Bifurcations in Continuous Piecewise Linear Differential Systems: Applications to Low-Dimensional Electronic Oscillators

ISBN-13: 9783031211348 / Angielski / Twarda / 2022 / 311 str.

Enrique Ponce; Javier Ros; El?sabet Vela

Opis

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.

Spis treści

Part I: Introduction

1 From linear to piecewise linear differential systems

1.1 Some caveats about the notation used in the book

1.2 A short review on linear systems in R2

1.2.1 Real and distinct eigenvalues

1.2.2 Complex eigenvalues

1.2.3 Nonzero double eigenvalues

1.2.4 A canonical form for affine systems

1.2.5 The case of vanishing determinant

1.3 Degenerate bifurcations in planar affine systems

1.4 The one-parameter Li´enard form

1.5 Computing the exponential matrix

1.6 Passing from linear to piecewise linear systems

1.7 Limit cycles in a continuous piecewise linear worked example

1.7.1 The right half-return map

1.7.2 The left half-return map

1.7.3 A bifurcation analysis

2 Preliminary results

2.1 A unified Li´enard form for continuous planar piecewise linear systems

2.2 Canonical forms for Lur´e systems in higher dimension

2.3 Some generic results about equilibria

2.3.1 Observable continuous piecewise linear systems with two zones

2.3.2 Observable symmetric continuous PWL systems with three zones

2.4 Analysis of periodic orbits through their closing equations

2.4.1 Observable continuous piecewise linear systems with two zones

2.4.2 Symmetric continuous PWL systems with three zones

2.5 Periodic orbits and Poincar´e maps in piecewise linear systems

2.5.1 Derivatives of transition maps

2.5.2 Continuous piecewise linear systems with two zones

2.5.3 Continuous piecewise linear systems with three zones

Part II: Planar piecewise linear differential systems

3 Continuousplanar systemswithtwo zones

3.1 Equilibria in continuous planar piecewise linear systems with two zones

3.2 Some preliminary results on limit cycles

3.3 The Massera’s method for uniqueness of limit cycles

3.4 General results about limit cycles

3.5 Refracting Systems

3.6 The bizonal focus-center-limit cycle bifurcation

4 Continuousplanar systemswiththree zones

4.1 Limit cycle existence and uniqueness

4.2 The focus-center-limit cycle bifurcation for symmetric systems

5 Boundary equilibriumbifurcations and limit cycles

5.1 Boundary Equilibrium Bifurcations in systems with two zones

5.2 Boundary Equilibrium Bifurcations in systems with three zones

5.3 Analysis of Wien bridge oscillators

6 Algebraically computable continuousPWLnodal oscillators

6.1 Preliminary results

6.2 Analysis of equilibria and periodic orbits

6.3 Application to a piecewise linear van der Pol oscillator

7 The focus-saddle boundary bifurcation

7.1 Preliminary results

7.2 Main results

7.3 Application to the analysis of memristor oscillators

Part III: Three-dimensional piecewise linear differential systems

8 The FCLC bifurcation in 3D systems with 2 zones

8.1 The generic focus-center-limit cycle bifurcation

8.2 The onset of asymmetric oscillations in Chua’s circuit

8.3 The degenerate FCLC bifurcation in systems with 2 zones

8.4 The degenerate FCLC bifurcation in Chua’s circuit

9 The FCLC bifurcation in 3D symmetric PWL systems

9.1 The symmetric focus-center-limit cycle bifurcation

9.2 The degeneration of the FCLC bifurcation in symmetric systems

10 The analogue of Hopf-pitchfork bifurcation

10.1 A one-parameter bifurcation analysis

10.2 The degenerate PWL Hopf-pitchfork bifurcation

10.3 The Hopf-pitchfork bifurcation in Chua’s circuit

10.4 An extended Bonhoeffer–van der Pol oscillator

11 Afterword

Appendices

A The piecewise linear characteristics of Chua’s diode

B The Chua’s oscillator

C Auxiliary results

Bibliography

O autorze

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.