LOCATION OF APPROXIMATIONS OF A MARKOFF THEOREM

Relative to the first two theorems of the well known Markoff Chain (J.W.S. Cassels, "An introduction to diophantine approximation" approximations are well located. Literature is silent on the question of location of approximations in reference to the other theorems of the Chain. Here we settle it for the third theorem of the Chain.

As regards (1.1) and (1.2) we have an ad-hoc idea of the j's satisfying them.In reference to T we know that one j must occur in {n, n+l, n+2} ngl.Relative to 9 then a j e {n n+l, n+2} These may be found T 2, we have a similar result if an+2 in Wright [3] or Prasad and Lari [4].
But the literature is surprisingly silent on such results in reference to T 3, T 4, etc.In this article we announce one such result in reference to T 3 in the following theorem:

MAIN RESULTS
THEOREM.an+2 2 and an+3 then .(8)(Vl)/5 for at least one j e {n, n+l, n+4} REMARK.Our method gives a way to try for similar results on T 4, T 5, etc.
This completes the proof of the theorem.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.