Riccati differential inequalities and dichotomies for linear systems (Q1849158)

Here, the authors consider finite-dimensional ordinary differential equations \(x'=A(t)x\), where \(A(t)\) is continuous, and provide criteria for them to possess a so-called \((\mu_1,\mu_2)\)-dichotomy. This dichotomy notion was introduced by the second author in [Trans. Am. Math. Soc. 283, 465--484 (1984; Zbl 0559.34049)] and includes ordinary and exponential dichotomies as special cases. The mentioned dichotomy criteria assume the existence of certain solutions to a Riccati differential inequality. In fact, the use of these Riccati inequalities leads to results extending earlier conditions expressed in terms of pairs of Lyapunov functions, and makes them easier to apply. Moreover, it provides a relaxation of the concept of diagonal dominance. Finally, as an application of their results, the authors give a proof for the roughness of \((\mu_1,\mu_2)\)-dichotomies.