The moments sliding vector field on the intersection of two manifolds (Q523091)

The paper is concerned with the piecewise smooth differential system \[ \dot x=f(x),\,\,\,f(x)=f_i(x),\,\,x\in R_i,\,\,i=1,\ldots,N,\,\,t\in [0,T], \] where \(R_i\subseteq \mathbb{R}^n\) are open, disjoint and connected sets with \(\mathbb{R}^n=\overline{\cup R_i}\), on each region \(R_i\) the function \(f\) is given by a smooth vector field \(f_i\), and \(\Sigma_i:=\{x\in\mathbb{R}^n,\,\,h_i(x)=0\}\), \(i=1,\ldots,p\), (\(2^p=N\)), with \(h_i\) smooth scalar valued functions (at least \(C^2\)), with specific interest in the case \(N=4\). The authors associate a quadrilateral to the attractivity configuration of \(\Sigma=\Sigma_1\cap\Sigma_2\), and then they investigate this geometrical configuration. They prove that the moments vector field (which belongs to the class of Filipov vector fields) is well-defined and smoothly exiting at generic first order exit points, and discuss other cases enjoying this property. Numerical experiments which support the main results are also presented.