Bifurcations in Continuous Piecewise Linear Differential Systems

The book is devoted to the qualitative study of differential equations defined by piecewise linear (PWL) vector fields, mainly continuous, and presenting two or three regions of linearity. The study focuses on the more common bifurcations that PWL differential systems can undergo, with emphasis on those leading to limit cycles. Similarities and differences with respect to their smooth counterparts are considered and highlighted. Regarding the dimensionality of the addressed problems, some general results in arbitrary dimensions are included. The manuscript mainly addresses specific aspects in PWL differential systems of dimensions 2 and 3, which are sufficinet for the analysis of basic electronic oscillators.
The work is divided into three parts. The first part motivates the study of PWL differential systems as the natural next step towards dynamic complexity when starting from linear differential systems. The nomenclature and some general results for PWL systems in arbitrary dimensions are introduced. In particular, a minimal representation of PWL systems, called canonical form, is presented, as well as the closing equations, which are fundamental tools for the subsequent study of periodic orbits. The second part contains some results on PWL systems in dimension 2, both continuous and discontinuous, and both with two or three regions of linearity. In particular, the focus-center-limit cycle bifurcation and the Hopf-like bifurcation are completely described. The results obtained are then applied to the study of different electronic devices.

In the third part, several results on PWL differential systems in dimension 3 are presented. In particular, the focus-center-limit cycle bifurcation is studied in systems with two and three linear regions, in the latter case with symmetry. Finally, the piecewise linear version of the Hopf-pitchfork bifurcation is introduced. The analysis also includes the study of degenerate situations. Again, the above results are applied to the study of different electronic oscillators.

A unique approach to piecewise linear (PWL) differential systems The bifurcation of periodic orbits is unveiled Including comprehensive analysis of some electronic oscillators

Auteur
Enrique Ponce was born in the very downtown of Seville, at his grandparent's home, on the afternoon of July's first Saturday, 1955. From his childhood, he is renowned for his ability to repair (sometimes, failing to repair) any mechanical or electrical device. After obtaining an electrical engineering degree, his interests turned into applied mathematics, an area in which he has become a full professor. Javier Ros was born in Almería in 1966. His main hobby is chess, where after some successes in his youth, he switched to correspondence chess where he qualified to play in the 33rd ICCF World Championship Final. His interest in mathematics led him, after a degree in engineering, to do his PhD in the area of applied mathematics, where he is currently an associate professor.

Elísabet Vela was born in Seville in 1982. Ever since she was little, she loved playing at being a teacher. From her great interest in teaching, added to her mathematical skills, it was easy to decide to study mathematics. After completing her doctoral thesis in applied mathematics she is currently an assistant professor in this subject.

Contenu
- Part I Introduction. - 1. From Linear to Piecewise Linear Differential Systems. - 2. Preliminary Results. - Part II Planar Piecewise Linear Differential Systems. - 3. Analysis of Planar Continuous Systems with Two Zones. - 4. First Results for Planar Continuous Systems with Three Zones. - 5. Boundary Equilibrium Bifurcations and Limit Cycles. - 6. An Algebraically Computable Bifurcation in Continuous Piecewise Linear Nodal Oscillators. - 7. The Focus-Saddle Boundary Bifurcation. - Part III Three-Dimensional Piecewise Linear Differential Systems. - 8. The Focus-Center Limit Cycle Bifurcation in 3D Continuous Piecewise Linear Systems with Two Zones. - 9. The FCLC Bifurcation in 3D Symmetric Continuous Piecewise Linear Systems. - 10. The Piecewise Linear Analogue of Hopf-pitchfork Bifurcation. - 11. Afterword.