Front Matter
1 Introduction
1.1 General Description of Differential Equations with Discontinuities
1.1.1 Impulsive Differential Equations
1.1.2 Differential Equations with Piecewise Constant Argument
1.1.3 Differential Equations with Discontinuous Right-Hand Sides
1.2 Nonautonomous Bifurcation
1.3 The Bernoulli Equations
1.4 Organization of the Book
2 Hopf Bifurcation in Impulsive Systems
2.1 Hopf Bifurcation of a Discontinuous Limit Cycle
2.1.1 The Nonperturbed System
2.1.2 The Perturbed System
2.1.3 Foci of the D-System
2.1.4 The Center and Focus Problem
2.1.5 Bifurcation of a Discontinuous Limit Cycle
2.1.6 Examples
2.2 3D Discontinuous Cycles
2.2.1 Introduction
2.2.2 The Nonperturbed System
2.2.3 The Perturbed System
2.2.4 Center Manifold
2.2.5 Bifurcation of Periodic Solutions
2.2.6 Examples
2.3 Periodic Solutions of the Van der Pol Equation
2.3.1 Introduction and Preliminaries
2.3.2 Theoretical Results
2.3.3 Center Manifold
2.4 Notes
3 Hopf Bifurcation in Filippov Systems
3.1 Nonsmooth Planar Limit Cycle from a Vertex
3.1.1 Introduction
3.1.2 The Nonperturbed System
3.1.3 The Perturbed System
3.1.4 The Focus-Center Problem
3.1.5 Bifurcation of Periodic Solutions
3.1.6 An Example
3.2 3D Filippov System
3.2.1 Introduction
3.2.2 The Nonperturbed System
3.2.3 The Perturbed System
3.2.4 Center Manifold
3.2.5 Bifurcation of Periodic Solutions
3.2.6 An Example
3.3 Notes
4 Nonautonomous Bifurcation in Impulsive Bernoulli Equations
4.1 The Transcritical and the Pitchfork Bifurcations
4.1.1 Introduction
4.1.2 Preliminaries
4.1.3 The Pitchfork Bifurcation
4.1.4 The Transcritical Bifurcation
4.2 Impulsive Bernoulli Equations: The Transcritical and the Pitchfork Bifurcations
4.2.1 Introduction and Preliminaries
4.2.2 Bounded Solutions
4.2.3 The Pitchfork Bifurcation
4.2.4 The Transcritical Bifurcation
4.2.5 Illustrative Examples
4.3 Notes
5 Nonautonomous Bifurcations in Nonsolvable Impulsive Systems
5.1 The Transcritical and the Pitchfork Bifurcations
5.1.1 Introduction
5.1.2 Preliminaries
5.1.3 Attractivity and Repulsivity in a Linear Impulsive Nonhomogeneous Systems
5.1.4 The Transcritical Bifurcation
5.1.5 The Pitchfork Bifurcation
5.2 Finite-Time Nonautonomous Bifurcations
5.2.1 Introduction and Preliminaries
5.2.2 Attractivity and Repulsivity in a Linear Nonhomogeneous Impulsive System
5.2.3 Bifurcation Analysis
5.2.4 An Example
5.3 Notes
6 Nonautonomous Bifurcations in Bernoulli Differential Equations with Piecewise Constant Argument of Generalized Type
6.1 Introduction and Preliminaries
6.1.1 Attraction and Stability
6.2 Bounded Solutions
6.3 The Pitchfork Bifurcation
6.4 The Transcritical Bifurcation
6.5 Illustrative Examples
6.6 Notes
References
