Roughness of a discontinuous system
Let \(G\subset R^ 2\) be a bounded domain divided by pointwise-continuous curves in a finite number of domains \(G_ 1,G_ 2,\ldots,G_ n\). Consider the dynamical system \[ \text{(A)}\quad x'=P(x,y),\quad y'=Q(x,y),\qquad (\tilde A)\quad x'=\tilde P(x,y),\quad y'=\tilde Q(x,y), \] whose right-hand sides belong to the class \(C^ 1(G_ i)\), \(i=1,\ldots,n\), with possible discontinuities on the boundaries of \(G_ 1,G_ 2,\ldots,G_ n.\) The set of all such systems endowed with metric \[ \rho_ 1(A,\tilde A)=\max_{1\leq i\leq n}(\| P-\tilde P\|_{C^ 1(\bar G_ i)}+\| Q-\tilde Q\|_{C^ 1(\bar G_ i)}) \] becomes a metric space \(R^ 1(\bar G)\). The system \((A)\in R^ 1(\bar G)\) has the roughness property in \(W\subset G\), if there exists a domain H, \(\bar W\subset H\subset \bar H\subset G\), such that for any \(\epsilon >0\) there exists a \(\delta >0\) such that \((\tilde A)\in R^ 1(\bar G)\) with \(\rho_ 1(A,\tilde A)<\delta\) implies the existence of \(\tilde H\) for which \((H,A)\) and \((\tilde H,\tilde A)\) are \(\epsilon\)-identical. The author gives necessary and sufficient conditions assuring the roughness of dynamical system (A) with the pointwise-continuous and the pointwise- smooth right-hand side.
scientific article; zbMATH DE number 3933540