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Complex Patterns for Strongly Interacting Dynamical Systems

Final Report Summary - COMPAT (Complex Patterns for Strongly Interacting Dynamical Systems)

This project was born to pursue a specific intuition, namely that apparently different mathematical models, characterized in any case by strongly nonlinear interactions and featuring a variational structure, do share many common paradigms and are therefore liable to a fundamentally unified mathematical treatment. This remarkable methodological unity could reflect some general natural laws. This idea has proved to be successful and has produced a remarkable series of new results in cases where (a) the interaction was the prevailing mechanism, (b) the equations were very far from being solved explicitly, and (c) the problems could not be seen in any extent as perturbations of simpler ones.

The basic object of our research were nontrivial solutions of differential models featuring strongly nonlinear interactions. Usually, in many of these circumstances, the emergence of self-organized structures is observed. Such patterns correspond to selected solutions of the differential system possessing special symmetries or shadowing particular shapes. We tried to describe, from the mathematical point of view, the main mechanisms involved in the aggregation process in terms of the global variational structure of the problem. Following this common thread, we dealt with

• attractive interactions: as in the classical N-body problem of Celestial Mechanics, where the balance between attraction and centrifugal effects produces solutions showing complex patterns. More precisely, we were interested in periodic and bounded solutions and parabolic trajectories} with the /bf final intent of proving density of periodic solutions and the occurrence of chaos.
• repulsive interactions: as in competition-diffusion systems, where pattern formation is driven by strongly repulsive forces. Our ultimate goal was to capture the geometry and analysis of the phase segregation, including its asymptotic aspects and the classification of the solutions to the related equation.

All the objectives initially set were achieved, but a scientific research cannot be defined as such if it does not produce also some disappointments and reserve some surprises. Among the most welcome surprises, the discovery of solutions with spiralling nodal sets for the limiting profiles of strongly competing asymmetric systems and that of the parabolic orbit that allows to connect two configurations at infinity for the problem of $N$-centers.

These two different problems have been addressed, like the others, with the same basic methodology, including the following steps.

• Asymptotic analysis. The study of the effect of singularities (or singular limits) and the infinity on the profiles. The monotonicity formulae, adjusted for the different cases, the blow-up analysis, the classification of the limiting solutions invariant by dilation.
• Analysis and classification of entire solutions. Entire solutions also carry transitions from one configuration to another: this is the case of parabolic trajectories in Celestial Mechanics and entire solutions of competition-diffusion systems. Entire solutions also heavily enter in the blow-up analysis, as they represent the limiting profiles in some scaling process.
• Gluing techniques. Having gathered different types of elementary solutions, the next step consists of gluing them to build more complex patterns. Gluing can be performed, once more, using global variational techniques, or other methods. This can be done, e.g. by the broken geodesics argument, in the case of trajectories of Classical Mechanics, or by other types of reductions, e.g. by solving optimal partition problems, as in the case of competition-diffusion systems.