order to illustrate the generality of the model. Finally, and in order to show more of the implementation we used for our own computer-based analysis tool, some of the most relevant numerical methods we used are introduced. It is noteworthy that the emphasis has been put on autonomous systems because the treatment of non-autonomous ones only requires a simplification (no time variation). This study brings a strong and general framework for the bifurcation analysis of nonlinear hybrid dynamical systems, illustrated by some results. Among them, some interesting local and global properties of the Alpazur Oscillator are revealed, such as the presence of a cascade of cusps in the bifurcation diagram. Our work resulted in the implementation of an analysis tool, implemented in C++, using the numerical methods that we chose for this particular purpose, such as the numerical approximation of the second derivative elements in the Jacobian matrix.
|年月日||必須||2009年 9月 30日|