On linear topological conjugacy of Lur´e systems

In this letter, we prove several results regarding linear conjugacy between Lur´e systems. For example, we prove that except for a measure zero set, two Lur´e systems are linearly conjugate if they share an equilibrium point and the eigenvalues of the Jacobian matrices are matched at every point. A corollary of that result is that piecewise-linear vector fields with parallel boundary planes are determined, up to linear conjugacy, by the boundary planes, the equilibrium points and the eigenvalues in each region