Perron Exponents Research Articles

Perron stability and its study at the first approximation

Natural concepts of Perron stability, asymptotic stability, and complete instability of the trivial solution to a differential system are introduced. Their features and relationships with Perron exponents and similar Lyapunov ones are indicated. An absolute and unique coincidence of research opportunities for Perron and Lyapunov stability and asymptotic stability at the first approximation is found.

Definition and Some Properties of Perron Stability

The natural notions of Perron stability, Perron asymptotic stability, and Perron complete instability of the zero solution of a differential system are introduced. Peculiar features of these notions are noted in the one-dimensional, autonomous, and linear cases. Their connections with Perron exponents and with their counterparts in the sense of Lyapunov are described. The complete coincidence of the possibilities for studying the Perron and Lyapunov stability and asymptotic stability in the first approximation is revealed.

On Simplified Formulas for the Central Exponents of Differential Systems With Non-Uniform Scales

We study the Vinograd–Millionshchikov central exponents, which represent the exact outer boundaries for the mobility of the extremal values of the Lyapunov and Perron exponents of a linear differential system under uniformly small perturbations of its coefficients. We prove the possibility of calculating those exponents using simplified formulas with expanding time scales and obtain concrete estimates for the central exponents with simplified ones, calculated in different scales: thick, expanding, slowly expanding and sparse.

What types of functions may be the Perron function of a diagonal discrete linear time-varying systems?

In this paper we investigate properties of the Perron exponents of diagonal discrete linear time-varying systems. We give necessary and sufficient conditions for a function / to be the Perron function of the considered systems.

On the Sequences Realizing Perron and Lyapunov Exponents of Discrete Linear Time-Varying Systems

We investigate properties of partial exponents (in particular, the Lyapunov and Perron exponents) of discrete time-varying linear systems. In the set of all increasing sequences of natural numbers, we define an equivalence relation with the property that sequences in the same equivalence class have the same partial exponent. We also define certain subclass of all increasing sequences of natural numbers, including all arithmetic sequences, such that all partial exponents are achievable on a sequence from this class. Finally, we show that the Perron and Lyapunov exponents may be approximated by partial exponents achievable on sequences in certain sense similar to geometric sequences.

On the Set of Perron Exponents of Discrete Linear Systems

Abstract The Lyapunov, Bohl and Perron exponents belong to the most important numerical characteristics of dynamical systems used in control theory. Properties of the first two characteristics are well described in the literature. Properties of the Perron exponents are much less investigated. In this paper we show an example of two-dimensional discrete-time linear system with bounded coefficients for which the set of Perron exponents constitutes an interval.

Bounds for characteristic exponents of discrete linear time varying systems

In this paper we introduce certain constants that characterize exponential stability or exponential unstability of discrete linear time varying systems. We also present bounds for Lyapunov and Perron exponents in terms of the introduced constants.

On the Perron exponents of discrete linear systems

This note studies properties of Perron or lower Lyapunov exponents for discrete time varying system. It is shown that for diagonal system of order s there are at most 2s-1 lower Lyapunov exponents. By example it is demonstrated that in non-diagonal case it is possible to have arbitrarily many different Perron exponents. Finally it is shown that the exponent is almost everywhere equal to the lower Lyapunov exponent of the matrices coefficient sequence.