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] [4]

For some -values, they obtained an attractive limit set made of bounded cloud of points as shown in Fig. 1 (). This could be the very the first “chaotic” solution to a piecewise-linear map reported with an explicit picture. In the original paper (1968), Mira mentioned that

La récurrence inverse de [l’application ci-dessus] permet, en prenant une condition initiale voisine de 0 (point double stable), de constater qu’il existe un cycle d’ordre très élevé, peut-être infini.

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.**

Voir ce site : Original paper

[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), *Doklady Akademii Nauk SSSR*, **76** (6), 1129-1132, 1950.

[3] **C. Mira**, Étude de la frontière de stabilité d’un point double stable d’une récurrence non linéaire autonome du deuxième ordre, *Proceedings of the IFAC Symposium on Pulse-rate and Pulse-number Signals in Automatic Control*, D 43-7II, 1968.Online

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