When a vector field in $\mathbb{R}^3$ is discontinuous on a smooth codimension one surface, it may be simultaneously tangent to both sides of the surface at generic isolated points (singularities). For a piecewise-smooth dynamical system governed by the vector field, we show that the local dynamics depends on a single quantity—the jump in direction of the vector field through the singularity. This quantity controls a bifurcation, in which the initially repelling singularity becomes the apex of a pair of parabolic invariant surfaces. The surfaces are smooth except where they intersect the discontinuity surface, and they divide local space into regions of attraction to, and repulsion from, the singularity.

Mike R. Jeffrey

Alessandro Colombo

SIAM Journal on Applied Dynamical Systems

SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS (SIADS) 发表关于动力系统的数学分析和建模及其在物理、工程、生命和社会科学中的应用的研究文章。 SIADS 仅以电子格式发布。

The SIAM Journal on Applied Dynamical Systems (SIADS) publishes research articles on the mathematical analysis and modeling of dynamical systems and their applications in physics, engineering, life sciences, and social sciences. SIADS is published exclusively in electronic format.

SIAM Journal on Applied Dynamical Systems (SIADS) 发表关于动力系统的数学分析、建模及其在物理、工程、生命科学和社会科学中的应用研究文章。SIADS 仅以电子格式发布。

The Two-Fold Singularity of Discontinuous Vector Fields

It is shown that the local dynamics depends on a single quantity—the jump in direction of the vector field through the singularity that controls a bifurcation, in which the initially repelling singularity becomes the apex of a pair of parabolic invariant surfaces.

Category	Quartile
数学	4
数学, 物理数学物理	3
数学, 应用数学	4

The Two-Fold Singularity of Discontinuous Vector Fields

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Category	Quartile
PHYSICS, MATHEMATICAL	2