Canonical Discontinuous Planar Piecewise Linear Systems

The family of Filippov systems constituted by planar discontinuous piecewise linear systems with two half-plane linearity zones is considered. Under generic conditions that amount to the boundedness of the sliding set, some changes of variables and parameters are used to obtain a Li´enard-like canonical form with seven parameters. This canonical form is topologically equivalent to the original system if one restricts one’s attention to orbits with no points in the sliding set. Under the assumption of focus-focus dynamics, a reduced canonical form with only five parameters is obtained. For the case without equilibria in both open half-planes we describe the qualitatively different phase portraits that can occur in the parameter space and the bifurcations connecting them. In particular, we show the possible existence of two limit cycles surrounding the sliding set. Such limit cycles bifurcate at certain parameter curves, organized around different codimension-two Hopf bifurcation points. The proposed canonical form will be a useful tool in the systematic study of planar discontinuous piecewise linear systems, in which this paper is a first step

The Focus-Center-Limit Cycle Bifurcation in Symmetric 3D Piecewise Linear Systems

The birth of limit cycles in 3D (three-dimensional) piecewise linear systems for the relevant case of symmetrical oscillators is considered. A technique already used by the authors in planar systems is extended to cope with 3D systems, where a greater complexity is involved. Under some given nondegeneracy conditions, the corresponding theorem characterizing the bifurcation is stated. In terms of the deviation from the critical value of the bifurcation parameter, expressions in the form of power series for the period, amplitude, and the characteristic multipliers of the bifurcating limit cycle are also obtained. The results are applied to accurately predict the birth of symmetrical periodic oscillations in a 3D electronic circuit genealogically related to the classical Van der Pol oscillator.Ministerio de Ciencia y Tecnología DPI2000-1218-C04-04Ministerio de Ciencia y Tecnología BFM2001-2668Ministerio de Ciencia y Tecnología BFM2003-00336Junta de Andalucía TIC-13

On the Takens-Bogdanov Bifurcation in the Chua’s Equation

The analysis of the Takens-Bogdanov bifurcation of the equilibrium at the origin in the Chua’s equation with a cubic nonlinearity is carried out. The local analysis provides, in first approximation, different bifurcation sets, where the presence of several dynamical behaviours (including periodic, homoclinic and heteroclinic orbits) is predicted. The local results are used as a guide to apply the adequate numerical methods to obtain a global understanding of the bifurcation sets. The study of the normal form of the Takens-Bogdanov bifurcation shows the presence of a degenerate (codimension-three) situation, which is analyzed in both homoclinic and heteroclinic cases

Bifurcation patterns in homogeneous area-preserving piecewise-linear maps

The dynamical behavior of a family of planar continuous piecewise linear maps with two zones is analyzed. Assuming homogeneity and preservation of areas we obtain a canonical form with only two parameters: the traces of the two matrices defining the map. It is shown the existence of sausage-like structures made by lobes linked at the nodes of a nonuniform grid in the parameter plane. In each one of these structures, called resonance regions, the rotation number of the associated circle map is a given rational number. The boundary of the lobes and a significant inner partition line are studied with the help of some Fibonacci polynomials.Postprint (author's final draft

Variable-Angle Phase-Shifted PWM for Multilevel Three-Cell Cascaded H-bridge Converters

Multilevel cascaded H-bridge converters have become a mature technology for applications where high-power medium ac voltages are required. Normal operation of multilevel cascaded H-bridge converters assumes that all power cells have the same dc voltage, and each power cell generates the same voltage averaged over a sampling period using a conventional phase-shifted pulse width modulation (PWM) technique. However, this modulation method does not achieve good results under unbalanced operation per H-bridge in the power converter, which may happen in grid-connected applications such as photovoltaic or battery energy storage systems. In the paper, a simplified mathematical analysis of the phase-shifted PWM technique is presented. In addition, a modification of this conventional modulation method using variable shift angles between the power cells is introduced. This modification leads to the elimination of harmonic distortion of low-order harmonics due to the switching (triangular carrier frequency and its multiples) even under unbalanced operational conditions. The analysis is particularized for a three-cell cascaded H-bridge converter, and experimental results are presented to demonstrate the good performance of the proposed modulation method

An efficient and accurate linearization approach for hydraulically actuated multibody systems with holonomic and nonholonomic constraints

Hydraulics is often used to actuate mechanisms in the applications of heavy machinery. In this work, a linearization approach for hydraulically driven multibody systems is presented. The approach allows linearizing the equations of motion of general multibody systems with holonomic and nonholonomic constraints, augmented with the hydraulic equations of the hydraulic subsystem. The derivation of this linearization approach is of interest in many applications, such as the performance of linear stability analyses. The procedure is tested with a three-dimensional multibody model of a hydraulically actuated four-bar mechanism. The validation of the approach is performed by means of the forward dynamics simulation of the linear and nonlinear systems. The results show the power of the approach, obtaining the linearized equations of motion around the equilibrium position of the four-bar mechanism multibody model in terms of the mechanical and hydraulic parameters. A comparison of the proposed procedure with a conventional counterpart approach is included, demonstrating the great accuracy and computational efficiency of the approach developed in this work

Bifurcación Silla-Nodo de Conos Invariantes en sistemas lineales a trozos Vía Bifurcación Foco-Centro-Ciclo Límite

En este trabajo se considera la existencia de conos invariantes en sistemas dinámicos continuos tridimensionales lineales a trozos, dada la relevancia que estas variedades invariantes tienen en la determinación de la estabilidad del origen en tales sistemas. Se recogen varios resultados de existencia de conos invariantes y se analiza una bifurcación silla-nodo de estas variedades invariantes. La relación biunívoca existente entre los conos invariantes y las ´orbitas periódicas de ciertos sistemas planos discontinuos (en particular, las que se generan en una bifurcación foco- centro-ciclo límite) constituye la herramienta fundamental en el estudio.Ministerio de Educaci´on y CienciaJunta de Andalucí

Families of symmetric periodic orbits in the three body problem and the figure eight

Ejemplar dedicado a: Actas de las VI Jornadas de Mecánica CelesteIn this paper we show a technique for the continuation of symmetric periodic orbits in systems with time-reversal symmetries. The geometric idea of this technique allows us to generalize the “cylinder” theorem for this kind of systems. We state the main theoretical result without proof (to be published elsewhere). We focus on the application of this scheme to the three body problem (TBP), taking as starting point the figure eight orbit [3] to find families of symmetric periodic orbits.DGYCIT/ Junta de Andalucía DGES PB98-1152DGYCIT/ Junta de Andalucía BFM-2003-0033

Continuation of Gerver's supereight choreography

Ejemplar dedicado a: Actas de las IX Jornadas de Mecánica CelesteIn [6] we developed a continuation technique for periodic orbits in reversible systems having some first integrals and corresponding symmetries. One of the applications was the continuation of Gerver’s supereight choreography when one or several of the masses are varied. In this note we give a more complete description of the families of periodic orbits which can be obtained in this way.Spanish Ministry of Education BFM2003-00336Spanish Ministry of Education MTM2006-00847University of Seville SAB2005-018