17 research outputs found
Some contributions to the analysis of piecewise linear systems.
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
18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems: Proceedings
Proceedings of the 18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems, which took place in Dresden, Germany, 26 – 28 May 2010.:Welcome Address ........................ Page I
Table of Contents ........................ Page III
Symposium Committees .............. Page IV
Special Thanks ............................. Page V
Conference program (incl. page numbers of papers)
................... Page VI
Conference papers
Invited talks ................................ Page 1
Regular Papers ........................... Page 14
Wednesday, May 26th, 2010 ......... Page 15
Thursday, May 27th, 2010 .......... Page 110
Friday, May 28th, 2010 ............... Page 210
Author index ............................... Page XII
Hybrid Bifurcations and Stable Periodic Coexistence for Competing Predators
We describe a new mechanism that triggers periodic orbits in smooth dynamical
systems. To this end, we introduce the concept of hybrid bifurcations: Such
bifurcations occur when a line of equilibria with an exchange point of normal
stability vanishes. Our main result is the existence and stability criteria of
periodic orbits that bifurcate from breaking a line of equilibria. As an
application, we obtain stable periodic coexistent solutions in an ecosystem for
two competing predators with Holling's type II functional response
Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion
In this tutorial, we discuss self-excited and hidden attractors for systems
of differential equations. We considered the example of a Lorenz-like system
derived from the well-known Glukhovsky--Dolghansky and Rabinovich systems, to
demonstrate the analysis of self-excited and hidden attractors and their
characteristics. We applied the fishing principle to demonstrate the existence
of a homoclinic orbit, proved the dissipativity and completeness of the system,
and found absorbing and positively invariant sets. We have shown that this
system has a self-excited attractor and a hidden attractor for certain
parameters. The upper estimates of the Lyapunov dimension of self-excited and
hidden attractors were obtained analytically.Comment: submitted to EP