本书主要讨论不同类型的自治和非自治不连续微分方程中的分岔。那些具有跳跃的微分方程既可以是右端点不连续的，也可以是在轨迹上不连续，或是方程解的区间常数近似的。本书的结果可以应用于各个领域，如神经网络、脑动力学、机械系统、天气现象、人口动力学等。毫无疑问，分岔理论应该进一步发展到不同类型的微分方程。在这个意义上，本书将是这个领域的首创。读者将从本书了解到该理论的最新成果，学会如何将该理论应用到不同类型的不连续微分方程的具体方法。此外，读者将学习到分析这些方程的非自治分岔情况的最新方法。
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 Bemoulli 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 Bifufcation
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
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