On the global asymptotic stability of difference equations satisfying a Markus-Yamabe condition

Abstract

We prove a global asymptotic stability result for maps coming from n-th order difference equation and satisfying a Markus-Yamabe type condition. We also show that this result is sharp.

Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem

“…We take, g 1 (y) := T 5,1 (0.6, y) · T Proceeding in an analogous way we obtain that g 2 (y) := T 5,2 (x, 2.3) · T 5,2 (x, 2.9) is a polynomial of degree 62 and it is negative for x ∈ [0. 6,1]. Hence the map f = (T 5,1 , T 5,2 ) satisfies the hypothesis of the PMT and there exists a solution of system (6) in B.…”

“…In this section we apply the PMT to give a simple proof of the following result, that in particular fixes the numerical counterexample presented in [6].…”

“…Proof of Proposition 3. We start noticing that in [6], it is proved that the maps (3) satisfy condition (2) if and only if |a| < 11664/3125 and b ∈ (3125a 2 /11664, 1). When a = 2b these conditions reduce to b ∈ (0, 2916/3125) .…”

“…To prove that there is a (unique) solution of system (6) in each box, and therefore there are 2-periodic orbits with 10 periodic points of minimal period 5 we apply the PMT. To illustrate the type of computations we deal with, we only show one of the computations.…”

“…To obtain simpler expressions and work more comfortably we will show that the hypotheses of the PMT are verified for a bigger box B =:= [0.6, 1] × [2.3, 2.9] , which has been obtained by visual inspection, see Figure 3, instead of using the actual box. It is easy to check that the only box of (7) contained in B is I 9,10 , and therefore if there is a solution of system (6) in B then it must be in I 9,10 , and be unique by construction. It corresponds to the intersection of the curves defined by the curves T 5,1 (x, y) = 0 (in blue) and T 5,2 (x, y) = 0 (in magenta).…”