Sistemes DinàmicsThe geography of integral curves in plane quadratic systems
Resum: We picture the space of all plane quadratic systems and all cofactors as an ocean and those that admit an algebraic integral curve as dry land. To get some idea about the geography of this land we use 3 approaches: (1) A random search over a finite field. This gives an heuristic overview. (2) Constructing systems in char 0 from singular curves using classification results from algebraic geometry and singularity theory and the solution of the inverse problem. This confirms the large features of the heuristic map. (3) For the smallest isolated islands of the map (those where neither the system nor the cofactor nor the curve nor a combination of them can be deformed non-trivially) we sample all systems of a particular normalization over a finite field and lift the solutions to normalized systems with rational coefficients using Hensel lifting (if possible). This yields several 100 systems over $Q$ each of which has a completely different geometry. Among them we rediscover Colin Christopher's famous system with a degree 12 curve. This is a work in progress, in particular we have run the experiments only up to degree 12 (characteristic 13) and even the data of these runs has not been completely analysed in all details.
From Generalized Controlled Nonlinear Oscillators to Lorenz-like Systems
Resum: Damped and driven oscillators are generally modeled with a nonautonomous second-order nonlinear ordinary differential equation including a sinusoidal driving forcing term, such as the forced Duffing equation and the forced Holmes-Rand equation. These equations have been extensively studied during the last century and the last two decades. In the early 1990s, Abarbanel, Rabinovich and Sushchik proposed replacing the sinusoidal forcing term with a "force controlled by the movements of the oscillator itself", i.e. by the product of two variables: the first being the solution of the oscillator itself, while the second is the solution of a first-order nonlinear ordinary differential equation. They referred to the resulting autonomous dynamical system of two coupled nonlinear ordinary differential equations as a "controlled nonlinear oscillator". To that end, they introduced a change of variables and parameters to transform the "controlled nonlinear oscillator" that corresponds to a particular case of the forced Duffing equation into the Lorenz system. The aim of this work is to show that their idea can be further generalized and applied to many other dynamical systems, including the forced Holmes-Rand equation, Chua's cubic circuit, Chen's system and the forced Helmholtz oscillator. It is proved that a certain class of three-dimensional dynamical systems can be rewritten into the form of "generalized controlled nonlinear oscillators", which can then be transformed into various Lorenz-like systems. Such a transformation could be very useful for the study of intermittent chaos.
TopologiaPre-monoidal $\infty$-operads
Resum: Pre-monoidal $\infty$-operads were introduced by Karlsson-Scheimbauer-Walde (under the name pre-coCartesian operads) to describe a class of $\infty$-operads that appears in the study of constructible factorisation algebras. A motivating example is the following: given an object x of a symmetric monoidal category C, one cannot expect the slice category C/x to be symmetric monoidal. However, if the unit is initial, there is always a natural pre-monoidal operad structure on C/x.
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