1968 : A one-dimensional noninvertible map

Christophe LETELLIER
Igor Gumowski & Christian Mira
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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]

      x_{n+1} = ( 1 - \lambda) x_n + y_n \\[0.1cm]
      y_{n+1} = y_n + f(x_n) 
\mbox{ where }
  f(x_n) = 
      - 2 \lambda x_n - 0.9 \lambda & & x_n < -0.5 \\[0.1cm]
      - \frac{\lambda x_n}{5} & \mbox{ if } & |x_n| < 0.5 \\[0.3cm]
      - 2 \lambda x_n + 0.9 \lambda & & x_n > -0.5  \, . 

For some \lambda-values, they obtained an attractive limit set made of bounded cloud of points as shown in Fig. 1 (\lambda = 2.30). 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.

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

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

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Fig. 2. Chaotic solution just before a boundary crisis.
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Budapest, 1968
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. online


Budapest, 1968
PDF · 1.4 Mo
22246 - 19/07/24

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