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Would chaotic dynamical systems be more beautiful if they were useless ?

Christophe LETELLIER
22/06/2023
René Lozi

Since the very beginning of their appearance in the history of humanity, research in mathematics has been guided by two different currents : theory and applications or in other words by beauty and utility. Around 5,000 years ago people in the Mesopotamia and Egypt began using arithmetic, algebra and geometry for commerce, trade, taxation and social activities. Later, in the 6th century BC, Greeks introduced mathematics as a demonstrative discipline. This dual stream of research still functions today in competition-cooperation mode.

I had the immense privilege of being student of Jean Alexandre Dieudonné, one of the founding members of the Bourbaki group. For him, the only need for humanity to do research in mathematics was ``For the honor of the human spirit’’ (as the great mathematician Karl Gustav Jacobi (1804-1851) has said before him).

It is widely accepted that the beginning of modern research on nonlinear dynamical systems is due to the initial work of Henri Poincaré on the three-body problem. Even if a real astronomical problem (will the Earth continue to orbit around the sun forever ?) is at the origin of his reflection, no practical application of his Méthodes nouvelles de la mécanique céleste has guided his mind.

The study of the frighteningly complicated solutions discovered by him continued quietly for almost 80 years in several directions including conservative and dissipative dynamical systems, differential and difference equations. We can cite among many, the pioneer works of Pierre Fatou (1878-1929) and Gaston Julia (1893-1978) related to one-dimensional maps with a complex variable, near a century ago ; those of Christian Mira and Igor Gumowski, who began their mathematical research in 1958 (the Gumowski-Mira map) ; the fractals introduced in 1967 by Benoît Mandelbrot (1924-2010) ; and of course the attractors of Otto E. Rössler.

The ``sexier’’ word chaos coined by James A. Yorke in 1975 and the butterfly effect revealed by Edward E. Lorenz in 1963 have brought global awareness of these concepts often not actually understood by the public. However, it is only at the beginning of 90s that the applications of chaotic properties of dynamical systems were introduced with the pioneering idea of synchronization by Louis M. Pecora and Thomas L. Carroll. Such concept was soon used (and improved) to transmit encrypted messages.

Since then, many applications have emerged in optimization (particle swarm optimization, differential evolution, \ldots), cryptography based chaos, generation of pseudo-random numbers, electronics (Chua circuit and memristors), etc.

Have these applications compromised the purity and beauty of theoretical research ? We will try to describe the evolution of this topic over the last 50 years.

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