Horseshoes for a Generalized Markus–Yamabe Example

Abstract

In this paper we present an example of a planar diffeomorphism satisfying the generalized Markus-Yamabe conditions, which has a horseshoe. This answers negatively a belief that generically they should be Morse-Smale.

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

“…Following the ideas of [1,7] and starting from the map (3), in this section we will construct a rational difference equation satisfying condition I and having a fixed point that is not GAS.…”

Stable Synchronous Propagation of Signals by Feedforward Networks

“…Quotient networks can have self-loops and multiple arrows, even if the original network does not. This feature is required to prove property (2); see [40,Section 8.10].…”

“…Definition 6.1 is motivated by the form of (3.9). Historically, it was conjectured for some time that transverse stability for a stable periodic orbit implies stability in the usual Floquet sense; see [2,5]. However, despite the terminology, this conjecture is false in general.…”