Minimising the complexity of complex dynamical systems While the general concepts of static and dynamic are relatively easy to understand, when it comes to complex dynamical systems and high-level mathematics things become – well – complex. A new theorem enables mathematicians to focus on less information to extract most of the behaviour of dynamical systems. Fundamental Research © Nuria Fagella_Anna Miriam Benini Formally, a dynamical system is one whose state evolves with time over a ‘state space’. The state space is also called a topological manifold, the multi-dimensional equivalent of a curved surface. For example, a circle is a 1D manifold (a line) embedded in two dimensions, where each arc of the circle locally represents a line. While a dynamical system is defined by a state space of integers or real numbers, a complex dynamical system has a corresponding complex manifold. With the support of the Marie Skłodowska-Curie programme, CoTraDy set out to study dynamical systems generated by 1D transcendental maps (non-polynomial such as exponential or trigonometric) acting on the complex plane. A mathematical journey in time – and state space According to project fellow Anna Miriam Benini, now of the University of Parma, and project coordinator Nùria Fagella of the University of Barcelona: “We were planning to investigate the consequences that a given combinatorics has for the dynamical behaviour of a given map.” Combinatorics is a sort of code from which one can theoretically determine all the dynamical features of a given map. Combinatorics enables mathematicians to group maps into classes in which all members have similar dynamics. Then, conversely, they can gain information about the dynamics of a map by knowing to which class of maps it belongs and its combinatorics. However, recovering the dynamics from the combinatorics is not always possible. Finding stability in chaos Benini continues: “What we managed to do was to link the combinatorics to the behaviour of a special set of points (called singular values) which are responsible for most of the dynamics of the map itself. As a result, we were able to obtain information about the equilibrium states of the dynamical system (the periodic points) and their relation to these points.” In other words, the team further simplified the ‘code’ or, rather, its interpretation by being able to extract the bulk of information about the dynamical behaviour from a subset of points without having to ‘connect all the dots’. A dynamic duo of female mathematicians powers innovation The work of Benini and Fagella provides important insight into combinatorics related to equilibrium states where the dynamical system returns over time. It could shed light beyond as well. Benini explains: “CoTraDy’s outcomes could open the door to better understanding of wandering domains, among the least understood phenomena in transcendental dynamics. These large sets of points move all together but never come back to themselves.” In addition, Benini is currently collaborating with others on the application of CoTraDy’s techniques to 2D transcendental dynamics, a broad field in which little is known. Fagella concludes: “Our proof extends previous results and also gives a direct approach to understanding the relation between combinatorics, singular values, and equilibrium points.” Complex dynamical systems in any dimension are often complexifications of real ones, sometimes motivated by models of real-world dynamical systems like population change or the stock market. CoTraDy’s outcomes could help us look at these real models from a complex point of view to explain phenomena that are not understandable otherwise. Keywords CoTraDy, dynamical system, combinatorics, complex, dynamics, map, state space, manifold, transcendental, singular value, equilibrium state
