Mathematical Handbook for Scientists and Engineers

Asymptotic Behaviour and Stability Problems in Ordinary Differential Equations. By L. Cesari. Pp. 271. DM. 68. 1959. (Springer, Berlin)

Differential Equations and Applications. By J. B. Scarborough. Pp. xiv, 479. 1965. (Warley Press Inc.)

On the number of upper Bohl exponents for diagonal discrete time-varying linear system

The Bohl exponents, similarly to the Lyapunov exponents, are among the most important numerical character- istics of dynamical systems used in control theory. Properties of the Lyapunov characteristics are well described in the literature. Properties of the Bohl exponents are investigated much less. In this paper we consider the so-called senior upper general exponent of a discrete linear time-varying system and present some alternative formulas for it. Moreover, we discuss relations between upper Bohl exponents of the perturbed system and the senior upper general exponent of the unperturbed system. Index Terms—time-varying systems, upper Bohl exponents, stability of linear systems, Lyapunov exponent, senior upper general exponent, perturbed system

The relations between the senior upper general exponent and the upper Bohl exponents.

The Perron effect is the effect in which the characteristic Lyapunov exponents of solutions of a differential system change sign from negative to positive when passing to a perturbed system. We show that this effect is realized on all nontrivial solutions of two two-dimensional systems: an original linear system with negative characteristic exponents and a perturbed system with small perturbations of arbitrary order m > 1 in a neighborhood of the origin, all of whose nontrivial solutions have positive characteristic exponents. We compute the exact positive value of the characteristic exponents of solutions of the two-dimensional nonlinear Perron system with small second-order perturbations, which realizes only a partial Perron effect.

On the perron sign change effect for Lyapunov characteristic exponents of solutions of differential systems

We obtain a general n-dimensional analog of the two-dimensional (partial) Perron effect of sign change of all arbitrarily prescribed negative characteristic exponents of an n-dimensional differential system of the linear approximation with infinitely differentiable bounded coefficients to the positive sign for the characteristic exponents of all nontrivial solutions of a nonlinear n-dimensional differential system with infinitely differentiable perturbations of arbitrary order m > 1 of smallness in a neighborhood of the origin and growth outside it. These positive exponents take n values distributed over n arbitrarily prescribed disjoint intervals and are realized on solutions issuing from nested subspaces R1 ⊂ R2 ⊂ ... ⊂ Rn.

Multidimensional analog of the two-dimensional perron effect of sign change of characteristic exponents for infinitely differentiable differential systems

We realize the Perron effect of change of values of characteristic exponents: for arbitrary parameters λ1 1, we prove the existence of a linear differential system $ \dot x $ = A(t)x, x ∈ R2, t ≥ t0, with bounded infinitely differentiable coefficients and with characteristic exponents λ1(A) = λ1 1), satisfying the condition ‖f(t, y)‖ ≤ const × ‖y‖m, y ∈ R2, t ≥ t0, and such that all nontrivial solutions y(t, c) of the perturbed system 

$$ \dot y = A(t)y + f(t,y), y \in R^2 $$

, have Lyapunov exponents λ[y(·, c)] = β1 for c1 = 0 and λ[y(·, c)] = β2 for c1 ≠ 0

Realization of the Perron effect whereby the characteristic exponents of solutions of differential systems change their values

We obtain a finite-dimensional Perron effect of change of values λ1 ≤ … ≤ λn 1 of smallness in a neighborhood of the origin and growth outside it. Each value βk is realized by all nontrivial solutions of the perturbed system issuing from the difference Rk|Rk−1 of embedded subspaces R1 ⊂ R2 ⊂ … ⊂ Rn.

N. A. Izobov

Papers

On the perron sign change effect for Lyapunov characteristic exponents of solutions of differential systems

Multidimensional analog of the two-dimensional perron effect of sign change of characteristic exponents for infinitely differentiable differential systems

Realization of the Perron effect whereby the characteristic exponents of solutions of differential systems change their values

Finite-dimensional Perron effect of change of all values of characteristic exponents of differential systems

On linear systems asymptotically equivalent under exponentially decaying perturbations