Some elements for a history of the dynamical systems theory.

Writing a history of a scientific theory is always difficult because it requires to focus on some key contributors and to "reconstruct" some supposed influences. In the 1970s, a new way of performing science under the name "chaos" emerged, combining the mathematics from the nonlinear dynamical systems theory and numerical simulations. To provide a direct testimony of how contributors can be influenced by other scientists or works, we here collected some writings about the early times of a few contributors to chaos theory. The purpose is to exhibit the diversity in the paths and to bring some elements-which were never published-illustrating the atmosphere of this period. Some peculiarities of chaos theory are also discussed.

Topological characterization versus synchronization for assessing (or not) dynamical equivalence.

Model validation from experimental data is an important and not trivial topic which is too often reduced to a simple visual inspection of the state portrait spanned by the variables of the system. Synchronization was suggested as a possible technique for model validation. By means of a topological analysis, we revisited this concept with the help of an abstract chemical reaction system and data from two electrodissolution experiments conducted by Jack Hudson's group. The fact that it was possible to synchronize topologically different global models led us to conclude that synchronization is not a recommendable technique for model validation. A short historical preamble evokes Jack Hudson's early career in interaction with Otto E. Rössler.

Is it ethical to heal a young white elephant from his physiological autism?

Forty years ago a causal therapy of autism was offered which has never been tried out by the therapeutic profession. It predictably is so effective that even members of other mirror-competent bonding species can be healed from their "physiological autism." Niklas Luhmann belonged to the therapy's supporters and Leo Szilard had anticipated it in fiction 30 years earlier. The Ottersberg Lectures on Philosophy revived it through the enthusiasm and cooperation of the youthful audience.

Hyperchaos of arbitrary order generated by a single feedback circuit, and the emergence of chaotic walks.

It is shown that hyperchaos of order m (i.e., with m positive Lyapunov exponents) can be generated by a single feedback circuit in n = 2m + 1 variables. This feedback circuit is constructed such that, dividing phase space into hypercubes, it changes sign wherever the trajectory passes from one hypercube into an adjacent one. Letting the negative diagonal elements in the Jacobian tend to zero, the dynamics becomes conservative. Instead of chaotic attractors, unbounded chaotic walks are then generated. Here we report chaotic walks emerging from a continuous system rather than the well known chaotic walks present in "Lorentz gas" and "couple map lattices."