Double-zero degeneracy and heteroclinic cycles in a perturbation of the Lorenz system

In this paper we consider a 3D three-parameter unfolding close to the normal form of the triple-zero bifurcation exhibited by the Lorenz system. First we study analytically the double-zero degeneracy (a double-zero eigenvalue with geometric multiplicity two) and two Hopf bifurcations. We focus on the more complex case in which the doublezero degeneracy organizes several codimension-one singularities, namely transcritical, pitchfork, Hopf and heteroclinic bifurcations. The analysis of the normal form of a Hopf-transcritical bifurcation allows to obtain the expressions for the corresponding bifurcation curves. A degenerate double-zero bifurcation is also considered. The theoretical information obtained is very helpful to start a numerical study of the 3D system. Thus, the presence of degenerate heteroclinic and homoclinic orbits, T-point heteroclinic loops and chaotic attractors is detected. We find numerical evidence that, at least, four curves of codimension-two global bifurcations are related to the triple-zero degeneracy in the system analyzed.We thank the reviewers for their careful reading of the manuscript and their very constructive remarks which have helped a lot to improve the presentation of the results. This work has been partially supported by the Ministerio de Economia y Competitividad, Spain (project MTM2017-87915-C2-1-P, co-financed with FEDER funds) , by the Ministerio de Ciencia, Innovacion y Universidades, Spain (project PGC2018-096265-B-I00, co-financed with FEDER funds) and by the Consejeria de Economia, Innovacion, Ciencia y Empleo de la Junta de Andalucia, Spain (FQM-276, TIC-0130, UHU-1260150 and P20_01160)

Double-zero degeneracy and heteroclinic cycles in a perturbation of the Lorenz system

In this paper we consider a 3D three-parameter unfolding close to the normal form of the triple-zero bifurcation exhibited by the Lorenz system. First we study analytically the double-zero degeneracy (a double-zero eigenvalue with geometric multiplicity two) and two Hopf bifurcations. We focus on the more complex case in which the double-zero degeneracy organizes several codimension-one singularities, namely transcritical, pitchfork, Hopf and heteroclinic bifurcations. The analysis of the normal form of a Hopf-transcritical bifurcation allows to obtain the expressions for the corresponding bifurcation curves. A degenerate double-zero bifurcation is also considered. The theoretical information obtained is very helpful to start a numerical study of the 3D system. Thus, the presence of degenerate heteroclinic and homoclinic orbits, T-point heteroclinic loops and chaotic attractors is detected. We find numerical evidence that, at least, four curves of codimension-two global bifurcations are related to the triple-zero degeneracy in the system analyzed.Ministerio de Economía y Competitividad MTM2017-87915-C2-1-PMinisterio de Ciencia, Innovación y Universidades - Fondos FEDER PGC2018-096265-B-I0Consejería de Economía, Innovación, Ciencia y Empleo - Junta de Andalucía FQM-276, TIC-0130, UHU-1260150 y P20_0116

Analytical study of a hovering magnetic system (Levitron)

