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
