文摘
In this paper, we consider the class of smooth sliding Filippov vector fields in \(\mathbb {R}^3\) on the intersection \(\Sigma \) of two smooth surfaces: \(\Sigma =\Sigma _1\cap \Sigma _2\), where \(\Sigma _i=\{x:\ h_i(x)=0\}\), and \(h_i:\ \mathbb {R}^3\rightarrow \mathbb {R}\), \(i=1,2\), are smooth functions with linearly independent normals. Although, in general, there is no unique Filippov sliding vector field on \(\Sigma \), here we prove that—under natural conditions—all Filippov sliding vector fields are orbitally equivalent to \(\Sigma \). In other words, the aforementioned ambiguity has no meaningful dynamical impact. We also examine the implication of this result in the important case of a periodic orbit a portion of which slides on \(\Sigma \). Keywords Filippov convexification Orbital equivalence Periodic orbit Stability