TY - JOUR

T1 - Sliding vector fields for non-smooth dynamical systems having intersecting switching manifolds

AU - Llibre, Jaume

AU - Da Silva, Paulo R.

AU - Teixeira, Marco A.

PY - 2015/2/1

Y1 - 2015/2/1

N2 - © 2015 IOP Publishing Ltd & London Mathematical Society. We consider a differential equation p = X(p), p ∈ ℝ3, with discontinuous right-hand side and discontinuities occurring on a set ∑. We discuss the dynamics of the sliding mode which occurs when, for any initial condition near p ∈ ∑, the corresponding solution trajectories are attracted to ∑. Firstly we suppose that ∑= H-1(0), where H is a smooth function and 0 ∈ ℝ is a regular value. In this case ∑ is locally diffeomorphic to the set F= {(x, y, z) ∈ ℝ3; z = 0}. Secondly we suppose that ∑ is the inverse image of a non-regular value. We focus our attention to the equations defined around singularities as described in Gutierrez and Sotomayor (1982 Proc. Lond. Math. Soc 45 97112). More precisely, we restrict the degeneracy of the singularity so as to admit only those which appear when the regularity conditions in the definition of smooth surfaces of ℝ3 in terms of implicit functions and immersions are broken in a stable manner. In this case ∑ is locally diffeomorphic to one of the following algebraic varieties: = {(x, y, z) ∈ ℝ3; xy = 0} (doublecrossing); = {(x, y, z) ∈ ℝ3; xyz = 0} (triple crossing); = {(x, y, z) ∈ ℝ3; z2-x2-y2 = 0} (cone) or = {(x, y, z) ∈ ℝ3; zx2-y2 = 0} (Whitney's umbrella).

AB - © 2015 IOP Publishing Ltd & London Mathematical Society. We consider a differential equation p = X(p), p ∈ ℝ3, with discontinuous right-hand side and discontinuities occurring on a set ∑. We discuss the dynamics of the sliding mode which occurs when, for any initial condition near p ∈ ∑, the corresponding solution trajectories are attracted to ∑. Firstly we suppose that ∑= H-1(0), where H is a smooth function and 0 ∈ ℝ is a regular value. In this case ∑ is locally diffeomorphic to the set F= {(x, y, z) ∈ ℝ3; z = 0}. Secondly we suppose that ∑ is the inverse image of a non-regular value. We focus our attention to the equations defined around singularities as described in Gutierrez and Sotomayor (1982 Proc. Lond. Math. Soc 45 97112). More precisely, we restrict the degeneracy of the singularity so as to admit only those which appear when the regularity conditions in the definition of smooth surfaces of ℝ3 in terms of implicit functions and immersions are broken in a stable manner. In this case ∑ is locally diffeomorphic to one of the following algebraic varieties: = {(x, y, z) ∈ ℝ3; xy = 0} (doublecrossing); = {(x, y, z) ∈ ℝ3; xyz = 0} (triple crossing); = {(x, y, z) ∈ ℝ3; z2-x2-y2 = 0} (cone) or = {(x, y, z) ∈ ℝ3; zx2-y2 = 0} (Whitney's umbrella).

KW - non-smooth dynamical system

KW - singular perturbation

KW - sliding vector field

U2 - https://doi.org/10.1088/0951-7715/28/2/493

DO - https://doi.org/10.1088/0951-7715/28/2/493

M3 - Article

VL - 28

SP - 493

EP - 507

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 2

ER -