Christophe LETELLIER
14/08/2011

Igor Gumowski & Christian Mira

**Christian Mira**

Before the emergence of chaos in electronic circuits during the late 70s and early 80s there is an important contribution by Igor Gumowski & Christian Mira (the “Toulouse Research Group”) whose a short story was given in Ref. [1] Stimulated by a paper by C. P. Pulkin [2] who showed that in one-dimensional noninvertible map infinitely many unstable cycles may lead to bounded complex iterated sequences, Gumowski & Mira studied the piecewise-linear map [3]

For some -values, they obtained attractive limit set made of bounded cloud of points as shown in Fig. 1 (). This could have been the first now called “chaotic” solution to a piecewise-linear map reported with an explicit picture. By these times, Gumowski & Mira indicated these types of behaviors as “Pulkin phenonmenon”. This is only ten years later that first chaotic solutions were widely investigated in electronic circuits.

**Fig. 1. Bounded-2 chaotic attractor solution to the piecewise-linear map as published in 1969**

Increasing slightly the value to 2.39, the chaotic solution observed is just before a boundary crisis unifying the two sets of points (Fig. 2).

**Fig. 2. Chaotic solution just before a boundary crisis.**

[1] **C. Mira**, I. Gumowski and a Toulouse Research Group in the “prehistoric times of chaotic dynamics”, *World Scientific Series in Nonlinear Science A*, **39**, 95-198, 2000.

[2] **C. P. Pulkin**, Oscillating iterated sequences (in Russian), *Dokl. Akad. Nauk. SSSR*, **76** (6), 1129-1132, 1950.

[3] **I. Gumowski & C. Mira**, Sensitivity Problems Related to Certain Bifurcations in Non-Linear Recurrence Relations, *Automatica*, **5**, 303-317, 1969.