El objetivo de este proyecto es llevar a cabo un estudio analítico del sistema de levitación magnética, conocido como Levitron. En primer lugar, se va a empezar haciendo una introducción explicando el funcionamiento del juguete, los elementos típicos que conforman el kit para jugar y su historia. En el capítulo 2 comienza el estudio analítico: se empieza describiendo el movimiento de la peonza como sólido libre, en ausencia de campo magnético externo y sometida únicamente a la acción de la gravedad, utilizando los ángulos clásicos de Euler. Sin embargo, veremos que el uso de estos ángulos produce una singularidad que nos obligará a presentar, en el capítulo 3, los ángulos de Tait-Byan para solucionarlo. En el capítulo 4 obtenemos las expresiones del campo magnético creado por la base, y en el capítulo 5 enunciamos el Teorema de Earnshaw y se estudia la estabilidad estática del sistema por medio de su energía potencial. En el capítulo 6, se obtienen las ecuaciones de movimiento utilizando la mecánica vectorial, y en el capítulo 7 se obtienen las ecuaciones del sistema empleando la formulación Lagrangiana y Hamiltoniana de la mecánica analítica. Una vez obtenidas las ecuaciones, se introducen variables adimensionales en el capítulo 8 para llevar a cabo la simulación numérica de los posteriores apartados. En el capítulo 9 estudiamos la estabilidad lineal del sistema, obteniendo la región de estabilidad en la que es posible la levitación y los modos normales de movimiento. En el capítulo 10, analizamos el acoplamiento lineal y no lineal y simulamos numéricamente la trayectoria considerando distintas condiciones iniciales que intentan reproducir situaciones reales que se dan a la hora de jugar. En el capítulo 11 presentamos las constantes de movimiento del sistema, y en el capítulo 12 se elabora un sencillo modelo para tener en cuenta el efecto de la fricción del aire sobre la peonza. Finalmente, en el capítulo 13 se añaden algunas instrucciones para dominar el juguete y se adjunta un diagrama de flujo donde se describen las situaciones habituales a las que un jugador se enfrenta cuando intenta hacer levitar la peonza. Por último, se adjuntan tres anexos: el primero contiene el ajuste experimental del campo magnético en el eje 01; el segundo está dedicado a los cuaterniones, incluyendo las propiedades más importantes y el sistema con las ecuaciones de movimiento en término de los parámetros de Euler; y en el tercero, se estudia la dinámica de tres sistemas que guardan analogías con el sistema de levitación magnética.The scope of this project is to perform an analytical study of the hovering magnetic system, known as Levitron. Firstly, we will start with an introduction where we explain how the toy works, the typical elements of the kit and its history. In chapter 2, the analytical study starts: we begin describing the free motion, in absence of the external magnetic field and solely submitted to the action of gravity, using the Classic Euler angles. Nevertheless, the use of these angles leads to a singularity that we need to avoid, so in chapter 3 we present the Tait-Bryan angles to solve this problem. In chapter 4 we obtain the expressions for the magnetic field generated by the base, and in chapter 5 we outline Earnshaw’s Theorem and study the static stability by means of the potential energy of the system. In chapter 6, we derive the equations of motion using vector mechanics, and in chapter 7 the same is done using the Lagrangian and Hamiltonian formulation of the analytical mechanics. Once the equations of the system are obtained, we define nondimensional variables in chapter 8 to perform the numerical simulations. In chapter 9 we study the linear stability of the system, obtaining the stability region for which stable hovering is possible and the normal modes. In chapter 10 we analyze the linear and nonlinear coupling, and numerically simulate the trajectory of the spinning top considering different initial conditions that reproduce real situations when one plays with the toy. In chapter 11, we present the constants of the motion of the system, and in chapter 12 a simple model considering air friction is shown. Finally, chapter 13 contains some instructions to master the toy and we attach a flowchart where we describe the usual situations that a player has to face to achieve levitation. Lastly, we include three addendums: the first contains the experimental adjustment of the magnetic field generated by the base in the 0�����1 axis; the second is dedicated to the quaternions, containing the main properties of its algebra and the set of equations of the system in terms of the Euler parameters; and finally, in the third we study the dynamics of three systems that share some of the characteristics of the hovering magnetic device.Universidad de Sevilla. Grado en Ingeniería de Tecnologías industriale

Progress in Surface Theory

Over the last 30 years global surface theory has become pivotal in the understanding of low dimensional global phenomena. At the same time surface geometry became a platform on which seemingly different areas of mathematics – such as geometric and topological analysis, integrable systems, algebraic geometry of curves, and mathematical physics – coalesce to produce far reaching ideas, conjectures, methods and results. The workshop hosted talks on the resolutions of famous conjectures in surface geometry, including the Willmore conjecture, and on exciting new progress in the understanding of moduli spaces of special surface classes

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

Robot Manipulators

Robot manipulators are developing more in the direction of industrial robots than of human workers. Recently, the applications of robot manipulators are spreading their focus, for example Da Vinci as a medical robot, ASIMO as a humanoid robot and so on. There are many research topics within the field of robot manipulators, e.g. motion planning, cooperation with a human, and fusion with external sensors like vision, haptic and force, etc. Moreover, these include both technical problems in the industry and theoretical problems in the academic fields. This book is a collection of papers presenting the latest research issues from around the world