Study of Singularities in Nonsmooth Dynamical Systems via Singular Perturbation

被引:44
作者
Llibre, Jaume [1 ]
da Silva, Paulo R. [3 ]
Teixeira, Marco A. [2 ]
机构
[1] Univ Autonoma Barcelona, Dept Matemat, E-08193 Barcelona, Catalonia, Spain
[2] IMECC UNICAMP, BR-13081970 Sao Paulo, Brazil
[3] IBILCE UNESP, Dept Matemat, BR-15054000 Sao Paulo, Brazil
来源
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS | 2009年 / 8卷 / 01期
关键词
regularization; vector fields; singular perturbation; discontinuous vector fields; sliding vector fields; DISCONTINUOUS VECTOR-FIELDS; INTERSECTING SWITCHING SURFACES; SLIDING MODES; STABILITY;
D O I
10.1137/080722886
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
In this article we describe some qualitative and geometric aspects of nonsmooth dynamical systems theory around typical singularities. We also establish an interaction between nonsmooth systems and geometric singular perturbation theory. Such systems are represented by discontinuous vector fields on R-l, l >= 2, where their discontinuity set is a codimension one algebraic variety. By means of a regularization process proceeded by a blow-up technique we are able to bring about some results that bridge the space between discontinuous systems and singularly perturbed smooth systems. We also present an analysis of a subclass of discontinuous vector fields that present transient behavior in the 2-dimensional case, and we dedicate a section to providing sufficient conditions in order for our systems to have local asymptotic stability.
引用
收藏
页码:508 / 526
页数:19
相关论文
共 14 条
  • [1] Alexander JC, 1999, HOUSTON J MATH, V25, P185
  • [2] Alexander JC, 1998, HOUSTON J MATH, V24, P545
  • [3] Broucke ME, 2001, COMPUT APPL MATH, V20, P51
  • [4] A singular approach to discontinuous vector fields on the plane
    Buzzi, Claudio A.
    da Silva, Paulo R.
    Teixeira, Marco A.
    [J]. JOURNAL OF DIFFERENTIAL EQUATIONS, 2006, 231 (02) : 633 - 655
  • [5] DiBernardo M, 2008, APPL MATH SCI, V163, P1, DOI 10.1007/978-1-84628-708-4
  • [6] Dumortier F, 1996, MEM AM MATH SOC, V121, P1
  • [8] Filippov AF., 2013, DIFF EQUAT+, VVol. 18
  • [9] Jones CKRT, 1995, LECT NOTES MATH, V1609, P44
  • [10] Global asymptotic stability for a class of discontinuous vector fields in R2
    Llibre, Jaume
    Teixeria, Marco Antonio
    [J]. DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL, 2007, 22 (02): : 133 - 146