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.
Author(s): Enrique Ponce, Javier Ros, Elísabet Vela
Series: RSME Springer Series, 7
Publisher: Springer
Year: 2022
Language: English
Pages: 316
City: Cham
Preface
Contents
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 ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (double struck upper R squared) /StPNE pdfmark [/StBMC pdfmarkR2ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
1.2.1 Real and Distinct Eigenvalues: ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (lamda 1 not equals lamda 2) /StPNE pdfmark [/StBMC pdfmarkλ1=λ2ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
1.2.2 Complex Eigenvalues
1.2.3 Nonzero Double Eigenvalues: ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (lamda 1 equals lamda 2 equals lamda not equals 0) /StPNE pdfmark [/StBMC pdfmarkλ1=λ2=λ=0ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
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 Liénard 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 Liénard Form for Continuous Planar Piecewise Linear Systems
2.2 Observable Piecewise Linear Luré Systems
2.3 Some Generic Results About Equilibria
2.3.1 Observable Continuous Piecewise Linear Systems with Two Zones
2.3.2 Observable Symmetric Continuous Piecewise Linear Systems with Three Zones
2.4 Analysis of Periodic Orbits Through Their Closing Equations
2.4.1 Closing Equations for Observable Continuous Piecewise Linear Systems with Two Zones
2.4.2 Closing Equations for Symmetric Continuous Piecewise Linear Systems with Three Zones
2.5 Periodic Orbits and Poincaré Maps in Piecewise Linear Systems
2.5.1 Derivatives of Transition Maps
2.5.2 Poincaré Maps in Continuous Piecewise Linear Systems with Two Zones
2.5.3 Poincaré Maps in Continuous Piecewise Linear Systems with Three Zones
Part II Planar Piecewise Linear Differential Systems
3 Analysis of Planar Continuous Systems with Two 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 First Results for Planar Continuous Systems with Three Zones
4.1 Limit Cycle Existence and Uniqueness
4.2 The Focus-Center-Limit Cycle Bifurcation for Symmetric Continuous Planar Piecewise Linear Systems with Three Zones
5 Boundary Equilibrium Bifurcations 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
5.3.1 Revisiting the Kriegsmann's Approach
6 An Algebraically Computable Bifurcation in Continuous Piecewise Linear Nodal Oscillators
6.1 Preliminary Results
6.2 Analysis of Equilibria and Periodic Orbits
6.3 An Example of Piecewise Linear Nodal van der Pol Oscillators
7 The Focus-Saddle Boundary Bifurcation
7.1 Some Results on Focus-Saddle Refracting Systems
7.2 Characterizing the Focus-Saddle Boundary Equilibrium Bifurcation
7.3 The Effect of Initial Conditions in a Memristor Oscillator
Part III Three-Dimensional Piecewise Linear Differential Systems
8 The Focus-Center-Limit Cycle Bifurcation in 3D Continuous Piecewise Linear Systems with Two Zones
8.1 The Generic Focus-Center-Limit Cycle Bifurcation
8.2 The Genesis of Asymmetric Oscillations in Chua's Circuit
8.3 The Degenerate Focus-Center-Limit Cycle Bifurcation in Continuous Piecewise Linear Systems with Two Zones in ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (double struck upper R cubed) /StPNE pdfmark [/StBMC pdfmarkR3ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
8.4 The Degenerate Focus-Center-Limit Cycle Bifurcation in Chua's Circuit
9 The FCLC Bifurcation in 3D Symmetric Continuous Piecewise Linear Systems
9.1 The Symmetric Focus-Center-Limit Cycle Bifurcation
9.2 The Degenerate Focus-Center-Limit Cycle Bifurcation in Symmetric Continuous Systems with Three Zones in ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (double struck upper R cubed) /StPNE pdfmark [/StBMC pdfmarkR3ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
10 The Piecewise Linear 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
A The Piecewise Linear Characteristics of Chua's Diode
B The Chua's Oscillator
C Some Auxiliary Results
Bibliography
