Algunas contribuciones al análisis de sistemas lineales a trozos.
|Alternative title||Some contributions to the analysis of piecewise linear systems.|
|Author/s||Amador Rodríguez, Andrés Felipe|
|Director||Ponce Núñez, Enrique
Ros Padilla, Francisco Javier
|Department||Universidad de Sevilla. Departamento de Matemática Aplicada II (ETSI)|
|Abstract||This thesis consists of two parts, with contributions to the analysis of dynamical systems in continuous time and in discrete time, respectively.
In the first part, we study several models of memristor oscillators of ...
This thesis consists of two parts, with contributions to the analysis of dynamical systems in continuous time and in discrete time, respectively. In the first part, we study several models of memristor oscillators of dimension three and four, providing for the first time rigorous mathematical results regarding the rich dynamics of such memristor oscillators, both in the case of piecewise linear models and polynomial models. Thus, for some families of discontinuous 3D piecewise linear memristor oscillators, we show the existence of an infinite family of invariant manifolds and that the dynamics on such manifolds can be modeled without resorting to discontinuous models. Our approach provides topologically equivalent continuous models with one dimension less but with one extra parameter associated to the initial conditions. It is possible so to justify the periodic behavior exhibited by such three dimensional memristor oscillators, by taking advantage of known results for planar continuous piecewise linear systems. By using the first-order Melnikov theory, we derive the bifurcation set for a three-parametric family of Bogdanov-Takens systems with symmetry and deformation. As an applications of these results, we study a family of 3D memristor oscillators where the characteristic function of the memristor is a cubic polynomial. In this family we also show the existence of an infinity number of invariant manifolds. Also, we clarify some misconceptions that arise from the numerical simulations of these systems, emphasizing the important role of invariant manifolds in these models. In a similar way than for the 3D case, we study some discontinuous 4D piecewise linear memristor oscillators, and we show that the dynamics in each stratum is topologically equivalent to a continuous 3D piecewise linear dynamical system. Some previous results on bifurcations in such reduced systems, allow us to detect rigorously for the first time a multiple focus-center-cycle bifurcation in a three-parameter space, leading to the appearance of a topological sphere in the original model, completely foliated by stable periodic orbits. In the second part of this thesis, we show that the two-dimensional stroboscopic map defined by a second order system with a relay based control and a linear switching surface is topologically equivalent to a canonical form for discontinuous piecewise linear systems. Studying the main properties of the stroboscopic map defined by such a canonical form, the orbits of period two are completely characterized. At last, we give a conjecture about the occurrence of the big bang bifurcation in the previous map.
|Citation||Amador Rodríguez, A.F. (2018). Algunas contribuciones al análisis de sistemas lineales a trozos.. (Tesis Doctoral Inédita). Universidad de Sevilla, Sevilla.|