Pseudoeigenvalues, Spectral Portrait of a Matrix and their Connections with Different Criteria of Stability

Estimation for the region of attraction of nonlinear difference equations

“…In this case for arbitrary solution of Eq. (3) the following inequality holds [11,12] where H is the spectral norm of H. Hence, by H = H * > 0, limit (2) holds for arbitrary initial conditions x(0) = x 0 . The zero solution of Eq.…”

“…The algorithm allows to obtain matrix (5) and the quality of stability ω(A) = H of A on a computer with a guaranteed accuracy. The survey of results in this direction is contained in [11,12].…”

Sensitivity of Schur stability of systems of linear difference equations with periodic coefficients

“…It is well-known in the literature that eigenvalue problem of non-self-adjoint matrices is an ill-possed problem (see, for example, [4,6]). For this reason, the parameters revealing the quality of the stability obtained by avoiding calculation of eigenvalues are preferred to investigate stability.…”

“…, N), then the matrix A is called to be an discrete−asymptotic stable matrix. Therefore, the system (2) is also called as discrete−asymptotic stable system (see, for example, [1,2,3,4]). …”

“…Then the matrix A is called as practically Schur stable (ω * −Schur stable) if ω(A) ≤ ω * holds. Otherwise the matrix A is called as ω * −Schur unstable matrix (see, for example, [1,4,7]). We should note here that the ω * practical Schur stability parameter determined by the user according to their physical problem.…”