The Dynamical System Generated by the Floor Function

On a two dimensional dynamical system generated by the floor function

“…We shall prove that fixed points of operator are just roots of ⌊λx⌋ = x. In [2] all limit points ω(x) of the floor function f (x) = ⌊λx⌋ found for ∀λ, x ∈ R. It was proved that the limit of {f n (x)} converges: to fixed points; to ∞, or -∞(case λ = −1 exception). For the case λ = −1 it was proved that Per…”

“…That's why fixed points given in parts 1-5 of Lemma took by solving ⌊λx⌋ = x (see e.g. [2]). But there may be some fixed points of g(x) = ⌊λ⌊λx⌋⌋ those are roots of ⌊λ⌊λx⌋⌋ = x and are not roots of ⌊λx⌋ = x.…”

“…In Theorem 2 of [2] was proved lim n→∞ u n = 0 for any λ ∈ (−1, 0) and for all x 0 ∈ R. That's why we have lim n→∞ A n (z) = (0, 0).…”

Dynamics of a population with two equal dominated species

“…Problems of like this appear, for example, in electrical circuits that have switches, mechanical devices in which components impact with each other or have free play, problems with friction, sliding or squealing, many control systems and models in the social and financial sciences where continuous change can trigger discrete actions [2]. See also motivating examples of a piecewise-smooth systems: generated by the floor function ( [19], [22]) and p-adic dynamical systems (see [1], [13], [18], [20]).…”