Is the dynamical system stable?

Abstract

Abstract. In this paper we consider stability in the Ulam-Hyers sense, and in other similar senses, for the five equivalent definitions of one-dimensional dynamical systems.Mathematics Subject Classification. Primary 39B12, 39B82; Secondary 37E05, 26A18.

Stability has many names

“…Equation (6) in the class of continuous functions from IxR to I is (see [16]) -stable (normally, strongly) only for I=R, -b-stable (normally, strongly), uniformly b-stable and restrictedly uniformly b-stable (normally) only for I being bounded or I=R, -inversely b-stable, absolutely b-stable, inversely uniformly b-stable, absolutely uniformly b-stable, superstable, uniformly superstable and inversely superstable only for I bounded, -inversely stable, absolutely stable and hyperstable for no I, -inversely hyperstable for every I. Equation (7) in the class of continuous functions from IxR to I is (see [16] and [17] Remark. The stability of Eq.…”

On the Normal Stability of Functional Equations

“…Moreover, the stability and the b-stability for equation (19) is the same as for the system (18) since the inequality |f (f (x)) − f (x)| + |f (x) − 1| ≤ δ implies |f (f (x)) − f (x)| ≤ δ and |f (x) − 1| ≤ δ for every δ > 0.…”

“…as well as uniformly b-stability (hence b-stability)" as the conclusion of [19,Corollary 3.8] after the proof of this corollary in the paper [19, p. 290] is not true for I unbounded (my mistake). Hence in the table in the end of the paper [19] (see [23] too) for the b-stability and uniform b-stability in the case of def. 2 in place of "for every I" must be "only for I bounded ".…”

“…1) The uniformly b-stable dynamical system is evidently restrictly uniformly b-stable (but not inversely). 2) If in the proof of Theorem 4.3 in [19] we assume that 2b−2a ≤ δ for a δ > 0, then we obtain the counter-example for the restricted uniform b-stability in the case of the def.1 and if the unbounded interval I is bounded from below (analogous from above). Proof for normal stability.…”

“…For I bounded the function Ψ(δ) = 10δ for δ ∈ (0, 2 5 ) and Ψ(δ) = |I| (the length of I) for δ ∈ [ 2 5 , +∞) is the measure of b-stability for which inf Ψ(0, +∞) = 0. Indeed, if δ ∈ (0, 2 5 ), then by [19,Corollary 3.8] for every function H : R × I → I such that…”