Dynamics on polytopes and polymatrix replicators

Evolutionary game theory provides many examples of ODEs and flows where the phase space is a 
polytope. A computational technique to analyze the asymptotic dynamics along the polytope's edges has been 
developed by Duarte (2011), and Alishah, Duarte and Peixe (2014, arXiv:1411.6227v2). 
A particular class of such ODEs defined on polytopes are the polymatrix replicators (see Alishah and 
Duarte(2015) and Alishah, Duarte and Peixe (2015)), which includes well known classes of 
evolutionary game dynamics, such as the symmetric and asymmetric games associated to replicator 
equations. We aim to expand and make these techniques user friendlier in order to disseminate their 
applicability.

Piecewise Linear Dynamics meets Economics and Biology

This project studies piecewise-smooth dynamical systems arising in real-world applications.
Piecewise-smooth dynamical systems are found in applications to problems in engineering, biology,
computer science, economics and management. We pretend to develop new methods to analyse the dynamics of a certain class of piecewise linear dynamical systems defined on polytopes. Using the novel methods, we pretend also to answer several standing open questions raised in the 90's concerning the dynamics of two basic models in manufacturing and logistics. Additionally, we want to study the dynamics of evolutionary game theory models from a piecewise-smooth dynamical systems perspective, and develop software for simulation and analysis of these models. From a pure mathematical side, this project also contributes to the development of the theory of piecewise-smooth dynamical systems which still today presents many challenging open problems.

New Trens in Lyapunov Exponents - FCT PTDC/MAT-PUR/29126/2017

This project deals primarily with Lyapunov exponents (LE), which measure the sensitivity of a dynamical system to initial conditions. The Lyapunov exponents will be studied in the framework of Ergodic Theory (ET), Mathematical Physics (MP) and the Theory of Dynamical Systems (DS). The applied strand of the project will focus on ranking algorithms for large sparse graphs, in the context of bibliometric analysis.